Class 10 Mathematics Compendium

MATHEMATICS COMPENDIUM

REAL NUMBERS Points to Remember : 1. Euclid’s division lemma : Given positive integers a and b, there exists whole numbers q and r satisfying a = bq + r, 0  r < b. 2. Euclid’s division algorithm : This is based on Euclid’s division lemma. According to this, the HCF of any two positive integers a and b, with a > b, is obtained as follows : Apply Euclid’s division lemma to find q and r where a = bq + r, 0  r < b. If r = 0, the HCF is b. If r  0, apply the Euclid’s lemma to b and r. Continue the process till the remainder is zero. The divisor at this stage will be HCF (a, b). 3. The fundamental theorem of arithmetic : Every composite number can be expressed (factorised) as a product of primes, and this factorisation is unique, except for the order in which the prime factors occur. 4. For any two positive integers a and b, HCF (a, b) × LCM (a, b) = a × b. 

HCF (a, b) 

ab and LCM(a, b)

LCM (a, b) 

a b HCF(a, b)

5. For any three positive integes a, b and c, we have

and

HCF (a, b, c) 

a  b  c  LCM (a, b, c) LCM (a, b)  LCM (b, c)  LCM (a, c )

LCM (a, b, c) 

a  b  c  HCF (a, b, c ) HCF (a, b)  HCF (b, c )  HCF (a, c)

6. Let a be a positive integer and p be a prime number such that p/a2, then p/a. 7. If p is a positive prime, then

p is an irrational number..

p 8. Let x be a rational number, whose decimal expansion terminates. Then we can express x in the form q , where p and q are co-prime and the prime factorisation of q is of the form 2n5m, where n, m are nonnegative integers. p be a rational number, such that the prime factorisation of q is of the form 2n5m, where n, m are q non-negative integers. Then x has a decimal expansion which terminates.

9. Let x 

10. Let x 

p be a rational number, such that the prime factorisation of q is not of the form 2n5m, where n, q

m are non-negative integers. Then x has a decimal expansion which is non terminating repeating (recurring). REAL NUMBERS

1

POLYNOMIALS Points to Remember : 1. Let x be a variable, n be a positive integer and a0, a1, a2, ......., an be constants. Then f ( x )  an x n  an 1 x n 1 

 a1 x , a0 is called a polynomial in variable x.

2. The exponent of the highest degree term in a polynomial is known as its degree. 3. Degree Name of Polynomial Form of the Polynomial 0 Constant Polynomial f(x) = a, a is constant 1 Linear Polynomial f(x) = ax + b, a  0 2 Quadratic Polynomial f(x) = ax2 + bx + c; a  0 3 Cubic Polynomial f(x) = ax3 + bx2 + cx + d; a  0 4. If f(x) is a polynomial and  is any real number, then the real number obtained by replacing x by  in f(x) at x =  and is denoted by f(). 5. A real number  is a zero of a polynomial f(x), if f() = 0. 6. A polynomial of degree n can have at most n real zeroes. 7. Geometrically, the zeroes of a polynomial f(x) are the x-coordinates of the points where the graph y = f(x) intersects x-axis. 8. For any quadratic polynomial ax2 + bx + c = 0, a  0, the graph of the corresponding equation y = ax2 + bx + c has one of the two shapes either open upwards like  or downwards like  , depending on whether a > 0 or a < 0. These curves are called Parabolas. 9. If  and  are the zeroes of a quadratic polynomial f(x) = ax2 + bx + c, a  0 then   

 

b coefficient of x  a coefficient of x 2

c constant term  a coefficient of x 2

10. If  are the zeroes of a cubic polynomial f(x) = ax3 + bx2 + cx + d, a  0 then   

b coefficient of x 2  a coefficient of x3

       

c coefficient of x  a coefficient of x3

d constant term  a coefficient of x3

11. Division Algorithm : If f(x) is a polynomial and g(x) is a non-zero polynomial, then there exist two polynomials q(x) and r(x) such that f ( x )  g ( x )  q( x )  r ( x ), where r(x) = 0 or degree of r(x) < degree of g(x). POLYNOMIALS

