MATHEMATICS

JUNIOR INTER
MATHEMATICS
I.P.E. IMPORTANT QUESTION
BANK
ANDHRA PRADESH
INDEX
CHAPTER NAME Page. No
1. SETS 1-6
2. RELATIONS AND FUNCTIONS 6-11
3. TRIGONOMETRIC FUNCTIONS 11-14
4. COMPLEX NUMBER AND QUADRATIC EQUATIONS 14-17
5. LINEAR INEQUALITIES 17-18
6. PERMUTATIONS AND COMBINATIONS 19-22
7. BINOMIAL THEOREM 22-25
8. SEQUENCES AND SERIES 25-28
9. STRAIGHT LINES 28-34
10. CONIC SECTIONS 34-42
11.INTRODUCTION TO THREE DIMENSIONAL GEOMETRY 42-44
12. LIMITS AND DERIVATIVES 44-49
13. STATISTICS 50-52
14. PROBABILITY 52-57

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1. SETS
4 MARKS QUESTIONS

***1. If A = {1, 2,3, 4}, B ={3, 4,5, 6},C ={5, 6,7,8}and D ={7,8,9,10},
Find (i) Au B (ii)A uC (iii)B uC (iv)B u D (v)A u B uC (vi)A u B u D (vii)B u C u
D
***2. If A ={3,5, 7,9,11}, B ={7,9,11,13},C = {11,13,15}and D = {15,17}, find
(i) A n
(ii) B (iii) A nC n (iv) A (v)B n D
B
nC D nC
(vi) A n(B uC)
(vii) A n
D
(viii) A n(B u ix)( A n B) n(B uC) (x) ( A u D) n ( B u C )
D)
***3. If A ={3, 6,9,12,15,18, 21}, B ={4,8,12,16, 20}, C ={2, 4, 6,8,10,12,14,16}, D ={5,10,15, 20} , find
(i) A – B
(ii) A – C (iii) A – (iv)B – (v) (vi)D – A
(vii) B – D A C–A
(viii) B – (xii) D – C
C
D (ix) C – B (x) D – B (xi) C –
D
***4. If U = {1, 2, 3, 4, 5, 6, 7,8, 9}, A = {2, 4, 6,8}B = {2, 3, 5, 7}.Verify that
( 1 1 1 1 1 1
i) ( A u B) = A n B (ii) ( A n B) = A u B
***5. Draw appropriate Venn diagram for each of the following:
1
(i) ( A u B) 1
(ii) A n (iii)( A n B)
1 1
(iii) A u B
1

, 1
,
B,
***6. Let U = {1, 2, 3, 4,5,6, 7,8,9}, A = {1, 2, 3, 4}, B ={2, 4, 6,8}, and C ={3, 4,5, 6}.
Find (i)
A1 (ii) B (iii) ( A u (iv) ( A u (v) ( A )
1
(vi) (B – C )
1

1 1 1
C) B) 1

**7. If A ={x : x is a natural number }, B={x : x is an even natural number}

C={x : x is a odd natural number }, and D ={x : x is a prime number} ,Find
(i) A n B (ii)A nC (iii)A n D (iv)B nC (v)B n D (vi)C n D
**8. Taking the set of natural numbers as the universal set, write down the complements of the
following sets:
i) {x : x is an even natural number} ii) {x : x is an odd natural number}
iii) {x : x is a positive multiple of 3} iv) {x : x is a prime number}
v) {x : x is a natural number divisible by 3 and 5} vi) {x : x is a perfect square}
vii) {x : x is a perfect cube} viii) {x : x +5 = 8}
ix) {x : 2x + 5 = 9 } x) {x : x  7}
xi) {x : x ϵ N and 2 x + 1 > 10 }

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**9. Show that for any set A and B,
A = (An B) u( A – B) and A u(B – A) = ( A u B)
**10. Let A and B be sets. If A n X = B n X =ф and A u X = B u X for some set X ,show that
A= B
**11. Let A, B and C be the sets such that A u B = A u C and A n B = A n C . Show that B = C
**12. Show that the following four conditions are equivalent:
i) A c ii) A – B iii) A u B = iv) A n B = A
B =ф B

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*13. Show that if A c B , the (C – B) c (C – A)
*14. Show that A n B = A n C need not imply B = C
2 MARKS QUESTIONS

***1. Which of the following are sets? Justify your answer.

i) The collection of all the months of a year beginning with the letter J.
ii) The collection of ten most talented writers of India
iii) A team of eleven best-cricket batsmen of the world
iv) The collection of all boys in your class
v) The collection of all natural numbers less than 100.
vi) A collection of novels written by the writer munshi Prem Chand.
vii) The collection of all even integers
viii) The collection of questions in this chapter.
ix) A collection of most dangerous animals of the world
***2. Write the following sets in roster form:
(i) A = {x : x is an int eger and – 3 x < 7}
(ii) B = {x : x is a natural number less than 6 }
(iii) C = {x : x is a two – digit natural number such that the sum of its digits is 8 }

(iv) D = {x : x is a prime number which is divisior of 60 }

(v) E= The set of all letters in the word TRIGONOMETRY

(vi) F= The set of all letters in the word BETTER
***3. Write the following sets in the set-builder form:
(i) {3, 6, (ii) {2, 4,8,16, (iii) {5, 25, 125, 625}
9,12} 32}

(iv) {2, 4, 6, (v) {1,4, 9, ...., 100} (vi){1, 4, 9,16, 25,...}

...............
}
***4. Write the solution set of the equation x2 + x – 2 = 0
***5. Match each of the set on the left described in the roster form with the same set on the right
described. In the set-builder form
(i) {P, R, I , N , C, A, L} (a) {x : x is a positive integer and is a divisor of 18}
2
(ii) {0} (b) {x : x is an integer and {x – 9 = 0}
(iii) {1, 2, 3, 6, 9,18} (c) {x : x is an integer and x +1 = 1}
(iv) {3, –3} (d) {x : x is a letter of the word ‘PRINCIPAL’}
***6. Write down all the subsets of the following sets
(i){a} (ii){a, b} (iii){1, 2,3} (iv)ф
***7. Write the following intervals in set-builder form:
(i)(–3, 0) (ii)[6,12] (iii)(6,12] (iv)[–23,5)

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***8. Let A= {1,2,3,4,5,6} B= {2,4,6,8}. Find A-B and B-A
***9. Let V= {a, e, i, o, u} B= {a, i, k, u} find V-B and B-V

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***10. Fill in the blanks to make each of the following a true statement:
1 1
(i) A u A = ... (ii)ф n A 1 1
(iii) A n A = (iv)U n A = ...
= ... ...
***11. Let U= {1,2,3,4,5,6,7,8,9,10} and A= {1,3,5,7,9} Find A1
***12. Let U= {1,2,3,4,5,6} A= {2,3} and B= {3,4,5} Find 1 1 1
A , B , A n B1 , A u B and hence show that
( 1 1 1
A u B) = A n B
***13. List all the subsets of the set {-1,0,1}
**14. If U ={a,b,c, d, e, f , g, h}, find the complements of the following sets:
(i) A ={a,b,
(ii)B ={d, e, f , (iii)C ={a, c, e, (iv)D ={ f , g, h, a}
c}
g} g}
**15. Let
A = {1, 2,{3, 4},5}, which of the following statements are incorrect and why?
(i) {3, 4} c A
(ii) {3, 4}ϵ A (iii) {{3, 4}} c (iv)1ϵ A (v)1 c A
(vi){1, 2,5} c A
фϵ A
(vii) {1, 2, 5}ϵ (ix)
A (viii) {1, 2,3} c
A
( x)фc A (xi) {ф} c A
A

**16. Which of the following pairs of sets are equal? Justify your answer
(i) X the set of letters in “ALLOY” and B the set of letters in “LOYAL”
2 2
(ii) A = {n : n ϵ Z and n  4} and B= {x : x ϵ R and x – 3x + 2 = 0}
*17. Match each of the set on the left in the roster form with the same set on the right described
in
set-builder form:
(i) {1, 2, 3, 6} (a) {x : x is a prime number and a divisor of 6}

(ii) {2, 3} (b) {x : x is aodd natural number less than 10}

(iii) {M,A,T,H,E,I,C,S} (c) {x : x is a natural number and divisor of 6 }

(iv) {1, 3, 5, 7, 9} (d) {x : x is a letter of theword MATHEMATICS }

*18. State whether each of the following statement is true or false. Justify your answer.
(i) {2,3, 4,5}and {3, 6}are disjoint sets.
(ii) {a, e,i, o, u} and {a,b, c, d} are disjoint sets.
(iii) {2, 6,10,14}and{3, 7,11,15} are disjoint sets.
(iv) {2, 6,10}and{3, 7,11} are disjoint sets.

1 MARK QUESTIONS (MULTIPLE CHOICE QUESTION)

3
1. List of elements of the set { x : x is an integer, x  50}
1) {2, 3, 4} 2) {1, 3, 4} 3) {1, 2, 3} 4) {1, 2, 4}

2. Set builder form of {1, 2, 3, 4} is

1) {x : x is a positive integer, 2) {x : x is a positive integer,

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3) {x : x is a positive integer,
30 }
1 < 20 }
< 15}
4) {x : x is a positive integer, 2x < 20}
3. Which of the following statements is true

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1) {a, b} c {x : x is a vowel in the English alphabet}

2) {1, 2, 3, 4} c {1, 2, 3}
3) {5}ϵ{4, 5, 6}
4) {2, 3}ϵ{1,{2, 3} 4, 5}
4. The set builder form of interval [–4, 5) is
1) {x : x is an integer –4  x < 5}
2) {x : x is an integer –4 < x < 5}
3) {x : x is an integer –4  x  5}
4) {x : x is an integer –4 < x  5}
5. A= {1, 2, 3, 4, 5}, B = {4, 5, 6, 7,8} , then ( A – B ) u( B – A)

1) {1, 2, 3, 4, 5, 6, 7,8} 2) {1, 2, 3, 6, 7,8}

3) {1, 2, 4, 5, 6}
4) {2, 4, 6,8}

6. If
A = {1, 2, 3, 4}, B = {2, 3, 4}, C = {3, 4, 5, A n( B u C )
6} then

1) {2, 3,
2) {1, 2, 3, 3) {3, 4, 5, 4) {1, 6}
4}
4} 6}
7. Which of the following is false?
1) ( A ')' = A 2) A u A' =ф

3) (A u B) ' = A 'n 4) ф' = U ( U is universal set)

B'
{2, 3} , then ф'n A =
8. If U = {1, 2, 3, 4, 5}
and

1) ф 2) U 3) A 4) {1, 4, 5}
9. U ={x : x is a natural number 1 < x < 15}
{x : x is a prime number 1 < x < 15} then
and
A' =
1) {2, 3, 5, 7,11,13} 2) {4, 6,8, 9,10,12,14}
3) {2, 3, 4, 5, 4) {2, 4, 6,8,10,12,14}
7,11,13}

10. Which of the following is not a subset of {a, b, c, d} ?

1) ф 2) {a, 3)
{d , 4)
{b, d}
b}
e}

List of elements of the set{x : x is a natural number, x  50}

3 3
11.

