17254093 Quantitative
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Aptitude test preparation for maths...
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From the desk of SOHAIL MERCHANT Quantitative Aptitude INDEX
ARITHMETIC 1. Real Number System
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03-12
2. *Ratio & Proportion
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13-19
3. **Percentages
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20-24
4. *Averages & Mixtures
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25-33
5. **Profit & Loss
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34-38
6. **Simple & Compound Interests
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39-43
7. *Time & Work
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44-49
8. *Time & Distance
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50-56
9. *Age Problems
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57-61
10. Plane Geometry
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62-80
11. **Mensuration
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81-91
GEOMETRY
COUNTING METHODS & PROBABILITY 12. Set Thoery
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91-97
13. *Permutations & Combinations
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98-103
14. *Probability
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104-110
15. Progressions
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111-117
16. Statistics
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118-124
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From the desk of SOHAIL MERCHANT ALGEBRA 17. Progressions
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126-132
18. Matrices
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133-142
19. Statements
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143-148
20. Sets
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149-154
21. Real Numbers, Rational Numbers & Law of Indices -------------------------- 155-160 22. Surds
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161-168
23. Linear Equations, Inequations & Modulus
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24. Polynomials, Remainder & Square Roots
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25. Quadratic Equations & Expressions
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178-180
26. Relations & Functions
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181-183
27. Derivatives & Limits
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184-187
28. Logarithms
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188-190
29. Binomial Theorem
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191-193
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From the desk of SOHAIL MERCHANT
NUMBER SYSTEMS In Hindu Arabic System, we use ten symbols 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 called digits to represent any number. This is the decimal system where we use the numbers 0 to 9. 0 is called insignificant digit. A group of figures, denoting a number is called a numeral. For a given numeral, we start from extreme right as Unit‘s place, Ten‘s place, Hundred‘s place and so on.
Ten Crore 108 Crores 107
Ten Lacs (million) 106
Lacs 105
Ten Thousand 104 Thousand 103
Hundred 102
Ten‘s 101
Units 100
Illustration 1 We represent the number 309872546 as shown below:
3
9
8
7
5
4
6
0
2
We read it as ―Thirty crores, ninety- eight lacs, seventy-two thousands five hundred and forty-six.‖ In this numeral: The place value of 6 is 6 ×1 = 6 The place value of 4 is 4 ×10 = 40 The place value of 5 is 5 ×100 = 500 The place value of 2 is2 ×1000 = 2000 and so on. The face value of a digit in a numbers is the value itself wherever it may be. Thus, the face value of 7 in the above numeral is 7. The face value of 6 in the above numeral is 6 and in the above numeral is 6 and so on. NUMBER SYSTEM Natural numbers Counting numbers 1, 2, 3, 4, 5,... are know as natural numbers. The set of all natural numbers, can be represented by N= {1, 2, 3, 4, 5,….} Whole numbers If we include 0 among the natural numbers, then the numbers 0, 1, 2, 3, 4, 5, … are called whole numbers. The set of whole number can be represented by W= {0, 1, 2, 3, 4, 5…} Clearly, every natural number is a whole number but 0 is a whole number which is not a natural number. INTEGERS All counting numbers and their negatives including zero are know as integers. The set of integers can be represented by Z or I = {…-4, -3, -2, -1, 0, 1, 2, 3, 4, …} Positive Integers The set I+ ={1, 2, 3, 4,…} is the set of all positive integers. Clearly, positive integers and natural numbers are synonyms. Negative Integers The set I- = {-1, -2, -3…} is the set of all negative integers. 0 is neither positive nor negative. Non-negative Integers The set {0, 1, 2, 3,…} is the set all non-negative integers. Rational Numbers The numbers of the form p/q, where p and q are integers and q ≠ 0, are known as rational numbers, e.g. 4/7, 3/2, 5/8, 0/1, -2/3, etc. The set of all rational numbers is denoted by Q. i.e. Q ={x:x =p/q; p,q belong to I, q≠0}. __________________________________________________________________________________________ -3
From the desk of SOHAIL MERCHANT Since every natural number ‗a‘ can be written as a/1, every natural number is a rational number. Since 0 can be written as 0/1 and every non-zero integer ‗a‘ can be written as a/1, every integer is a rational number. Every rational number has a peculiar characteristic that when expressed in decimal form is expressible rather in terminating decimals or in non-terminating repeating decimals. For example, 1/5 =0.2, 1/3 = 0.333…22/7 = 3.1428704287, 8/44 = 0.181818…., etc. The recurring decimals have been given a short notation as 0.333…. = 0.3 4.1555… = 4.05 0.323232…= 0.32. Irrational Numbers Those numbers which when expressed in decimal from are neither terminating nor repeating decimals are known as irrational number, e.g. √2, √3, √5, π, etc. Note that the exact value of is not 22/7. 22/7 is rational while π irrational number. 22/7 is approximate value of π. Similarly, 3.14 is not an exact value of it. Real Numbers The rational and irrational numbers combined together are called real numbers, e.g.13/21, 2/5, -3/7, 3, 4 + 2, etc. are real numbers. The set of all real numbers is denote by R. Note that the sum, difference or product of a rational and irrational number is irrational, e.g. 3+ √2, 4-√3, 2/3-√5, 4√3, -7√5 are all irrational. Even Numbers All those numbers which are exactly divisible by 2 are called even numbers, e.g.2, 6, 8, 10, etc., are even numbers. Odd Numbers All those numbers which are not exactly divisible by 2 are called odd numbers, e.g. 1, 3, 5, 7 etc., are odd numbers. Prime Numbers A natural number other than 1 is a prime number if it is divisible by 1 and itself only. For example, each of the numbers 2, 3, 5, 7 etc., are prime numbers. Composite Numbers Natural numbers greater than 1which are not prime, are known as composite numbers. For example, each of the numbers 4, 6, 8, 9, 12, etc., are composite numbers. Note: 1. 2. 3.
4.
The number 1 is neither a prime number nor composite number. 2 is the only even number which is prime Prime numbers up to 100 are: 2, 3, 5, 7, 11, 13, 17, 19,23, 29, 31, 37, 41,43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, i.e. 25 prime numbers between 1 and 100. two numbers which have only 1 as the common factor are called co-primes or relatively prime to each other, e.g. 3 and 5 are co-primes. Note that the numbers which are relatively prime need not necessarily be prime numbers, e.g. 16 and 17 are relatively prime although 16 is not a prime number.
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From the desk of SOHAIL MERCHANT ADDITION AND SUBTRACTION (SHORT-CUT METHODS) The method is best illustrated with the help of following example: Illustration 2 54321 – (9876+8976+7689) = ? Step 1 Add 1st column:
54321 9876 8967 7689 27789
6+7+9 = 22 To obtain 1 at unit‘s place add 9 to make 31. In the answer, write 9 at unit‘s place and carry over 3. Step 2 Add 2nd column: 3+7+6+8=24 To obtain 2 at tens place add 8 to make 32. In the answer, write 8 at ten‘s place and carry over 3. Step 3 Add 3rd column: 3 + 8 + 9 + 6 = 26 To obtain 3 at hundred‘s place, add 7 to make 33. In the answer, write 7 at hundred‘s place and carry over 3. Step 4 Add 4th column: 3 + 9 + 8 + 7 = 27 To obtain 4 at thousand‘s place add 7 to make 34. In the answer, write 7 at thousand‘s place and over 3. Step 5 5th column: To obtain 5 at ten-thousand‘s place add 2 to it to make 5. In the answer, write 2 at the ten-thousand‘s place. 54321 – (9876 + 8967 + 7689) = 27789. Common Factor A common factor of two or more numbers is a number which divides each of them exactly. For example, 4 is a common factor of 8 and 12. Highest common factor Highest common factor of two or more numbers is the greatest number that divides each one of them exactly. For example, 6 is the highest common factor of 12, 18 and 24. Highest Common Factor is also called Greatest Common Divisor or Greatest Common Measure. Symbolically, these can be written as H.C.F. or G.C.D. or G.C.M., respectively. Methods of Finding H.C.F. I. Method of Prime Factors Step 1 Express each one of the given numbers as the product of prime factors. [A number is said to be a prime number if it is exactly divisible by 1 and itself but not by any other number, e.g. 2, 3, 5, 7, etc. are prime numbers] Step 2 Choose Common Factors. Step 3 Find the product of lowest powers of the common factors. This is the required H.C.F. of given numbers. Illustration 1 Find the H.C.F. of 70 and 90. Solution 70 = 2 5 7 90 = 2 5 9 Common factors are 2 and 5. H.C.F. = 2 5 = 10.
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From the desk of SOHAIL MERCHANT Illustration 2 Find the H.C.F. of 3332, 3724 and 4508 Solution 3332 = 2 2 7 7 17 3724 = 2 2 7 7 19 4508 = 2 2 7 7 23 H.C.F. = 2 2 7 7 = 196. Illustration 3 Find the H.C.F. of 360 and 132. Solution 360 = 23 32 5 132 = 22 31 11 H.C.F. = 22 31 = 12. Illustration 4 If x = 23 35 59 and y = 25 37 511, find H.C.F. of x and y. Solution The factors common to both x and y are 23, 35 and 59. H.C.F. = 23 35 59. II. Method of Division A. For two numbers: Step 1 Greater number is divided by the smaller one. Step 2 Divisor of (1) is divided by its remainder. Step 3 Divisor of (2) is divided by its remainder. This is continued until no remainder is left. H.C.F. is the divisor of last step. Illustration 5 Find the H.C.F. of 3556 and 3444. 3444 )3556 (1 3444 112 ) 3444 ( 30 3360 84 ) 112 ( 1 84 28 ) 84 ( 3 84 B. For more than two numbers: Step 1 Any two numbers are chosen and their H.C.F. is obtained. Step 2 H.C.F. of H.C.F. (of(1)) and any other number is obtained. Step 3 H.C.F. of H.C.F. (of (2)) and any other number (not chosen earlier) is obtained. This process is continued until all numbers have been chosen. H.C.F. of last step is the required H.C.F. Illustration 6 Find the greatest possible length which can be used to measure exactly the lengths 7 m, 3 m 85 cm, 12 m 95 cm. Solution Required length = (H.C.F. of 700, 385, 1295) cm = 35 cm. Common Multiple A common multiple of two or more numbers is a number which is exactly divisible by each one of them. For Example, 32 is a common multiple of 8 and 16. 8 4 = 32 16 2 = 32. Least Common Multiple The least common multiple of two or more given numbers is the least or lowest number which is exactly divisible by each of them. For example, consider the two numbers 12 and 18. Multiples of 12 are 12, 24, 36, 48, 72, … __________________________________________________________________________________________ -6
From the desk of SOHAIL MERCHANT Multiple of 18 are 18, 36, 54, 72, … Common multiples are 36, 72, … Least common multiple, i.e. L.C.M. of 12 and 18 is 36. Methods of Finding L.C.M. A. Method of Prime Factors Step 1 Resolve each given number into prime factors. Step 2 Take out all factors with highest powers that occur in given numbers. Step 3 Find the product of these factors. This product will be the L.C.M. Illustration 7 Find the L.C.M. of 32, 48, 60 and 320. Solution 32 = 25 1 48 = 24 3 60 = 22 3 5 320 = 26 6 L.C.M. = 26 3 5 = 960.
B. Method of Division Step 1 The given numbers are written in a line separated by common. Step 2 Divide by any one of the prime numbers 2, 3, 5, 7, 11, … which will divide at least any two of the given nu8mbers exactly. The quotients and the undivided numbers are written in a line below the first. Step 3 Step 2 is repeated until a line of numbers (prime to each other) appears. 1 Find the product of all divisors and numbers in the last line which is the required L.C.M. Illustration 8 Find the L.C.M. of 12, 15, 20 and 54. Solution 2 12, 15, 20, 54 2 6, 15, 10, 27 3 3, 15, 5, 27 5 1, 5, 5, 9 1, 1, 1, 9 L.C.M. = 2 2 3 5 1 1 1 9 = 540. Note: Before finding the L.C.M. or H.C.F., we must ensure that all quantities are expressed in the same unit. Some Useful Short-Cut Methods 1. H.C.F. and L.C.M. of Decimals Step 1 Make the same number of decimal places in all the given numbers by suffixing zero(s) if necessary. Step 2 Find the H.C.F./L.C.M. of these numbers without decimal. Step 3 Put the decimal point (in the H.C.F./L.C.M. of step 2) leaving as many digits on its right as there are in each of the numbers. 2. L.C.M. and H.C.F. of Fractions L.C.M = L.C.M. of the numbers in numerators H.C.F. of the numbers in denominators H.C.F. = H.C.F. of the numbers in numerators L.C.M. of the numbers in denominators 3. Product of two numbers = L.C.M. of the numbers H.C.F. of the numbers
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From the desk of SOHAIL MERCHANT 4. To find the greatest number that will exactly divide x, y and z. Required number = H.C.F. of x, y and z. 5. To find the greatest number that will divide x, y and z leaving remainders a, b and c, respectively. Required number = H.C.F. of (x – a), (y – b) and (z – c). 6. To find the least number which is exactly divisible by x, y and z. Required number = L.C.M. of x, y and z. 7. To find the least number which when divided by x, y and z leaves the remainders a, b and c, respectively. It is always observed that (x – a) = (y – b) = (z – c) = k (say) Required number = (L.C.M. of x, y and z) – k. 8. To find the least number which when divided by x, y and z leaves the same remainder r in each case. Required number = (L.C.M. of x, y and z) + r. 9. To find the greatest number that will divide x, y and z leaving the same remainder in each case. (a) When the value of remainder r is given: Required number = H.C.F. of (x – r), (y – r) and (z – r). (b) When the value of remainder is not given: Required number = H.C.F. of (x – y), (y – z) and (z – x) 10. To find the n-digit greatest number which, when divided by x, y and z. (a) leaves no remainder (i.e. exactly divisible) Step 1 L.C.M. of x, y and z = L L ) n – digit greatest number ( Step 2 remainder = R Step 3 Required number = n-digit greatest number – R (b) leaves remainder K in each case Required number = (n-digit greatest number – R) + K. 11. To find the n-digit smallest number which when divided by x, y and z (a) leaves no remainder (i.e. exactly divisible) Step 1 L.C.M. of x, y and z = L L )n-digit smallest number( Step 2 remainder = R Step 3 Required number = n-digit smallest number + (L – R). (b) leaves remainder K in each case. Required number = n-digit smallest number + (L – R) + k.
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From the desk of SOHAIL MERCHANT EXERCISE 1.