MATHEMATICS–X

PAIR OF LINEAR EQUATIONS IN TWO VARIABLES Points to Remember : 1. A pair of linear equations in two variables x and y can be represented as follows : a1x + b1y + c1 = 0; a2x + b2y + c2 = 0, where a1, a2, b1, b2, c1, c2 are real numbers such that a12  b12  0, a22  b22  0. 2. Graphically, a pair of linear equations a1x + b1y + c1 = 0, a2x + b2y + c2 = 0 in two variables represents a pair of straight lines which are : (i) Intersecting, if

a1 b1  a2 b2

here, the equations have a unique solution, and pair of equations is said to be consistent. (ii) parallel, if

a1 b1 c1   a2 b2 c2

here, the equations have No solution, and pair of equations is said to be inconsistent. (iii) Coincident, if

a1 b1 c1   a2 b2 c2

here, the equations have infinitely many solutions, and pair of equations is said to be consistent. 3. A pair of linear equations in two variables can be solved by the : (i) Graphical method (ii) Algebraic Methods; which are of three types : (a) Substitution method (b) Elimination method (c) Cross-multiplication method

ILLUSTRATIVE EXAMPLES Example 1. Draw the graph of linear equation 2x + 3y = 7. Solution.

7  2x 3 Give atleast two suitable values to x to find the corresponding value of y.

2x + 3y = 7  3y = 7 – 2x  y 

7  2(2) 3  1 3 3 7  2(5) 3   1 If x  5, y  3 3

If x  2, y 

PAIR OF LINEAR EQUATIONS IN TWO VARIABLES

25

QUADRATIC EQUATIONS Points to Remember : 1. A quadratic equation in the variable x is of the form ax2 + bx + c = 0, where a, b, c are real numbers and a  0. 2. A real number a is said to be a root of the quadratic equation ax2 + bx + c = 0, if a2 + b + c = 0. The zeroes of the quadratic polynomial ax2 + bx + c and the roots of the quadratic equation ax2 + bx + c = 0 are the same. 3. If ax2 + bx + c, a  0 is factorisable into a product of two linear factors, then the roots of the quadratic equation ax2 + bx + c = 0 can be found by equating each factor to zero. 4. The roots of a quadratic equation can also be found by using the method of completing the square. 5. Quadratic Formula (Shreedharacharya’s rule) : The roots of a quadratic equation ax2  bx  c  0 are given by

b  D , where D = b2 – 4ac is known as discriminant. 2a

6. A quadratic equation ax2 + bx + c = 0 has (i) Two distinct real roots, if b2 – 4ac > 0 i.e. D > 0 (ii) Two equal roots (i.e., coincident roots), if b2 – 4ac = 0 i.e. D = 0 (iii) No real roots, if b2 – 4ac < 0 i.e. D < 0. 7.

x 2  a 2  x  a or x   a

8.

x2  a 2   a  x  a

QUADRATIC EQUATIONS

MATHEMATICS–X

ARITHMETIC PROGRESSION Points to Remember : 1. A sequence is an arrangement of numbers or objects in a definite order. 2. A sequence a1, a2, a3, ......., an, ...... is called an Arithmetic Progression (A.P) if there exists a constant d such that a2 – a1 = d, a3 – a2 = d, a4 – a3 = d, ....., an – an–1 = d and so on. The constant d is called the common difference. 3. If ‘a’ is the first term and ‘d’ the common difference of an A.P., then the A.P. is a, a + d, a + 2d, a + 3d.... 4. The nth term of an A.P. with first term ‘a’ and common difference ‘d’ is given by an  a  (n  1)d . 5. The sum to n terms of an A.P. with first term ‘a’ and common difference ‘d’ is given by n Sn  [2a  (n  1)d ] 2 n Also, S n  [a  l ] , where l = last term. 2

