ICSE Mathematics Formulae Booklet 2016-2017

LIST OF FORMULAE, ICSE MATH (X), By Rakesh Kushwaha Sir (M.Sc.; B.Ed.): 09224190389/0993035 0748

I.C.S.E. (2016 – 2017)

Mathematics(X) By

B.Sc.(Hons);M.Sc.(Mathematics); M.A.(Lings, English & Economics); B.Ed.(Sp. Edu.); GNIIT; S.B.T.C.

    

Ex-Vice Principal (M.V.M. PUBLIC SCHOOL, ALIGARH) Ex-Co-ordinator, Ex-Co-ordinator, I.C.S.E. SCHOOL , SOUTH MUMBAI EXAMINER, ICSE MATHEMATICS H.O.D. MATHEMATICS, I.C.S.E. SCHOOL , SOUTH MUMBAI Ex-Facilitator Ex-Facilitator (IBDP mathematics)

 Address: Cluster III, Poonam Estate, Estate, Mira Road Road (East)

Contact:

9224190389/9930350748 1 E-mail: [email protected]

LIST OF FORMULAE, ICSE MATH (X), By Rakesh Kushwaha Sir (M.Sc.; B.Ed.): 09224190389/0993035 0748

ICSE MATHEMATICS (X)

 

There will be one paper of 2  hours duration carrying 80 marks and Internal Assessment of 20 marks. The paper will be divided into two Sections. Section I (40 marks), Section II (40 marks). Section I: It will consist of compulsory short answer questions. Section II: Candidates will be required to answer four out of seven questions. UNITS & CHAPTERS

1. COMMERCIAL ARITHMETIC 

Compound Interest (Paying back in equal installments in stallments not included)



Sales Tax and Value Added Tax



Banking (Saving Bank Accounts and Recurring Deposit Accounts)

Shares and Dividends (Brokerage and fractional shares not included) 2. ALGEBRA 



Linear Inequations



Quadratic Equations and Solving Problems



Ratio and Proportion



Remainder and Factor Theorems ( f   f ( x)  x) not to exceed degree 3)

Matrices 3. CO-ORDINATE GEOMETRY 



Reflection



Distance and Section Formulae

Equation of a Straight Line 4. GEOMETRY 



Symmetry



Similarity



Loci (Locus and Its Constructions)



Circles



Tangents and Intersecting Chords

∆

Constructions (tangents to circle, circumscribing & inscribing circle on & reg. hexagon) 5. MENSURATION 



Circumference andAreaof a circle (Area of sectors of circles other than semi-circle and quarter-circle not included)

Surface Area and Volume (of solids) 6. TRIGONOMETRY 



Trigonometrical Identities and Trigonometrical Tables

Heights and Distances (Cases involving more than 2 right angled 7. STATISTICS 

∆

 excluded)



Graphical Representation (Histogram and Ogives)



Measures of Central Tendency (Mean, Median, Quartiles and Mode)



Probability 2 E-mail: [email protected]

LIST OF FORMULAE, ICSE MATH (X), By Rakesh Kushwaha Sir (M.Sc.; B.Ed.): 09224190389/0993035 0748

GENERAL INSTRUCTIONS:  Mathematics needs Continuous Practice right from the start of the year.  Solve each and every ever y question by yourself with understanding. Do similar questions from other

question banks as well. If you understand one sum, you can do hundred similar sums. Tea chers. Ask them for more guidance, academic acad emic help etc.  Be in regular touch with your Teachers.  Remember that without consistent practice you cannot be perfect in Mathematics.  Be prepared for the worst correction. Don‟t give any chance to the examiner to cut your marks for

silly mistakes or careless work.  Don‟t use short cut ways to get answer , make habit of writing all required steps. You never know

which step has to be marked for giving marks. Marking scheme is changed every year. b e substituted  Read the question carefully, write the data first and convert into the same unit if to be for further calculation. v ery common error.  Copy the digits carefully. Check the digits again at the end. Misreading is very  Write 0 and 6 clearly. It has been observed that your 6 is taken as 0 and 0 is taken as 6 by you

only.  If the value is to be substituted in between the sum (e.g., value of r), don‟t keep in decimal form

or round off form. Keep it in fraction form onl y or else this may lead to wrong answer.  If the answer is to be written up to two decimal places, find up to three decimal places and round

off at the end. sometimes it‟s taken as minus sign which provides you  Make proper equal to sign for every step, sometimes wrong answer.  Do Small calculation also (roughly/not mentally) parallel or adjacent to the sum only. (You are

suggested to do it to gain marks, e.g., 12 × 6, 19 + 22, 90 –  90 –  65  65 etc. have been found wrong due to carefree attitude of students.)  Reading time must be used to make the right choice in section „B‟.  Don‟t spend more than 5 or 6 minutes on average for a sum. This will give you spare time for a

quick review at last. If you get stuck, stop and move on. steps. If   Never cross out an answer until you revise it at last. You may get sum marks for correct steps. you attempt a sum twice or thrice, don‟t cancel anything. anything. Let the examiner decide which one is correct. This is very useful tool if you are not confident about your answer.

