1.

REAL NUMBERS
Key concepts :
1) Division algorithm

Given positive integers a and b, there exist unique pair of integers q

and r satisfying a = bq + r, 0 ≤ r < b

1) We define loga x = n, if a n = x, where a and x are positive numbers

and a 1.

2) Let a, x and y be positive real numbers with a 1. Then loga xy =

loga x + loga y

3) Let a, x and y be positive real numbers with a 1. Then loga = loga

x - loga y

4) If a, x and y are positive real numbers and a 1, then

MATHEMATICS a) loga x m = m loga x

b) =N

c) loga 1 = 0

d) loga a = 1

Long answer Questions

1) Use division algorithm to show that the square of any positive

integer is of the form 3p or 3p + 1

2) Use division algorithm to show that the cube of any positive integer
is of the form 9 m, 9m + 1 or 9m + 8.

3) Prove that 2 + 3 is irrational.

4) Prove that 3+2 is irrational.

5) Use Euclid’s division lemma to show that square of any positive

integers is of the form 5n or 5n+1 or 5n+4.

6) If x2+y2=27xy, then show that log =

7)Show that is an irrational number

4 5

8) Use Euclid’s division lenman to show that square of any positive 2. SETS
integers is of the form 7m or 7m+1 or 7m+6 Key concepts :

9) Show that is an irrational number 1) Set of Natural numbers. N= {1,2,3,4,5...... -}.

2) set of whole numbers w = {0, 1, 2, 3, 4,.......}
Short Answer Questions 3) Set of Integers. Z = {----3-2-1, 0, 1,2,3-.....}
4) Set of Even numbers E ={2,4,6,8,10..........}
1) Find the HCF of the following by using Euclid’s algorithm.
5) set of odd numbers. =O {1,3,5,7,9,- --}
a) 50 and 70 b) 300 and 550 6) set if Prime numbers P = {2,3,5,7,11}
7) set of composite numbers. C = {4,6,8,9,10,- --}
2) Find the HCF and LCM of 12 and 18 by the prime factorization
method. 8) Empty set is denoted by the letter (Phi). Universal set is denoted
by the letter (mu).
3) Express the number 3825 as a product of its prime factors. 9) If each element of set A is in set B then A is called the subset of B.
it is denoted by ACB (A is subset of B)
4) Find the LCM and HCF of 8,9 and 25 by the prime factorization
method. 10) Union of sets: The union if two sets A and B is written as AUB(A
union B).
5) Show that 3 is irrational A ={x/x A or x/x B}
11) Intersection of sets: The intersection of two sets A and B is written
6) Sove log2 32=x
as A B (A intersection B). A B = {X/X A and X B}
7) Expand log 12) Disjoint sets: If the sets A and B have no common elements then
A and B are called disjoint sets. If A and B are disjoint sets then
8) Find the value of A
13) Difference of sets: The set of elements which are in A but not in B
9) Is log3 81 rational or irrational? Justify your answer.
is called the difference of A and B.
10) Expand log10 385 A-B={x/x
14) The number of elements of a set is called the cardinal number of
11) Show that log =log2 that set. The cardinal number of set À' is denoted by n (A)
15) n (AUB) = n (A) + n (B) - n(A B).
12) Expand loga3b2c5 16) If the sets A and B have same elements then A and B are called
13) if x2+y2=7xy then prove that 2log(x+y)= logx +logy+2log3 equal sets. It is denoted by A=B.
17) If the number of elements of a set one countable, then it is called
14)Write 2log3+3log5-5log2 as a single logarithm finite set. Otherwise it is an infinite set.
15) find whether is a terminating decimal or not without actual
Answer the following questions:
division.
1) Write the set of letters of TELANGANA in roaster form.
16) Write 0.0875 in form. 2) Write the set of solutions of the equation x²+x-2=0 in roaster form.
3) Write the set B = { 1/10,1/100,1/1000.........} in set builder form.
4) Nikhita Said A= {x/x<1 and X>3} is an empty set, Do you agree?
Give reasons?
5) If A= {0, 1, 2, 3, 5} and A B = {2,3} then find A-B.
6) Write all the subsets of a set D={p, i, n}
6 7
 7) A and B are disjoint sets. N(AUB) = 12, n (A) = 7 and n (1B) = x+2. 3. POLYNOMIALS
find the value of "x":
8) A = {x/x is a Prime number less than 7} and B is a set it first three Key concepts :
prime numbers. is A=B? Give reasons. 1. Polynomials are classified as Linear, Quadratic and Cubic
9) If P = {x/x is an even number and less than 10}, Q = {x / x is according to their degree.
natural number less than 5} then find PUQ, P Q, P-Q and Q-P. Also 2. Degree of the Linear polynomial is 1
draw the venn diagrams. 3. Degree of the Quadratic Polynomials is 2
10) If B = {1,3,5} then find B and B B and comment. 4. Degree of the Cubic Polynomials is 2
11) A = {x/x is an odd number and x≤9} and B = {x/x=3n+2, n W and 5. The values which satisfy the polynomial is called as zero of the
n<5}. Find AUB, A B, A-Band B-A. Polynomial.
12) From the following Venn diagram write the sets A and B. Also 6. Linear polynomial has 1 zero
verify n(AUB) = n (A) + n(B) - n(A B). 7. Quadratic polynomial has 2 zeros
8. Cubic Polynomial has 3 zeros
9. Zeros of the polynomial can be formed by
i. Factorization
13) A={x/x is an even number between 1 and 18}, and B = {x/x is a ii. Graphically
perfect square between 1 and 30} find i) (AUB) - (A B) ii) (A-B) U 10. Relation between coefficients and zeros of Quadratic polynomial
(B-A) and what do you notice? (ax2+bx+c )
i.
Multiple Choice Questions: ii.
1 If A ⊂B then A-B is ( )
11. Relation for Cubic polynomial is (
A) B) A C) B D) AUB
i.
2) It E is the set of vowels in English alphabets, then the number of
subsets Of E ( ) ii.
A) 16 B) 32 C) 48 D) 64. iii.
3) If A = {3, 5, 7, 11} then the cardinal number of A ( )
12. Polynomials of higher degree can be divided by polynomials of
A) 8 B) 16 C) 12 D) 4
lower degree and Euclid’s division rule can be verified.
Very Short Answer Questions