6. Sum of first n natural numbers = 1 + 2 + 3 + ..... n 

n(n  1) . 2

ARITHMETIC PROGRESSIONS

71

TRIANGLES Points to Remember : 1. Two figures having the same shape but not necessarily the same size are called similar figures. 2. All the congruent figures are similar but the converse is not true. 3. Two polygons having the same number of sides are similar, if (i) their corresponding angles are equal and (ii) their corresponding sides are proportional (i.e. in the same ratio) 4. Basic proportionality theorem (Thales theorem) : If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, then the others two sides are divided in the same ratio. 5. Converse of Thales’ theorem : If a line divides any two sides of a triangle in the same ratio, then the line is parallel to the third side of the triangle. 6. The line drawn from the mid-point of one side of a triangle is parallel to another side bisects the third side. 7. The line joining the mid-points of two sides of a triangle is parallel to the third side. 8. AAA similarity criterion : If in two triangles, corresponding angles are equal, then the triangles are similar. 9. AA similarity criterion : If in two triangles, two angles of one triangle are respectively equal to the two angles of the other triangle, then the two triangles are similar. 10. SSS similarity criterion : If in two triangles, corresponding sides are in the same ratio, then the two triangles are similar. 11. SAS similarity criterion : If one angle of a triangle is equal to one angle of another triangle and the sides including these angles are in the same ratio, then the triangles are similar. 12. The ratio of the areas of two similar triangles is equal to square of the ratio of their corresponding sides. 13. If the areas of two similar triangles are equal, then the triangles are congruent. 14. If a perpendicular is drawn from the vertex of the right angle of a right triangle to the hypotenuse, then the triangles on both sides of the perpendicular are similar to the whole triangle and also to each other. 15. Pythagoras Theorem : In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. 16. Converse of Pythagoras Theorem : If in a triangle, square of one side is equal to the sum of the squares of the other two sides, then the angle opposite to first side is a right angle.

87

CO-ORDINATE GEOMETRY Points to Remember : 1. The abscissa and ordinate of a given point are the distances of the point from y-axis and x-axis respectively. 2. The coordinates of a point on the x-axis are of the form (x, 0) and of a point on the y-axis are of the form (0, y). 3. Distance Formula : The distance between two points P(x 1, y 1) and Q(x 2, y 2) is given by PQ  ( x2  x1 )2  ( y2  y1 ) 2 4. Distance of a point P(x, y) from the origin O(0, 0) is given by OP  x 2  y 2 . 5. 6. 7. 8. 9.

Three points P, Q and R are collinear, then PQ + QR = PR or PQ + PR = QR or PR + RQ =PQ. A ABC is an isosceles triangle, if AB = BC or AB = AC or BC = AC. A ABC is an equilateral triangle, if AB = BC = AC. A ABC is a right-angle triangle, if AB2 + BC2 = AC2 or AB2 + AC2 = BC2 or BC2 + AC2 = AB2. Section Formula : The co-ordinates of the point which divides the join of points P(x1, y1) and Q(x2, y2)

mx  nx1 my2  ny1  internally in the ratio m : n are  2 . , m  n   mn 10. Mid-point Formula : The co-ordinates of the mid-point of the line segment joining the points P(x1, y1) and Q(x2, y2) are  x1  x2 , y1  y2  . 2   2 11. The coordinates of the centroid of a triangle formed by the points P(x1, y1), Q(x2, y2) and R(x3, y3) are

 x1  x2  x3 y1  y2  y3  ,   3 3   12. The area of the triangle formed by the points P(x1, y1), Q(x2, y2) and R(x3, y3) is given by

1 .| x1 ( y2  y3 )  x2 ( y3  y1 )  x3 ( y1  y2 ) | 2 1 | x1 y2  x2 y3  x3 y1  x2 y1  x3 y2  x1 y3 | 2 13. The points P(x1, y1), Q(x2, y2) and R(x3, y3) are collinear if area of PQR = 0 i.e. x1 (y2 – y3) + x2(y3 – y1) + x3(y1 – y2) = 0. 

14. The area of a quadrilateral formed by the points P(x1, y1), Q(x2, y2), R(x3, y3) and S(x4, y4), taken in order is given by : 1 | x1 y2  x2 y3  x3 y4  x4 y1  x2 y1  x3 y2  x4 y3  x1 y4 | 2

CO-ORDINATE GEOMETRY

117

INTRODUCTION TO TRIGONOMETRY Points to Remember : 1. If ABC is a right triangle right angled at B and BAC = , 0°    90°, we have : Base = AB, Perpendicular = BC and, Hypotenuse = AC here, sin =

BC Perpendicular  AC Hypotenuse

;

cos =

tan =

BC Perpendicular  AB Base

;

cosec  

AC Hypotenuse  AB Base

;

cot  

sec  

2. We have, cosec  

C

AB Base  AC Hypotenuse AC Hypotenuse = BC Perpendicular

B

90°



A

AB Base  BC Perpendicular

1 1 1 , sec   , cot   sin  cos  tan 

Also, tan   sin  , cot   cos  cos  sin  3. Values of various Trigonometric ratios :  T-ratio