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1) {1, 8, 24} 2) {1, 6,12,15,18, 21, 24}
3) {1, 8, 27} 4) {1, 4, 9,16, 25, 36, 49}

12. The set builder form of {2, 5,10,17} is

1) {x : x is a positive integer,
20} 2) {x : x is a positive integer,
1 < 20}
3) {x : x is a positive integer, 2
x –1 < 4) {x : x is a positive integer, 2x < 20}
20}
13. If
{1, 2, 3}, B = {2, 3, 4}, C = {4, 5, 6} , then ( A u B ) u C =

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1) {2, 2) {1, 2, 3, 4, 5, 3) {2, 4, 4) {1, 3, 5, 7}
3} 6} 6}

14. If a set has n elements then the total number of subsets of A is

1) n 2) n2 3) 2n 4) 2n
15. Given the sets 1, 3, 5 , B =
{ } {2, 4, 6} and C = {0, 2, 4, 6,8} , which of the following may be
considered as universal set for all the three sets A, B and C
1) {0,1, 2, 3, 4, 5, 2) ф 3) {0,1, 2, 3, 4, 5, 6, 7,8, 4) {1, 2, 3, 4, 5, 6, 7,8}
6} 9,10}

1- MARK QUESTIONS:
16. let
A= {1, 2, 3, 4, 5, 6}.Insert the appropriate symbol ϵ or  in the blank spaces:

(i) 5 . . . A (ii) 8 . . . A (iii) 0 . . . A

(iv) 4 . . . A (v) 2 . . . A (vi) 10 . . . A
17. Which of the following are examples of the null set
(i) Set of odd natural numbers divisible by 2
(ii) Set of even prime numbers
(iii) { x:x is a natural number, x < 5 and x > 7}
(iv) {y : y is a point common to any two parallel lines}
18. Which of the following sets are finite or infinite?
(i) The set of months of a year
(ii) {1, 2, 3, ….}
(iii) {1, 2, 3, …., 99, 100}
(iv) The set of positive integers greater than 100
(v) The set of prime numbers less than 99
19. State whether each of the following set is finite or infinite
(i) The set of lines which are parallel to the x-axis
(ii) The set of letters in the English alphabet (iii) The set of numbers which are multiple of 5
(iv) The set of animals living on the earth
(v) The set of circles passing through the origin (0,0)
20. In the following, state whether A= B or not:
(i) A= {a, b, c, d} B= {d, c, b, a}
(ii) A= {4, 8, 12, 16} B= { 8, 4, 16, 18}
(iii) A= { 2, 4, 6, 8, 10} B={x : x is positive even integer and
10 }
(iv) A= { x:x is a multiple of 10}, B={10, 15, 20, 25, 30, . . .}
21. Are the following pair of sets equal? Give reasons
(i) A = {2,3}, B= {x : x is a solution of x2+5x+6=0}\
(ii) A= { x : x is a letter in the word FOLLOW}
B= { y : y is a letter in the word WOLF}
22. From the sets given below, select equal sets:

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A= {2, 4, 8, 12}, B= {1, 2, 3, 4}, C={4, 8, 12, 14}, D={3, 1, 4, 2}
E= {-1,1}, F= {0, a}, G={1, -1} H={0,1}
23. Make correct statements by filling in the symbols c or ą in the blank spaces:

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(i) {2, 3, 4}...{1, 2, 3, 4, 5}
(ii) {a, b, c } {b, c, d}
(iii) { x: x is a student of Class XI of your school}….{ x: x student of your school}

(iv) { x : x is a circle in the plane}….{ x : x is a circle in the sample plane with radius 1 unit}
(v) { x : x is a triangle in a plane} …. { x : x is a rectangle in the plane}
(vi) { x : x is an equilateral triangle in a plane} …{ x : x is a triangle in the same plane}
(vii) { x : x is an even natural number} … { x : x is an integer}
24. Examine whether the following statements are true or false:
(i){a, b} ą {b, c, a}
(ii ){a, e} c {x : x is a vowel in the English

alphabet} (iii ){1, 2, 3} c {1, 3, 5}

(iv) {a} c {a, b, c}
(v) {a}ϵ{a, b, c}
(vi) {x : x is an even natural number less than 6} c {x : x is a natural numnber which divides
36}
25. Write the following as intervals:
(i){x : x ϵ R,–4 < x  6} (ii){x : x ϵ R,–12 < x < –10}
(iii){x : x ϵ R, 0  x < 7} (iv){x : x ϵ R,3  x  4}
26. If X = {a, b, c, d} and Y = { f , b, d , g} ,
Find i) X – Y
ii) Y – X iii) X nY
27. If R is the set of real numbers and Q is the set of rational numbers, then what is R-Q?

2. RELATIONS AND FUNCTIONS

8. Marks Questions
***1. If f and g are real valued functions defined by f ( x) = 2x –1 and g ( x) = x2 then find

i)
(3 f – 2g )( ( f 
(x
ii)
( fg )( x) iii) iv)
(f + g + 2 )( x)
x)
vi)
( 2 + f )( )
v)
(2 f )( x)  
g
x)
***2. If f ( x) = x2 and g ( x ) = x find the following function
I) f + II) f – III) f . g IV) 2 f
g g
V) f + f
3
VI) ( for x ϵ N )
g
***3. If f = {(4, 5),(5, 6 ) , ( 6, – 4)} and g = {(4,– 4 ) , ( 6, 5),(8, 5)} then find
i) f + ii) f – g iii) 2 f + iv) f + 4
g 4g
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f
v) fg vi)
g
***4. Determine the quadratic function ‘ f ’ defined by f ( x ) = ax2 + bx + c f (0) = 6, f ( 2 ) = and
. If 1
f (–3) = 6

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3x – 2, x>3
***5. If the function f is defined by 
f( x) = x2 – –2  x  2 then find the values if exists of
2, x < –3
 2x +1,

f (4) , f (2.5), f ( –2 ) , f ( –4 ) , f ( 0 ) , f ( –7 ) , f (1), f (9)
***6. The function f is defined by
1– x, x<0

f ( x) = 1, x=0
x +1, x > 0

Draw the graph of f ( x)
4- Marks Questions
***1. Let f ( x) = x2 and g ( x) = 2x +1 be two real functions. Find
( f 
( f + g )( x ) , ( f – g )( x ) , ( fg )( x ) , (
 
x) g
 
***2. Let f , g : R → be defined, respectively by f ( x) = x +1, g ( x) = 2x – 3 . f+ f– and
f R Find g, g
g

***3. Let f ( x)
= and g ( x ) = x be two functions defined over the set of non-negative real numbers.
x (f
Find ( f + g )( x ) , ( f – g )( x ) , ( fg )( x) and ( x) .
 
g
***4. The function ‘ t ’ which maps temperature in degree Celsius into temperature in degree Fahrenheit
9C
is defined by t (C ) = +
32
5
Find i) t (0) ii)
t (28) iii)
t (–10)
iv)
The value of C, when t (C ) = 212 v) t (–28)
***5. Which of the following relations are function? Give reasons. If it is a function, determine its
domain and range.
i) {(2,1),(5,1)(8,1),(11,1), (14,1),(17,1)}
ii){(2,1),(4, 2 ) , ( 6, 3),(8, 4),(10, 5),(12, 6),(14, 7)}
iii) {(1, 3),(1, 5 ) , ( 2, 5)}
***6. Let f = {(1, 1), (2, 3), (0,–1),(–1, – 3)} be a function from Z to Z defined by f ( x) = ax + b ,
for some integers a, b . Determine a, b
***7. Let A = {1, 2, 3} , B = {3, 4} and C = {4, 5, 6} . Find
i) A ( B n C ) ii) ( A B) n ( A C )
iii) A ( B u C iv) ( A B) u ( A C )
)
***8. Let A = {1, 2}, B = {1, 2, 3, 4}, C = {5, 6} and D = {5, 6, 7, 8} . Verify that
i) A
( B n C ) = ( A B) n ( A C ) ii) A  C is a subset of B  D
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***9. Let A ={1, 2, 3,.................,14}. Define a relation R from A to A by
R = {( x, y ) : 3x – y = 0, where, x, y ϵ A}. Write down its domain, codomain and range.
***10. Define a relation R on the set N of natural numbers by

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R = {( x, y ) : y = x + 5, x is anatural number less than 4; x, yϵ N} . Depict this relationship
using roster from. Write down the domain and the range.
***11. The fig shows a relationship between the sets P and Q. Write this relation

**12. Let A = {1, 2, 3, 4, 6}. Let R be the relation on A defined by

{(a, b) : a, b ϵ A, bis exactly divisibleby a}.
i) Write R in roster form
ii) Find the domain of R
iii) Find the range of R
**13. Let A = {1, 2, 3, 4, 5, 6}. Define a relation R from A to A by R = {( x, y ) : y = x +1}
i) Depict his relation using an arrow diagram
ii) Write down the domain, codomain and range of R.
**14. The Fig Shows a relation between the sets P and Q. Write this relation (i) in set-builder from, (ii)
in roster form. What is its domain and range?

 x2 , 0  x  3
**15. The relation f is defined by f ( x = 
3x, 3  x  10
 x2 , 0  x  2
The relation g is defined by g ( x = 
) 3x, 2  x  10
Show that f is a function and g is not a function
2- Marks Questions

1. i) If
f (1.1) – f (1)
) = x2 , find
(
1.1–1
f (5) – f (1)
find the value)of
ii) if = x3
)
5 –1
2. Find the domain of the following functions
x2 + 2x +1 x2 + 3x + 5
f (x = 2 ii) 2(
=
i) x – 8x x – 5x + 4
) )
+12
3. Find the domain and range of the following real functions.
i) iii) f ( x) = (– x –1)x ) =

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ii) iv) ( x) = )= x –1

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4. Let A = {9, 10, 11, 12, 13} f : A → N be defined by f ( n ) = the highest prime factor of n .
and let
Find the range of f
5. Let R be the set of real numbers. Define the real function
f : R → R by f ( x) = x +10 , and sketch the graph of this function
6. A function f is defined by f ( x) = 2x – 5 . Write down the value of

i) f (0) ii) f (7) iii) f (–3)

7. The Cartesian product A A has 9 elements among which are found (–1, 0) and (0, 1) . Find the
set A and the remaining elements of A A
8. If P = {a, b, c} and Q = {r} , form the sets P  Q and Q  P , are these two products equal?
9. Determine the domain and range of the relation R defined by R = {( x, x + 5) : x ϵ{0,1, 2, 3, 4,
5}}.
10. Let N be the set of natural numbers. Define a real valued function
f : N → N by f ( x) = 2x +1 . Using this definition, complete the table given below

x 1 2 3 4 5 6 7
y f (1) = ... f ( 2 ) = ... f (3) = ... f ( 4 ) = ... f (5) = ... f (6) = ... f ( 7 ) = ...
11. Define the function f : R → R by y = f ( x) = x2 , x ϵ R . Complete the Table given below by
using
this definition. What is the domain and range of this function Draw the graph of f
x -4 -3 -2 -1 0 1 2 3 4
y = f ( x) = x2