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LIST OF FORMULAE, ICSE MATH (X), By Rakesh Kushwaha Sir (M.Sc.; B.Ed.): 09224190389/0993035 0748

COMMERCIAL ARITHMETIC Compound Interest:  Simple interest is computed on the principal.  Compounded interest is computed on the sum of the Principal and Interest previously earned. In

other words, the interest also earns interest.  A=P+I  S.I. =

  ×  × 100 st

st

 S.I. for 1  year = C.I. for 1  year if compounded annually.  S.I. for 1year

≠

 C.I. for 1year if compounded half yearly. (C.I. > S.I.) th

C. I. of n year + Int. on it for 1 year ; R% =  C. I. for (n + 1) year = C.I. th

 − 

( 2

1) ×100

 Amount in (n + 1) year = Amount in n year + Int. on it for 1 year; R% =

    

 %, where T = 1yr

  −   ( 2

1) ×100 1

 %

      C.I. = P1 +  − 1   1 +   1 +  ; when rates for successive years are different. A = P 1 +   ;when the interest is compounded half-yearly. A = P 1 +   1 +   , If the time is 2  years and the rate is compounded yearly. A = P 1 +   , V = Initial Value, V = Final Value For Growth: V= V 1 +   For Depreciation: V = V 1 −

 A=P 1+ 

1

100

100

1

2

3

100

100

100

 ×2

2 × 100 2

1

100

1

2 × 100

0

2

0

100

0

100

 Round off Amount (money) up to two decimal places.(61.166 = 61.17, 440.2 = 440.20)  Skip decimal values (after round off) if the amount is calculated to the nearest rupee. nd

 C.I. of 2  year and C.I. of 2 years are two different terms.

Sales Tax and Value Added Tax:  The price at which an Article is marked : List Price/Marked Price/Printed Price/Quoted Price  Sale Price = M.P. –  M.P. –  Discount,  Discount, Discount is calculated on M.P.  Sales Tax is calculated after deducting the discount (on the discounted price).

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LIST OF FORMULAE, ICSE MATH (X), By Rakesh Kushwaha Sir (M.Sc.; B.Ed.): 09224190389/0993035 0748

 Sales Tax =

 Sale-price =  Sale-price =

  

100

 −  −   −  

 Sale-price =

 S.P. =

Rate Rate of Sales Sales Tax Tax ×Sal ×Sales es Pric Price e

100+Pro 100+Profit fit %

× C.P.

100

100 Loss Loss %

× C.P.

100

100 discount  100

100 d%

100+r%

100

100

 VAT paid by a person =

× M.P.

× M.P. , where d = discount, d iscount, r = rate of sales tax

pric price e Adde Added d by the the pers person on ×VAT% ×VAT% 100

 = Rate % (S.P. –  (S.P. –  C.P.)  C.P.)

sale –  Tax  Tax paid on the purchase  VAT = Tax recovered(charged) on the sale – 

Banking: 1. SB Account:

a. Withdrawal = Debit  b. Deposit = Credit c. Steps for calculation of interest: th

i. Find the minimum balance of each month between 10  day and the last day. ii. Add all the balances. This is the Equivalent Monthly Principal for 1 month. iii. Calculate the SI on the Equivalent E quivalent Monthly Principal with T =

1 12

 years. th

iv.  No interest is paid for the month in which the account is opened after 10  day or closed on any day (principal for that month is taken as zero). v. If the Amount Received on closing is asked, add the interest to the LAST BALANCE (actual amount available in your account) and not to the Equivalent Monthly Principal. Amount Received on closing the account = Closing Balance + S.I. 2. RD Account:

a. Qualifying sum (P) =  b. I =

 

P ×n n+1 ×r 2 ×12 ×100

c. M.V. = P ×



 ; T =

  , where x where x =  = monthly deposit, n = no. of months   years ; P = monthly monthl y deposit, n = no. of months, r = rate% n n+1 2

n n+1

2 ×12



+ I ; Maturity Value = Total deposit (monthly deposit × ) + Interest

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LIST OF FORMULAE, ICSE MATH (X), By Rakesh Kushwaha Sir (M.Sc.; B.Ed.): 09224190389/0993035 0748