1. If p( ) = , then find the values of p(0), p(-1) and p(1).

2. Write a quadratic polynomial whose zeroes are 2 and 3.
3. Write a quadratic polynomial whose sum and product of zeroes are

and 4 respectively.
4. If one zero of the polynomial is 2, then find the volue of
‘K’.

8 9

Short Answer Questions 4. PAIR OF LINEAR EQUATIONS IN TWO VARIABLES

1. If one zero of the polynomial p(x)=( ) is reciprocal Key concepts:

of the other, then find the value of ‘k’. 1. Pair of Linear Equations in standard form are denoted by
2. Find the zeroes of the polynomial +12 verify the relation . a1x + b1y + c1 = 0 ----------(1) and
. a2x + b2y + c2 = 0 ----------(2)
between zeroes and coefficients.
2. Relation between coefficients and system of linear equations is
3. Complete the following table for the polynomial p(x)= given by
(i) If , then the equations are consistent, and they have
-2 -1 0 1 2
only on solution.
(ii) If , then the equations are inconsistent and they
+4
have no solution.
+4
(iii) If then the equations are mutually dependent, and
P
have infinite solutions.
3. There are three methods of solving pair of Linear equations.
Long Answer Questions
(i) Elimination Method
1. Verify 1,-1,-3 are the zeroes of the cubic polynomial (ii) Substitution Method
and check the relation between zeroes and coefficients. (iii) Graphical Method
(4) Equations reducible to Linear equation in two variables.
2. Draw the graph of p( )= and find its zeroes from the
(5) Word problems
graph. Solve the following :

Multiple Choice Questions: (1) Show that the following pair of Lines are consistent
(2)
1. Among the following, the expression that is not a polynomial is ( )
(3) If the pair of Linear equations are
A) B) inconsistent, then find the value of ‘k’
(4) Give an example of a linear equation in two variables which is
C) D) mutually dependent with
2. Degree of polynomial p(x)= + is ( ) (5) Reduce the equations into standard form
(6) Find one equation each, which has unique solution, infinite solutions
A) 2 B) -5 C) -3 D) 7
and no solution with
3. If sum of zeroes of the polynomial P( )= -3k +4 -5 is 6, then (7) 3 bags and 4 pens together cost Rs. 257/- whereas 4 bags and 3
the value of ‘k’ is ( ) pens cost Rs. 324/- from two linear equations for the above
information.
A) 2 B) 4 C) -2 D) -4
(8) Find the value of ‘k’ for which the equations
4. The coefficient of term in the polynomial P( )= -5 -3 is has infinitely many solutions.
( ) (9) The sum of two numbers is 8. If their sum is four times their
A) 2 B) -5 C) -3 D) 7 difference, then find the numbers.
(10) Complete the table for the given equations
10 11
 (6) The line x=0 represent ( )
x 0 x 1 (A) x-axis (B) y –axis (C) Origin (D) None
y 0 y 2 (7) The line y-k is ( )
(A) parallel to y-axis (B) parallel to x-axis (C) represent y-axis (D) None
(8) Solution of the equations 3x+y-3=0 and 2x-y+8=0 is ( )
(11) Solve the pair of equations x+y=2 and x-y=0 by elimination method (A) x= -1, y=6 (B) x=6, y=-1 (C) x=6, y=1 (D) x=1, y=-6
(12) Solve 2x-y=5 and 3x+y=15 by substitution method. (9) If a pair of lines are inconsistent, then the lines are ( )
(13) Draw the graph of linear equations 2x+3y-1=0 and 3x-4y+19=0 and (A) Parallel (B) Coincident (C) Intersecting (D) None
find the solution from the graph. (10) The age of the father is 3 years more than 3times the son’s age. 3
(14) 4 Tables and 3 chairs together cost Rs. 2250 and 3 tables, 4 chairs years later the age of the father will be 10 years more than twice the age
cost Rs. 1950. Find the cost of 2 chairs and 1 table. of the son. Their present ages are ( )
(15) A number consists of two digits whose sum is 5 when the digits are (A) 30, 10 (B) 45, 12 (C) 33, 10 (D) 55, 16
reversed, the number exceeds the original number by 9. Find the (11) The value of ‘k’ if 2x+3y=5 and 4x+6y=k has infinitely many
number. solutions ( )
(16) Solve the pair of equations by reducing them to a pair of linear (A) 5 (B) 6 (C) 20 (D) 10
equations (12) One solution of (x,y) in the equation 3x-4y=7 is ( )
(A) (-1,1) (B) (0,1) (C) (1,-1) (D) (2,-2)

(17) Solve the pair of equations by reducing them to a pair of linear

equations.