( x2  
12. Let  x, 2
:xϵ R be a function from R into R. Determine the range of f .
 1+ x  J
13. Let A = {1, 2, 3, 4}, B = {1, 5, 9,11, and f = {(1, 5 ) , (2, 9 ) , ( 3, 1),(4, 5 ) , (2, 11)}. Are the
15, 16 }
following true?
i) f is a relation from A to B
ii) f is a function from A to B justify your answer in each case
14. Find the range of each of the following functions
i) f ( x) = 2 – 3x, x ϵ R, x > 0
ii) = x2 + 2, x
) real number.
iii) f
( x) = x, is a real number
x
15. Let R be a relation from N to N defined by R = {(a, b) : a, b ϵ N and a = b } . Are the
2

following true?
i) (a, a) ϵ R , for all a ϵ N ii) (a, b ) ϵ R , implies (b, a) ϵ R
iii) (a, b ) ϵ R,(b, c) ϵ R implies (a, c ) ϵ R
16. Let f be the subset of Z  Z M e Choice Questions
Justify your answer. ult
ipl
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 MATHEMATICS
defined by f= {(ab, a + b) : a, b ϵ Z } . Is f a function from Z to Z?
( y
1. If 2x +1, = (3, 3) then the value of x & y are
 
 2 
3
1) 1, 6 2) 2, 2 3) 3, 3 4) 7,
2
2. If A = {1, 2, 3}, B = {a, b} , then A B =

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 MATHEMATICS
1){(1, a )( 2, b)(3, a)} 2) {(a,1)(b, 2 )( a, 3)(b, 3)}
3) {(1, a)(1, b )( 2, a )( 2, b)(3, a)(3, 4) {(1, a )(1, b )( 2, b)(3, b)}
b )}
3. A relation can be represented as
1) Roster method 2) Set-builder method 3) An arrow diagram 4) In above three forms
4. If A = {1, 2, 3, 4} and a relation R from A to A is defined by R = {( x, y) : y = 2x} then co-
domain of
R is
1) {2, 4} 2) {1, 2, 3, 4} 3) {1, 3} 4) {1, 4}
5. If x then f 0 = ……
)= ()
x
1) 1 2) -1 3) undefined 4) 0
1
6. The domain of the function is
x2 – 25
1) (–, – 5) u(5, ) 2)
(–, – 5]u[5, ) 3)
(–, – 5]u(5,  ) 4) (–, – 8) u[5, )
7. Range of the function
) = x2 , x ϵ R is
1) [0, ) 2)
(–, ) 3)
(–, 0) 4)
(0, )
8. A function) is defined by ) = x2 + 2x – 7 the value of ) = ………
1) 2 2) -7 3) 8 4) 9
9. Let f = {(0,1),(1, 3),(2, 5)} be a linear function from {0,1, 2} to f ( x) = ……
N then
1) 2 x – 1 2) 2 x + 3) x2 –1 4) x2 +1
10. The domain of 1
x–x is
1) R 2) [0, ) 3) (–, 0] 4) Z
11. Let f = (1,1), (2, 3),(0,–1),(–1, – 3) be a linear function from Z into Z, f
{ } ( x) =
then
1) 2 x + 2) x2 –1 3) 2 x – 4) x2 +1
1 1
12. Let→ R defined as ) = x4 choose the correct answer.
1) f is one-one onto 2) f is many-one onto
3) f is one-one but not onto 4) f is neither one-one nor onto
13. Let f : R → R be defined as f ( x) = 3x . Choose the correct answer.
1) f is one-one onto 2) f is many-one onto
3) f is one-one but not onto 4) f is neither one-one nor onto
1

If f : R → R be given f ( x ) = (3 – x ) then fof ( x ) is

3
14. 3

1) x3 2) x3 3) x 4) (3 – x3 )
15. Let  4 
– – →R function defined as )= . The inverse of f is the map g :
  3x + 4
3
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 MATHEMATICS
Range  J 4 
f →R– – given by
 
 3 J
3y 4 4 3y
1) g ( y ) = 2) g ( y ) = 3) g ( y ) = 4) g ( y ) =
y y

3–4y 4 – 3y 3–4y 4–3y

One word Questions
(x 2 (5 1
16. If + 1, y – = , , then find the values x, y
   
of 3 3 3 3
   
17. If the set A has 3 elements and the set B = {3, 4, 5} , then find the number of elements in ( A B )
=

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18. If G = {7, 8} and H = {5, 4, 2} , then find G  H and H  G
19. If A = {–1, 1} , then find A A A
20. If= (a, x ) , ( a, y ) , ( b, x ) , ( b, y ) . Then find set A and set B
{ }
21. Let A = {1, 2} and B = {3, 4} . Write A B . How many subsets will A B have? List them.
22. If P = {1, 2} , then find the set P  P  P

23. Write the relation R = {(x, x ): x is a primenumberlessthan10}in roster form

3

24. let
{x, y, z} and B = {1, 2} . Find the number of relations from A to B.
3. TRIGONOMETRIC FUNCTIONS

8 MARKS QUESTIONS
3 3π x x and x
***1. If tan x = ,π< x < ,find the value of sin , tan [MODEL PAPER-2]
4 2 cos 2 2 2
x x x 4
I) Find sin , cos and tan , if tan x = – , in quadrant II
x
2 2 2 3
x x x 1
II) Find sin , cos and tan , if cos x = – , x in quadrant III
2 2 2 3
x x x 1
III) Find sin , cos and tan , if sin x = , x in quadrant II
2 2 ( π 2 2( π4 3
***2. Prove that cos2 x + cos2 x + + cos x – = [MODEL PAPER-2]
   3 2
3
   
***3. Prove that sin x + sin 3x + sin 5x + sin 7x = 4 cos x cos 2x [MODEL PAPER-1]
sin 4x
***4. Prove that cos 6x = 32 cos6 x – 48 cos4 x +18 cos2 x –1
(sin 7 x + sin 5x) + (sin 9x + sin
***5.
3x) Prove that = tan 6x
(cos 7 x + cos 5x) + (cos 9x +
cos 3x)
Cos 4 x + Cos 3x + Cos 2 x
***6. Prove that = Cot 3x
Sin 4 x + Sin 3x + Sin 2 x
***7. Prove that Sin 2 x + 2 Sin 4 x + Sin 6 x = 4 Cos 2 x Sin 4 x
***8. Prove that Cot4x(Sin 5x +Sin 3x) = Cotx(Sin 5x – Sin 3x)
( 3π   ( 3π  
***9. Prove that Cos + x Cos(2π+ x) Cot – x + Cot(2π+ x) = 1
     
2 2
  L   
***10. Prove that Cot x Cot 2 x – Cot 2 x Cot 3 x – Cot 3 x Cotx = 1
cos 7 x + cos 5x
**11. I) Prove that = cot x
sin 7x – sin 5x
sin 5x – 2 sin 3x + sin x
II) Prove that cos 5x – cos x = tan x
π 7π π 3
**12. I) Prove that 2 Sin 2 + Cos ec 2 Cos2 =
6 6 3 2
π 5π π
II)Prove that Cot 2
+ Cos ec + 3 Tan =6 2

6 6 6
3π π π
**13. I) Prove that 2 Sin 2 + 2 Cos 2 + 2 Sec 2 = 10
(π4  4 3
Tan +x 2

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  1+ Tan x 
4

II) Prove that  = 
(π
Tan  4 – x  L1– Tan x 

 

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**14. I) Prove that Cos + x – Cos – x = – 2 Sin x
   
4 4
   
Cos 9 x – Cos 5 x Sin 2 x
II) Prove that =
Cos10 x
–
Sin 17 x – Sin 3 x
Sin 5 x + Sin 3
**15. I) Prove that = Tan 4 x
x
Cos 5 x + Cos 3 = Tan 2 x
x
Sin x + Sin 3 x
II) Prove that
Cos x + Cos 3
x
Sin x – Sin x–y
**16. I) Prove that = Tan
y 2
Cos x + Cos
y
II) Prove that Sin x – Sin 3x
= 2 Sin x
Sin 2 x – Cos 2 x
(
4 tan x 1– tan2 x )
**17. I) Prove that tan 4x = 2 4
1– 6 tan x + tan x
II) Prove that Cos4x =1–8Sin2 x Cos2 x
x–y
**18. I) Prove that (cos x – cos y)2 + (sin x – sin y)2 = 4 sin2
2
II) Prove that sin 3x + sin 2x – sin x x 3x
= 4 sin x cos cos
2 2
**19. I) Prove that (sin 3x + sin x) sin x + (cos 3x – cos x) cos x = 0
x+y
II) Prove that (cos x + cos y)2 + (sin x – sin y)2 = 4 cos2
2
**20. I) If sin x = 3 , cos y = –
where x and y both lie in second quadrant, find the value of
12
5 13
sin(x + y)
x 9x 5x
II) Prove that cos 2x cos – cos 3x cos = sin 5x sin
2 2 2
π
**21. I) Find the value of Tan
8
π 9π 3π 5π
II) Prove that 2 cos cos + cos + cos =0
13 13 13 13
2 MARKS QUESTIONS
–1
1. If Cos x = , x lies in third quadrant, then find the other five Trigonometric functions
2
[MODEL PAPER-1]
3
2. If Sin x = , x lies in Second quadrant, then find the other five Trigonometric functions
5
3
3. If Cot x = , x lies in third quadrant, then find the other five Trigonometric functions
4

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13
4. If Sec x = , lies in fourth quadrant, then find the other five Trigonometric functions
x
5 x lies in second quadrant, then find the other five Trigonometric functions
–5
5. If Tan x =
, 12
π π 5π π
6. Prove that 3sin sec – 4 sin cot = 1
6 3 6 4
0
7. Find the value of sin15
13π
8. Find the value of tan
12

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sin(x + y) tan x + tan y
9. Prove that sin(x – y) = tan x – tan y
10. Show that tan(π3x tan 2x tan πx = tan 3x – tan 2x – tan x
 ( 
11. Prove that cos + x + cos – x = 2 cos x
   
4 4
   
12. Prove that Sin(n + 1) x Sin(n + 2) x + Cos (n +1) x Cos(n + 2) x = Cos x
(π  (π  (π  (π 
Cos
13. Prove that
 – x Cos – y  –Sin – x  Sin – y  = Sin( x + y)
π 4 4 4 4
Cos( + x) Cos(–x)   
14. Prove that = Cot2 x
Sin(π– π
x) Cos( + x)
2
15. Prove that Sin2 6x –Sin2 4x =Sin 2x. Sin10x
16. Prove that Cos2 2x–Cos2 6x =Sin4x Sin8x
17. Evaluate sin (π+ x)sin (π– x) cos ec2 x . [MODEL PAPER-2]
18. If in two circles, arcs of the same length subtends angles 6 0 and 7 5 0 at the center, find the
0

ratio of their radii?