The product of 4 consecutive positive integers greater than 1 is always divisible by (1) 24 (2) 48 (3) 72 (4) 120
2.
Which of the following sets of numbers are relative primes? (1) 51, 85 (2) 26, 65 (3) 57, 76 (4) 29, 75 (1) (2) (3) (4)
Only (2) Only (4) Both (1) and (3) All the above
3.
What is the cube root of 28 327 54? (1) 22 32 3 (2) 24 315 (3) 23 310 (4) 630
4.
What is the cube root of 22 32 42 6282 926? (1) 29 63 (2) 211 32 (3) 25 33 (4) 29 33
5.
How many of the following numbers 212,253,362,182 are divisible by 33? (1) 1 (2) 2 (3) 3 (4) 4
6.
13 +23+33+43+53 = (1) 202 (2) 152 (3) 252 (4) 122
7.
How many digits are required for numbering the pages of a book containing 1000 pages? (1) 1000 (2) 2892 (3) 3000 (4) 3126
8.
What is the total number of divisors of 1200? (1) 15 (2) 14 (3) 30 (4) 60
9.
What is the total number of divisors of 5040, including one and itself?
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From the desk of SOHAIL MERCHANT (1) (2) (3) (4)
54 60 27 25
10. A number when divided by 391 gives a remainder of 49. Find the remainder when it is divided by 13. (1) 10 (2) 9 (3) 8 (4) Cannot be determined 11. The least number that should be added or subtracted from 13218 to make it a perfect square is (1) 7 should be added. (2) 4 should be subtracted (3) 8 should be added (4) 3 should be subtracted 12. Find the value of K, if the number 1233K5 is divisible by 125. (1) 7 (2) 2 (3) 3 (4) 5 13. If the number 24P890 is divisible by 9, find the value of P. (1) 1 (2) 4 (3) 3 (4) 2 14. Evaluate : (75-74)/14 (1) 6 73 (2) 3 74 (3) 3 73 (4) 3 74 15. Find the sum of divisors of 480. (1) 1516 (2) 1512 (3) 1526 (4) 1412 16. What is the units digit of 65351? (1) 1 (2) 3 (3) 7 (4) 9 17. What is the digit in the units place of the product 23 49 5136? (1) 1 (2) 3 (3) 7 (4) 4.9
18. How many number between 100 and 300 are divisible by 11 (1) 11 __________________________________________________________________________________________ - 10
From the desk of SOHAIL MERCHANT (2) 10 (3) 12 (4) 18 19. The units digit in the sum 364102 + 364101 is (1) 4 (2) 6 (3) 0 (4) 8 20. Which of the following is a multiple of 8? (1) 468210 (2) 469828 (3) 4692304 (4) 4695028 21. The sum of first ‗r‘ even numbers is (1) r2 (2) r (r+1) (3) r2 + 2r (4) r(r-1) 22. The number 2837393449 is divisible by (1) 5 (2) 7 (3) 9 (4) 11 23. Which of the following numbers is exactly divisible by 11? (1) 27184 (2) 68039 (3) 587247 (4) 92939 24. The least number which when divided by 2, 3, 4, 5 or 6 leaves a remainder of 1 in each case is (1) 162 (2) 121 (3) 221 (4) 61 25. Find the least natural number which when divided by 18,24 and 30 leaves remainders 14, 20, and 26 respectively. (1) 256 (2) 356 (3) 456 (4) 326 26. Which of the following numbers, when divided by 10 leaves a remainder of 5, when divided by 20 leaves a remainder of 15, and when divided by 30 leaves a remainder of 25? (1) 135 (2) 165 (3) 115 (4) 65 27. Find the greatest number that will divide 55, 127 and 175 leaving the same remainder in each case. (1) 24 (2) 16 ______________________________________________________________________ - 11 -
From the desk of SOHAIL MERCHANT (3) 18 (4) 15 28. Find the greatest possible length which can be used to measure exactly the lengths 7 m, 3.85 m, and 12.95 m. (1) 30 cm (2) 35 cm (3) 45 cm (4) 38 cm 29. Find the greatest number of 4 digits and the least number of 5 digits, such that they have 144 as their HCF. (1) 9999, 10000 (2) 9936, 10080 (3) 9946, 10070 (4) 9956, 10090 30. Find the least positive whole number that should be added to 1515 to make it a perfect square. (1) 3 (2) 4 (3) 6 (4) 5 31. How many numbers are there in between 1 and 50 that are divisible by 3 or 4? (1) 28 (2) 24 (3) 30 (4) 32 32. A person distributes chocolates to some children. If the gives 3 chocolates to each child, he is left with 1 chocolate. If he give 4 chocolates to each child, still he is left with 1 chocolate. But if he gives 5 chocolates to each child, he is left with none. Find the least possible number of chocolates. (1) 13 (2) 26 (3) 25 (4) 11 33. 1996 question papers are to be packed in bundles so that the number of question papers in each bundle should be equal to the total number of bundle. If the objective is to pack a maximum possible number of question papers in a bundle, how many question papers have to be left out from packing? (1) 60 (2) 90 (3) 70 (4) 80 34. Three men start together to walk along a road at the same rate. The length of their strides are respectively 68 cms, 51 cms and 85cms. How far will they go before they are ―in-step‖ again? (1) 102 m (2) 1020 m (3) 10.2m (4) 150 m 35. A supervisor was employed on the condition that he will paid highest wages per day. The money to be paid was Rs.1184 as per contract. But was finally paid Rs.1073. For how many days did he actually work? (1) 29 (2) 35 (3) 37 (4) 31
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From the desk of SOHAIL MERCHANT RATIOS, PROPORTIONS AND VARIATION A ratio is a comparison of two quantities by division. It is a relation that one quantity bears to another with respect to magnitude. In other words, ratio means what part one quantity is of another. The quantities may be of same kind or different kinds. For example, when we consider the ratio of the weight 45 kg of a bag of rice to the weight 29kg of a bag of sugar we are considering the quantities of same kind but when we talk of allotting 2 cricket bats to 5 sportsmen, we are considering quantities of different kinds. Normally, we consider the ratio between quantities of the same kind. If a and b are two numbers, the ratio of a to b is a/b or a +b and is denoted by a : b. The two quantities that are being compared are called terms. The first is called antecedent and the second term is called consequent. For example, the ratio 3 : 5 represents 3/5 with antecedent 3 and consequent 5. Note: 1. A ratio is a number, so to find the ratio of two quantities, they must be expressed in the same units. 2. A ratio does not change if both of is terms are multiplied or divided by the same number. Thus, 2/3= 4/6 = 6/9 etc. TYPES OF RATIOS 1. 2. 3.
4.
5.
6.
Duplicate Ratio The ratio of the squares of two numbers is called the duplicate ratio of the two numbers. For example, 32/42 or 9/16 is called the duplicate ratio of ¾. Triplicate Ratio The ratio of the cubes of two numbers is called the triplicate ratio of the two numbers. For example, 33/43 or 27/64 is triplicate ratio of ¾. Sub-duplicate Ratio The ratio of the square roots of two numbers is called the sub-duplicate ratio of two numbers. For example, 3/4 is the sub- duplicate ratio of 9/16. Sub-duplicate Ratio The ratio of the cube roots of two numbers is called the sub-triplicate ratio of two numbers. For example, 2/3 is the sub-triplicate ratio of 8/27. Inverse Ratio or Reciprocal Ratio If the antecedent and consequent of a ratio interchange their places, the new ratio is called the inverse ratio of the first. Thus, if a : b be the given ratio, then 1/a : 1/b or b : a is its inverse ratio. For example, 3/5 is the inverse ratio of 5/3. Compound Ratio The ratio of the product of the antecedents to that of the consequents of two or more given ratios is called the compound ratio. Thus, if a :b and c:d are two given rations, then ac : bd is the compound ratio of the given ratios, For example, if ¾, 4/5 and 5/7 be the given ratios, then their compound ratios is 345/ 457, that is, 3/7. PROPORTION The equality of two ratios is called proportion. If a/b = c/d, then a, b, c and d are said to be in proportion and we write a : b: : c: d. This is read as ―a is to b as c is to d‖. For example, since ¾ = 6/8, we write 3; 4: : 6: 8 and say 3, 4, 6 and 8 are in proportion. Each term of the ratio a/b and c/d is called a proportional. a, b, c, and d are respectively the first, second, third and fourth proportionals Here, a, d are known as extremes and b, c are known as means. SOME BASIC FORMULAE 1.
2.
If four quantities are in proportion, then product of Means = product of Extremes For example, in the proportion a : b: : c: d, we have bc = ad. From this relation we see that if any three of the four quantities are given, the fourth can be determined. Fourth proportional If a: b: :c :x, x is called the fourth proportional of a, b, c. We have, a/b = c/x or, x = b×c/a Thus, fourth proportional of a, b, c is b × c / a.
Illustrational 1 Find a fourth proportional to the numbers 2, 5, 4. __________________________________________________________________________________________ - 13
From the desk of SOHAIL MERCHANT Solution Let x be the fourth proportional, then 2 : 5 : : 4 : x or 2/5 = 4/x. x = 5 × 4 /2 = 10. 3. Third proportional If a: b: : c: x, x is called the third proportional of a, b. We have, a/b= b/x or x= b2/a. Thus, third proportional of a. b is b2/a Illustration 2 Find a third proportional to the numbers 2.5, 1.5 Solution Let x be the third proportional, then 2.5 : 1.5 : :1.5 : x or 2.5/1.5= 1.5/x. x = 1.5 × 1.5/2.5 = 0.9 4. Mean Proportional If a: x: : x: b, x is called the mean or second proportional of a, b. We have, a/x =x/b or x2 = ab or x = ab Mean proportional of a and b is ab. We also say that a, x, b are in continued proportion Illustration 3 Find the mean proportional between 48 and 12. Solution Let x be the mean proportional. Then, 48 : x : : x : 12 or, 48/x = x/12 or, x 2= 576 or, x=24. 5. If a/b = c/d, then (i) (a + b)/b = (c +d)/d (Componendo) (ii) (a – b)/b = (c-d)/d (Dividendo) (iii) (a + b)/a-b = c +d/c-d = (Componendo and Dividendo) (iv) a/b = a + c/b+d = (a –c)/b-d Illustration 4 The sum of two umber is c and their quotient is p/q. Find the numbers. Solution Let the numbers be x, y. Given: x + y = c …(1) and, x/y = p/q …(2) x/ x+y = p/p+q x/c = p/p+q [Using (1)] x = pc/p +q. SOME USEFUL SHORT-CUT METHODS 1.
(a) If two numbers are in the ratio of a: b and the sum of these numbers is x, then these numbers will be ax/ a + b and bx/ a+b, respectively. or If in a mixture of x liters of, two liquids A and B in the ratio of a: b, then the quantities of liquids A and B in the mixture will be ax / a + b litres and bx/ a + b litres, respectively. (b) If three numbers are in the ratio a : b: c and the sum of these numbers is x, then these numbers will be ax / a + b + c , bx / a + b + c and cx / a + b + c, respectively. Explanation Len the three numbers in the ratio a: b: c be A, B and C. Then, A = ka, B = kb, C =kc and, A + B + C = ka + kb + kc = x k(a+b+c) = x k = x / a + b+ c. A = ka = ax / a + b+ c. B = kb = bx / a + b+ c. C = kc = cx / a + b+ c. Illustration 5 Two numbers are in the ratio of 4 : 5 and the sum of these numbers is 27. Find the two numbers. Solution Here, a = 4, b = 5, and x = 27. The first number = ax / a + b = 4 27 / 4+5 = 12. and, the second number = bx / a + b = 5 27 / 4+5 = 15. Illustration 6 Three numbers are in the ratio of 3: 4 : 8: and the sum of these numbers is 975. Find the three numbers. Solution Here, a = 3, b = 4, c = 8 and x = 975 __________________________________________________________________________________________ - 14
From the desk of SOHAIL MERCHANT The first number = ax / a + b+ c = (3 975)/ 3 + 4 + 8 = 195. The second number = bx / a + b+ c = (4 975)/ 3 + 4 + 8 = 260. and, the third number = cx / a + b+ c = (8 975)/ 3 + 4 + 8 = 520. 2. If two numbers are in the ratio of a : b and difference between these is x, then these numbers will be a) ax/ a-b and bx/ a-b, respectively (where a > b). b) ax/ a-b and bx/ a-b, respectively (where a < b). Explanation Let the two numbers be ak and bk. Let a > b. Given : ak – bk = x (a – b)k = x or k = x / (a-b). Therefore, the two numbers are ax / a-b and bx/ a-b. Illustration 7 Two numbers are in the ratio of 4 : 5. If the difference between these numbers is 24, then find the numbers. Solution Here, a = 4, b = 5 and x = 24. The first number = ax/ b-a = 4 24/5- 4 = 96 and, the second number = bx/ b-a = 5 24 / 5-4 = 120. 3. (a). If a : b = n1 : d1 and b : c = n2 : d2, then a : b : c = (n1n2) : (d1 n2) : (d1 d2). (b). If a : b = n1 : d1, b : c = n2 : d2, and c : d = n3 : d3 then a : b : c : d= (n1 n2 n3) : (d1 n2 n3 ) : (d1 d2 n3 ) : (d1 d2 d3 ). Illustration 8 If A : B = 3 : 4 and B : C = 8 : 9, find A : B : C. Solution Here, n1 = 3, n2 =8, d1 =4 and d2 = 9. a : b : c = (n1n2) : (d1n2) : (d1d2) = (38) : (48) : (49) = 24 : 32 : 36 or, 6: 8 : 9. Illustration 9 If A : B = 2 : 3, B:C = 4 : 5 and C : D = 6 : 7, find A :D. Solution Here, n1 = 2, n2 = 4, n3 = 6, d1 = 3, d2 = 5 and d3 = 7. A : B : C : D = (n1n2n3) : (d1n2n3) : (d1 d2 n3) : (d1 d2 d3) = (2 4 6) : (3 4 6) : (3 5 6) : (3 5 7) = 48 : 72 : 90: 105: or, 16: 24 : 30 ; 35. Thus, A : D = 16 : 35. 4. (a) The ratio between two numbers is a : b. If x is added to each of these numbers, the ratio becomes c : d. The two numbers are given as: ax(c – d) / ad – bc and bx(c- d) / ad –bc. Explanation Let two number be ak and bk. Given : ak +x / bk+x = c/d akd +dx= cbk + cx k(ad –bc) = x(c –d) k =x(c-d)/ ad – bc. Therefore, the two numbers are ax(c-d) / ad-bc and bx(c-d) / ad- bc (b) The ratio between two numbers is a : b. if x is subtracted from each of these numbers, the ratio becomes c : d. The two numbers are given as: ax(d-c) / ad-bc and bx(d-c) / ad- bc Explanation Let the two numbers be ak and bk. Given : ak-x/bk-x = c/d akd-xd = bck-xc k(ad-bc) = x(d-c) k = x(d-c)/ad-bc. Therefore, the two numbers are ax(d-c)/ad-bc and bx(d-c)/ad-bc Illustration 10 Given two numbers which are in the ratio of 3 : 4, If 8 is added to each of them, their ratio is changed to 5 : 6. Find two numbers. Solution We have, __________________________________________________________________________________________ - 15
From the desk of SOHAIL MERCHANT a : b = 3 : 4, c : d = 5 : 6 and x = 8. The first number = ax(c – d)/ ad –bc = 3 8(5-6) / (3 6- 4 5) = 12. and, the second number = bx(c – d)/ ad –bc = 4 8(5-6) / (3 6- 4 5) = 16. Illustration 11 The ratio of two numbers is 5 : 9. If each number is decreased by 5, the ratio becomes 5: 11. Find the numbers. Solution We have, a : b = 5: 9, c: d = 5: 11 and x =5. The first number = ax (d – c)/ ad –bc = 5 5(11-5)/ (511- 9 5) = 15. and, the second number= bx(d-c)/ad-bc 95(11-5)/(511-95)= 27. 5. (a) If the ratio of two numbers is a: b, then the numbers that should be added to each of the numbers in order to make this ratio c : d is given by ad-bc/ c-d. Explanation Let the required number be x. Given: a+x/ b+x = c/d ad+ xd = bc + xc x(d-c) =bc –ad or, x =ad-bc/c-d. (b)If the ratio of two numbers is a : b, then the number that should be subtracted from each of the numbers in order to make this ratio c : d is given by bc-ad/c-d. Explanation Let the required number be x. Given: a-x/ b-x = c/d ad- xd = bc - xc x(c-d) =bc –ad or, x = bc-ad/c-d. Illustration 12 Find the number that must be subtracted from the terms of the ratio 5 : 6 to make it equal to 2 : 3. Solution We have a : b= 5 : 6 and c: d =2 : 3. The required number = bc-ad/ c-d = 6 2-53/ 2-3 =3. Illustration 13 Find the number that must be added to the terms of the ratio 11 : 29 to make it equal to 11 : 20. Solution We have, a : b= 11 : 29 and c: d =11: 20. The required number = ad-bc/ c-d = 11 20-2911/ 11-20 = 11. EXERCISE 1.