Multiple choice questions

(1) Which of the following is the condition for intersecting lines ( )
(A) (B) (C) (D) None
(2) Standard form of the equation is ( )
(A) 2x+3y-6=0 (B) 3x+2y=6 (C) 2x-3y+6=0 (D) 2x+3y-3=0
(3) To solve the equations 2x-3y+1=0 and 3x+5y-2=0 by elimination
method and to eliminate ‘y’ coefficient, First equation should be
multiplied by ( )
(A) -3 (B) 2 (C) 1 (D) 5
(4) If 2x+ky-6=0 and 3x-2y+5=0 have unique solution then ’k’ is not
equal to ( )
(A) -4/3 (B) -3/4 (C) 1 (D) 4
(5) The pair of linear equations -5x+2y=8 and 2x-5y-3=0 have ( )
(A) No solution (B) One solution (C) Two solutions (D) Many solutions

12 13

5. QUADRATIC EQUATIONS Long Answer Questions

Key concepts:
1. The perimeter of a right-angled triangle is 30 cm and it's hypotenuse is 13
Quadratic Equation: A quadratic equation in the Variable x is an equation of cm. find the remaining two sides.
the form ax² + bx + c = 0 where a, b, c are real numbers a 0
2. Sum of the areas of two Squares is 1076 sq. m. If the difference of their
A real number is said to be a root of the quadratic equation, if it satisfies the perimeters is 24 m. then find the sides of the two Squares.
equation
3. Sum of squares of two Consecutive positive integers is 100 Find those
Discriminant: b2-4ac is called discriminant.b2-4ac determines whether the numbers by using quadratic equations.
quadratic equation ax² + bx + c=0 has real roots are not.
4. Sum the present ages of two friends is 18 years. Five years ago, Product of
2 their ages was 15. find their present ages.
The roots of a quadratic equation ax + bx + c =o (a 0) are given by
provided b2-4ac≥ Multiple choice questions

Short Answer Questions 1. Quadratic equation of the following is ( )

2 2
1. If b -4ac > 0 in ax + bx + c then what can you say about nature of roots (A) (B) x²-6x²+9x=4 (C) (D) (x+1)(x+2) (x-3)=0
of the equation? (a 0)
2. Quadratic equation of the following whose roots are equal ( )
2. Find the Value of ‘k’ if x2+kx+9=0 has two equal real roots.
2 2 2 2
A) x -6x+9=0 (B) x -4x+3= 0 (C) x -4=0 (D) x +4x-5= 0
3. Write the nature of the roots of the equation 3x2-6x+4=0.
3. In a quadratic equation ax2+bx+c=0 if b2-4ac <0, then its roots are ( )
4. Find the sum and product of the roots of x2-4 3x+9=0
(A) Real and distinct (B) real and equal (C) Imaginary (D None
5. Find the roots of the quadratic equation x2-5x+6=0
4. If x2-qx+x=0 q, r R and q r has distinct real roots then ( )
6. Check whether 2 and & 5/2 are the roots of the quadratic equation
2 2 2 2
(A) q <4r (B) q >4r (C) q =4r (D) q +4r
2x2-9x+10=0 or not.

Short Answer Questions

1. Is it possible to design a rectangular park whose length is twice of its

breadth and area 450 sq.m. ? If so, find it's length and breadth.

2 The sum of a number and its reciprocal is 17/4 find the number.

3. Write the quadratic equation whose roots are 1+√2 and 1-√2.

4) Shiram Said that (x+2) (x-1) = (x+3)(x-4) is a quadratic equation. Do you

agree? Give reasons.

5) Find the roots of quadratic equation x2-4x+3=0 by 'Completing Square"

method.
14 15
 6. PROGRESSIONS Very Short answer questions
Key concepts:
1. In an A.P. the nth term an = 3n-1. Find the common difference.
“ Arithmetic Progression (AP) :
2. Is 3,6,9,12……..A.P.? Give reasons
Arithmetic Progression is a List of numbers in which each term is
3. If a=3 and 10th term of an A.P. is 21 then find ‘d’
obtained by adding a fixed number to the preceding term except the
4. In a GP first term is 2 and common ratio is 3 Find first 3 terms.
first term.
5. Find the 11th term of A.P. 24,20,16…….
This fixed number is called the common difference of the AP
6. Find 10th term of the G.P. 64,32,16,……
Let us denote the first term of an AP by a, second term by a2 …….nth
7. Is 3,6,12…….G.P.? Give reasons.
term by an and common difference by “d” then the AP becomes a 1,
a2,a3……………..an Short answer Questions
So a2-a1 = a3-a2=………….an-an-1=d
Nth term of an A.P.: 1. In a GP first three terms are x, x+4, x+12, find x.
 The nth term of an A.P. with first term ‘a’ and common difference ‘d’ is 2. Find the sum of 10 terms of the AP 2,7,12, ……..
given by an = a+(n-1)d an is also called the general term of the A.P. 3. How many three digit numbers are divisible by 9
Sum of n terms of A.P. : 4. Which term of AP 3,7,11,15,…..is 59
5. In an AP 4th term is 18 and 9th term is 43. Find the 21st term.
 Sn = (a+an) where ‘a’ is the first term and an is the last term.
 Or Long answer Questions.
 Sn= where ‘a’ is the first term and ‘d’ is the common 1. In a G.P. 3rd term is 12 and 6th term is 96. Find the 8th term.
difference. 2. Find the sum of all 3-digit numbers which are divisible by ‘5’
 Sum of first ‘n’ positive integers Sn= 3. 3rd and 6th term of a GP are and respectively. Prove that
Geometric Progression (G.P.) the 10th term of GP is
Geometric progression is a list of numbers in which first term is non-
4. Find the sum of all multiples 8 between 100 and 200
zero and each succeeding term is obtained by multiplying preceding
term with a fixed non-zero number. This fixed number is called the Multiple Choice Questions.
common ratio.
1. In an AP 1,-2,-5,-8………… d= ( )
Let us denote the first term of G.P. by ‘a’ , second term by
A. -1 B. -3 C. -2 D. -7
a2…………nth term by an common ratio by ‘r’ Then the GP becomes
2. If the common difference of an AP is 3 then a8-a3 = ( )
a1,a2,a3………an
A. 15 B.12 C. 10 D.5
So
3. Common ratio of GP , 2, 2 , ……..is ( )
General form of G.P. :
A. 2 B.1 C. D.4
The general form of G.P. is a, ar, ar2……….where ‘a’ is the first term ‘r’
4. sum of first 20 natural numbers is ( )
is the common ratio
A. 100 B.210 C. 110 D.325
Nrh term of G.P.
5. In a GP 3rd term is 8 and the 5th term is 32. Its common ratio is
The nth term of a G.P. with first term ‘a’ and common ratio ‘r’ is given
A. 4 B.6 C. 8 D.2
by an= arn-1 an is also called general term of G.P.