19. Find the angle in radian through which a pendulum swings if its length is 75cm and the tip
describes an arc of length
i) 10 cm ii) 15cm iii) 21cm
–3
20. If Cos x = , x lies in the third quadrant, find the values of other five trigonometric functions
5
–5
21. If Cot x =
, 12 x lies in second quadrant, find the values of other five trigonometric functions.
MULTIPLE CHOICE QUESTIONS
1. The value of cos 5π is
1) 0 2)1 3)-1 4) None of these
0 0 0 0
2. The value of cos1 cos 2 cos 3 .....cos179 is [MODEL PAPER-1]
1
1) 2) 0 3)1 4)-1
2
3. If sinθ+ cos ecθ= 2 then sin2θ+ cos ec2θ is equal to
1)1 2)4 3)2 4) None of these
1 1
4. If Tanθ= Tanф= the value of θ+ф is
2 3
π π
1) 2) π 3)0 4)
6 4
1– tan2 150
5. The value of 2 0
is
1+ tan 15
1)1 2) 3
3 3) 4)2
2
6. The value of sin(450 +θ) – cos(450 –θ) is
1) 2 cosθ (π  2) 2(πsinθ 3)1 4)0
7. The value of cot +θ cot –θ is
   
4 4
   
1) -1 2) 0 3) 1 4) None of these
8. cos 2θcos 2ф+ sin2 (θ–ф) – sin2 (θ+ф) is equal to

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1) 2) cos 3) sin 2(θ– 4) cos 2(θ–ф)
2(θ+ф) ф)

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9. The value of sin 50 – sin 70 + sin10 is equal to
0 0 0

1
1)1 2) 0 3)
2 4)2
10. If sinθ+ cosθ= 1 then the value of sin 2θ is equal to
1
1) 1 2) 3) 0 4) -1
2
1 MARK QUESTIONS
11. Convert 400 201 in to radian measure.
12. Convert 6 radians into degree measure
13. Find the radius of the circle in which a central angle of 600 intercepts an arc of length 37.4cm
22
(use π= )
7
14. The minute hand of a watch is 1.5cm long. How far does its tip move in 40 minutes?
(use π= 3.14 )
15. Find the radian measures corresponding to the following degree measures.
i) 250 ii) –470301 iii) 2400 iv) 5200
22
16. Find the degree measures corresponding to the following radian measures. (use π= )
7
i) 11 5π 7π
ii) –4 iii) iv)
16 3 6
17. A wheel makes 360 revolutions in one minute. Through how many radians does it turn in one
second?
18. Find the degree measure of the angle subtended at the centre of a circle of radians 100 cm by
22
an arc of length 22 cm (useπ = )
7
31π
19. Find the value of Sin
3
20. Find the value of Cos(–17100 ) [MODEL PAPER-2]
21. Find the values of 19π ( –11π
i) Sin 765o ii) Cos ec (–1410o iii) Tan iv) Sin
)  
3 3
( –15π  
v) Cot =1
 
 4
22. Find the value of
i) sin ii) tan150 iii) cot150 iv) cos v) sin1050 vi) tan 750
750 750
vii) cot viii) cos(1050 ) ix ) x) cot1050 xi) cos150 xii) sin150
750
tan1050
π π π –1
23. Prove that Sin2 + Cos2 – Tan2 =
6 3 4 2

4. COMPLEX NUMBERS
4 Marks Questions
(3 + i 5 )( 3 - i 5 )
***1. Express the following expression in the form of a + ib,
(3 + 2i ) - 3 (
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-i 2 )
***2. Reduce ( 1
2 ( 3 - 4i  to the standard form.
 -  
 1 - 4i 1 + i 5 + i
 

28 | P a g e
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***3. Find the conjugate of

( 3 - 2i )( 2 + 3i
)
(1 + 2i )( 2 - i
)
***4. Find the real numbers x and y if is ( x - iy )( 3 + 5i ) conjugate of -6 - 24i

***5. If ( x + iy)
3
= u + iv , then show
u
+
v
(
= 4 x2 - y2
x y
that
)
***6. If x - iy a - ib a2 + b2
that (x + y ) = 2 2
2 2

= c - id prove c +d
***7. Show that the four points in the Argand plane represented by the complex numbers 2 + i ,
4 + 3i , 2 + 5i , 3i , are the vertices of a square.
***8. Show that the points in the Argan plane represented by the complex numbers.-2 + 7i ,
-3 1 7
+ i , 4 - 3i , ( 1 + i) are the vertices of a rhombus.
2 2 2
***9. The points P, Q denote the complex numbers z1 , z 2 in the argand diagram. O is the origin. If
1
z z 2 + z1z 2 = 0 , then show that 3POQ = 900
π
***10. If the complex number z has argumentθ , 0 < θ < and satisfy the equation z - 3i = 3 . Then
( 6 2
prove that cotθ - =i
 
 z

(x + 1 )
2
(x + i 2 2
prove that a + b 2
=
2
***11. If a + ib
= )2x2 + 1 (2x + 1) 2
2

a + ib
***12. If x + iy = , Prove that x2 + y2 = 1
a - ib
***13. Let Z1 = 2 - i, Z2 = -2 + i find
Re( Z1Z2 
i)
ii) im ( 1 
  Z Z1
Z
 1
  1 
Z1 + Z2 + 1 Z1 - Z2 + 1
***14. Z1 = 2 - i, Z2 = 1 + i
find

2- Mark Questions
1. Find the multiplicative inverse of the following complex numbers
i) 4 - 3i + 5
ii) 3i
iii) 2 – iv) -i
3i
2. Express the following in the form of a + ib
i) 5 + i 2 ii) i-33 iii)  1 + i 7  + 4 + i 1  -  -4 + i iv) [1 - i]
4

1-i 
3     
 L 3 L 3  L3 
2 3 3
L 
1   1 

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vi) -2 - i
v)  + 3i L 3 
L 3


1+i 1-i
3. Find the modules of -
1-i 1+i
4. Express (-3 + -2 )( 2 3 - i) in the form of a + ib
5. If 4x + i (3x - y ) = 3 + i (-6) , where x, y are real numbers, the find the value of x, y

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6. If (a + ib)(c + id)(e + if )(g + ih) = A + iB Then show that
(a ( + b )( c
2 2
m
2
+ d2 )( e 2 + f 2 )( g 2 + h2 ) = A2 + B 2
1+i
7. If   = 1 , then find the least positive integral value of m
 1-i
β-α
8. If α and β are different complex numbers with β = 1 , then find 1 - αβ

9. Find the number of non-zero integral solutions of the equation 1 - i x = 2x

10. For any two complex numbers Z
and1 , prove that (Z1Z2 ) =
Re (Z1 ) Re (Z2 ) - im(Z1 )im (Z2 )
MULTIPLE CHOICE QUESTIONS
1. The value of i-999 is
1) 1 2) -1 3) i 4) -i
2. The multiplicative inverse of 1 + i is
1 1
1) (1 - i 2) (1 + i 3) 1 - i 4) i
) )
2 2
3. The modules of 5 + 4i is
1) 41 2) -41 3) 4) 41 -41
4. The value of -25 × -9 is
1) 15 2) -15 3) 15 i 4) None
2-i
5. The conjugate of is
1 - 2i
1) 4 + 3i 4 - 3i
5 2) 4 - 3i 3) 4) 1
5
6. The value of (z + 3) z + 3 ( ) is equivalent to

1) z + 3 2 2) z - 3 3) z2 + 3 4) None
7. Let x, y ϵR, then x+iy is a non real complex number if
1) x = 2) y = 3) x s 4) y s 0
0 0 0
8. Ifa + ib = c + id , then
1) a2 + c2 = 0 2) b2 + c2 = 3) b2 + d 2 = 4) a2 + b2 = c2 + d 2
0 0
9. The sum of the series i +i2 +i3 +.........+ upto 1000 terms are in G.P
1) 1 2) 0 3) -1 4) -2
i+z
10. The complex number z which satisfies the condition = 1 lies on
i-z
4) The line x+y = 1
2 2
1) Circle x + y = 1 2) The X-axis 3) The Y-axis
1 Mark Questions
11. Express the following in the form
3
of a + ib
(1  ( -1  3  -3 
i) (-5i )  i  ii) (-i )( 2i )  i iii) (5 - 3i
8  8  iv) 5i  i
) L5 
v) i9 + i19 vi) i-39 vii) 3[7 + 7i] + [7 + 7i]
1 2  5
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
28 MATHEMATICS
(1 
(1 - i) - (-1 + 6i) ix)  + i  - 4 + i 
viii)
x) i18 +   
L5 5 L 2 L  i  
12. Solve each of the following equations
i) 2x2 +x+1 = 0 ii) x2 + 3x+ 9 = 0 iii) -x2 +x- 2 = 0

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iv) x2 + 3x+ 5 = 0 v) x2 - x+ 2 = 0 vi) 2x2 + x +2 = 0
vii) 3 1
- 2x + 3 0 viii) x2 + x + =0 ix) x2 +1 = 0
2 2
x) x2 + 3 = 0
5. LINEAR INEQUALITIES
4 MARKS QUESTIONS
***1. Solve the following inequalities
i) 3( x – 2) 5(2 – x) x (5x – 2) (7x – 3)
 ii)
5 3 4 3< 5 –

iii) 2 x – 1 ) (3 x – 2 ) (2 – x ) iv) 5 ( 2x – 7) – 3 ( 2x + 3)  0, 2x +19  6x + 47

3 ( 4  5 –

***2. Solve the following inequalities and represent the solution graphically on number line
i) x (5x – 2) (7 x –
3)  – ii) 3x – 7 > 2 ( x – 6), 6 – x >
2 3 5 11– 2x

iii) 2 ( x –1) < x + 5 , 3( x + 2) > 2 3x – 4 x +1

 –1 v) 3 x – 7 < 5 + x , 11 – 5 x  1
iv)
–x 2 4
5 – 3x
***3. Solve i) –5  8 3(x – 2 3x +11
ii) –15  0 iii) 7   11
2
)5 2
7x
iv) –3  4 –  18
2
***4. Find all pairs of consecutive odd natural numbers, both of which are larger than 10, such
that their sum is less than 40
***5. Find all pairs of consecutive odd positive integers both of which are smaller than 10 such
that their sum is more than 11.
***6. Find all pairs of consecutive even positive integers, both of which are larger than 5 such that
their sum is less than 23.
***7. In an experiment, a solution of hydrochloric acid is to be kept between 300 and 350 Celsius.
What is the range of temperature in degree Fahrenheit if conversion formula is given by
5
C = ( F – 32), where C and F represent temperature in degree Celsius and degree
9
Fahrenheit respectively
***8. A solution is to be kept between 680 F and 770 F. What is the range in temperature in degree
9
Celsius (c) if the Celsius/Fahrenheit (F) conversion formula is given by F = C + 32?
5
***9. Solve 5x – 3 < 3x + 1 when
i) x is an integer ii) x is a real number
***10. How many litres of water will have to be added to 1125 litres of the 45% solution of acid so
that the resulting mixture will contain more than 25% but less than 30% acid content?
***11. A solution of 8% boric acid is to be diluted by adding a 2% boric acid solution to it. The
resulting mixture is to be more than 4% but less than 6% boric acid. If we have 640 litres of
the 8% solution. How many litres of the 2% solution will have to be
added?
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***12. A manufacture has 600 litres of a 12% solution of acid. How many litres of a 30% acid
solution must be added to it so that acid content in the resulting mixture will be more than
15% but less than 18%?
***13. The marks obtained by a student of class XI in first and second terminal examination are 62
and 48, respectively. Find the minimum marks he should get in the annual examination to
have an average of at-least 60 marks
***14. The longest side of a triangle is 3 times the shortest side and the third side is 2 cm shorter
than the longest side. If the perimeter of the triangle is at least 61 cm, find the minimum
length of the shortest side.
***15. A man wants to cut three lengths from a single piece of board of length 91cm. The second
length is to be 3cm longer than the shortest and the third length is to be twice as long as the
shortest. What are the possible lengths of the shortest board if the third piece is to be at least
5cm longer than the second?
MA
***16. IQ of a person is given by the formula IQ = 100
CA
Where MA is mental age and CA is chronological age. If 80  IQ  140 for a group of 12
years old children, find the range of their mental age
**17. A man wants to cut three lengths from a single piece of board of length 91cm. The second
length is to be 3cm longer than the shortest and the third length is to be twice as long as the
shortest. What are the possible lengths of the shortest board if the third piece is to be at least
5cm longer than the second?
*18. Solve the following system of inequalities graphically
5x + 4 y  40......(1)

x  2..................(2)

y  3..................(3)

*19. Solve the following system of linear inequalities graphically.

x + y  5.....(1)
x – y  3.....(2)
*20. Solve the following system of inequalities
8x + 3 y  100........(1)

x  0....................(2)

y  0....................(3)
*21. Solve the following system of inequalities graphically
x + 2 y  8...