2.
Divide Rs.1870 into three parts in such a way that half of the first part, one-third of the second part and one-sixth of the third part are equal. 1. 241, 343, 245 2. 400, 800, 670 3. 470, 640, 1160 4. None Divide Rs.500 among A, B, C and D so that A and B together get thrice as much as C and D together, B gets four times of what C gets and C gets 1.5 times as much as D. Now the amount c gets? 1. 300 2. 75 3. 125 4. None
3.
If 4 examiners can examine a certain number of answer books in 8 days by working 5 hours a day, for how many hours a day would 2 examiners have to work in order to examine twice the number of answer books in 20 days. 1. 6 2. 1/2 3. 8 4. 9
4.
In a mixture of 40 liters, the ratio of milk and water is 4:1. How much water much be added to this mixture so that the ratio of milk and water becomes 2:3
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From the desk of SOHAIL MERCHANT 1. 20 litres
2. 32 litres
3. 40 litres
4. 30 litres
5.
If three numbers are in the ratio of 1:2:3 and half the sum is 18, then the ratio of squares of the numbers is: 1. 6:12:13 2. 1:2:4 3. 36:144:324 4. None
6.
The ratio between two numbers is 3:4 and their LCM is 180. the first number is: 1. 60 2. 45 3. 15 4. 20
7.
A and B are tow alloys of argentums and brass prepared by mixing metals in proportions 7:2 and 7:11 respectively. If equal quantities of the two alloys are melted to form a third alloy C, the proportion of argentums and brass in C will be: 1. 5:9 2. 5:7 3. 7:5 4. 9:5
8.
If 30 men working 7 hours a day can do a piece of work in 18 days, in how many days will 21 men working 8 hours a day do the same work? 1. 24 days 2. 22.5 days 3. 30 days 4. 45 days
9.
The incomes of A and B are in the ratio 3:2 and their expenditure are in the ratio 5:3. If each saves Rs.1000, then, A‘s income is 1. 3000/2. 4000/3. 6000/4. 9000/-
10. If the ratio of sines of angles of a triangle is 1:1:2, then the ratio of square of the greatest side to sum of the squares of other two sides is 1. 3:4 2. 2:1 3. 1:1 4. Cannot be determined 11. Divide Rs.680 among A, B and C such that A gets 2/3 of what B gets and B gets 1/4 th of what C gets. Now the share of C is? 1. 480/2. 300/3. 420/4. None 12. A, B, C enter into a partnership. A contributes one-third of the whole capital while B contributes as much as A and C together contribute. If the profit at the end of the year is Rs.84, 000, how much would each received? 1. 24,000, 20,000, 40,000 2. 28,000, 42,000, 14,000 3. 28,000, 42,000, 10,000 4. 28,000, 14,000, 42,000 13. The students in three batches at AMS Careers are in the ratio 2:3:5. If 20 students are increased in each batch, the ratio changes to 4:5:7. the total number of students in the three batches before the increases were 1. 10 2. 90 3. 100 4. 150 14. The speeds of three cars are in the ratio 2:3:4. The ratio between the times taken by these cars to travel the same distance is 1. 2:3:4 2. 4:3:2 3. 4:3:6 4. 6:4:3
15. Rs.2250 is divided among three friends Amar, Bijoy and Chandra in such a way that 1/6th of Amar‘s share, 1/4th of Bijoy‘s share and 2/5th of chandra‘s share are equal. Find Amar‘s share. __________________________________________________________________________________________ - 17
From the desk of SOHAIL MERCHANT 1. 720/-
2.1080/-
3. 450/-
4. 1240/-
16. After an increment of 7 in both the numerator and denominator, a fraction changes to ¾. Find the original fraction. 1. 5/12 2. 7/9 3. 2/5 4. 3/8 17. The difference between two positive numbers is 10 and the ratio between them is 5:3. Find the product of the two numbers. 1. 375 2. 175 3. 275 4. 125 18. If 30 oxen can plough 1/7th of a field in 2 days, how many days 18 oxen will take to do the remaining work? 1. 30 days 2. 20 days 3. 15 days 4. 18 days 19. A cat takes 5 leaps for every 4 leaps of a dog, but 3 leaps of the dog are equal to 4 leaps of the cat. What is the ratio of the speed of the cat to that of the dog? 1. 11:15 2. 15:11 3. 16:15 4. 15:16 20. The present ratio of ages of A and B is 4:5. 18 years ago, this ratio was 11:16. Find the sum total of their present ages. 1. 90 years 2. 105 years 3. 110 years 4. 80 years 21. Three men rent a farm for Rs.7000 per annum. A puts 110 cows in the farm for 3 months, B puts 110 cows for 6 months and C puts 440 cows for 3 months. What percentage of the total expenditure should A pay? 1. 20% 2. 14.28% 3. 16.66% 4. 11.01% 22. 10 students can do a job in 8 days, but on the starting day, two of them informed that they are not coming. By what fraction will the number of day required for doing the whole work get increased? 1. 4/5 2. 3/8 3. 3/4 4. 1/4 23. A dishonest milkman mixed 1 liter of water for every 3 liters of milk and thus make up 36 liters of milk. If he now adds 15 liters of milk to the mixture, find the ratio of milk and water in the new mixture. 1. 12:5 2. 14:3 3. 7:2 4. 9:4 24. Rs.3000 is distributed among A, B and C such that A gets 2/3rd of what B and C together get and C gets ½ of what A and B together get. Find C‘s share 5 1. 750/2. 1000/3. 800/4. 1200/25. If the ratio of the ages of Maya and Chhaya is 6:5 at present, and fifteen years from now, the ratio will get changed to 9:8, then find Maya‘s present age. 1. 24 years 2. 30 years 3. 18 years 4. 33 years 26. If Rs.58 is divided among 150 children such that each girl and each boy gets 25 p and 50 p respectively. Then how many girls are there? 1. 52 2. 54 3. 68 4. 62 27. If 391 bananas were distributed among three monkeys in the ratio ½:2/3:3/4, how many bananas did the first monkey get? 1. 102 2. 108 3. 112 4. 104 __________________________________________________________________________________________ - 18
From the desk of SOHAIL MERCHANT 28. A mixture contains milk and water in the ratio 5:1. On adding 5 liters of water, the ratio of milk to water becomes 5:2. the quantity of milk in the mixture is: 1. 16 litres 2. 25 litres 3. 32.5 litres 4. 22.75 litres 29. A beggar had ten paise, twenty paise and one rupee coins in the ratio 10:17:7 respectively at the end of day. If that day he earned a total of Rs.57, how many twenty paise coins did he have? 1. 114 2. 171 3. 95 4. 85 30. Vijay has coins of he denomination of Re.1, 50 p and 25 p in the ratio of 12:10:7. The total worth of the coins he has is Rs.75. Find the number of 25 p coins that Vijay has 1. 48 2. 72 3. 60 4. None 31. If two numbers are in the ratio of 5:8 and if 9 be added to each, the ratio becomes 8:11. Now find the lower number. 1. 5 2. 10 3. 15 4. None 32. A cask contains a mixture of 49 liters of wine and water in the proportion 5:2. How much water must be added to it so that the ratio of wine to water may be 7:4? 1. 3, 5 2. 6 3. 7 4. None 33. A cask contains 12 gallons of mixture of wine and water in the ratio 3:1. How much of the mixture must be drawn off and water substituted, so that wine and water in the cask may become half and half. 1. 3 litres 2. 5 litres 3. 6 litres 4. None of these 34. The total number of pupils in three classes of a school is 333. the number of pupils in classes I and II are in the ratio 3:5 and those in classes II and III are in the ratio 7:11. Find the number of pupils in the class that had the highest number of pupils. 1. 63 2. 105 3. 165 4. 180
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From the desk of SOHAIL MERCHANT PERCENTAGES Introduction The term per cent means per hundreds or for every hundred. It is the abbreviation of the Latin phrase per centum. Scoring 60 per cent marks means out of every 100 marks the candidate scored 60 marks. The term per cent is sometimes abbreviated as p.c. The symbol % is often used for the term per cent. Thus, 40 per cent will be written as 40%. A fraction whose denominator is 100 is called a percentage and the numerator of the fraction is called rate per cent, e.g. 5/100 and 5 per cent means the same thing, i.e. 5 parts out of every hundred parts. 1. To Convert a fraction into a per cent: to convert any fraction l/m to rate per cent, multiply it by 100 and put % sign, i.e. l/m 100% 2. To Convert a Percent into a Fraction: To convert a per cent into a fraction , drop the per cent sign and divide the number by 100. 3. To find a percentage of a given number: x % of given number (N) = x/100 N. Some useful shortcut methods 1. (a) if A is x% more than that of B, then B is less than that of A by
x 100 x 100% (b) If a is x% less than that of B, then B is more than that of A by
x 100 x 100% 2. If a is x% of C and B is y% of C, then A = x/y 100% of B. 3. (a) If two numbers are respectively x% and y% more than a third number, then the first number is
100 x 100% 100 y
of the second and the second is
100 y 100% of the first. 100 x
(b) If two numbers are respectively x% and y% less than a third number, then the first number is
100 x 100% of the first. 100 y 4. (a) If the price of a commodity increases by P%, then the reduction in consumption so as not to increase the expenditure is
P 100% . 100 P
(b) If the price of a commodity decreases by p%, then the increase in consumption so as not to decrease the expenditure is
P 100% . 100 P
5. If a number is changed (increased/decreased) successively by x% and y%, then net% change is given by (x+y+(xy/100))% which represents increase or decrease in value according as the sign is +ve or –ve. If x or y indicates decrease in percentage, then put –ve sign before x or y, otherwise +ve sign.
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From the desk of SOHAIL MERCHANT 6. If two parameters A and B are multiplied to get a product and if A is changed (increased/decreased) by x% and another parameter B is changed (increased/decreased) by y%, then the net% change in the product (A B) is given (x+y+(xy/100))% which represents increase or decrease in value according as the sign in +ve or –ve. If x or y indicates decrease in percentage, then put –ve sign before x or y, otherwise +ve sign. 7. If the present population of a town (or value of an item) be P and the population (or value of item) changes at r% per annum, then
r (a) Population (or value of item) after n years = P1 100 P (b) Population (or value of item) n years ago = n r 1 100
n
where r is +ve or –ve according as the population (or value of item) increase or decreases. 8. If a number A is increased successively by x% followed by y% and then by z%, then the final value of A will be
x y z A1 1 1 100 100 100 In case a given value decreases by any percentage, we will use a negative sign before that. 9. In an examination, the minimum pass percentage is x%. If a student secures y marks and fails by z marks, then the maximum marks in the examination is
100( y z) . x
10. In an examination x% and y% students respectively fail in two different subjects while z% students fail in both the subjects, then the percentage of students who pass in both the subjects will be (100-(x+y-z))%. EXERCISE 1.
What is 20% of 50% of 75% of 70? 1. 5.25 2. 6.75
3. 7.25
4. 5.50
2.
Ram sells his goods 25% cheaper than Shyam and 25% dearer than Bram. How much percentage is Bram‘s goods cheaper than Shyam‘s? 1. 33.33% . 50% 3. 66.66% 4. 40%
3.
In an election between 2 candidates, Bhiku gets 65% of the total valid votes. If the total votes were 6000. What is the number of valid votes that the other candidate Mhatre gets if 25% of the total votes were declared invalid? 1. 1625 2. 1575 3. 1675 4. 1525
4.
In a medical certificate, by mistake a candidate gave his height as 25% more than normal. In the interview panel, he clarified that his height was 5 feet 5 inches. Find the percentage correction made by the candidate from his stated height to his actual height. 1. 20 2. 28.56 . 25 4. None
5. __________________________________________________________________________________________ - 21
From the desk of SOHAIL MERCHANT 6.
Arjit Sharma generally wears his father‘s coat. Unfortunately, his cousin Shaurya poked him one day that he was wearing a coat of length more than his height by 15%. If the length of Arjit‘s father‘s coat is 120 cm then find the actual length of his coat. 1. 105 2. 108 3. 104.34 4. 102.72
7.