16 17

7. CO-ORDINATE GEOMETRY 12. A(-1,8), B(5,-2) are the two vertices of is (2,3) then find the 3rd
Key concepts : vertex ‘c’
13. Find the area of a triangle whose vertices are (3,-5)(-2,7) and (-1,-4).
 Distance between two points A(x1,y1), B(x2,y2)
14. Prove that (-6,4),(-2,2) and (2,0) are the collinear.
√
 Section formula A(x1,y), B(x2,y2) Ratio m1:m2 ( ) ( )
Long Answer Questions

 Ratios for trisection points of a line segment are (1) 1:2 (2) 2:1 1. Prove that the points (2,3) (4,5) and (7,2) are the vertices of a right
 Centroid of a triangle whose vertices are A(x1,y1) B(x2,y2) and (x3,y3) angled triangle.
G=( ) 2. Show that the points (-8,2),(-2,6),(4,10) and (-2,-4) taken in order
are the vertices of a parallelogram.
 Midpoint of a line segment ( ) 3. If points (-8,2),(-2,6),(4,0) and (-2,-4) taken in order are the vertices
 Area of triangle whose vertices are of a parallelogram.
I 4. Find the ratio in which the line segment joining the points (-6,4)
 If area of triangle is zero, then the vertices of the triangle are and (-6,2) is divided by (-2,2).
collenear. 5. Find the point of trisection of the line segment joining (2,3) and
 Area of triangle – Heron’s formula (5,6),
6. Find the points of which divides the line segment joining (5,4) and
 Heron’s formula to find area of triangle is √
(-1,-2) into three equal parts.
 Slope of the straight line 7. A(2,y), B(-2,2) and C(3,1) are the vertices of if area of is
Slope = 7sq units then fine the value of ‘y’.
8. Find the value of ‘x’, for which the points (-1, 1), (5, 4) and (x, -1)
Very Short Questions are collinear.
1. Find the distance between origin and (3,4) Multiple choice questions.
2. Find the mid-point of line segment joining (8,-6) and (-4,10) 1. The slope of the line passing through (2,3) and (1,-2) is ( )
3. Write the formula to find the area of triangle by Heron’s formula A. 5 B. -5 C. 1/5 D.-1/5
and explain terms in it. 2. Which of the following point is on the x-axis. ( )
4. Find the slope of line passing through (7,4) and (2,-1) A. (2,2) B. (-2,0) C. (0,2) D. (3,2)
5. Find the centroid of the triangle whose vertices are (-3,7),(6,5) and 3. Which of the following point is on the y-axis ( )
(9,0) A. (5,-5) B. (3,0) C. (0,-3) D. (10,7)
6. Find the distance between (6,0) and (0,-8)
4. If A=(0,4), B=(5,0), C=(0,0) then area of is (in sq. units)( )
7. If the distance between two points (5,8) and (-3,y)is 10 units, then A. 5 B. 4 C. 0 D. 10
find the value of ‘y’. 5. Which of the following is the mid-point of line segment joining
8. Find the diameter of a circle whose centre (4,5) and passing
A. B. (
through (0,2)
C. D.
9. Find the point on the y-axis which is equidistance from (3,6) and
(4,5) 6. The centroid of a triangle divides its median in which ratio
10. Find the point which divides the line segment joining the points A. 2 : 3 B. 2 : 1 C. 3 : 2 D. 1 : 3
(-4,-6) and (4,-2) is the ratio 1:3 7. The distance between is
11. The end points of a diameter of a circle are (2,6) and (4,-2) then A. B. )
find the centre of the circle. C. D.

18 19
 8. SIMILAR TRIANGLES Very Short Answer Questions :
Key concepts : 1. The corresponding sides of two similar triangles are 14cm and 8cm
 Similarity of triangle respectively, if perimeter of first triangle is 56cm, then determine
 Basic proportionality theorem (BPT) (THALES THEOREM) the perimeter of second triangle.
If a line is drawn parallel to one side of a triangle to intersect the 2. The areas of two similar triangles 108 sq.cm and 75 sq.cm, if the
other two sides is distinct points, then the other two sides are altitude of the bigger triangle is 6 cm, then find the corresponding
divided in the same ratio. altitude of the second triangle.
 Converse of ‘BPT’ 3. PQR XYZ, if PQ=8cm, QR=4cm and XY=5cm then find the value
If a line divides two sides of a triangle in the same ratio, then the of YZ
line is parallel to the third side.
 Criteria for similarity of triangle.
i. ‘AAA’ criteria. 6