0°

30°

45°

60°

90°

sin 

0

1 2

1

3 2

1

cos 

1

3 2

1 2

1 2

0

tan 

0

1

3

not defined

cosec 

not defined

sec 

1

cot 

not defined

1 3 2

2

2 2

2 3

2

3

1

INTRODUCTION TO TRIGONOMETRY

3 2

1 3

1

not defined

0

MATHEMATICS–X

4. The value of sin  increases from 0 to 1 as  increases from 0 ° to 90°. Also, the value of cos  decreases from 1 to 0 as  increases from 0° to 90°. 5. If  is an acute angle, then sin (90° – ) = cos , cos (90° – ) = sin  tan (90° – ) = cot , cot (90° – ) = tan  sec (90° – ) = cosec , cosec (90° – ) = sec  6. Basic trigonometric identities : (i) sin 2   cos 2   1 or 1  cos 2   sin 2  or 1  sin 2   cos 2  (ii) 1  tan 2   sec2  or sec2   tan 2   1 (iii) 1  cot 2   c osec2  or cosec2 – cot2 = 1

133

SOME APPLICATIONS OF TRIGONOMETRY Points to Remember : 1. The line of sight is the line drawn from the eye of an observer to the point in the object viewed by the observer.

lin eo fs

ig ht

2. The angle of elevation of an object viewed, is the angle formed by the line of sight with the horizontal when it is above the horizontal level, i.e. the case when we raise our head to look at the object.

angle of elevation

3. The angle of depression of an object viewed, is the angle formed by the line of sight with the horizontal when it is below the horizontal level, i.e. the case when we lower our head to look at the object.

angle of depression Li ne of sig ht 4. The height or length of an object or the distance between two distant objects can be determined with the help of trigonometric ratios. 5. The observer is taken as a point unless the height of the observer is given.

AREAS RELATED TO CIRCLES Points to Remember : 1. A circle is a collection of points which moves in a plane in such a way that its distance from a fixed point always remains the same. The fixed point is called the centre and the fixed distance is known as radius of the circle. 2. Area and circumference of a circle : If r is the radius of the circle, then (i) circumference = 2r = d where d = 2r (diameter) (ii) Area = r 2 

d 2 4

(iii) Area of a semicircle  (iv) Area of a quadrant 

1 2 r 2

1 2 r 4

3. Area of a circular ring : If R and r (R > r) are radii of two concentric circles, then area enclosed by the two circles. =  (R2 – r2) 4. Number of revolutions completed by a rotating wheel 

Distance moved circumference

5. If a sector of a circle of a radius r contains an angle of . Then, (i) length of the arc of the sector 

(ii) Perimeter of the sector  2r  (iii) Area of the sector 

  2 r 360

O

 2 r 360

  r 2 360

r



r

A

B

(iv) Area of the segment = Area of the corresponding segment – Area of the corresponding triangle. 

 1    r 2  r 2 sin , or r 2  r 2 sin cos 360 2 360 2 2

6. (i) In a clock, a minute hand rotates through an angle of 6° in one minute.

 1  (ii) In a clock, an hour hand rotates through an angle of   in one minute. 2 AREAS RELATED TO CIRCLES

MATHEMATICS–X

SURFACE AREAS AND VOLUMES Points to Remember : 1. Cuboid (i) Volume = lbh (ii) Curved surface area = 2h (l + b) (iii) Total surface area = 2 (lb + bh + lh) (iv) Diagonal  l 2  b 2  h 2

2. Cube (i) Volume = a3 (ii) Curved surface area = 4a2 (iii) Total surface area = 6a2 (iv) Diagonal  3 .a

3. Cylinder (i) Volume = r2h (ii) Curved surface area = 2rh (iii) Total surface area = 2r (r + h)

4. Hollow Cylinder r

(i) Volume =  h (R2 – r2) (ii) Curved surface area = 2  h (R + r)

h R

(iii) Total surface area = 2h (R + r) + 2 (R2 – r2) = 2 (R + r) (h + R – r) SURFACE AREAS AND VOLUMES