Shares and Dividend:  The total money invested by the company is called its capital stock.  The capital stock is divided into a number of equal units. Each unit is a called a share.   Nominal Value is also called Register Value, Printed Value, and Face Value.  The FV of a share always remains the same, while its MV goes on changing.  The part of the profit of a compan y which is distributed amongst the shareholders is known as

dividend.  If the MV = NV, the share is said to be at par.  If the MV >NV, the share is said to be at premium.  If the MV ,< ,

≥ ≤  and

 are called signs of inequality.

becomes –  ve  ve and vice versa.  On transferring + ve term becomes –   If each term is multiplied or divided by + ve number, the sign of inequality inequalit y remains the same.  The sign of inequality reverses:  If each term is multiplied or divided by same negative number.  If the sign of each term on both the sides of an inequation is changed.  On taking reciprocals of both sides, in case both the sides are positive or negative.

≤ ∈

 Always, write the solution set for the inequation, e.g.,{ x : x 3, x

 N }, }, solution set = {1, 2, 3}

6 E-mail: [email protected]

LIST OF FORMULAE, ICSE MATH (X), By Rakesh Kushwaha Sir (M.Sc.; B.Ed.): 09224190389/0993035 0748  To represent the solution on a number line:  Put arrow sign on both the ends of the line and keep extra integers beyond the range.  Use dark dots on the line for each element of N, W and Z. (WIN)  For Q, R: mark range with solid circle

(for

≥ ≤ or

), hollow circle

(for < and >.)

Intersection ( only common elements of the sets). Common range for R  “and” means Intersection ( Union (all elements of the sets without repetition).  “or” means Union (all   Don’t use decimal values e.g., 1.66 or 1.33 on number line, use fraction form 1

2 3

etc only.

1

 Show the number on the number line which you are representing, e.g., 1 . 2

 Solution must be written inside curly brackets { } only.  Show the fraction

 1

1 2

 on number line or use an arrow to indicate the written value.

form only.   Remember that solution set for R and Q is always written in set-builder form

Quadratic Equations: 1. Quadratic equation is an equation with one variable, the highest power of the variable is 2. 2. Some useful results: 2

2

2

a) (a + b)  = a  + b  + 2ab 2

2

2

b) (a - b)  = a  + b  - 2ab 2

2

c) a  –  b  b = (a + b) (a –  (a –  b)  b) 2

2

d) (a + b)  - (a - b)  = 4ab 3

3

2

2

e) a  + b  =(a + b)(a - ab+ b  ) 3

3

2

2

 f) a  - b  =(a - b)(a + ab+ b  ) 3

3

3

 g) (a + b)  = a  + b  + 3ab(a + b) 3

3

3

h) (a - b)  = a  - b  - 3ab(a - b) 2

2

2

2

i) (a + b + c)  = a  + b  + c  + 2ab + 2bc + 2ca 3

3

3

2

2

2

 j) a  + b  + c  –  3abc  3abc = (a + b + c) (a  + b  + c  –  ab –   ab –  bc –   bc –  ca)  ca) 7 E-mail: [email protected]

LIST OF FORMULAE, ICSE MATH (X), By Rakesh Kushwaha Sir (M.Sc.; B.Ed.): 09224190389/0993035 0748 3. Steps for solving quadratic equation by b y factorization: (PSI system) a. Clear all fractions and brackets if necessary. 2

 b. Bring it to the form ax  + bx + c = 0 by 0 by transposing terms. c. Factorize the expression by splitting the middle term as a sum of product of a and c. 4. Discriminant (D) =

 −  2

 4

a. if D > 0, then the roots are real and unequal  b. if D = 0, then the roots are real and equal c. if D  angle



Angle of depression

 Angle of elevation



Bigger angle is always inside and smaller is outside.



Height is vertical length and distance is horizontal length.

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LIST OF FORMULAE, ICSE MATH (X), By Rakesh Kushwaha Sir (M.Sc.; B.Ed.): 09224190389/0993035 0748



=

 =  =     



tan



Remember the values of tan 30°, 45° and 60°

   1

 45° ,

         1

3

 30°,

3

. It‟s very easy to remember. Just remember

1

  . 3

 60°

 Diagram is must in Heights and Distances. Mark angles co rrectly. Outside angle is always  

 

smaller. Angle of elevation decreases when distan ce increases. Make habit of writing the standard values of tan  as per given in the question. E.g., you get 1 mark just for writing tan 45° = 1. You may easily get 3 marks in heights and distances sums: 1 mark for correct diagram, 1mark for the value of tan 30°, 45° or 60°and 1 mark for writing correct formula with correct substitution. There is only one mark for big b ig calculation. So don‟t omit this important topic. Use log table to write the values of angles (22°, 32°, 48 etc) other than standard angles in finding heights and distances. Keep unknown side as numerator while using tan , to avoid division by big decimal number. Multiplication is always easier. Use complementary angle if required.