In two triangles if corresponding angles are equal, then their 4. =4/3 and AC=10.5cm then fine AE?
corresponding sides are in proportion and hence the triangles are 5. A man 150 cm tall casts 225cm shadow. At the same instance, a
similar. tower casts a shadow of 30cm. Find the height of the tower.
ii. ‘SSS” criteria Short Answer Questions
In two triangles, if corresponding sides are proportional, then their 1. Two poles of heights 5m and 11m stand on a playground. If the
corresponding angles are equal and hence the triangles are similar. distance between the feet of the poles is 8m, then find the distance
iii. ‘SAS’ Criteria between their tops.
If one angle of triangle is equal to one angle of the other triangle 2. DEF is an isosceles right angled triangle right angled at F prove
and the including sides of these angles are proportional, then the that DE2=2DF2
two triangles are similar. 3. The hypotenuse of a right angled triangle is 3cm more than twice of
 Perimeters of similar triangles the shortest side and the third side is 3cm less than the thrice the
The ratio of the perimeters of two similar triangles is equal to the shortest side, then find the sides of the triangle.
ratio of their corresponding sides. 4. Draw a line segment of length 6.5cm and divide it in the ratio 2:3.
 Areas of similar triangles 5. In the DE//BC, if AD=x+1. BD=3x-1 AE=x and CE=4x-6,
The ratio of the areas of two similar triangles is equal to the ratio of then find the value of ‘x’
the squares of their corresponding sides.
Long Answer Questions:
 If the areas of two similar triangles are equal then they are
1. Construct a triangle if AB=5cm, BC=8cm and AngleB=600,
congruent.
then construct a triangle similar to it whose sides are ¾ of the
 PYTHAGORAS THEOREM (BAUDHAYANA THEOREM)
corresponding sides of the first triangle.
In a right triangle, the square of length of the hypotenuse is equal
2. Construct an isosceles triangle whose base is 6cm and altitude
to the sum of the squares of lengths of the other two sides.
4cm, then draw another triangle whose sides are 3/2 times the
 Converse of Pythagoras:- In a triangle, if square of the length of
corresponding sides of given triangle.
one side is equal to the sum of squares of the lengths of the other
3.Construct a triangle , if PQ=6cm ∠P =550, ∠Q=650, then
two sides, then the angle opposite to the first side is a right angle
construct a triangle similar to it whose sides are 3/5 of the
and the triangle is a right angled triangle.
corresponding sides of the first triangle.

20 21

4. In a trapezium ABCD, AB//CD, E and F are the points on non- 9. CHAPTER TANGENTS SECANTS TO A CIRCLE
parallel sides AD and BC respectively. Such that EF//AB show Key concepts :
that .  A straight line that intersects a circle in two points called a secant
5. In trapezium PQRS, PQ//RS, Diagonal PR and Qs intersect at ‘O’ line.
show that  A straight line that touches the circle at one point is called a tangent
line to the circle.
6. Prove that the ratio of areas of two similar triangle is equal to the
 The point where tangent meets the circle is called point of tangency.
squares of the ratio of their corresponding medians.
(point of contact)
7. YP and ZQ are the medians of triangle XYZ, right angled at X. Prove
 The tangent at any point of a circle is perpendicular to the radius
that
through the point of contact.
8. In an equilateral triangle ABC , Dis a point on side BC Such that
 Find length of the tangent, we can use Pythagoras theorem.
BD=1/3BC Prove that 9AD2=7AB2
 There is no tangent to circle passing through a point inside the circle.
9. ABC is a right angled triangle right angled at B, Let D and E be any
 There is one and only one tangent to a circle at a point on the circle.
point on AB AE2+CD2=AC2+DE2.
 There are exactly two tangents to a circle through appoint outside the
Multiple choice Questions :
circle.
1. Which of the following pair of figures are not similar ( )
 The lengths of tangents drawn from an external point to a circle are
A) A pair of Squares B) A pair of Circles
equal
C) A pair of Equilateral triangles D) A pair of Rectangles.
 A secant divides the circle into two segments i.e. one is minor
2. In the given figure BC//DE, =________ ( )
segment and second is major segment.
A) B) C) D)  Diameter divides the circle into two equal halves i.e. semi-circle.
3. In the given figure QR//XY, if PX=4.8cm XQ=4cm and PY= 6cm, Very Short Answer Questions
then value of YR is ( ) 1. AP and PB are two tangents a circle with centre’o’, if angle
A)6cm B) 5cm C) 4cm D) 10cm AOB=1200 then find angle APB?
4. If , perimeters of is 64cm and is 48cm, if 2. A tangent drawn from a point 10cm away from the center of circle
AB=8cm then PQ=_____ ( ) with radius 6cm then find length of tangent?
A) 8cm B) 7cm C) 6cm D) 5cm 3. A right triangle circumscribe a circle given in the figure CR=8cm,
5) If Area of triangles is 32sq cm and area of BP=4cm, AQ=6cm, find the perimeter of ?
is 98 sq.cm, If QR=7c, then BC=_____ ( ) Short Answer Questions
A)49cm B) 7cm C) 4cm D) 16cm 1. Two concentric circles of radii 10cm and 8cm drawn. Find the
6) If a ladder 17m long reaches a window of building 15m above the length of the chord of the larger circle, which touches the smaller
ground then distance from the foot of the ladder to the building is circle
A) 10m B) 9m C)8m D) 7m 2.Prove that the tangents to circle at the end points of diameter are
parallel.
3. Calculate the length of tangent from a point 17cm away from the
center of a circle of radius 8cm.
4. From the given figure a quadrilateral ABCD is drawn to
circumscribe a circle If AP=3cm, BQ=2cm, RC=4cm, SD=6cm, then
find the perimeter of quadrilateral.