***In a mixture of 80 liters of milk and water, 25% of the mixture is milk. How much water should be added to the mixture so that milk becomes 20% of the mixture? 1. 20 liters 2. 15 liters 3. 25 liters 4. None
8.
50% of a% of b is 75% of b% of c. Which of the following is c? 1. 1.5a 2. 0.667a 3. 0.5a 4. 1.25a
9.
***A landowner increased the length and the breadth of a rectangular plot by 10% and 20% respectively. Find the percentage change in the cost of the plot assuming land prices are uniform throughout his plot. 1. 33% 2. 35% 3. 22.22% 4. None
10. The height of a triangle is increased by 40%. What can be the maximum percentage increase in length of the base so that the increase in area is restricted to a maximum of 60%? 1. 50% 2. 20% 3. 14.28% 4. 25% 11. The length, breadth and height of a room in the shape of a cuboid are increased by 10%, 20% and 50% respectively. Find the percentage change in the volume of the cuboids. 1. 77% 2. 75% 3. 88% 4. 98% 12. The salary of Amit is 30% more than that of Varun. Find by what percentage is the salary of Varun less than that of Amit? 1. 26.12% 2. 23.07% 3. 21.23% 4. None 13. ***The price of sugar is reduced by 25% but in spite of the decrease, Aayush ends up increasing his expenditure on sugar by 20%. What is the percentage change in his monthly consumption of sugar? 1. +60% 2. –10% 3. +33.33% 4. 50% 14. The price of rice falls by 20%. How much rice can be bought now with the money that was sufficient to buy 20 kg of rice previously? 1. 5kg 2. 15 kg 3. 25 kg 4. 30 kg 15. 30% of a number when subtracted from 91, gives the number itself. Find the number. 1. 60 2. 65 3. 70 4. None 16. ***At an election, the candidate who got 56% of the votes cast won by 144 votes. Find the total number of voters on the voting list if 80% people cast their vote and there were no invalid votes. 1. 360 2. 720 3. 1800 4. 1500 17. The population of a village is 1,00,000. The rate of increase is 10% per annum. Find the population at the start of the third year? 1. 1, 33,100 2. 1, 21, 000 3. 1, 20, 000 4. None
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From the desk of SOHAIL MERCHANT 18. the population of the village of Gavas Is 10, 000 at this moment. It increases by 10% in the first year. However, in the second year, due to immigration, the population drops by 5%. Find the population at the end of the third year if in the third year the population increases by 20%. 1. 12, 340 2. 12, 540 3. 1, 27, 540 4. 12, 340 19. A man invests Rs.10,000 in some shares in the ratio 2:3:5 which pay dividends of 10%, 25% and 20% (on his investment) for that year respectively. Find his dividend income. 1. 1900 2. 2000 3. 2050 4. 1950 20. *In an examination, Mohit obtained 20% more than Sushant but 10% less than Rajesh. If the marks obtained by Sushant is 1080, find the percentage marks obtained by Rajesh if the full marks is 2000. 1. 86.66% 2. 72% 3. 78.33% 4. None 21. In a class, 25% of the students were absent for an exam. 30% failed by 20 marks and 10% just passed because of grace marks of 5. Find the average score of the class if the remaining students scored an average of 60 marks and the pass marks are 33 (counting the final scores of the candidates). 1. 37.266 2. 37.6 3. 37.8 4. 36.93 22. Ram spends 20% of his monthly income on his household expenditure. 15% of the rest on books, 30% of the rest on clothes and saves the rest. On counting, he comes to know that he has finally saved Rs.9520. Find his monthly income. 1. 10000 2. 15000 3. 20000 4. None 23. Hans and Bhaskar have salaries that jointly amount of Rs.10,000 per month. They spend the same amount monthly and then it is found that the ratio of their savings is 6:1. Which of the following is Hans‘s salary? 1. 6000/2. 5000/3. 4000/4. 3000/24. The population of a village is 5500. If the number of males increases by 11% and the number of females increases by 20%, then the population becomes 6330. Find the population of females in the town. 1. 2500 2. 3000 3. 2000 4. 3500 25. Vicky‘s salary is 75% more than Ashu‘s. Vicky got a raise of 40% on his salary while Ashu got a raise of 25% on his salary. By what percent is Vicky‘s salary more than Ashu‘s? 1. 96% 2. 51.1% 3. 90% 4. 51.1% 26. On a shelf, the first row contains 25% more books than the second row and the third row contains 25% less books then the second row. If the total number of books contained in all the rows is 600, then find the number of books in the first row. 1. 250 2. 225 3. 300 4. None 27. An ore contains 25% of an alloy that has 90% iron. Other than this, in the remaining 75% of the ore, there is no iron. How many kilograms of the ore are needed to obtain 60 kg of pure iron? 1. 250kg 2. 275 kg 3. 300 kg 4. 266.66 kg
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From the desk of SOHAIL MERCHANT 28. Last year, the Indian cricket team played 40 one-day cricket matches out of which they managed to win only 40%. This year, so far it has played some matches, which has made it mandatory for it to win 80% of the remaining matches to maintain its existing winning percentage. Find the number of matches played by India so far this year? 1. 30 2. 25 3. 28 4. Insufficient information 29. In the recent, climate conference in New York, out of 700 men, 500 women, 800 children present inside the building premises, 20% of the men, 40% of the women and 10% of the children were Indians. Find the percentage of people who were not Indian? 1. 73% 2. 77% 3. 79% 4. 83% 30. Ram sells his goods 20% cheaper than Bobby and 20% dearer than Chandilya. How much percentage is Chandilya‘s goods cheaper/dearer than Bobby‘s 1. 33.33% 2. 50% 3. 42.85% 4. None 31. Out of the total production of iron from hematite, an ore of iron, 20% of the ore gets wasted, and out of the remaining iron, only 25% is pure iron. If the pure iron obtained in a year from a mine of hematite was 80, 000 kg, then the quantity of hematite mined from that mine in the year is 1. 5, 00, 000 kg 2. 4, 00, 000 kg 3. 4, 50, 000 kg 4. None 32. Recently, while shopping in Patna Market in Bihar, I came across two new shirts selling at a discount. I decided to buy one of them for my little boy Sherry. The shopkeeper offered me the first shirt for Rs.42 and said that it usually sold for 8/7 of that price. He then offered me the other shirt for Rs.36 and said that it usually sold for 7/6th of that price. Of the two shirts which one do you think is a better bargain and what is the percentage discount on it? 1. First shirt, 12.5% 2. second shirt, 14.28% 3. Both are same 4. None 33. 4/5th of the voters in Bellary promised to vote for Sonia Gandhi and the rest promised to vote for Sushma Swaraj. Of these voters, 10% of the voters who had promised to vote for Sonia Gandhi did not vote on the Election Day, while 20% of the voters who had promised to vote for Sushma Swaraj did not vote on the Election Day. What is the total no. of votes polled if Sonia Gandhi got 216 votes? 1. 200 2. 300 3. 264 4. 100 34. Ravana spends 30% of his salary on house rent, 30% of the rest he spends on his children‘s education and 24% of the total salary he spends on clothes. After his expenditure, he is left with Rs.2500. What is Ravana‘s salary? 1. 11, 494, 25/2. 20, 000/3. 10, 000/4. 15, 000/35. The entrance ticket at the Minerva theatre in Mumbai is worth Rs.250. When the price of the ticket was lowered, the sale of tickets increased by 50% while the collections recorded a decrease of 17.5%. Find the deduction in the ticket price. 1. 150/2. 112.5/3. 105/4. 120/36. In the year 2000, the luxury car industry had two car manufacturers—Maruti and Honda with market shares of 25% and 75% respectively. In 2001, the overall market for the product increased by 50% and a new player BMW also entered the market and captured 15% of the market share. If we know that the market share Maruti increased to 50% in the second year, the share of Honda in that year was: 1. 50% 2. 45% 3. 40% 4. 35% __________________________________________________________________________________________ - 24
From the desk of SOHAIL MERCHANT Averages & Mixtures Whenever we are asked the marks scored by us in any examination, we usually tell the marks in percentage, taking the percentage of total marks of all subjects. This percentage is called average percentage. Also, in a class, If there are 100 students, instead of knowing the age of individual student, we usually talk about average age. The average or mean or arithmetic of a number of quantities of the same kind is equal to their sum divided by the number of those quantities. For example, the average of 3, 11, 15, 18,19, and 23 is 3 + 9 +11+ 15+ 18+ 19+ 23+ /7 = 98/7 = 14. SOME BASIC FORMULAE 1. 2. 3.
Average = sum of quantities/ Number of quantities Sum of quantities = Average Number of quantities Number of quantities = Sum of quantities/ Average
Illustration 1 A man purchased 5 toys at the rate of Rs 200each, 6 toys at the rate of Rs 250each and 9 toys at the rate of Rs 300 each. Calculate the average cost of one toy. Price of 5 toys = 200 5 = Rs 1000 Price of 6 toys = 250 6 = Rs 1500 Price of 9 toys = 300 9 = Rs 2700 Average price of 1 toy = 1000 + 1500 + 2700/ 20 = 5200/20 = Rs 260. Illustration 2 The average marks obtained by 200 students in a certain examination are 45. Find the total marks. Solution
Solution Total marks = Average marks Number of students = 200 45 = 900. Illustration 3 Total temperatures for the month of September is 8400C, If the average temperature of that month is 280C, find of how many days is the month of September. Solution Number of days in the month of September = Total temperature/ Average temperature = 840/28 = 30days. SOME USEFUL SHORT–CUT METHODS 1.
Average of two or more groups taken together a) If the number of quantities in two groups be n1 and n2 and their average is x and y, respectively, the combined average (average of all of then put together) is n1x +n2y / n1 + n2 Explanation No. of quantities in fist group = n1 Their average = x Sum = n1 x No. of quantities in second group = n2 Their average = y Sum = n2 y No. of quantities in the combined group = n1+n2 Total sum (sum of quantities of first group and second group) = n 1x+n2y Combined Average = n1x+n2y./ n1 +n2. b). If the average of n1 quantities is x and the average of n2 quantities out of them is y, the average of remaining group (rest of the quantities) is __________________________________________________________________________________________ - 25
From the desk of SOHAIL MERCHANT n2x – n2y/ n1- n2. Explanation No. of quantities = n1 Their average = x Sum = n1x No of quantities taken out – n2 Their average = y Sum = n2y Sum of remaining quantities = n1x – n2y No. of remaining quantities = n1 – n2 Average of remaining group = n1x – n2y/ n1 – n2 Illustration 4 The average weight of 24 students of section A of a class is 58 kg whereas the average weight of 26 students of section B of the same class is 60. 5 kg. Find the average weight of all the 50 students of the class. Solution Here, n1 = 24, n2 = 26, x = 58 and y = 60.5. Average weight of all the 50 students = n1x+n2y/ n1 +n2 = 24 ×58 + 26×60.5 / 24+26 = 1392 +1573/ 50 = 2965/ 50 =59.3kg. Illustration 5 Average salary of all the 50 employees including 5 officers of a company is Rs 850. If the average salary of the officers is Rs 2500, find of the class. Solution Here, n1 = 50, n2 =5, x = 850and y = 2500. Average salary of the remaining staff = n1x-n2y/ n1-n2 = 50850 -52500 / 50-5 = 42500-12500/ 45 = 30000/ 45 = Rs 667(approx) 2. If x is the average of x1, x2, …, xn, then a) The average of x1 + a, x2 + a, …., xn + a is x +a. b) The average of x1 - a, x2 - a, …., xn - a is x -a. c) The average of ax1, ax2,….,axn is ax, provided a ≠ 0. d) The average of x1 / a, x2 / a, …., xn / a isx /a, provided a ≠ 0. Illustration 6 The average value of six numbers 7, 12, 17, 24, 26 and 28 is 19. If 8 is added to each number, what will be the new average? Solution The new average = x +a. = 19+8 = 27. Illustration 7 The average value of x numbers is 5x. If x – 2 is subtracted from each given number, what will be the new average? Solution The new average =x -a. = 5x- (x-2) = 4x +2. Illustration 8 The average of 8 numbers is 21.If each of the numbers multiplied by 8, find the average of a new set of numbers. Solution The average of a new set of numbers = ax = 8 21 = 168. 3. The average of n quantities is equal to x. If one of the given quantities whose value is p, is replaced by a new quantity having value q, the average becomes y, then q = p+n(y-x) Illustration 9 The average weight of 25 persons is increased by 2 kg when one of them whose weight is 60kg, is replaced by a new person. What is the weight of the new person? __________________________________________________________________________________________ - 26
From the desk of SOHAIL MERCHANT Solution The weight of the new person = p + n(y-x) = 60 + 25(2)= 110kg 4.
a). The average of n quantities is equal to x. When a quantity is removed, the average becomes y. The value of the removed quantity is n(x- y)+y. b) The average of n quantities is equal to x. When a quantity is added, the average becomes y. The value of the new quantity is n(y-x)+y
Illustration10 The average are of 24 students and class teacher is16 years, If the class teacher‘s age is excluded, the average age reduces by 1 year. What is the age of the class teacher? Solution The age of class teacher = n(x- y) + y = 25(16 – 15) + 15 = 40 years. Illustration 11 The average age of 30 children in a class is 9 years. If the teacher‘s age be included, the average age becomes 10years. Find the teacher‘s age. Solution The teacher‘s age = n(y- x) + y = 30(10 – 9) +100 = 40 years. 5.
a).The average of first n natural numbers is (n +1) /2 b). The average of square of natural numbers till n is (n +1)(2n+1)/6. c). The average of cubes of natural numbers till n is n(n +1) 2/4 d). The average of odd numbers from 1 to n is (last odd number +1) / 2 e). The average of even numbers from 1 to n is (last even number + 2) / 2.