STATISTICS Statistics:



∑ 



Arithmetic mean of non-tabulated data:  =  =



Arithmetic mean of tabulated data(Direct Method):  =  =



Arithmetic mean by Short-cut Method:  =  =













∑  ; x = mid value v alue (C.I.) ∑

∑  + A ; A = assumed mean , d = x = x –   –  A  A ∑ ∑ ×   + A ; i= class width , t =  Arithmetic mean by Step-deviation Method:  =  =  ∑    term For raw data, if n is odd, Median = 



+1 2

    For raw data, if n is even, Median =   From ogive, Median =   term. 2

term term +

   2

+1

term

2

2

From ogive Lower quartile, Q1 =

 4

term 21

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LIST OF FORMULAE, ICSE MATH (X), By Rakesh Kushwaha Sir (M.Sc.; B.Ed.): 09224190389/0993035 0748

  3



From ogive Upper quartile, Q3 =



Inter Quartile Range, IQR = Q3 –  Q  Q1



Semi Inter Quartile Range =



Mode is the variate which has the maximum frequency.



The class with maximum frequency is called the moda l class. (e.g., 20 –  20 –  30)  30)



To estimate mode from histogram: draw two straight lines from the corners of the rectangles on either

4

term

 – 3

1

2

sides of the highest rectangle to the opposite co rners of the highest rectangle. Through the point of intersection of the two straight lines, draw a vertical line to meet the x-axis at the point M (say). The variate at the point M is the required mode. 

To find median of grouped data, draw ogive (cumulative frequency curve).  For finding mean of raw data, be very careful in counting number of data (n).  Use kink (if required), show scale and label the axes to get marks. Don‟t dare to dare to change the scale

if already given in the question.  For finding median, Q1, Q3 or number of desired range (below or above) a bove) using ogive,

 perpendiculars are must. No mark is allotted without perpendiculars.  Write the answer in decimal form not in fraction.  In Ogive, it should be the smooth curve without halt.

Probability: 

Probability is a measure of uncertainty.



An Experiment is an action which results in some (well-defined) outcomes.



Sample space is the collection of all possible outcomes of an experiment. n(S)



An Event is a subset of the sample space associated with a random experiment. n(E)



An Event occurs when the outcome of an experiment satisfies the condition mentioned in the event. 22 E-mail: [email protected]

LIST OF FORMULAE, ICSE MATH (X), By Rakesh Kushwaha Sir (M.Sc.; B.Ed.): 09224190389/0993035 0748 

The outcomes which ensure the occurrence of an event are called favorable outcomes to that event.



The probability of an event E, written as P(E), is defined as P (E) =



P(E) =



The value of probability is always between 0 and 1.



The probability of sure (certain) event is 1.



The probability of an impossible event is 0.



An elementary event is an event which has one (favorable) outcome from the sample space.



A Compound event is an event which has more than one outcome from the sample space.



If E is an event, then the event „not E‟ is complementary event of E and denoted by E.



0



P(E) + P(E) = 1



In a pack (deck) of playing pla ying cards, there are 52 cards which are divided into 4 suits of 13 cards each –  each – 

 

( ) ( )



≤  ≥  ( )

spades (

         

 1

), hearts (

), diamonds (

) and clubs (

). Spades and clubs are black in colour,

while hearts and diamonds are of red colour. The cards in each suit are ace, king, queen, jack, 10, 9, 8, 7, 6, 5, 4, 3, 2. Kings, queens and jacks are called face (picture/court) cards. The cards bearing number 10, 9, 8, 7, 6, 5, 4, 3, 2 are called numbered cards. Thus a pack of playing cards has 4 aces, 12 face cards and 36 numbered cards. The aces together with face cards (= 16).are called cards of honour. 

When a coin is tossed, it may show head h ead (H) up or tail (T) up. Thus Th us the outcomes are: {H, T}.



When two coins are tossed simultaneously, then the o utcomes are: {HH, HT, TH, TT}. [n(S) = 2 ]



When a die is thrown once the outcomes are: {1, 2, 3, 4, 5, 6}.



When two dice are thrown simultaneously, then the outcomes are: {(1, 1), (1, 2)…….(6, 6)}. 6)}.

n

 Write the answer in simplified form.



9

30

=

3 10

n

[n(S) = 6 ]



 Don‟t forget to write sample space. At times, there is special 1 mark for it, which is ignored by

most of the students.

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