(1) 2x + y  8 (2)
x  0..................(3)

y  0..................(4)

*22. Solve the inequalities graphically in two dimensional plane 3x + 2y 12, x 1, y  2
*23. Solve the inequalities graphically in two dimensional plane 2x + y  6,3x + 4y 12

35 | P a g e
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6. PERMUTATIONS AND COMBINATIONS
4 MARKS QUESTIONS
***1. Find the sum of all 4 digited numbers that can be formed using the digits 1, 2, 4, 5, 6
without repetition
***2. Find the sum of all 4-digited numbers that can be formed using digits 0, 2, 4, 7, 8 without
repetition.

***3. Prove that 4n

C 1.3.5.....(4n –1)
={1.3.5.......(2n –1)}2
2n
2nC n
4
34
***4. Simplify C5 (38– r ) 4
+r Σ
=0
C
n
***5. Prove that C r + nCr –1 = n+1Cr
***6. Prove that for 3  r  n ,
(n – (n – 3)C ( n – 3) C
( )C r + 3.
n–3 =
+ 3. + (MODEL PAPER -1)
3)
(r – ( r – 2) (r – n
C
C 1) 3) r
***7. In how many ways can one select a cricket team of eleven from 17 players in which only 5
players can bowl if each cricket team of 11 must include exactly 4 bowlers?
***8. A question paper is divided into 3 sections A, B, C containing 3, 4, 5 questions respectively.
Find the number of ways of attempting 6 questions choosing at-least one from each section.
***9. A committee of 7 has to be formed from 9 boys and 4 girls. In how many ways can this be
done when the committee consists of
(i) exactly 3 girls? (ii) at-least 3 girls? (iii) at-most 3 girls?
***10. A committee of 3 persons is to be constituted from a group of 2 men and 3 women. In how
many ways can this be done? How many of these committees would consist of 1 man and 2
women?
***11. In an examination, a question paper consists of 12 questions divided into two parts i.e., part
I and part II, containing 5 and 7 questions, respectively. A student is required to attempt 8
questions in a selecting at-least 3 from each part. In how many ways can a student select the
questions?
***12. A committee of 3 persons is to be constituted from a group of 2 men and 3 women. In how
many ways can this be done? How many of these committees would consist of 1 man and 2
women?
***13. How many numbers greater than 1000000 can be formed by using the digits
1, 2, 0, 2, 4, 2, 4? (MODEL PAPER -2)
***14. Find the number of words with or without meaning which can be made using all the letters
of the word AGAIN. If these words are written as in a dictionary, what will be the 50th
word?
***15. Find the number of different 8-letter arrangements that can be made from the letters of the
word DAUGHTER so that
(i) all vowels occur together (ii) all vowels do not occur together.
***16. How many words, with or without meaning can be made from the letters of the word
‘MONDAY,’ assuming that no letter is repeated, if
(i) 4 letters are used at a time.
(ii) all letters are used at a time.
(iii) all letters are used but first letter is a vowel.
**17. Find the number of numbers that are greater than 4000 which can be formed using the
digits 0,2,4,6,8 without repetition.
**18. In how many ways can 5 girls and 3 boys be seated in a row so that no two boys are
together?
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**19. If a set A has 12 elements, find the number of subsets of A having
(i) 4 elements (ii) Atleast 3 elements (iii) Atmost 3 elements

37 | P a g e
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**20. How many words, with or without meaning each of 2 vowels and 3 consonants can be
formed from the letters of the word ‘DAUGHTER’?
**21. How many words, with or without meaning, can be formed using all the letters of the word
‘EQUATION’ at a time so that the vowels and consonants occur together?
**22. Find the number of ways of arranging 10 students A1, A2, …. , A10 in a row such that
(i)
A1, A2, A3 sit together.
(ii)
A1, A2, A3 sit in a specified order.
(iii)
A1, A2, A3 sit together in a specified order.
**23. How many 3-digit numbers can be formed from the digits 1, 2, 3, 4 and 5 assuming that
(i) repetition of the digits is allowed?
(ii) repetition of the digits is not allowed?
**24. 9 different letters of an alphabet are given. Find the number of 4 letter words that can be
formed using these 9 letters which have
(i) No letter is repeated (ii) At-least one letter is repeated.
**25. Find the number of 4-digit numbers that can be formed using the digits 1, 2, 3, 4, 5, if no
digit is repeated. How many of these will be even?
**26. Find the number of ways of arranging the letters of the word ASSOCIATIONS. In how
many of them
(i) all the three S’s come together. (ii) the two A’s do not come together.
**27. How many words, with or without meaning, each of 3 vowels and 2 consonants can be
formed from the letters of the word INVOLUTE ?
**28. In how many of the distinct permutations of the letters in ‘MISSISSIPPI’ do the four I’s not
come together?
**29. Find the number of ways of selecting 9 balls from 6 red balls, 5 white balls and 5 blue balls if
each selection consists of 3 balls of each colour.
**30. Find the number of 4 letter words, with or without meaning, which can be formed out of the
letters of the word ROSE, where the repetition of the letters is not allowed.
**31. Given 4 flags of different colours, how many different signals can be generated, if a signal
requires the use of 2 flags one below the other?
**32. How many numbers lying between 100 and 1000 can be formed with the digits 0, 1, 2, 3, 4,
5, if the repetition of the digits is not allowed?
**33. Find the number of 4-digit numbers that can be formed using the digits 0, 1, 2, 3, 4, 5 which
are divisible by 6 when repetition of the digits is allowed.
**34. There are 8 railway stations along a railway line. In how many ways can a train be stopped
at 3 of these stations such that no two of them are consecutive?
**35. How many 2 digit even numbers can be formed from the digits 1, 2, 3, 4, 5 if the digits can
be repeated?
**36. Find the number of different signals that can be generated by arranging at least 2 flags in
order (one below the other) on a vertical staff, if five different flags are available.
**37. Find the number of arrangements of the letters of the word INDEPENDENCE. In how
many of these arrangements,
(i) do the words start with P
(ii) do all the vowels always occur together
(iii) do the vowels never occur together?
(iv) do the words begin with I and end in P?
**38. In how many ways can the letters of the word ‘PERMUTATIONS’ be arranged, if the
(i) words start with P and end with S.
(ii) vowels are all together.
(iii) there are always 4 letters between P and S?
**39. A class contains 4 boys and g girls. Every Sunday, five students with at-least 3 boys go for a
picnic. A different group is being sent every week. During the picnic, the class teacher gives
each girl in the group a doll. If the total number of bolls distributed is 85, find g.
**40. The English alphabet has 5 vowels and 21 consonants. How many words with two different
vowels and two different consonants can be formed from the alphabet?

38 | P a g e
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**41. From a class of 25 students, 10 are to be chosen for an excursion party. There are 3 students
who decide that either all of them will join or none of them will join. In how many ways can
the excursion party be chosen?
**42. What is the number of ways of choosing 4 cards from a pack of 52 playing cards? In how
many of these
(i) four cards are of the same suit,
(ii) four cards belong to four different suits,
(iii) are face cards,
(iv) two are red cards and two are black cards,
(v) cards are of the same colour?
**43. A group consists of 4 girls and 7 boys. In how many ways can a team of 5 members be
selected if the team has (i) no girl ? (ii) at least one boy and one girl ? (iii) at least 3 girls ?
2 MARKS QUESTIONS
1. If n p 3 = 1320 , find n.
2. If n P7 = 42. nP5 , find n.
(n+1)
3. If P5 : nP6 = 2 : 7 , find n.
12
4. If P5 + 5. 12 P4 =
13
rP , find ‘r’
18
5. If P
P(r –1) 9(r :–17) , find r. (MODEL PAPER -1)
n –1
6. Find n, if P3 : n 4 = 1 : 9 (MODEL PAPER -2)
P
7. Find r if (i) 5Pr = 2 . 6 P (ii) 5Pr = 6 P
r –1 –1

8. If 10. C 2 = 3.n (n+1)

C3 , find n.
2n 2n
9. Determine n, if (i) C3 : nC2 = 12 :1 (ii) C3 : nC3 = 11 :1
15 15
10. If C2r–1 = C2r+4 , find r.
12
11. If Cr +1 = 12C3r –5 , find r.
n 13
12. If C5 = nC6 , then find Cn
n n
13. If C9 = C8 , find C17 .
n

14. Find the number of ways of arranging the letters of the words.
(i) MATHEMATICS (ii) SINGING (iii) PERMUTATION
(iv) COMBINATION (v) INTERMEDIATE
15. Find the number of permutations of the letters of the word ALLAHABAD.
16. Find the number of diagonals of a polygon with 12 sides.
17. In how many ways can a team of 3 boys and 3 girls be selected from 5 boys and 4 girls?
18. Find the number of ways of selecting 3 vowels and 2 consonants from the letters of the
word ‘EQUATION’
19. A bag contains 5 black and 6 red balls. Determine the number of ways in which 2 black and
3 red balls can be selected.
20. Find the number of functions from a set A containing 5 elements into a set B containing 4
elements.
21. Find the number of ways of arranging the letters of the word TRIANGLE so that the
relative positions of the vowels and consonants are not disturbed
22. How many 3-digit even numbers can be formed from the digits 1, 2, 3, 4, 5, 6 if the digits can
be repeated?
23. How many 4-letter codes can be formed using the first 10 letters of the English alphabet, if
no letter can be repeated?
24. How many 5-digit telephone numbers can be constructed using the digits 0 to 9, if each
number starts with 67 and no digit appears more than once?
25. A coin is tossed 3 times and the outcomes are recorded. How many possible outcomes are
there?
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26. Given 5 flags of different colours, how many different signals can be generated, if each
signal requires the use of 2 flags, one below the other?
27. A man has 4 sons and there are 5 schools within his reach. In how many ways can he admit
his sons in the schools so that no two of them will be in the same school.
28. How many 3-digit numbers can be formed by using the digits 1 to 9, if no digit is repeated?
29. How many 4-digit numbers are there with no digit repeated?
30. How many 3-digit even numbers can be made using the digits 1, 2, 3, 4, 6, 7, if no digit is
repeated?
31. From a committee of 8 persons, in how many ways can we choose a chairman and a vice-
chairman assuming one person cannot hold more than one position?
32. Find the number of 4-digit numbers that can be formed using the digits 1, 2, 4, 5, 7, 8
when repetition is allowed.
33. Find the number of 5 letter words that can be formed using the letters of the word RHYME,
if each letter can be used any number of times.
34. If there are 25 railway stations on a railway line, then how many types of single second class
tickets must be printed, so as to enable a passenger to travel from one station to another.
35. Determine the number of 5 card combinations out of a deck of 52 cards, if there is exactly
one ace in each combination.
36. In how many ways can a student choose a program of 5 courses, if 9 courses are available
and 2 specific courses are compulsory for every student?
37. In a class there are 30 students. If each student plays a chess game with each of the other
student, then find the total number of chess games played by them.
38. If n persons are sitting in a row, find the number of ways of selecting two persons, who are
sitting adjacent to each other.
39. It is required to seat 5 men and 4 women in a row so that the women occupy the even places.
How many such arrangements are possible?
40. In how many ways can the letters of the word ‘ASSASSINATION’ be arranged so that all
S’s are together?
41. If the different permutations of all the letters of the word ‘EXAMINATION’ are listed as in
a dictionary, how many words are there in this list before the first word starting with E?
42. How many 6-digit numbers can be formed from the digits 0, 1, 3, 5, 7 and 9 which are
divisible by 10 and no digit is repeated?
43. Determine the number of 5-card combinations out of a deck of 52 cards if each se lection of 5
cards has exactly one king. 0
44. How many 4-digit numbers can be formed by using the digits 1 to 9 if repetition of digits is
not allowed? n
P 5
45. Find the value of n such that (i) nP5 = 42. nP3, n > 4 (ii) 4 = ,n>4
n–1 P 3
4
46. Find r, if 5.4Pr = 6 5Pr–
1.
47. In how many ways can 4 red, 3 yellow and 2 green discs be arranged in a row if the discs of
the same colour are indistinguishable ?
7. BINOMAIAL THEOREM
4. MARKS QUESTONS
***1. Prove that 2.C + 5.C + 8.C +.....+ (3n + = (3n + 4) 2n–1 (MODEL PAPER)
2).C
0 1 2 n
***2. Prove that C + 3.C + 5.C +. .+ (2n = (2n + 2).2n–1
+1).C
0 1 2 n