Illustration 12 Find the average of first 81natural number. Solution The required average = n + 1/ 2 = 81 + 1 /2 = 41. Illustration 13 What is the average of squares of the natural numbers from 1 to 41? Solution The required average = (n+1)(2n+1)/ 6 = (41+1)(241+1)/ 6 = 42 83/ 6 = 3486/ 6 = 581 Illustration 14 Find the average of cubes of natural numbers from 1 to 27. Solution The required average = n(n +1)2 / 4 = 27(27+1)2 / 4 27 28 28 / 4 = 21168 / 4 = 5292. Illustration 15 What is the average of odd numbers from 1 to 40? Solution The required average = last odd number + 1/ 2 = 39 +1/ 4 =20. Illustration 16 What is the average of even numbers from 1 to 81? Solution The required average = last even number + 2/ 2 = 80+2 = 41. 6. a).If n is odd: The average of n consecutive numbers, consecutive even numbers or consecutive odd numbers is always the middle number. b). If n is even: The average of n consecutive numbers, consecutive even numbers or consecutive odd numbers is always the average of the middle two numbers. c). The average of first n consecutive numbers is (n+1). d). The average of first n consecutive odd numbers is n. e). The average of squares of first n consecutive even number is2 (n+1)(2n+1) / 3. __________________________________________________________________________________________ - 27
From the desk of SOHAIL MERCHANT f). The average of squares of consecutive even number till n is (n+)(n+2) / 3. g). The average of squares of squares of consecutive odd numbers till n is n(n+2)/ 3. h). If the average of n consecutive numbers is m, then the difference between the smallest and the largest number is 2(n-1). Illustration 17 Find the average of 7 consecutive numbers 3, 4, 5, 6, 7, 8, 9. Solution The required average= middle number=6. Illustration 18 Find the average of consecutive odd numbers 21, 23, 25, 27, 29, 31, 33, 35. Solution The required average = average of middle two numbers = average of 27 and 29 = 27+29 / 2 = 28. Illustration 19 Find the average of first 31 consecutive even numbers. Solution The required average = (n+1) = 31+ 1= 32. Illustration 20 Find the average of first 50 consecutive odd numbers. Solution The required average = n = 50. EXERCISE 1.
The average of 13 papers is 40. The average of the first 7 papers is 42 and of the last seven papers is 35. Find the marks obtained in the 7th paper? (A) 23 (B) 38 (C) 19 (D) None of these
2.
The average age of the Indian cricket team playing the Nagpur test is 30. The average age of 5 of the players is 27 and that of another set of 5 players, totally different from the first five, is 29. If it is the captain who was not included in either of these two groups, then find the age of the captain. (A) 75 (B) 55 (C) 50 (D) Cannot be determined
3.
A bus goes to Ranchi from Patna at the rate of 60 km per hour. Another bus leaves Ranchi for Patna at the same times as the first bus at the rate of 70 km per hour. Find the average speed for the journeys of the two buses combined if it is known that the distance from Ranchi to Patna is 420 kilometers. (A) 64.615 kmph (B) 64.5 kmph (C) 63.823 kmph (D) 64.82 kmph
4.
A train travels 8 km in the first quarter of an hour, 6 km in the second quarter and 40 km in the third quarter. Find the average speed of the train per hour over the entire journey. (A) 72 km/h (B) 18 km/h (C) 77.33 km/h
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From the desk of SOHAIL MERCHANT (D) 78.5 km/h 5.
The average weight of 6 men is 68.5 kg. If I is known that Ram and Tram weigh 60 kg each, find the average weight of the others. (A) 72.75 kg (B) 75 kg (C) 78 kg (D) None of these
6.
The average score of a class of 40 students is 52. What will be the average score of the rest of the students if the average score of 10 of the students is 61. (A) 50 (B) 47 (C) 48 (D) 49
7.
The average age of 80 students of IIM, Bangalore of the 1995 batch is 22 years. What will be the new average if we include the 20 faculty members whose average age is 37 years? (A) 32 years (B) 24 years (C) 25 years (D) None of these
8.
Out of the three numbers, the first is twice the second and three times the third. The average of the three numbers is 88. The smallest number is (A) 72 (B) 36 (C) 42 (D) 48
9.
The sum of three numbers is 98. If the ratio between the first and second is 2 : 3 and that between the second and the third is 5 : 8, then the second number is (A) 30 (B) 20 (C) 58 (D) 48 10. The average height of 30 girls out of a class of 40 is 160 cm and that of the remaining girls is 156 cm. The average height of the whole class is (A) 158 cm (B) 158.5 cm (C) 159 cm (D) 157 cm 11. The average weight of 6 persons is increased by 2.5 kg when one of them whose weight is 50 kg is replaced by a new man. The weight of the new man is (A) i65 kg (B) 75 kg (C) 76 kg (D) 60 kg __________________________________________________________________________________________ - 29
From the desk of SOHAIL MERCHANT 12. The average age of A, B C and D five years ago was 45 years. By including X, the present average age of all the five is 49 years. The present age of X is (A) 64 years (B) 48 years (C) 45 years (D) 40 years 13. The average salary of 20 workers in an office is Rs. 1900 per month. If the manager‘s salary is added, the 11 average salary becomes Rs. 2000 per month. What is the manager‘s annual salary? (A) Rs. 24, 000 (B) Rs. 25,200 (C) Rs. 45,600 (D) None of these 14. The average weight of a class of 40 students is 40 kg. If the weight of the teacher be included, the average weight increases by 500 gm. The weight of the teacher is (A) 40.5 kg (B) 60 kg (C) 62 kg (D) 60.5 kg 15. In a Infosys test, a student scores 2 marks for every correct answer and loses 0.5 marks for every wrong answer. A student attempts all the 100 questions and scores 120 marks. The number of questions he answered correctly was (A) 50 (B) 45 (C) 60 (D) 68 16. The average of the first ten natural numbers is (A) 5 (B) 5.5 (C) 6.5 (D) 6 17. The average of the first ten whole numbers is (A) ****4.5 (B) 5 (C) 5.5 (D) 4 18. The average of the first ten even numbers is (A) 18 (B) 22 (C) 9 (D) 11
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From the desk of SOHAIL MERCHANT 19. The average weight of a class of 30 students is 40 kg. If, however, the weight of the teacher is included, the average become 41 kg. The weight of the teacher is (A) 31 kg (B) 62 kg (C) 71 kg (D) 70 kg 20. 30 oranges and 75 apples were purchased for Rs. 510. If the price per apple was Rs. 2, then the average price of oranges was (A) Rs. 12 (B) Rs. 14 (C) Rs. 10 (D) Rs. 15 21. A batsman made an average of 40 runs in 4 innings, but in the fifth inning, he was out on zero. What is the average after fifth innings? (A) 32 (B) 22 (C) 38 (D) 49 22. The average weight of a school of 40 teachers is 80 kg. If, however, the weight of the principle be included, the average decreases by 1 kg. What is the weight of the principal? (A) 109 kg (B) 29 kg (C) 39 kg (D) None of these 23. The average age of Ram and Shyam is 20 years. Their average age 5 years hence will be (A) 25 years (B) 22 years (C) 21 years (D) 20 years 24. The average of 20 results is 30 and that of 30 more results is 20. For all the results taken together, the average is (A) 25 (B) 50 (C) 12 (D) 24
25. The average of 5 consecutive numbers is 18. The highest of these numbers will be (A) 24 (B) 18 (C) 20 (D) 22
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From the desk of SOHAIL MERCHANT 26. Three years ago, the average age of a family of 5 members was 17 years. A baby having been born, the average of the family is the same today. What is the age of the baby? (A) 1 years (B) 2 years (C) 6 months (D) 9 months 27. Varun average daily expenditure is Rs. 10 during May, Rs. 14 during June and Rs. 15 during July. His approximate daily expenditure for the 3 months is (A) Rs. 13 approximately (B) Rs. 12 (C) Rs. 12 approximately (D) Rs. 10 28. A ship sails out to a mark at the rate of 15 km per hour and sails back at the rate of 20 km/h. What is its average rate of sailing? (A) 16.85 km (B) 17.14 km (C) 17.85 km (D) 18 km 29. The average temperature on Monday, Tuesday and Wednesday was 41 0C and on Tuesday, Wednesday and Thursday it was 40 0C. If on Thursday it was exactly 39 0 C, then on Monday, the temperature was (A) 42 0C (B) 46 0C (C) 23 0C (D) 26 0C 30. The average of 20 results is 30 out of which the first 10 results are having an average of 10. The average of the rest 10 results is (A) 50 (B) 40 (C) 20 (D) 25 31. A man had seven children. When their average age was 12 years a child aged 6 years died. The average age of the remaining 6 children is (A) 6 years (B) 13 years (C) 17 years (D) 15 years 32. The average weight of 35 students is 35 kg. If the teacher is also included, the average weight increases to 36 kg. The weight of the teacher is (A) 36 kg (B) 71 kg (C) 70 kg (D) 45 kg __________________________________________________________________________________________ - 32
From the desk of SOHAIL MERCHANT 33. The average of x, y and z is 45. x is as much more than the average as y is less than the average. Find the value of z. (A) 45 (B) 25 (C) 35 (D) 15 34. The average salary per head of all the workers in a company is Rs. 95. The average salary of 15 officers is Rs. 525 and the average salary per head of the rest is Rs. 85. Find the total number of workers in the workshop. (A) 660 (B) 580 (C) 650 (D) 46
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From the desk of SOHAIL MERCHANT PROFIT AND LOSS Business transactions have now-a-days become common feature of life. When a person deals in the purchase and sale of any item, he either gains or loses some amount generally. The aim of any business is to earn profit. The commonly used terms in dealing with questions involving sale and purchase are: Cost Price The cost price of an article is the price at which an article has been purchased. It is the abbreviated as C.P. Selling Price The selling price of an article is the price at which an article has been sold. It is abbreviated as S.P. Profit or Gain If the selling price of an article is more that the cost price, there is a gain or profit. Thus, Profit or Gain = S.P- C.P. Loss If the cost price of an article is greater than the selling price, the suffers a loss. Thus, Loss = C.P- S.P. Note that profit and loss are always calculated with respect to the cost price of the item. Illustration 1. (i)If C.P. = Rs. 235, S.P. = Rs. 240, then profit = ? (ii) If C.P. = Rs. 116, S.P. = Rs. 107, then loss = ? Solution (i) Profit = S.P.- C.P. =Rs. 240- 235 =Rs.5. (ii) Loss = C.P.- S.P. = Rs. 116- 107 =Rs.9. SOME BASIC FORMULAE 1.
Gain on Rs. 100 is Gain per cent Gain% = (Gain 100)/C.P Loss on Rs. 100 is Loss per cent Loss% = (Loss 100)/C.P
Illustration 2 The cost price of a shirt is Rs. 200 and selling price is Rs. 250. Calculate the % profit. Solution We have, C.P. = Rs. 200, S.P = Rs. 250. Profit = S.P.- C.P. = 250- 200 =Rs.50. Profit% = profit 100/ C.P = 50 100/ 250 = 25% Illustration 3 Anu bought a necklace for Rs. 750 and sold it for Rs. 675. Find her percentage loss. Solution Here, C.P. = 750, S.P. = Rs. 675. Loss= C.P- S.P. = 750-675 = Rs. 75. Loss% = Loss 100/ C.P = 75 100/ 750 = 10% 2. When the selling price and gain% are given: C.P = 100S.P / (100+Gain%) 3. When the cost and gain per cent are given; S.P = (100+Gain%)C.P/ 100 4. When the cost and loss per cent are given: S.P = (100-Loss%)C.P / 100 5. When the selling price and loss per cent are given: C.P =(100)S.P / (100-Loss%) Illustration 4 Mr. Sharma buys a cooler for Rs. 4500. For how much should he so that there is a gain of 8%? Solution We have, C.P. = Rs. 4500, gain% = 8% S.P = (100+Gain%/100)C.P. = (100+ 8/ 100) 4500 108/100 4500 = Rs. 4860 __________________________________________________________________________________________ - 34
From the desk of SOHAIL MERCHANT Illustration 5 By selling a fridge Rs. 7200, Pankaj loses 10%. Find the cost price of the fridge. Solution We have, S.P. = Rs. 7200, loss% = 10%. C.P =(100/100-Loss%)S.P. = (100/100-10) 7200 100/90 7200= Rs. 8000. Illustration 6 By selling a pen for Rs. 99, Mohan gains 12 ½ %. Find the cost price of the pen. Solution Here, S.P. = Rs. 99, gain% = 12 ½% or 25/2%. C.P =(100/100+Gain%)S.P. = (100/100+25/2) 99 = (1002/ 225) 99 =Rs. 88 SOME USEFUL SHORT-CUT METHODS 1.
If a man buys x items for Rs. y and sells z items for Rs. w, then the gain or loss per cent made by him is (xw/zy -1) 100%.
Explanation S.P. of z items = Rs. w S.P. of x items = Rs. w/z x Net profit =w/z x-y. % profit = w/z x-y/y 100% i.e.(xw/zy -1) 100, which represent loss, if the result is negative. Note: In the case of gain per cent the result obtained bears positive sign whereas in the case of loss per cent the result obtained bears negative sign. How to remember: 1. 2.
Cross-multiply the numbers connected by the arrows (xw and zy) Mark the directon of the arrows for crossmultiplicaton. The arrow going down forms the numerator while the arrow going up forms the denominator (xw/ zy).
Illustration 7 If 11 oranges are bought for Rs. 10 and sold at 10 for Rs. 11 what is the gain loss%? Solution % profit= (xw/zy -1) 100% = (1111/1010-1) 100% = 21/100 100% = 21% Illustration 8 A fruit seller buys apples at the rate of Rs 12 per dozen and sells them at the rate of 15 for Rs.12. Find his percentage gain or loss. Solution % gain or loss = (xw/ zy -1) 100% = (12 12/15 12 -1) 100% = -36/144 100% = -25% Since the sign is –ve, there is a loss of 25%. 2. If the cost price of m articles is equal to the selling price of n articles, then % gain or loss = ( m-n/n) 100 [If m > n, it is gain and if m n, m, n N, and x > x if ______ a) x > 1 b) x = 1 c) 0 < x < 1 d) x < 0 8. 0 < x < 1; m, n N; m > n; then m n a) x > x m n b) x = x m n c) x < x d) none of these m 9. (ab) = _____ m a) ab m b) a b m m c) a b d) abm
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From the desk of SOHAIL MERCHANT m
10.
a _____ b m
11.
12.
13.
14.
15.
a) a /b m b) a/b m m c) a /b d) a/bm m n m+n x x x = X ; then _____ a) m, n, N b) m, n Z c) m, n, Q d) m, n, R m-n n-m a x a = _______ a) 1 b) 0 (m-n)2 c) a m-n-m-n d) a m-n n-p p-m x x x x x = _____ a) 1 b) 0 (m-n)(n-p)(p-m) c) x 2m-2n-2p d) x 3/2 1/2 1/2 a xa xa 5/2 a) a 3/8 b) a 5/2 c) 3x 5/2 d) 2x
xm n x
m n
xn p x
n p
xp m x
pm
a) 1 b) 0 m n p m n n p p q p q n
c) a d) none of these 1/p 16. a expressed in radical form = _______ a) b) c) d)
a p 1 a p a ap
p/q
17. a = _____ p a) a /q b) ap/q c) d)
p
aq
q
ap
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From the desk of SOHAIL MERCHANT 18.