22 23
Long answer Questions: 10. MENSURATION
1. Draw a circle of radius 3cm, From a point 5cm. away from its
Centre. Construct the pair of tangents to the circle and measure Key Concepts.
their lengths Verify by using Pythagoras theorem. 1. Solid is 3-D shaped
2. Construct tangents to a circle of radius 6cm from a point on the 2. Volume is defined as the space occupied with inn the boundaries of
concentric circle of radius 8cm. an object in three dimensional space.
3. Draw a circle of radius 5cm from a point 7.5cm away from its 3. Volume is also known as the capacity of the object.
centre, construct the pair of tangents to the circle. 4. Volume measured in cubic units.
4. Draw a pair of tangents to a circle of radius 6cm which are inclined 5. The lateral surface area of an object is sum of the areas of all sides
to each other at an angle 600 of the object excluding base and top when they exist.
5.Find the area of the shaded region in the adjacent figure, where 6. The total surface area of an object is the sum of the areas of all
ABCE is a square of side 7cm and semi circles are drawn with each sides of the object including base and top.
side of the square as diameter. 7. LSA and TSA are measured in square units.
6. In the figure OACB is a quadrant of circle with centre’o’ and radium
7cm. If OD=4cm find the area of the shaded region. Section-I
7.A triangle PQR is drawn to circumscribe a circle of radius 3cm, 1. Find the curved surface area of a cylinder whose radius is 7 cm
such that the segment QX and XR into which QR is divide by the and height is 14 cm ( )
points of contact X are of length 3cm and 4cm respectively, find
2. “A conical solid block is exactly fitted inside the cubical box of side
the sides PQ and PR.
‘a’, then the volume of conical solid block is . Is the statement
8. Find the area of segments shaded in figure BC =12cm, AC=5cm
and AB is diameter of the circle with Centre’O’ true? Justify your answer.
Multiple Choice Questions : 3. If the radius of the hemisphere is 7 cm, then find its volume.
1. The line intersecting a circle in two points is called? ( ) 4. If a cylinder and a cone are of same radius and same height, then
A) Tangent B) Secant C) Diameter D) Direct tangent how many cones full of milk can fill the cylinder? Explain the
2. How many tangents can we draw to a circle at appoint on the reasons.
circle. ( ) 5. How much cloth is required to set up a conical shape tent with
A) One B) Two C) Three D) Zero height 8 m and radius 7m.
3. How many tangent can we draw to a circle through a point outside 6. If a cylinder, cone and hemi sphere are on the same base and
the circle ( ) having the same height, then find the ratio of their volumes.
A) 1 B) 3 C) 2 D) 4 7. In a right circular cone, the height is three times the radius of the
4. The angle between a tangent to a circle and the radius drawn at the base. If its volume is 134.75 , then find the area of the base.
point of contact is ( ) 8. The curved surface area of a sphere is 616 Find its diameter.
A) 45o B) 90o C) 60o D) 30o Section-II
5. The area of sector whose radius is 14cm and angle subtended at
1. Find the volume of a square pyramid whose base is a square of
centre is 90o ( )
side 5 cm and height 30 cm.
A) 154 sq.cm B) 77 sq.cm C) 108 sq.cm C) 308 sq.cm
2. Find the lateral surface area and total surface area of a sphere of
radius 7cm.
3. Find the height of a cylinder of radius 3.5 cm and lateral surface
area is 154 .

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4. A solid sphere and a solid hemisphere have the same total surface 11. TRIGONOMETRY
area, and then find the ratio of their volumes. Key Concepts.
5. A metallic hemisphere of radius 4.2 cm is melted and recast into 1. Sides of a right angled triangle.
the shape of a cylinder of radius 6 cm. find the height of the 2. Pythagoras theorem:
cylinder. (Hypotenuse)2=(opp.side)2+(Adj. side)2
6. If radius of a cylinder and a cone are equal and height of cone is 3. Trigonometric Ratios- Definitions.
double of that of cylinder, then find the relation between their a.
volumes in the form of a ratio.
7. If three spheres of radius 3 cm, 4 cm and 5cm are melted and cast b.
in to a large sphere, then find the radius of the large sphere so c.
formed.
d.
Section-III
e.
1. How many spherical balls each 7 cm in diameter can be made out
of a solid lead cube whose edge measures 66 cm. f.
2. The area of a sector shaped canvas cloth is 264 . With this 4. Values of trigonometric ratios for 300, 45o,60o,900 and 00
canvas cloth, if a right circular conical tent is erected with the
00 300 450 600 900
radius of the base is 7 cm, then find the height of the tent. ( )
3. A cylindrical tank has two hemispheres at its two ends. The length Sin 0 1
of axis at its centre is 11 m and radius of a hemisphere is 3.5 m.
then find the capacity of the tank in liters. Cos 1 0
4. How many silver coins 1.75 cm in diameter and 2 mm thickness,
need to be melted to form a cuboid of 5.5 cm X 10 cm X 3.5 cm. Tan 0 1
5. The ratio of the radii of two spheres is 3:2, find the ratio of their (i)
volumes and (ii) surface areas. Cosec 2 1

Sec 1 2

Cot 1 0

5. Complementary angles (90- )

Sin(90- )=cos
cos(90- )=sin
tan(90- )=cot
cosec(90- )= sec
sec(90- ) = cosec
cot(90- ) = tan