***3. If P and Q are the sum of odd terms and the sum of even terms respectively in the expansion
of (x + a)n then prove that
(i) P2 –Q2 = (x2 – a2)n (ii) 4PQ = (x + a)2n –(x – a)2n

41 | P a g e
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n
***4. If the 2nd ,3rd , and 4th terms is the expansion of (a + x ) are respectively 240, 720, 1080,
find the x, a and n

42 | P a g e
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***5. If ‘n’ is a positive integer and ‘x’ is any non-zero real number, then prove that
2 n+1
x x + xn (1+ x) –1
C0 + C1. + C2 ....+ C .
. 2 n
n +1
( n +1) x
3 =
C C C
***6. If ‘n’ a positive integer then prove that C0 + 1 + 2 +.... + 2nn+1=
–1
2 3 n +1 n +1
***7. If (1 + x + x ) =0 a +1 a x +
n
2 2 2n
2 a x + ......2 + a x , then prove that
n
(i) a + a + a +...... + a = 3n
0 1 2 2n

3n +1
(ii) a0 + a2 + a4 +. + a2n
= 2
3n –1
(iii) a1 + a3 + a5 +...... + a2n–1 =
2
(iv) a + a + a + a +. .= 3n–1
0 3 6 9

***8. If (1 + 3x – 2x ) =0 a +1 a x +
10
2 2 20
2 a x + ..........
20 + a x prove that
i) a + a + a +............. + a = 210
0 1 2 20
ii) a – a + a –.............+ a = 410
0 1 2 20

***9. Show that 9 – 8n – 9 is divisible by 64, whenever n is positive integer.

n+1

***10. Using binomial theorem prove that 50n – 49n – 1 is divisible by 492 for all positive integers ‘n’
( 3 25
***11. Find the term independent of x in the expansion of +5 x
 
3 x 
(
**12. Using binomial theorem. Prove that 6 – 5n always
n
leaves remainder 1 when divided by 25.
4
x 2
**13. Expand using Binomial Theorem 1+ – ,xs0.
2 x

 
**14. Find the expansion of (3x2 – 29x + 3a 2 ) using binomial theorem.
3

1.3.5.......(2n –
2n
**15. Show that the middle term in the expansion of (1+ x) n

is 1)
(2x)
n!

*16. Prove that, for any real numbers a, d

a.C + ( a + d ).C + (a + 2d ).C +....... + (a + nd ).C = (2a + nd ).2n–1
0 1 2 n

*17. Theorem:
n State and prove binomial theorem for the positive integral index.
(a + b) = nC an + nC an–1b + nC an–2b2 + ..... + nC an–rbr + .... + nC bn
0 1 2 r n

2 MARK QUESTIONS(
4 5
3 (2 7 
1. Expand i)  x +  , x s 0 (BOARD MODEL PAPER) ii)  x + y 
2

 x 3 4 
2. Write down and simplify
( 2x9
(i) 6th term in  (ii) 7th term in (3x – 4 y )
10

 
3 2  8
(3p ( 3a

14
(iii) 10th term in  – 5q 
43 | P a g e
 MATHEMATICS
(iv) rthterm in  +  (1  r  8)
 4 7 5 7 
( x3  (x
10

44 | P a g e
 MATHEMATICS
(v) Middle term in  3 –  (vi) Middle term in  +9y
 6  3 

45 | P a g e
 MATHEMATICS
12 term in ( x – 2 y
(vii) 4th
) (
(viii) 13th term in 9x 18 ,xs0
–
 1 
 3 x
3. using binomial theorem, evaluate each of the following
(96 3 ii)
( 102)
5
iii)
( 101)
4
iv)
(
5
99)vii (1.1)
10000
viii) (98)
5
i)
)
4. Using binomial theorem, indicate which number is larger
1000000
i) (1.01) or 10000 ii) 1000.
10000
(1.1)
(3 )– )
6 6
5. Evaluate + 2 – 2

( 3

( ) ( )
4 4
6. Find the value of a2 a2 –1 + a 2 – a2 –1 (BOARD MODEL PAPER)
+
6 6 2
7. Find ( x +1) + ( x –1) , Hence or otherwise evaluate 2
+1 )+
6
–1 )
6

( (
4 4
Find ( a + b ) – ( a – b ) , Hence evaluate 3
)– )
4 4
8. + 2 – 2
( (
3
n

9. Prove that Σ3
r =0
r n
C r = 4n .

10. Prove that C o+ 2.C1 + 4.C2 + 8.C

3
+... + n
2n.C
5
11. Find an approximation of (0.99) using the first three terms of its expansion.
12. If a and b are distinct integers, prove that a – b is a factor of a n – bn , whenever n is a
positive integer.
n
[Hint Write an = (a – b + b) and expand]
MULTIPLE CHOICE QUESTIONS
4
( y 3

1. The 3rd term of 3x – is
 2 6  2
(–y 
3
( – y2 
2 2
i) 4 C (3x)   ii) 4
(3x)  
2
C
2
 6   6 
2 2
4 ( – y3  ( – y2 
C 2( )  C 2( ) 
iii) 3x  iv) 4
3x 
 6   6 
2. C + 3.C + 32.C +..... + 3n.C =
0 1 2 n

1) 2n 2) 3n 3) 4n 4) 5n
6
3. The middle term in the expansion of (4 + 2x ) is
1) 11240 2) 10240 x3 3) 12240 x4
x2
46 | P a g e
 MATHEMATICS
4) 10340 x4
4. The coefficient of 2 3
2 in the expansion of ( x + 1) ( y + 1) is
1) 3 2) 5 3) 2 4) 10
20
5. In the expansion of (1 + x ) , if the coefficient of rth and ( r + terms are equal, then r is
th
4)
1) 7 2) 8 3) 9 4) 10
6. The remainder when 848 is divided by 63 is
1) 4 2) 2 3) 1 4) 7
1000
7. The expansion of ( x + y ) is
1000
1000 r –1000 r 1000

Σ Σ
1000 1000–r r
1) r =0
r
Cx 2) r Cx
r =0
y y

47 | P a g e
 MATHEMATICS
999
1000 r –1000 r 999

3) rΣ Σ
1000 1000–r r
=0
Cx r
4) Cr x
r =0

8. y
The coefficient y
of the first and the last terms of the expansion ( x+y are
n
)
1) 2 2) 3 3) n 4) 1
2
9. The expansion of ( xy + 2) is
1) x2 + y2 + 2) x2 y2 + 4xy + 3) xy2 + 4 + 4) x2 y2 + 2xy + 4
4 4 2xy
91
10. The term C x89 belongs to
2
( x–
90 90
1) x89 2) ( x – 2) 3) 4) ( x + 1)
91
1)
18
11. The coefficient of x8 y10 in the expansion of ( x + y ) is (MODEL PAPER)
1) 18C 2) 18
P 3) 2 18
4) None of these
8 10

1-MARK QUESTIONS
12. Expand each of the following expressions
5
i) (1– 2x)
5
(2 ( x (
ii) iii) ( 2x – iv)  v) x
2 x x 
+ 
6
– 3) +

 x 6  3 
7 ( 2 p 3q 
vi) ( 4x + 5y) vii)  – 
 5 7 
(
13. Find the number9 of terms in the expansion of 9
3a b  14 ( 3a b 
(i)  + (ii) ( 3 p + 4q (BOARDMODEL PAPER) (iii)  + 
4
 2 ) 4 2
   
8. MARKS QUESTONS
***1. Find the sum of the sequence 7, 77, 777, 7777, ... to n terms.
***2. Find the sum to n terms of the sequence, 8, 88, 888, 8888…
***3. I) Find the sum of 5 + 55 + 555 +.......... n terms
II) Find the sum of 0.6 + 0.66 + 0.666 +…………..
n+1 n+1
***4. Find the value of n so that a +b may be the geometric mean between a and b.
an + bn
***5. A man deposited Rs 10000 in a bank at the rate of 5% simple interest annually. Find the
amount in 15th year since he deposited the amount and also calculate the total amount after
20 years.
***6. A farmer buys a used tractor for Rs 12000. He pays Rs 6000 cash and agrees to pay the
balance in annual instalments of Rs 500 plus 12% interest on the unpaid amount. How
much will be the tractor cost him?
***7. Shamshad Ali buys a scooter for Rs 22000. He pays Rs 4000 cash and agrees to pay the
balance in annual instalment of Rs 1000 plus 10% interest on the unpaid amount. How
much will the scooter cost him?