19.
m
a n _____ m/n
a) b) c) d)
a n/m a n-m a 1/n m.a
mn
x ____
a)
m
b)
n
xn
c)
xm mn x
d)
xn
m
20. If x 12 x then a) p/q = 12 b) pq = 12 c) q/p = 12 d) p – q = 12 x x+3 21. 2 x _____ = 2 a) 3 p q
b)
x 3 x 3
c) 2 d)
x x 3
m+k = x
22. x m – 3, then k = ______ a) 3 b) -3 3 c) x -3 d) x 2 3 23. (3x ) = ______ 2 a) 27x 5 b) 27x 6 c) 27x 6 d) 3x -1/4 24. 5 = _____ a) b) c)
1 54 1 4 5 4 5 4 5
d) x x+2 25. 3 = 81; then 3 = a) 83 b) 162 c) 729 d) 90
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From the desk of SOHAIL MERCHANT Exercise-2 x
x-2
1. 5 = 625; What is the value of 5 ? a) 623 b) 625/2 c) 25 d) 600 x 2x 2. If 3 = 8, what is the value of 3 ? a) 16 b) 64 c) 48 d) 24 x x 3. 4 = 1024; then 2 = _____ a) 512 b) 32 c) 1022 d) Can‟t be found x+3 x-3 4. 3 = 729; then 3 = _____ a) 0 b) 1 c) 27 d) 81 1/4 5. 81 = _____ a) 3 b) -3 c) 1/3 d) – 1/3 3/2
6. If x = 0.2, then x _____ a) 0.8 b) 8 c) .08 d) .008 1/2 7. (0.0064) = _______ a) .8 b) .008 c) .08 d) .0008 1/4 -1/3 8. 16 x 8 = ______ a)
1 1 4 3
2 2
1 1 3 4
b) c)
1
d)
128
9.
3 4
1 12
x _____ x
a) b)
x 1 x
c)
12
d)
3 4
x
__________________________________________________________________________________________ - 157 -
From the desk of SOHAIL MERCHANT
x x ; then x = x
x x
10. If x = a) 0 b) 1 c) 3/2 d) 3 11.
1 1 x 2 a) b) c) d)
+
1 1 x 2
=
0 1 2 x 2 +x 2 3
64 = a) 4 b) ¼ c) 8 d) 1/16 2 3 13. If a = 0.04; then a = a) 0.02 b) 00.8 c) 0.08 d) 0.008 12.
x = 0.3, then x = 0.9 0.009 0.1 0.09
14. If a) b) c) d)
81 4 = 3
15.
a) 27 b) 1/27 c) -243/4
81
d)
2/3
3 4
16. (0.027) = a) 0.018 b) 0.009 c) 0.09 d) 0.006 1
17.
1
81 4 125 3 625 8 a) b) c) d)
4.2 3.6 3.5 3.1 3
18. If x = a) b) c) d)
27 2 ; then x = 125
9/25 3/5 9/5 9/125
__________________________________________________________________________________________ - 158 -
From the desk of SOHAIL MERCHANT x
19. If 2 a) b) c) d) 20.
a
–2 1 27 3 2
2 3
x
a
a)
x-1
a
1 3
x
= 4; then x =
=
2 9
2
2 1
b) a 3 3 2 c) a d) a -m 21. x = (m N) a) –m . x
1 xm
b)
c) x . d) x
m
1 m
12
22.
a a 3 9
a) a -15 b) a 15 c) a 15 d) 1/a 0 23. a (a 0) = a) a b) 0 c) d) 1 x+3 x 24. 2 is obtained by multiplying 2 with a) 2 b) ½ c) 8 d) 1/8 a x 4a 25. If (3 ) = 3 , then x = a) 3 b) 4 c) 3a d) 4a
__________________________________________________________________________________________ - 159 -
From the desk of SOHAIL MERCHANT SURDS ____________________________________________________________________________________ Exercise-1 1. A surd is ______ a) any irrational number b) irrational root of a rational number c) irrational root of a nutural number d) irrational root of an irrational number 2.
n
a
is a surd if and only if _______ n
a a) b) n is rational c) a is rational and n a is irrational d) a is rational 3. Which of the following number is a surd? a)
3
b)
3
c)
64 16 27 3 8 5 32
d) 4. Which of the following is not a surd? a) b) c) d) 5.
8 16 4 8 5 8 3
n
a is a surd of order : a) a b) n c) d)
a n n a
6. Which of the following is a quadratic surd? a) b) c)
3
a2 a 9 4
d) 3 7. Two surds x and y are said to be like if ______ a) x and y are natural numbers b) x/y is a natural number c) x/y is a rational number d) x/y is irrational
8. The product of two quadratic like surds is a __________________________________________________________________________________________ - 160 -
From the desk of SOHAIL MERCHANT a) like surd with them b) an irrational number c) a rational number d) a natural number 9. If the product of two surds is a rational number, each of them is called a ______ of the other. a) rational factor b) like surd c) rationalizing factor d) factor
x
10. The rationalizing factor of the sured a) b)
3
c) d) x
x x
x2
11. The rationalizing factor of a)
3
b) c)
3
d)
2
3
a2
is _____
a2 a2 a a2
12. The rationalizing factor of
a b
is ______
a b b a a b a b
a) b) c) d)
13. What is the rationalizing factor of a)
x y
b)
2 x 3 y
c)
3 x 2 y
d)
x y
14. What is the rationalizing factor of a)
3
a 3 b
b)
3
a 2 3 ab 3 b 2
c)
3
a 2 3 b2
2 x 3 y
3
a 3 b
15. Which is the rationalizing factor of a a) b) c)
is ______
?
?
2 b
a2 b a 2b 2a b
__________________________________________________________________________________________ - 161 -
From the desk of SOHAIL MERCHANT d)
2a b
16. Which is the rationalizing factor of a)
3
x2 3 y2
b)
3
x 2 3 xy 3 y 2
c)
3
x2 3 y2
d)
3
x 2 3 xy 3 y 2
17. What is the rationalizing factor of a)
x 2 x3 y 3 y 2
b)
x3 y
c)
x 3 y2
d)
x2 3 y
3
x 3 y ?
x3 y
?
18. Two quadratic Surds are said to be conjugate to each other if _______ a) Their sum is rational b) Their product is rational c) if their sum of product is rational d) Their sum and product are both rational 19. What is the surd conjugate to a) b) c) d)
a b
?
a b a b a b 2a b
20. What is the surd conjuagate to
2a 3 b
3a 2 b b) 2a 3 b c) 3a 3 b d) 2a b 1 = _____ 2 3 a) 2 3 b) 3 2 c) 3 2 d) 2 5 6 a)
21.
22. If the denominator of the surd a) 5 2 b) 1 c)
1 5 2 6
is rationalized, the resulting fraction equals ______
6
5 12
__________________________________________________________________________________________ - 162 -
From the desk of SOHAIL MERCHANT 25 6 a b a b a b a) a b b) d) 23.
c) a + b d) a - b 24.
1
1
x y x 3 y 3
a) b) c) d)
2 3
2 3
2 3
1 3
2 3
2 3
2 3
1 3
x y
1 3
2 3
1 3
2 3
x x y y x y
x x y y a 2 ab b
25. a) b) c) d)
a b a b a b a b
__________________________________________________________________________________________ - 163 -
From the desk of SOHAIL MERCHANT EXERCISE-2 1. If
1 1 ;b 2 3 2 3
a
then ab =
a) 1 b) 0
2 d) 2 5 2 If 74
3 3 3 is written in the form of a-b 3 3
c)
2.
a) b) c) d) 3.
then value of b is
5 7 2 6
5 2 3 a b 3 ; then a = 74 3 a) b) c) d)
11 12 9 -11
4. Rationalising factor of a)
5
x2
is
7
x7
b) c)
x
7
7
x5
5
5
x d) 5. If the sum and product of two surds is a rational number, then they are _______ surds. a) similar b) dissimilar c) conjugate d) none 6. If a) b) c) d)
x
3 2 3 2
1 x x
then
3 5 2 10
1 1 3 2 3 5 3 5 3
7. a) b) c) d)
1 0 -1 2
8. The simplest form of
3
768 is
__________________________________________________________________________________________ - 164 -
From the desk of SOHAIL MERCHANT a) b)
4 3 12 8 3 12 16 3 12
c) d) none 9.
3 2 4 2 6 6 3 6 2 3 2 a) 2 b) 2 5 c) 0 d) 1
10.
11.
30 12 6 a) 2 3 3 2 b) 3 2 2 3 c) 3 5 5 3 6 5 d)
3 2
a) b) c) d)
=
3 32 6
12. a) b) c) d) 13.
52 6
0 1 -1 2
2 1 2 3 1
3
1 4
1 4
3 3 1 2 3 1
8 2 15 3 5 a) 5 3 b) 3 5 c) d) none
3 5 3 5
14. a)
10 8
b) c) 0 d) -2
15.
4x y 4 xy
__________________________________________________________________________________________ - 165 -
From the desk of SOHAIL MERCHANT
16.
a)
2 x y
b)
2 x y
c)
y 2 x
d)
x y
m n 2 mn m n a) b) m – n c) m + n d)
17.
m n
92 6 4 2 4 3 a) 2 5 6 b) 2 2 3 3 8 c) d) None
6 7 27 4 35
18. a) b) c) d) 19.
5 5 5 1
2 2 1 5
21 4 5 8 3 4 15 a) 2 3 2 5 b) 3 3 5 c) 3 3 2 5 d) None
7
20.
16 6 7 16 6 7
a) a rational b) multiple of c) 0 d) None 21.
7
12 68 48 2 a) b)
2 2 2 2 4 2
c) d) none
__________________________________________________________________________________________ - 166 -
From the desk of SOHAIL MERCHANT 98 50 8
22. a) b) c) d)
0 -1 1 2
1
23.
12 140 a) b) c) d)
1 8 60
a)
25.
1 10 84
1 2 -1 0
24. The rationalizing factor of
b)
3
c)
3
d)
3
81
81 81 9 9
2x 1 2 x 2 x 6 x 2 x 3 a) x 2 x 3 b) x 3 x 2 c) x 3 x 2 d)
__________________________________________________________________________________________ - 167 -
From the desk of SOHAIL MERCHANT LINEAR EQUATIONS, INEQUATIONS & MODULUS ____________________________________________________________________________________ Exercise-1 1. The graph of 3x – 5y + 16 > 0 is _____ a) A line b) The half plane on the origin side of the line 3x – 5y + 16 = 0 c) The half plane on the side of the line not containing the origin d) The line 3x – 5y + 16 = 0 and the original side of it. 2. The graph of 2x – 3y + 5 ≥ 0 is _____ a) The region on the original side of the line 2x – 3y + 5 > 0 b) The line 2x -3y + 5 = 0 and the region on the origin side of it. c) The line 2x – 3y + 5 = 0 and the side of it not containing the origin d) The region on the side of the line not containing the origin. 3. The intersection of the graph of x – 2y + 3 ≥ 0 and the line x – 2y = 0 is _____ a) ø b) The line x – 2y + 3 = 0 c) The half plane on the origin side of the line x – 2y + 3 = 0 d) The half plane on the non-origin side of the line x – 2y + 3 = 0 4. The intersection of the graphs x + y – 5 > 0 and x + y – 5 < 0 is _____ a) The line x + y – 5 = 0 b) The half plane on the origin side of the line x + y – 5 = 0 c) ø d) The half plane on the non-origin side of the line x + y – 5 = 0 5. A point belonging to the region 5x – 4y – 9 > 0 is _____ a) (4, 5) b) (5, 4) c) (6, 4) d) (6, 7) 6. A point which satisfies 3x + y > 6 is …….. a) (1, 0) b) (2, -1) c) (1, 3) d) (2, 1) 7. A point in the region 2x – 3y < 5 is _____ a) (-1, -1) b) (3, -3) c) (-2, -4) d) (0, -4) 8. A point satisfying x + y ≤ 4 is _____ a) (2, 4) b) (3, 1) c) (3, 2) d) (3, 4) 9. The point which does not lie on the straight line x + y = 6 is _____ a) (3, 3) b) (2, 4) c) (3, 2) d) (4, 2) 10. A point which lies in the are represented by x + y > 3 is a) (1, 1) b) (1, 0) c) (1/3; 1/3) d) (1, 3) 11. A point which satisfies the condition x + 2y + 5 ≥ 0 is _____ __________________________________________________________________________________________ - 168 -
From the desk of SOHAIL MERCHANT
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
a) (2, -1) b) (1, -4) c) (2, -4) d) (-3, 0) A point which lies in the area 3x + y – 5 ≤ 0 is _____ a) (1, 2) b) (1, 3) c) (3, -1) d) (4, 0) Point (2, 3) lies in the area represented by a) x + y < 2 b) x + y > 5 c) 2x + y < 6 d) x + 2y > 7 (3, 4) belongs to the area represented by _____ a) 3x – 4y + 8 ≤ 0 b) 3x – 4y + 7 ≤ 0 c) 3x + 4y + 9 ≤ 0 d) 3x – 4y > 0 Which of the following points lies on the same side of the line 4x + 3y – 25 = 0 as (3, 4)? a) (4, 3) b) (3, 5) c) (4, 4) d) (2, 3) Which of the following points lies on the side of 2x + 3y – 12 = 0 not containing (3, 3)? a) (1, 1) b) (2, 3) c) (1, 4) d) (-1, 5) Which pair of the points lie on the same side of the line 5x – 2y + 5 = 0? a) (2, 1), (1, 2) b) (-1, 1), (3, 2) c) (5, 2), (-5, -2) d) (-1, -1), (-2, -1) Which of the following pairs of points lie on different sides of the line 5x + 3y – 15 = 0. a) (1, 6), (2, 1) b) (2, 2), (1, 4) c) (0, 0), (-2, 2) d) (5, 3), 6, 2) A point which lies on the some side of the line 3x + 5y – 30 = 0 containing the origin is _____ a) (0, 0), (1, 1) b) (2, 1), (0, 2) c) (1, 0), (0, 1) d) (-1, 0), (0, 0) A point which lies on the same side of the line 3x + 5y – 30 = 0 containing the origin is _____ a) (10, 1) b) (8, 2) c) (7, -1) d) (7, 3) 3x + 2y should not be less than 50. This can be represented as _____ a) 3x + 2y > 50 b) 3x + 5y = 50 c) 3x + 2y ≥ 50 d) 3x + 5y ≤ 50