26 27
 6. Trigonometric identities.
a. Sin2 +Cos2 =1 7) If sec then sec ( )
b. Sec2 -Tan2 =1 A) B) C) D)
c. Cosec2 Cot2 =1
8) If sin(A+B)=0 then A= ( )
7. Expressing all trigonometric ratios in terms of any one ratio.
A) B B) -A C) -B D) 0
Solve the following problems:
9) 9sin2 2 ( )
1) if Sin = , find the value of cos A) 1 B) 8 C) 10 D) 9
2) In AngleB=900 AB=12cm, BC=5cm AngleC= then find Sin 10) if 8tan x =15, then cotx = ( )
and Cos A) 8/15 B) 15/8 C) 17/8 D)8/17
3) Find the value of sin300.Cos600+Cos300.Sin600 11) Value of cos 10.cos 20.cos 30 ...............xcos1800 ( )
4) Simplify :2tan230+cot260/sec245-tan245 A) 1 B) 0 C) -1 D) 2
5) Express ‘cos in terms of tan 12) cos230-sin230= ( )
6) If sin(A-B)=1/2 and TanA=1, A,B are acute angels find the value of A) 1 B) 0 C) sin450 D) cos600
angleB. 13) If sin , then cos ( )
7) Show that Tan 480.Tan 160.Tan420.Tan740=1
A) 9/41 B) 41/40 C) 9/40 D)40/9
8) Show that cot +tan =Sec .cosec
14) in terms of sin ( )
9) Show that (sin 2 +(sin 2 =2

10) If 5sin , find the value of cosec2 -cot2 A) B) C) D)

11) Show that = 15) If sin(A-B)=1/2 , cos(A+B)=0 then A= ( )

A) 300 B) 900 C) 00 D) 600
12) If cosec , then show that Sec 16) 1+tan2 = ( )
13) If a cos and , prove that A) sec2 B) -sec2 C) cot2 D) cos2
a2+b2=m2+n2. 17) sin2 +sin2(90- = ( )
14) If sin , then find the value of tan . A) -1 B) cos2 C) 1 D) 0
15) Determine the value of ‘x’ if 2cosec230+x.sin260-tan230=10 18)if sec -tan =3, then sec +tan = ( )
Choose the correct answer A) 3 B) 2 C) 2/3 D) 1/3
1) (cosec ( )
A) 1 B) 0 C) ½ D) -1
2) Simplified value of 4sin230+1/3tan260 is ( )
A) 1 B) 2 C) 3 D) 4
3) Which of the following is a correct identity? ( )
A) Sin2-cos2=1 B) Sec2-tan2=1 C) Cot2-cosec2=1 D) Sec2 +cos2=1
4) If SinA=3/5 and CosA =4/5, then TanA= ( )
A) 1 B) 4/3 C) 3/4 D) 1/5
5) If cotA=TanB, then A+B= ( )
A) 300 B) 600 C) 00 D) 900
6) = ( )
A) 1 B) 2 C) 0 D) -1

28 29

12. APPLICATIONS OF TRIGONOMETRY Choose the correct answer (MCQ)

Key concepts : (1) If the ratio the length of a pole and the shadow 1:√3. Then the
(1) The knowledge of trigonometry can be used to find heights and angle of elevation of the Sun is ( )
distances in real life situations. (A) 60 (B) 90° (C) 30 0 (D)45
(2) Angle of elevation: (2) If the angle of depression of an object from 75m high tower is
The angle formed by the line of sight with the horizontal, when it is object is 30, then the distance from the base of tower is ( )
above the horizontal level Called the angle of elevation. (A) 50√3 m (B) 25√3 m (C) 150 m (D) 75
(3) Angle of Depression: (3) The length of the string of a kite of the flying at 100 m above the
The angle formed by the line of sight with the horizontal when it is ground with the elevation of 60° is ( )
below the horizontal level is Called the angle of depression. (A) 200√3 (B) 200 m (C) 20 m (D) 20
(4) If the height and length of the shadow of a man are Same, then
Solve the following: the elevation of son is ( )
(A) 45 (B) 30 (C) 60 (D) 90
1. A person observes the top of a tower at an angle of elevation 30°, (5) From the adjacent figure BC = ( )
from a point 50m away from the foot of the tower. Draw a suitable A) B) 1√3 (C) √3/h D)
diagram for this problem. (6) A ladder of 10 m length touches a wall at height of 5 m. The angle
2. A boat has to cross a river It Crosses the river by making an angle made by it with the horizontal is ( )
of 300 with the bank of the river due to stream and travels a distance (A) 90° (B) 30° (C) 450 (D) 600
of 300 m to reach the other side of the river. Draw the diagram for (7) From the adjacent figure of Value of ( )
this data. (A) 90° (B) 60° (C) 45° (D) 0°
3) If the length of the shadow of a 25 m high pole is also 25m, find the (8) If the angles of elevation of a tower from two points distant a and
angle of elevation of Sun rays. 'b' (a>b) from a its foot and in the same straight line from it are tower
4) From the top of building of height 25 m, a Ram has observed the is 300 and 600 then the height of the tower is ( )
top and bottom of another building with an angle of elevation 45 0 and A) (B) (c) a-b √
angle of depression 60° respectively Find the height of The second (9) A ladder makes. an angle of 60° with the ground, when placed
building. against a wall. If the foot of the ladder is 2m away from the wall, then
5) The pillars of equal height are standing. On either side of the road length. of ladder in (metres) is ( )
which is 100 m wide From a point in between the pillars, the angles (A) 4 /√3 (B) 4√3 (C) (D) 4
of elevation of the top of Pillars is 600 and 300 respectively. Find (10) The distance between two poles is ‘a’ metres, the height of one
height of the pillars and distance of the point from one pillar. pole is double that of the other If the angles of elevation of the tops of
6) A Man observes the top of a tower from a Point with an angle of poles from the mid-point of their Joining, are Complementary, then
elevation 60°, After walking 100m away from the tower, he once again the height of the smaller one ( )
observes the top with an angle of elevation 45. Find the height of the (A) a metres (B) metres (c) metres (D) 2a metres
tower.
7) A vertical tower is surmounted by a Vertical flag staff. At a Point
on the plane 70 m away from the tower, an observers notices that the
angles of elevation of the top and bottom of the flag staff are
respectively. 60° and 45°. Find the height of the flag staff and height
of the tower also.