48 | P a g e
 MATHEMATICS
***8. The sum of three consecutive terms in G.P. is 56. If we subtract 1, 7, 21 from these numbers
in that order, we obtain an arithmetic progression. Find the numbers.
13
***9. The sum of first three terms of G.P is and their products is – 1. Find the common ratio
2
and the terms.
***10.
( )( )
If a, b, c and d are in G.P. show that a 2 + b2 + c2 b 2 + c2 + d 2 = (ab + bc + cd )
2

***11. The sum of two numbers is 6 times their geometric mean, show that numbers are in the

49 | P a g e
 MATHEMATICS

( )(
ratio 3 + 2 2 : 3 – 2 2 )
***12. If A and G be A.M. and G.M., respectively between two positive numbers, prove that the
numbers are A ±( A + G )( A –. G )

***13. If a and b are the roots of x2 – 3x + p = 0 and c, d are roots of x2 –12x + q = 0 , where a, b, c,
d,
form a G.P. Prove that (q + p): (q-p)= 17:15.
***14. The ratio of the A.M and G.M. of two positive numbers a and b, is m:n. Show
That a : b = (m + m2 – n2 ) : (m – m2 – n2 )

***15. If a, b, c, d are in G.P, prove that (an + bn ),(bn + cn ),(cn + d n ) are in G.P.

***16. If f is a function satisfying f ( x+y) = f ( x) f ( y ) for all x, y ϵ N such t

n

f (1) = 3 and Σ f ( x) = 120 , find the value of n.

x=1

**17. Find four numbers forming a geometric progression in which third term is greater than the
first term by 9, and the second term is greater than the 4th by 18.
**18. If the pth , qth and rth terms of a G.P. are a, b and c, respectively. Prove that aq-rbr-pcp-q = 1
**19. If the first and the nth term of a G.P. are a ad b, respectively, and if P is the product of n
terms, prove that P2= (ab)n.
**20. If A.M. and G.M. of two positive numbers a and b are 10 and 8, respectively, find the
numbers.
**21. Show that the ratio of the sum of first n terms of a G.P. to the sum of terms from
th th 1
(n +1) to (2n) term is
n
r
**22. If a, b, c, d and p are different real numbers such that
(a 2
+ b2 + c2 ) p2 – 2 (ab + bc + cd ) p + (b 2
+ c2 + d 2 )  0 then show that a, b, c and d
are in G.P.
*23. A person writes a letter to four of his friends. He asks each one of them to copy the letter
and mail to four different persons with instruction that they move the chain similarly.
Assuming that the chain is not broken and that it costs 50 paise to mail one letter. Find the
amount spent on the postage when 8th set of letters is mailed.
*24. 150 workers were engaged to finish a job in a certain number of days. 4 workers dropped
out on second day, 4 more workers dropped out on third day and so on. It took 8 more days
to finish the work. Find the number of days in which the work was completed.
1 MARK QUESTIONS:
1. What is the 20th term of the sequence defined by an = ( n –1)(2 – n )( 3 + n) ?

2. Write the first five terms of the sequences whose nth term is

i) an
=n
n ii) a n = 2n iii) an = 2n –
36
+1
3. Find the indicated terms in each of the sequences whose nth terms are:
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n
iv) (an – 2) ; a
n–1
i) a n–3:a ,a ii) a ; iii) –1a) n3; a
2
a =
n 17 24
n n 7 n 9 n 20
2 n+3

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th
4. Find the 20 term of the series 2x + 4x + 6x + ....
4 6 8

5. Find the 10th and nth terms of the G.P. 5, 25 , 125,… .

6. Which term of the G.P., 2,8,32, ... up to n terms is 131072?
7. In a G.P., the 3rd term is 24 and the 6th term is 192.Find the 10th term.
8. A person has 2 parents, 4 grandparents, 8 great grandparents, and so on. Find the number
of his ancestors during the ten generations preceding his own
9. Find the 12th term of a G.P. whose 8th term is 192 and the common ratio is 2.
10. Which term of the following sequences:
1 1 1 1
(a) 2, 2 2, 4, ..... is 128? (b ) 3 , 3, 3 3,..... is 729? (c) , , , ..... is ?
3 9 27 19683
2 7
11. For what values of x, the numbers – , x, – are in G.P?
7 2
12. Find the sum to n terms in the geometric progression1, –a, a2 ,–a3....n terms (if a s –1)
13. Find sum of infinite terms in G.P
1 1 20 80 -3 3 -3
1, , , ........ ii) 6, 1.2, 2.4, . . . . iii) 5, , , iv) G.P , , , ..........
3 i)
9 4 16 64
.....
7 49
MULTIPLE CHOICE QUESTION 1MARK
1. Which of the following is finite sequence
1) 48, 24, 12,....... 2)1,2,3,....... 3) 2,4,6,8,10 4) 2,3,5,7,11,13,….
2. Which of the following relation gives Fibonacci sequence?
1) an = an-1 + an-2 , n > 2 2) an-1 = an + an-2 , n > 2
3) an-2 = an + an-1 , n > 2 4) an = an+1 + an-2 , n > 2
3. The first term of Fibonacci sequence is
1) 0 2)1 3)2 4) 3

4. If an = 4n + 6 , then 15thterm of the sequence is

1)6 2)10 3)60 4)66

5. A series can also be denoted by symbol……..
1) πan 3) фa
2) Σ 4) θan
n
an
6. The sum of first five terms of series 2+4+6 +................is
1) 14 2) 16 3) 20 4) 30
4

7.
n=1
Σ 2n + 3 = .....

1) 5 2) 12 3) 21 4) 32
8. If an+1=an.r then the sequence is called …….
1) Arithmetic progression 2) geometric progression
3) Harmonic progression 4) special progression

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9. The nth term of G.P. is
1) a n = a + (n - 1)d 2) a = a + nd 3) a = a.rn- 4) a n = arn
n n
1
10. If r=1 in G.P. then the sum of first n terms is

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a
1) na 2)
3) (n - 1)a 4) (n + 1)a
n
11. If 4,8,16, 32,…. Are in G.P. then the sum up to 5th term is
1) 16 2) 64 3) 128 4) 124
1 1
12. The sum of series 1+ + +. .upto 6 terms is
2 4
63 2) 32 3) 26 4) 53
1) 32 63 53 26
13. The geometric mean of 3 and 12 is …….
1) 4 2) 6 3) 9 4)12
14. If A.M. of two numbers is
15 and their G.M. is 6 then the two numbers are
2
1) 6 and 8 2)13 and 3 3) 24 and 6 4) 27 and 3
15. If A means arithmetic mean and G means geometric mean then which of the following is
true?
1) A > G 2) A > G 3) G > A 4) G > A
16. The ratio of A.M and G.M of two positive numbers a and b is 5:3, Then the ratio of a to b is
1) 9:1 2) 3:5 3) 1:9 4) 3:1
9. STRAIGHT LINES
8 MARKS QUESTIONS
π
***1. A straight lines through Q ( 3, makes
an angle with the positive direction of the X-
2 ) 6

axis. If the straight line intersects the line 3x – 4y +8 = 0 at P, find the distance PQ.

***2. A straight line through Q (2,

makes an angle with the negative direction of the X-axis.
3) 3π
4
If the straight line intersects the line x + y – 7 = 0 at P, find the distance PQ.
***3. A line is such that its segment between the lines 5x – y + 4 = 0 and 3x + 4y – 4 = 0is

bisected at the point (1,5) . Obtain its equation.

***
4. If p and q are the lengths of perpendiculars from the origin to the lines x cos θ –y sin θ = k
cos 2θ and x sec θ+ y cosec θ = k, respectively, prove that p2 + 4q2 = k2
0
***5. A straight line through P (3, 4) makes an angle of 60 with the positive

direction of the X-axis. Find the coordinates of the points on the

line which are 5 units away from P.
***6. Find the equations of the straight lines passing through the point (-3, 2) and making an

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angle of 450 with the straight line 3x – y + 4 = 0 .

***7. Find the equation of the lines through the point (3, 2) which make an angle of 45° with the
line x-2y =3.

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***8. Find the point on the straight line 3x + y + 4 = 0 which is equidistant from the points(–5, 6)) and

(3, 2) .
***9. Two lines passing through the point (2, 3) intersects each other at an angle of 60°. If slope of
one line is 2, find equation of the other line.
***10. Show that the lines x – 7y – 22 = 0,3x + 4y + 9 = 0and 7x + y –54 = 0 form a right
angled isosceles triangle.
***11. Show that the straight lines x + y = 0,3x + y – 4 = 0and x +3y – 4 = 0 form an isosceles
triangle.
***12. A straight line meets the coordinate axes in A and B. Find the equation of the straight line,
when

(i) AB is divided in the ratio 2:3 at (-5,2) (ii) AB is divided in the ratio 1:2 at (-5,4)

(iii) (p,q) bisects AB

***13. Find the distance of the line 4x + 7y + 5 = 0 from the point (1, 2) along the line 2x – y = 0.
**14. Find the direction in which a straight line must be drawn through the point (–1, 2) so that
its point of intersection with the line x + y = 4 may be at a distance of 3 units from this point.
**15. A ray of light passing through the point (1, 2) reflects on the x-axis at point A and the reflected
ray passes through the point (5, 3). Find the coordinates of A.
**16. If sum of the perpendicular distances of a variable point P (x, y) from the lines x+y–5=0 and
3x–2y+7=0 is always 10. Show that P must move on a line.
**17. Show that the path of moving point such that its distance from two lines 3x – 2y = 5 and
3x + 2y = 5 are equal is a straight line.
**18. Prove that the product of the lengths of the perpendiculars drawn from the points
x y
( a – b ,0)and
2 2 2 2
)
a – b , 0 to the line
a
cosθ+
b
2
sinθ= 1 is b .

(–
*19. Show that the area of the triangle formed by the lines y = m1x + c1, y = m2 x + c2
and
(c 2 2
–c ) (c –c )
x = 0 is 1 2
(or) 2 1
.
2 m1 – m2 2 m1 – m2

4 MARKS QUESTIONS
***1. Find equation of the line passing through the point (2, 2) and cutting off intercepts on the
axes whose sum is 9.
***2. Find the equation of a line that cuts off equal intercepts on the coordinate axes and passes
through the point (2, 3).

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***3. Find the equation of the straight line passing through (–4, 5 ) and cutting off equal and non-
zero intercepts on the coordinate axes.

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***4. Find the equation of the line passing through the point of intersection of the lines 4x + 7y–
3= 0 and 2x – 3y + 1 = 0 that has equal intercepts on the axes.
***5. Find the equations of the lines, which cut-off intercepts on the axes whose sum and product
are 1 and – 6
***6. The slope of a line is double of the slope of another line. If tangent of the angle between them
1
is , find the slopes of the lines.
3
π 1
***7. If the angle between two lines is and slope of one of the lines is , find the slope of other
4 2
line.
***8. Show that the lines 2x + y – 3 = 0,3x + 2y – 2 = 0and 2x – 3y – 23 = 0 are concurrent and

find the point of concurrency.

***9. If the lines 2x + y – 3 = 0,5x + ky –3 = 0and 3x – y – 2 = 0 are concurrent, find the value of
k.

***10. Find the value of p, if the lines 3x + 4y = 5, 2x + 3y = 4 and px + 4y = 6are concurrent.

***11. Find the value of p so that the three lines 3x + y – 2 = 0, px + 2y – 3 = 0 and 2x – y – 3 = 0
may intersect at one point.
***12. If 3a + 2b + 4c = 0 then show that the equation ax + by + c = 0 represents a family of

concurrent straight lines and find the point of concurrency.

***13. If p is the length of perpendicular from the origin to the line whose intercepts on the axes are

a and b, then show that 1 1 1

2
= 2
+ 2
p a b
***14. A triangle of area 24 sq. units is formed by a straight line and the co-ordinate axes in the
first quadrant. Find the equation of that straight line if it passes through (3,4).
***15. Find the coordinates of the foot of perpendicular from the point (–1, 3) to the line
3x – 4y – 16= 0.
***16. Find the foot of the perpendicular drawn from (3,0) upon the straight line 5x +12y – 41= 0.