22. 3x + 2y would be at most 50. This can be represented as _____ a) 3x + 2y < 50 __________________________________________________________________________________________ - 169 -
From the desk of SOHAIL MERCHANT b) 3x + 2y ≤50 c) 3x + 5y > 50 d) 3x + 2y = 50 23. 3x should not exceed 5y by more than 10. This can be represented as _____ a) 3x – 5y ≤ 10 b) 3x – 5y = 10 c) 3x – 5y < 10 d) 3x – 5y > 10 24. A chair costs Rs. 50 and a table costs Rs. 150. The cost of x chairs and y tables should not exceed R. 5000. How do you represent this? a) 50x + 150 y = 5000 b) 50x - 150 y < 5000 c) 50x + 150 y ≥ 5000 d) 50x + 150 y ≤ 5000 25. If an iso-profit line coincides with a boundary of the feasible polygonal region, we have ____ a) 1 solution b) more than 1 solution c) Infinite number of solutions d) no solution
__________________________________________________________________________________________ - 170 -
From the desk of SOHAIL MERCHANT EXERCISE-2 1. The objective function sought to be maximized is 3x + 4y. The vertices of the polygonal region representing the solution set are (0, 0), (0, 1), (8, 0), (4, 8). At which vertex has it the minimum value? a) (0, 0) b) (8, 0) c) (4, 8) d) (0, 1) 2. The objective function sought to be minimized is 5x + 2y. The vertices of the polygonal region representing the solution the solution set are (0, 16), (4, 18) and (12, 0). At which vertex has it the minimum value? a) (0, 16) b) (12, 0) c) (4, 8) d) none of these 2 3. The solution set which satisfies the inequations x – 4x + 3 < 0 a) (1, 4) b) (1, -4) c) (1, -3) d) (-4, 3) 4. x < 0; y > 0; (x, y)lies in the quadrant a) Q1 b) Q2 c) Q3 d) Q4 5. The point which does not lie in the region 2x – 3y > 5. a) (1, 1) b) (3, -3) c) (-2, -4) d) (0, -4) 6. The inequations with the solution set 1 < x < 3 is 2 a) x + 4x + 3 > 0 2 b) x – 4x + 3 < 0 2 c) x – 4x – 3 = 0 2 d) x – 4x + 4 > 0 7. The point which belongs to the region indicated by the inequations x + 3y < -5 is a) (2, 1) b) (-2, -1) c) (-3, 1) d) (-3, -1) 8. Which of the following inequations represents the region containing the points (1, 2) and (2, 1) a) x + y < 2 b) x + y > 5 c) 2x + y < 6 d) x + 2y > 7 9. If |x + 1| < 6, then „x‟ belongs to the set a) {x / - 7 < x < 5} b) {x / - 7 < x ≤ 5} c) {x / - 7 ≤ x < 5} d) {x / - 7 ≤ x ≤ 5} 2 10. x – 4x + 3 < 0, then the value of x lies between: a) 1 and 2 b) 1 and 3 c) 1 and -4 d) 4 and 3 11. If x > 0, y < 0, then (x, y) lies in : __________________________________________________________________________________________ - 171 -
From the desk of SOHAIL MERCHANT
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
a) Q1 b) Q2 c) Q3 d) Q4 The inequation which represents the shaded region is a) 2x + 3y > 6 b) 2x + 3y < 6 c) 3x + 2y < 6 d) 3x + 2y > 6 If | -3x | < 6, then a) -3 < x < 3 b) -1 < x < 1 c) -2 < x < 2 d) -2 > x > 2 The point which is in the region satisfying the inequation 2x – 3 + 5 < 0 a) (1, 1) b) (-1, 1) c) (2, -1) d) (-1, 2) x = k is the solution of the following inequations : a) x k ; y k b) x k ; y k c) x k ; y k d) x k ; x k The graph of the inequality x > 0 is a) + ve Y – axis b) + ve X – axis c) - ve Y – axis d) - ve X – axis If | x | = a (a > 0), then x + a) a b) –a c) a or –a d) a are a If | 5x – 1| > 9, then which of the following belongs to the solution set: a) 1 b) 0 c) 2 d) 3 2 The graph of y = mx is a a) parabola b) hyperbola c) ellipse d) straight line The solution of | x | = -4 a) 4 b) -4 c) 4 or -4 d) no solution If x < 0; y < 0 then (x, y) lies in a) Q1 b) Q2 c) Q3 d) Q4 The figure given below represents a) 3x + 2y 6
__________________________________________________________________________________________ - 172 -
From the desk of SOHAIL MERCHANT b) 3x + 2y 6 c) 3x + 2y > 6 d) 3x + 2y < 6 23. The value of 2x + 5y should not be less than 75. This is represented by the following inequation a) 2x + 5y < 75 b) 2x + 5y 75 c) 2x + 5y 75 d) 2x + 5y > 75 24. Solution of | x – 2 | > 6 a) x < 8 : x > 4 b) x > 8 c) x > 8 : x < -4 d) x > 8 : x > -4 25. If 12 – 3x > 0, then a) x = 4 b) x > 4 c) x < 4 d) x < -4
__________________________________________________________________________________________ - 173 -
From the desk of SOHAIL MERCHANT POLYNOM., REMAINDER, & SQUARE ROOTS ____________________________________________________________________________________ n
n-1
n-2
1. a0x + a1x + a2x + …+ an-1 x + an is a polynomial in x of the nth degree, if n is a whole number and … 0. a) a0 b) an c) a1 d) a2 n n-1 n-2 2. a0x + a1x + a2x + …+ an is called the zero polynomial if … a) a0 = a1 = a2 = . . . = an b) a0 + a1 + … + an = 0 c) a0 = a1 = a2 = … = an = 0 d) a0 = 0 3. Which of the following is not a polynomial in x? 2 a) x 2 b) 2x + 3x - 5 2 c) x d) 5x 4. Which of the following is a polynomial in x? -1/2 a) x 1/2 b) x c) 2/x d) x/2 th th 5. f(x) and g(x) are polynomials of the 8 and the 4 degrees respectively in x. What is the degree of f(x) / g(x) ? a) 12 b) 4 c) 2 d) any degree less than 8 th nd 6. f(x) and g(x) are polynomials of the 10 and the 2 degree respectively. What is the degree of the remainder when f(x) is divided by g(x)? a) 8 b) 2 c) 0 d) 1 or 0 7. What is the remainder when f(x), a rational integral function in x is divided by x – a? a) a b) f(a) c) –f(a) d) f(-a) 8. What is the remainder when f(x) rational integral function in x divided by (ax + b)? a) f(-b) b) f(b/a) c) f(-(b/a)) d) f(-a) 9. If f(x) a rational integral function in x, is divided by (x + a), the remainder is . . . a) f(a) b) f(-a) c) f(1/a) d) f 2 10. If ax + bx + c is divisible by x-1, then … a) a + b = 0 b) a + c = 0 c) a + b + c = 0 d) a + b = c 11. if f(x) is divided by 4x – 5, the remainder is . . . __________________________________________________________________________________________ - 174 -
From the desk of SOHAIL MERCHANT
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
a) f(4/5) b) f(- 4/5) c) f(5/4) d) f(- 5/4) If f(x) is divided by 5x + 4, the remainder is . . a) f(4/5) b) f(- 4/5) c) f(- 5/4) d) f(5/4) 2 What is the remainder when 3x – 4x + 9 s divided by x + 4? a) 41 b) 73 c) 1 d) 3 3 What is the remainder when x – 3x+ + 3x - 5 is divided by x - 1? a) 4 b) -46 c) -6 If the sum of the coefficients of a polynomial in x is zero, then . . . is one of its factors. a) x b) x – 1 c) x + 1 d) x - 2 3 2 The condition for x + 1 being a factor of ax + bx + cx + d . . . is one of its factors. a) a + b = c + d b) a + b + c + d =0 c) a + b + c = d d) a + c = b + d 2 4 3 2 x – 1 is a factor of ax + bx + cx + dx + e if . . . a) a + b + c + d + e = 0 b) a + c + e = b + d c) a + c + e = b + d = 0 d) a + b = c + d + e 2 If ax -5x+6 is divisible by x – 2, then a = . . . a) 1 b) 2 c) 6 d) 3 2 2 x – 3 is a factor of 3x – x + px + q if . . . a) p + q = 72 b) b) p + q = 72 c) 3p + q = -72 d) q – 3p = 72 3 2 Which of the following is a factor of x – 6x + 12x – 8? a) x – 2 b) x + 1 c) x + 2 d) x - 3 If f(a/b) = 0, then a factor of f(x) is _____ a) ax – b b) bx – a c) ax + b d) bx + a
22. The sum of the coefficients of the odd power terms of x and the sum of the coefficients of the even power terms of x are equal. Then ____ is a factor of the expression. __________________________________________________________________________________________ - 175 -
From the desk of SOHAIL MERCHANT a) x – 1 b) x c) x + 1 2 d) x - 1 23. If f(2/3) = 0, then ____ is a factor of f(x). a) 2x – 3 b) 2x + 3 c) 3x – 2 d) 3x + 2 2 24. One of the factors of x + 19x – 20 is ____ a) x + 1 b) x – 1 c) x – 1 d) x + 2 25. If 2x2 + 9x + k is divisible by x – 3, then k = a) 11 b) 7 c) 9 d) -45
__________________________________________________________________________________________ - 176 -
From the desk of SOHAIL MERCHANT QUADRATIC EQUATIONS & EXPRESSIONS ____________________________________________________________________________________ 2
1. ax + bx + c = 0 is a quadratic polynomial equation in x if a) a 0 b) b 0 c) a = b 0 d) c = 0 2. The number of roots of a quadratic polynomial equation in a single variable is ____ a) 1 b) 3 c) 2 d) infinity 2 3. The roots of the quadratic equation ax + bx + c = 0 are ____ a)
b)
c)
d)
b b 2 4ac b b 2 4ac ; 2a 2a 2 2 b b 4ac b b 4ac ; 2a 2a 2 b b 4ac b b 2 4ac ; 2a 2a 2 b b ac b b 2 ac ; 2a 2a 2
4. A root of the equation 13x – 22x – 8 = 0 is a) 3 b) 2 c) 1 d) -2 2 5. A root of 25x + 57x + 32 = 0 is ____ a) 5 b) 1 c) 2 d) -1 2 6. One of the roots of the equation x – 2x – 1 = 0 is ____ a) b) c)
2 1 32 3 1 2 1
d) 2 7. One of the roots of the quadratic equation x + 2x – 1 = 0 is ____ a) b) c) d)
2 1 3 1 2 1 32
__________________________________________________________________________________________ - 177 -
From the desk of SOHAIL MERCHANT 2
8. One of the roots of the equation x – 4x – 1 = 0 is _____ a) b) c)
2 5 2 1 2 2 5 2
d) 2 9. If and are the roots of the equation ax + bx + c = 0, then + = _____ a) b) c) d)
b a
c a b a c a
10. If and are the roots of the equation px + qx + r = 0, then = _____ a) p/r b) –p/r c) r/p d) –r/p 2 2 2 11. If and are the roots of the equation ax + bx + c = 0, then + = _____ 2 a) b – 4ac 2 2 b) (b -2ac) / a 2 c) (ab -2c) / a 2 d) (ab +2c) / a 2 12. If one of the roots of the equation 3x – 2x – k = 0 is 1, then k = ______ a) 1 b) 3 c) -1 d) -2 2 13. If the roots of the equation ax + bx + c = 0 are a and 1/a, then _____ a) a = c b) b = c c) a = b d) a + c = 0 14. If the roots of the equation x2 + ax + b = 0 are reciprocals of each other, then _____ a) a = 1 b) b = 1 c) +a = -1 d) b = -1 15. The sum of the roots of the equation 3x2 – 5x + 9 = 0 is ______ a) 3 b) -5 c) 5/3 d) -3 2 16. The sum of the roots of the equation x + ax + b = 0 is 0 if _____ a) a = 0 b) b = 0 c) a = b d) if a or b = 0 2
2
17. If one root of 3x – kx + 2 = 0 is 2/3, then k = _____ __________________________________________________________________________________________ - 178 -
From the desk of SOHAIL MERCHANT a) b) c) d)
5 1 -5 -2 2
18. kx – 5x – 18 = 0 has -2 as one of its roots, then k = _____ a) 1 b) 2 c) 3 d) 4 2 3 2 2 3 19. If and are the roots of the equation x – px + q = 0 then + is _____ 2 a) p q 2 b) pq c) pq 2 d) –pq 2 2 2 20. If r and s are the roots of the equation ax + bx + c = 0, the value of 1/r + 1/s = _____ 2 a) b – 4ac 2 b) (b – 4ac)/2a 2 2 c) (b – 4ac)/c 2 2 d) (b – 2ac)/c 2 21. Sum of the roots of the equation 6x = 1 is _____ a) 1 b) 2 c) 0 d) -4 2 22. Product of the roots of the equation px + qx – r = 0 is -1, if _____ a) p = r b) p + r = 0 c) q = r d) p = q 2 2 23. Sum of all roots of the equation 4x – 8x + 13x – 9 = 0 is ____ a) 8 b) 2 c) -2 d) -8 24. If p and q are the roots of the equation x2 + px + q = 0, then _____ a) p = 1, q = 2 b) p = 2, q = 1 c) p = 1, q = -2 d) p = -2, q = 1 2 25. x + bx + a = 0 and x2 + ax + b = 0 have a common root; then a) a b b) a + b = 1 c) a + b + 1 = 0 d) a – b = 1
.