30 31
 13. PROBABILITY 9) 20 defective pens are added to 150 good pens. one pen is drawn at
Key concepts : random, what is the probability that the pen is a good one.
10) There are cards in the box with numbers 1 to 100 written on them. If
1) The Probability of an Event E is P(E) = one card is picked randomly from the box. What is the Probability that
the card is
2) The sum of Probability of an event is P(E). The probability its
a) a perfect square (b) a composite number and c) a two digit number.
complementary event is P( ) is 1.
11) one card is drawn from a well shuffered deck of cards. Calculate the
P(E) + P(E) = 1.
Probability that the card will be
3) If the probability of an event is P(E), then. 0≤ P(E) ≤1.
a) a multiple of 2 b) Factor of 8
12) Two dice are thrown at the same time, what is the Probability that
4)Coins:
a) the sum of two numbers appearing on the top of the dice is an even
when a coin is tossed, the total no of possible outcomes =2
number.
when two coins are tossed, total no.of posible outcomes=4
b) the sum of two numbers appearing on the top of the dice is composite
when three coins are tossed, the total no of possible outcomes=8
number.

5) Dice:
Multiple Choice Questions
When a single dies is thrown, the no of possible outcomes = 6
when two dies are thrown, the no of possible outcomes=36
For what value of x, is the probability of an event.
6) Playing cards (52)
A)1 B) 2 C) 3 D) 4
Black (26) Red (26) Spade (13) clubs (13)
Hearts (13) Diamond (13)
A dice is thrown once. The probability of getting a prime number is

A, 2, 3, 4, 5, 6, 7, 8, 9, 10, J, Q, K. A) B) C) D)

(7) J, Q, K are the face cards,

1) If P(E)= then find the value of P( ).

3) Ramya said," the probability of an event is 5/3” Do you agree? Give

reasons.
4) There are cards in a box with english alphabets written on them. If a
card is picked randomly find the Probability of not gelling a vowel card.
5) Find the Probability of perfect square numbers from a first 50 natural
numbers.
6) A card is drawn at random from well shuffled deck of cards, Find the
Probability of not getting red queen.
7) Three coins are tossed. Find the Probability to get.
a) At least two heads b) All are tails.
8) Two dice are thrown at the same time. What is the Probability that the
sum of the two numbers appearing on the top of the dice is less than 10.

32 33

14. STATISTICS Find the mean daily wages of the workers of the factory by using an
Key concepts : appropriate method.
1) The mean for grouped data 3) Find the median age of 100 residents of a colony from the following
1) The direct method: x= data.
Age (in Years) 0-10 10-20 20-30 30-40 40-50 50-60 60-70
2) The assumed mean method: x=a+ No. of persons 10 15 25 25 10 10 5
3) The step deviation method: x=a+ xn
4) If the mean of the following frequency distribution is 50, then find
2) The median for grouped data the value of ‘k’.
Median = l + [ ]xn Class 0-20 20-40 40-60 60-80 80-100
Frequency 17 20 32 K 19
3) The mode for grouped data
Mode = l + xn 5) Find the mode of the following data
4) While drawing ogives boundaries are taken and x-axis and Class 1-3 3-5 5-7 7-9 9-11
cumulative frequencies are taken on y- axis Frequency 6 7 9 2 1
Short Answer Questions
1) Write the formula for arithmetic mean of assumed method of a 6) The distribution below gives the weights of 30 students of a class.
grouped data and explain each term of it. Find the median weight of the students.
2) Write the formula for median of a grouped data and explain each Weights
40-45 45-50 50-55 55-60 60-65 65-70 70-75
term of it. (in Kgs)
3) Write the formulas for mode of a grouped data and explain each No.of
2 3 8 6 6 3 2
term of it. Students
4)The median of observations 8,-2,5,3,-1,4,6,7 is 4.5 is it Correct? 7) The following data gives the information on the observed life times
Justify your answer. (in hours) of 225 electrical components.
5) Find the mean of first seven composite numbers? Life time (
0-20 20-40 40-60 60-80 80-100 100-120
6) Find the mode of the data 6,8,3,6,3,7,4,6,7,3,6,3. in hours
Frequency 10 35 52 61 38 29
7) Find the median of the data
Determine the model life times of the components
8) If the mean of 10,8,7,6,9,8,x.5 is 8 then find ‘x’. 8) The following data gives the information on the observed life span
9) if the mode of 7,8,8,7,9,5,1,11,x. is 8 the find ‘x’ (in hours) of 90 electrical components.
Long answer questions Life span 100-
1) Find the mean for the following data 0-20 20-40 40-60 60-80 80-100
(in hours) 120
Frequency 8 12 15 23 18 14
Class interval 0-10 10-20 20-30 30-40 40-50 50-60
Frequency 3 8 13 15 8 3 Draw both ogives for the above data.
9) The following table gives the mark obtained by 100 students in an
2) The daily wages of 80 workers of a factory. examination in mathematics subject.
Daily wages Draw ogive graph of less than and greater than cumulative
500-600 600-700 700-800 800-900 900-1000 frequencies.
(Rs)
Number of Marks 50-55 55-60 60-65 65-70 70-75 75-80
12 17 28 4 9
workers No.of Students 2 8 12 24 38 16

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