MATHEMATICS–X

5. Cone (i) Volume 

1 2 r h 3

(ii) slant height, l  h 2  r 2 (iii) curved surface area   r l (iv) Total surface area   r (l  r ) 6. Sphere (i) Volume  4 r 3 3 r

(ii) Total surface area = 4  r2

7. Spherical Shell (i) Volume

4 ( R 3  r 3 ) 3

r

(ii) Surface area (outer) = 4R2

R

8. Hemi-sphere (i) Volume  2 r 3 3

r

(ii) Curved surface area = 2r2 (iii) Total surface area = 3r2 9. Frustum of a cone :

r

(i) Volume  1 h(r 2  R 2  rR ) 3

l

h

(ii) Slant height (l )  h 2  ( R  r ) 2 (iii) Curved surface area =  l (r + R)

R

(iv) Total surface area = l (r + R) +  (r2 + R2)

SURFACE AREAS AND VOLUMES

215

STATISTICS Points to Remember : 1. Mean of ungrouped data : x 

x1  x2  .......  xn 1 n   xi where x1, x2, ....., xn are n observations. n n i 1

2. Mean of grouped data : (i) Direct method : If the values of the variables be x1, x2, ....., xn and their corresponding frequencies are f1, f2, ......, fn then the mean of the data is given by n

x

f1 x1  f 2 x2  ...... f n x n  f1  f 2  ........ f n

 f i xi i 1 n

 fi i 1

n

fd i

(ii)

Short-cut method : x  A 

i 1 n

f

i

,

where A is assumed mean and, di  xi  A

i

i 1

(iii) Step-deviation method :  n    f i ui  x  A  h.  i 1n ,   fi   i 1 

where, A is assumed mean, h = class width x A and ui  i h

3. Mode is the value of the variable which has the maximum frequency. 4. The mode of a continuous or grouped frequency distribution : mode = l 

f1  f 0  h, where 2 f1  f 0  f 2

l = lower limit of the modal class f1 = frequency of the modal class f0 = frequency of the class preceding the modal class. f2 = frequency of the class following the modal class. h = width of the modal class 5. The median is the middle value of a distribution i.e. median of a distribution is the value of the variable which divides it into two equal parts. 6. Median for individual series (ungrouped data) : Let n be the number of observations. (i) Arrange the data in ascending or descending order. STATISTICS

239

 n 1  (ii) (a) If n is odd, then median = value of   2 

(b) If n is even, then median =

1  value of 2

th

observation.

 n  th  n th    +   1  observations.  2    2 

7. Median for continous grouped data : N F Median  l  2  h, where f l = lower limit of the median class f = frequency of the median class h = width of the median class F = cumulative frequency of the class preceding the median class. n

N   fi i 1

8. Emperical formula : Mode = 3 Median – 2 Mean 9. Cumulative frequency distribution can be represented graphically using curve, known as ‘ogive’ of the less than type and of the more than type. 10. The median of grouped data can be obtained graphically as the x-coordinate of the point of intersection of the two ogives for the data.

PROBABILITY Points to Remember : 1. In the experimental approach to probability, we find the probability of the occurence of an event by actually performing the experiment a number of times and adequate recording of the happening of event. 2. In a theoretical approach to probability, we try to predict what will happen without actually performing the experiment. 3. An outcome of a random experiment is called an elementary event. 4. The theoretical (classical) probability of an event E, written as P(E), is defined as P(E)=

Number of outcomes favourable to E Number of all possible outcomes of the experiment

5. The probability of an impossible event is 0, and that of sure event is 1. 6. The probability of an event E is a real number P(E) such that 0  P(E)  1. 7. An event having only one outcome is called an elementary event. The sum of the probabilities of all the elementary events of an experiment is 1. 8. For any event E, P(E) + P( E ) = 1, where E stands for ‘not E’. 9. Total possible outcomes, when a coin is tossed n times, is 2n. 10. Total possible outcomes, when a die is thrown n times, is 6n. 11. Playing Cards (Total 52)

13



Spade ( ) (Black coloured)

13 Club () (Black coloured)

13 Heart () (Red coloured)

13 Diamond ( ) (Red coloured)

The cards in each suit are ace, king, queen, jack and number cards 2 to 10. Kings, queens and jacks are called face cards.