__________________________________________________________________________________________ - 179 -
From the desk of SOHAIL MERCHANT RELATIONS AND FUNCTIONS ____________________________________________________________________________________ 1. Which of the following is an ordered pair? a) {5,8} b) 5,8 c) (5,8) d) {(5,8)} 2. A = {p, q, r, s}; B = {c, d}. Which of the following is a relation from A to B? a) {(p, q), (p, c), (q, d), (r, c), (r, s)} b) {p, c), (q, d), (r, c), (s, d)} c) {(r, s), (p, p), (q, d), (s, c)} d) None of the above 3. R is a relation from A to B. Then the domain of the relation is _____ a) A b) B c) a subset A d) a subset of B 4. A = {(-2,8), (3,-4), (5,10), (6,4), (-9,2)}. The domain of a is _____ a) {-2,3,5,6,-9} b) {8,-4,10,4,2} c) {-2,3,4,21} d) {8,3,5,2} 5. P = {(a, q), (I, j), (k, l)}. The range of P is a) {a, i, k} b) {q, j, l} c) {a, j, k} d) (q, j, a) 6. P = {5,6,7}, Q = (6,8). R is a relation from P to Q defined by R = {(x, y)/x < y}. Then R = a) {(5,6), (6,8), (5,8)} b) {(8,7), (6,7), (5,6), (6,8)} c) {(5,6), (5,8), (6,8), (7,8)} d) {(5,5), (6,6), (7,8), (6,8)} 7. A is a subset of B; C is a subset of D. Then a) A x B C x D b) A x C B x D c) A x D B x C d) AC BD 8. A = {-3,5,8}. Any relation in A cannot contain morethan ____ ordered pairs a) 6 b) 7 c) 8 d) 9 9. B = {11,12,13}. R is a relation in B defined by {(x, y) / (y b and b>a, then a = b a, b R. This relation is _____ a) anti symmetric b) symmetric c) reflexive d) transitive 12. „Greater than‟ in the set of natural numbers is __ relation. a) reflexive b) transitive c) symmetric d) equivalence 13. An example for „transitive‟ relation is ____ a) being subset of b) being to in a set of lines in a plane c) being not equal to in a set of numbers d) none of these 14. For a relation to be an equivalence relation it must be a) reflexive b) symmetric c) reflexive, symmetric and transitive d) transitive 15. An example for equivalence relation is ____ a) is a brother of b) is congruent to c) is to d) is wife of 16. A relation is a function if no two ordered pairs ____ a) have the same first co-ordinate b) have the some second co-ordinate c) have the some elements d) have a common element 17. Which of the following relations is a function? a) {(a, b), (b, c), (c, d)} b) {(a, b), (a, c), (a, d)} c) {(c, a), (c, b), (c, c)} d) {(c, a), (c, d), (b, d)} 18. Which of the following is a function? a) f = {(1, 2), (1, 3), (1, 4)} b) g = {(1, 2), (2, 3), (3, 4)} c) f = {(x, y), / x = 2, y = (1, 2, 3)} d) g1 = {a, ) (a, c)} 19. A = {1, 2, 3}; B = {p, q, r, s}, then which of the following is a function from B to A? a) {(q, 1), (p, 2), (r, 3)} b) {(p, 2), (q, 1), (r, 3), (q, 2)} c) {(p, 1), (q, 1), (r, 1) d) {(p, 1), (r, 1), (q, 2), (s, 3)} 20. F: x 3 Which of the following is correct? a) f(1) = 0 b) f(2) = 0 c) f(3) = 0 d) f(1) = 3 21. f: R R and g : R R, f(x) = x + 2, g(x) = 2 – x, then fog and gof a) do not exist b) are equal c) are inverse functions d) none of these 22. R is the set of real numbers. f: R R, f(x) = x , then f is ______ 2
__________________________________________________________________________________________ - 181 -
From the desk of SOHAIL MERCHANT a) one-one but not onto b) onto but not one-one c) one-one and onto d) Neither one-one nor onto 23. In which of the following functions f(a) = f(-a)? a) x + 2 2 b) x + 4 2 c) x + x – 2 2 d) x + x + 2 24. f is an identity function defined by f(x) = x, then f(12) = _____ a) 0 b) 12 c) 1 d) None of these 25. If a function has its inverse also a function, then it is a) one-one b) one-one and onto c) onto d) one-one and into
__________________________________________________________________________________________ - 182 -
From the desk of SOHAIL MERCHANT DERIVATIVES & LIMITS ____________________________________________________________________________________ n
1. d/dx (x ) = n-1 a) nx x-1 b) –x.n c) x/n . n-1 d) xn x 2. d/dx(e ) = x-1 a) e x b) e x c) 2e d) None 3. d/dx (ax) = x a) a . log a a b) x log x a c) x log a x d) a log x 4. d/dx (log x) = 2 a) 1/x b) x long x c) 1/x 2 d) -1/x 5. d/dx (sin x) = a) cos x b) – cos x c) sin 2x d) None 6. d/dx (sin x) = a) –sin x b) sinx 2 c) cos x d) cos 2x 7. d/dx (tan x) = a) 2 sec x 2 b) sec x 2 c) 2 sec x d) secx. tanx 8. d/dx (cot x) = 2 a) –cosec x 2 b) –sec x c) cosec x cos x d) None 9. d/dx (sec x) = 2 a) sec x b) 2secx 2 c) tan x d) sec x tan x 10. d/dx (cosec x) = a) –cosecx cot x 2 b) –cos x 2 c) –cosec x d) 2 sin x
11. d/dx (k) = __________________________________________________________________________________________ - 183 -
From the desk of SOHAIL MERCHANT a) 1 b) k c) 0 d) -1 2 12. d/dx(1/x ) = 2 a) -1/x 3 b) -2/x
x c) 2 d) 2x x)= 1 2 x 1 2 x
13. d/dx( a) b)
c) x
3/2
2 x x d 1 = dx x 1 a) 2 x 1 b) 2x x 1 c) 2x 2 d)
14.
d) None 2 15. d/dx(x – 5x) = a) 2x – 5 b) 2x + 5 c) -2x – 5 2 d) 5x – 2x 2 16. d/dx (7x – 5x + 6) = a) 2x – 5 b) 14x – 5 2 c) 14x – 6 2 d) 14x + 6 – 5x 3 17. d/dx (2x – 8x + 4) = 2 a) 6x – 8 2 b) 6x + 8 2 c) 8x + 5 2 d) 3x – 8x + 1 6 5 2 18. d/dx (x – 6x + 5x + 2) = 5 4 a) 6x + 5x + 2x 5 4 b) 6x – 30x + 10x 5 4 c) 6x – 5x – 2x 5 4 d) 6x + 30x – 10x
__________________________________________________________________________________________ - 184 -
From the desk of SOHAIL MERCHANT 19.
20.
3 d 2 3 2 5 x x 4 dx 3 a) 15 x 2 x 15 2 b) x 3 2 3 x 15 3 2 c) x 2 3 x 15 2 2 d) x 2 3 x d 4 3 x dx x x
5
a) b)
3 6x 2 x 1 6 5 2 x x2 5
c) d)
6 x 5x 2 5 3 6x 2 2 x -7
2
21. d/dx (x + x – x) = a) b)
7 2x 1 x8 7 2x 1 x8 -7
c) 7x -2x+1 d) None 22.
d 4 dx x -3/2
a) -2x 3/2 b) 2/x c) d)
23.
2 x x 2x x
d (2 x ) dx
__________________________________________________________________________________________ - 185 -
From the desk of SOHAIL MERCHANT 1 x 1 b) x 1 c) x x d) x x d x5 x2 x dx x2 a)
24.
a) b) c) d) 25.
2
2
3x – 1/x 2 -2 3x + x 2 3x – (1/x ) 2 x – (1/3x )
d 2 log x dx a) b) c) d)
2
-2/x 3 -2/x 2/x -2/x
__________________________________________________________________________________________ - 186 -
From the desk of SOHAIL MERCHANT LOGARITHMS ____________________________________________________________________________________ 3
1. The logarithmic form of 5 = 125 is a) log5 125 = 3 b) log3 5 = 125 c) log125 3 = 5 d) log3 125 = 5 3 2. The logarithmic form of 4 = 64 is a) log64 4 = 3 b) log3 64 = 4 c) log4 64 = 3 d) log3 4 = 64 b 3. The logarithmic form of a = c is a) loga b = c b) loga c = b c) logb c = a d) logc c = b 4. The Exponential form of log2 32 = 5 is 5 a) 2 = 32 32 b) 2 = 5 2 c) 5 = 32 4 d) 2 = 16 5. The exponential form of log10 1000 = 3 is 3 a) 10 = 1000 2 b) 10 = 0.001 = -3 10 c) 3 = 1000 4 d) 10 = 10000 6. Which of the following is true? a) log3 81 = 5 b) log10 0.001 = -3 c) log12 144 = 3 d) log2 1/6 = -3 7. log, p + log, q = a) logrq p b) logp qr c) logp pr d) logr pq 8. When (r, s 0), logp (r/s) = a) logp r-logp s b) logp s-logp r c) logr p-logr s d) logp r-logs p y 9. logx x = a) x logz y b) y logz x c) z logy x d) y logx z 10. loga a= a) 1 b) 0 c) can‟t be determined d) -1
11. log5 625 = __________________________________________________________________________________________ - 187 -
From the desk of SOHAIL MERCHANT
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
a) 1 b) 2 c) 3 d) 4 log 24 = a) log 2 + log 12 b) log 3 + log 8 c) log 4 + log 6 d) All of the above 5 log 2 + 2 log 5 = a) log 800 b) log 54 c) log 49 d) log 60 log 1944 = a) 3 log 6 + 2 log 3 b) 6 log 3 + 3 log 2 c) 4 log 3 + 2 log 4 d) 3 log 4 + 5 log 2 log 6912 = a) 2 log 4 + 5 log 3 + 2 log 3 b) 4 log 2 + 3 log 3 + 2 log 4 c) 5 log 3 + 3 log 5 + 2 log 6 d) None log 25/81 = a) 2 log 5 – 4 log 3 b) 3 log 4 – 5 log 2 c) 3 log 5 – 2 log 5 d) 3 log 4 + 4 log 5 – log2 2 log 2 + log 6 + log 7 = a) log 84 b) log 72 c) log 96 d) log 108 log 3 + log 5 – 3 log 2 = a) log (20/8) b) log (25/9) c) log (15/8) d) log (15/9) log ab/cd = a) log a + log c – log b – log d b) log a + log b – log c – log d c) log c – log a + lob b – log d d) None log50 1 = a) 1 b) -1 c) 0 d) 2 y x logb a = a) x/y logb a b) a/b logy x c) y/x logb a d) a/b logx y
x log x y
22.
__________________________________________________________________________________________ - 188 -
From the desk of SOHAIL MERCHANT a) x b) y c) xy d) x/y 23. loga x . logb b. logy c = a) logb a b) logy c c) logy x d) logx y 24. log5 343 x log6 5 x log7 6 = a) 3 b) 4 c) 5 d) 6 25.
log a 3 b 5 a) b) c) d)
3
5
log a + log b 3log a + 5 log b 3/2 log a + 5/2 log b None
__________________________________________________________________________________________ - 189 -
From the desk of SOHAIL MERCHANT BINOMIAL THEOREM ____________________________________________________________________________________ 1.
n
Cr = ____ a) n(n – r) b) n(n – 1) _ (n – r) c) n(n – 1)(n – 2) __ (n – r + 1) d)
2.
n! r (n r )!
5
C3 = ____ a) 5 b) 5 3 c) 10 d) 5 4 3 11 3. The number of terms in the expansion of (x + y) is ____ a) 10 b) 12 c) 11 d) 13 4. If the number of terms in the expansion of
5.
6.
7.
8.
2 3x 3x
n
is 16, the value of n + ____
a) 16 b) 15 c) 17 d) 8 2 5 The number of terms in the expansion of (3-2x ) is a) 5 b) 6 c) 7 d) 8 m The (r + 1) th term in the expansion of (a + y) is ____ m r-m r a) Cr. a .y m r m-r b) Cr . a y m m r c) Cr a y m m-r r d) Cr a . y n The (r + 1)th term in the expansion of (ax + b) is ____ n n-r r n-r a) Cr . x a b n r n-r r b) Cr . x a b n n-r r r c) Cr . a b x n n-r r n-r d) Cr . a b x rd 5 The 3 term in the expansion of (3x + 5y) is _____ 2 2 a) 10 x y 3 2 b) 675 x y 3 2 c) 6750 x y 3 3 d) 6750x y th
9. The 5 term in the expansion of a) b) c) d)
6
2
1 x x
6
is
C4 1/x C5 1/x 6 2 C2 . x 6 C3 6
__________________________________________________________________________________________ - 190 -
From the desk of SOHAIL MERCHANT n
10. The (r + 1)th term (general term) in the expansion of (x + y) is ____ n n-r+1 r+1 a) Cr . x y n n-r-1 r b) Cr+1 . x y n n-2 r r c) Cr . xy b x n n-r r d) Cr . x y 7 11. The coefficient of x4 in the expansion of (2x + 3) is _____ 7 3 4 a) C4 . 2 .3 7 4 3 b) C4 . 2 .3 7 7 7 c) C4 . 2 .3 7 1 7 d) C4 . 2 .3 2
12. The coefficient of x in the expansion of 6
2
4
1 2x 3x
6
is _____
a) C2 . 2 . 1/3 6 4 2 b) C2 . 2 . 1/3 6 2 2 c) C2 . 2 .1/3 6 4 4 d) C2 . 2 .1/3 m 13. The (r + 1)th term in the expansion (a + x) ____ is m r-3 r a) Cr . a .y m m-r r b) Cr a .x m m r c) Cr a .x m m-r m-r d) Cr a .x 4 14. The binomial coefficients successively in the expansion of (x + y) are ____ a) 1, 4, 6 b) 1, 5, 10, 5, 1 c) 1, 3, 3, 1 d) 1, 4, 6, 4, 1 15. The middle term in the expansion of a) b) c)
6
C2 C4 6 C3
6
6
3
16.
1 x x
5 2 3 3 d) 6C0 2x b x x 6 2 2 The middle term in the expansion of x is ____ x
a) 160x3 3 b) -160x 3 c) -240x 3 d) 240x 4 17. The middle term in the expansion of (a + b) is _____ 2 2 a) 6a b 3 2 b) 4a b 3 c) 6ab 2 2 d) a b 18. The last term in the expansion of a) b) c) d)
2
2 x x
5
is _____
2/x 5 10/x 5 32/x 32/x
__________________________________________________________________________________________ - 191 -
From the desk of SOHAIL MERCHANT 19. The middle term in the expansion of a) b) c) d)
4
4
is ______
81x 4 1/16x 27/2 2 -54x
20. The middle term in the expansion of a) b) c) d)
1 3x 2x
th
1 x2 x
8
is its _____
4 term th 5 term th 6 term rd 3 term th
21. The 4 term in the expansion of
4x 2 3 3 2x
7
is _____
5
a) 480 x 5 b) -480x 5 c) -325x 5 5 d) -35.2 /3 x n 22. If the coefficient of x2 in the expansion of (2 + x) is 240, the value of n = _____ a) 3 b) 4 c) 5 d) 6 4 4 23. (1 + x) – (1 – x) = _____ 2 4 a) 2 + 6x + x 2 4 b) 2 + 12x + 2x 2 3 4 c) 2 + 8x + 12x 8x + 2x 3 d) 8x + 8x 24. The term independent of x in the expansion of
2 x x
6
is
a) 80 6 b) C2 c) 160 6 2 d) C3.2
__________________________________________________________________________________________ - 192 -
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