***17. Find the image of the given points with respect to given straight line
(i) (3,8)….
x +3y = (ii) (1,2)… 3x + 4y –1= 0.
7
***18. Assuming that straight line work as the plane mirror for a point, find the image of the point
(1, 2) in the line x – 3y + 4 = 0
***19. In the triangle ABC with vertices A (2, 3), B (4, –1) and C (1, 2), find the equation and length
of altitude from the vertex A.
***20. Find the equation of the right bisector of the line segment joining the points (3, 4) and (-1, 2)

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***21. The vertices of ΔPQR are P (2, 1), Q (–2, 3) and R (4, 5). Find equation of the median through
the vertex R.

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***22. If a portion of a straight line intercepted between the axes of coordinates is bisected at (2p,
2q) write the equation of the straight line.
***23. P (a, b) is the mid-point of a line segment between axes. Show that equation of the line is
x y
+ =2.
a b
***24. Find the equation of the line passing through (–3, 5) and perpendicular to the line through
the points (2, 5) and (–3, 6).
***25. Line through the points (-2, 6) and (4, 8) is perpendicular to the line through the points (8,
12) and (x, 24). Find the value of x.
***26. Find the equations of the straight lines passing through (1,3) and (i) parallel to (ii)
perpendicular to the line passing through the points (3,-5) and (-6,1).

(
***27. Find the values of k for which the line ( k – 3) x – 4 – k 2 ) y+k 2
– 7k + 6 = 0

(a) Parallel to the x-axis, (b) Parallel to the y-axis,

(c) Passing through the origin.
x y
***28. What are the points on the y-axis whose distance from the line + = 1 is 4 units.
3 4
***29. Find the equation of the line parallel to y-axis and drawn through the point of intersection of
the lines x – 7y + 5 = 0 and 3x + y = 0 .

***30. Show that the distances of the point (6, –2) from the line 4x + 3y =12 is half the distance

of the point (3, 4) from the line 4x –3y =12 .

***31. If the lines x = 3x + 1 and 2 y = x + 3 are equally inclined to the line y = mx + 4 . Find the value
of
m
***32. A line perpendicular to the line segment joining the points (1, 0) and (2, 3) divides it in the
ratio 1:n. Find the equation of the line.
1. MARK QUESTIONS
1. Write the equations for the x and y-axes.
2. Find the slope of the lines:
(a) Passing through the points (3, -2) and (-1, 4)
(b)Passing through the points (3,-2) and (3,4)
(c)Making inclination of 60 0 with the positive direction of x-axis.
1
3. Find the equation of the line which passes through the point (–4, 3) with slope .
2
4. Find the equation of the line, which makes intercepts -3 and 2 on the x-and y-axis
respectively.
5. Find the condition for the points (a, 0 ) , ( h, k ) and (0, b) where a.bs 0 to be collinear.

6. By using concept of equation of line prove that the points (3, 0 ) , ( –2, – 2) and (8, 2)
are colinear
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7. Find the distance between following parallel lines
i) 3x – 4 y = 12, 3x – 4 y = 7 ii) 5x – 3 y – 4 = 0 , 10x – 6 y – 9 = 0
iii) 15x + 8 y – 34 = 0,15x + 8 y + 31 = 0

8. Find the point on x -axis which is equidistant from the points (7, 6) and (3, 4)

9. Find the value of k , if the straight lines y – 3kx + 4 = 0 and (2k –1) x – (8k –1) y – 6 = 0
are perpendicular
10. Reduce the following in to slope – intercept form and find their slopes and y -intercepts
i) x + 7 y = 0 ii) 6x + 3y – 5 = 0
11. Reduce the following equations into intercept form and find the their intercepts on the axes
i) 3x + 2 y –12 =
ii) 4x – 3y = iii) 3y + 2 = 0
0
6
12. Find the area of triangle formed by the straight lines and coordinate axis
i) x – 4 y + 2 = 0 ii) 3x – 4 y +12 = 0
1
13. Write the equation of the lines for which tanθ=
2 where θ is inclination of the lines and
–3
i) y -intercept is
2 ii) x -intercept is 4

MULTIPLE CHOICE QUESTIONS

1. The equation of the straight line making an intercept of 3 units on the Y-axis and inclined at
450 to the X-axis is (Model paper – II)
1) y = x –1
2) y = x + 3) y = 45x + 4) y = x + 45
3 3
2. The angle made by the line segment joining (5, 2)(6, –15) at (0, 0) is

π π π
1) 2) 3)
6 4 2 4) π
3. The lines 2x + 3y = 6, 2x + 3y = 8 cut the X-axis at A, B respectively. A line 'l ' drawn

through the point (2, 2) meets the X-axis at ' C ' in such a way that abscissa of A, B, C are in
arithmetic progression then the equation of the line 'l ' is
1) 2 x + 3 y = 2) 3x + 2 y = 3) 2 x – 3 y = 4) 3x – 2 y = 10
10 10 10

4. The area (in sq. m) of the triangle formed by the lines x = 0; y = 0 and 3x + 4y =12 is
1) 3 2) 4 3) 6 4) 12
5. The area of the triangle formed by the axes and the line x cosα + y secα = 2

1) 4 2) 3 3) 2 4) 1

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6. If the area of the triangle formed by the lines x = 0; y = 0;3x + 4 y = a ( a > 0)is1; then a=

1) 6 2) 26 3) 4 6 4) 6 2

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7. Equation of the straight line passing through the image of (3, 6) in the line 2x – y = 5 and
perpendicular to the line 3x + 4 y = 15 is
1) 4 x – 3 y = 2) 4 x – 3 y = 3) 4 x – 3 y = 4) 4 x – 3 y = 16
10 7 5
8. A straight line through the point
3, 4 ) is such that its intercept between the axes bisected
at A its equation is
1) 4 x + 3 y = 2) 3x + 4 y = 3) x + y = 4) 3x – 4 y + 7 = 0
24 25 7
9. If the straight lines4 – 3x;ay = x +10 and 2y + bx + 9 = 0
represent the three consecutive
sides of a rectangle, then ab =
1 4) 1
1) 18 2) -3 3) –
2 3
10. The ends of a rod length 'l ' moves on the co-ordinate axes the locus of the point on the rod
which divides it in the ratio 1 : 2 is

1) 36x2 + 9y2 =
2) 36x2 + 9y2 3) 9x2 + 36y2 = 4) 9x2 + 36y2 = l2
4l2
= l2 4l2
11. Two equal sides of an isosceles triangle are
y + 3 = 0, x + y – 3 = 0 its third side

posses through the point (1, –10) the equation of the third side is

1) 3x + y + 7
=0 2) x – 3 y + 29 = 3) 3x + y + 3 = 4) 3x + y – 3 = 0
0 0

12. The perpendicular distance from

2) to this straight line 12 x + 5 y = 7 is

15 2) 12 5 7
1) 13 13 3) 4)
13 13
13. If p and q are the perpendicular distance from the origin to the straight lines
x sec θ – y cosec θ = aand xcos θ + yθ = a 2θ

1) 4 p2 + q2 =
2) p2 + q2 = 3) p2 + 2q2 = 4) 4 p2 + q2 = 2a2
a2 a2 a2
14. The equation of the line parallel to 5x + 12 y = 1 and at distance of 3 units from (1, 0)is

1) 5x + 12 y + 34 =
0 2) 12 x – 5 y + 34 = 0 3) 12 x – 5 y – 44 4) 5x + 12 y + 44 = 0 15.The
=0
area (in sq. units) of the circle which touches the lines 4 x + 3 y = 15 and 4 x + 3 y = 5 is

1) 4π
2) 3π 3) 2π 4) π
16. If
2,–1 ) and B (6,5) two points the ratio in which the foot of the perpendicular from
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(4,1) to AB divides it is

1) 8 : 15
2) 5 : 3) –5 : 8 4) –8 : 5
8
17. A straight line which makes equal intercepts on positive X and Y axes and which is at a

distance 1 unit from the origin intersects the straight line y = 2x + 3 +

at (2 x1, y1 ) then
2 x1 + y1 =

1) 3 2 2
2) –1 3) 1 4) 0
+

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18. If a, b, c are in AP the lines ax + by + c = 0 passes through the fixed point

1) (1, 2)
2) (–1, 2) 3) (1, –2) 4) (–1, –2)

19. If a, b, c are in AP then the lines ax + by + c = 0

1) passes through a fixed point 2) from an equilateral triangle

3) from a rhombus 4) from a square
20. The point on the line 2 x – 3 y = 5 which is equidistant from (1, 2) and (3, 4) is
1) (2,3) 2) (4,1) 3) (1, –1) 4) (4, 6)

10. CONIC SECTIONS

10.1. CIRCLES
8 MARK QUESTIONS
***1. Find the equation of a circle which passes through (2,-3) and (-4, 5) and having the centre on
4x+3y+1=0
***2. Find the equation of a circle which passes through (4, 1), (6, 5) and having the centre on
4x+3y-24=0.
***3. Find the equation of the circle which passes through the points (2,–2) and (3, and whose
4)

centre lies on the line x + y = 2

***4. Find the equation of circle passing through each of the following three points
I) (0,0), (2,0), (0,2) (ii) (3,4), (3,2), (1,4) III)(2,1), (5,5), (-6,7) (Model Paper-I)
**5. Find the equation of the circle passing through the points (4,1) and (6,5) and whose centre is
on the line 4x+y=16.
**6. Find the equation of the circle passing through the points (2,3) and (-1,1) and whose centre is
on the line x-3y-11=0.
**7. Find the equation of the circle with radius 5 whose centre lies on x-axis and passes through
the
Point (2, 3)
**8. Find the equation of the circle passing through (0,0) and making intercepts a and b on the
coordinate Axis.
4 MARK QUESTIONS
**1. If the abscissae of points A,B are the roots of the equation x2 + 2ax – b2 = 0 and ordinates of
A,B are roots of y2 + 2 py – q2 = 0 then find the equation of a circle for which AB is a
diameter
**2. Show that A (3,-1) lies on the circle x2 + y2 – 2x + 4 y = 0 . Also find the other end of the
diameter Through A.
**3. Show that A(-3,0) lies on the circle x2 + y2 + 8x +12y +15 = 0 and find the other
end of diameter through A.

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**4. Find the equation of the circle whose centre lies on the X-axis and passing through

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(-2, 3) and (4, 5)
2 MARK QUESTIONS
1. Find the centre and the radius of the following circles

( x + 5)
2 2
i) + ( y – 3) = 36
ii) x2 + y2 + 8x +10 y – 8 = 0
2 2
iii) x + y – 4x – 8y – 45 = 0 2 2
iv) x + y – 8x +10 y –12 = 0

2 2
v) 2x + 2 y – x =
0 vi) m2 (x2 + y2 ) – 2cx – 2mcy = 0
2. Find the equation of the circle passing through the origin and having the centre at (-4,-3)
3. Find the equation of the circle passing through (2,-1) having the centre at (2, 3)
4. Find the equation of the circle passing through (-2, 3) having the centre at (0, 0)
5. Find the equation of the circle passing through (3, 4) and having the centre at (-3, 4)
6. Find the equation of a circle with centre (2,2) and passes through the point (4,5)

7. Find the value of a if 2x2 + ay2 –3x + 2y –1= 0 represents a circle and also find its radius

8. Find the values of a,b, if ax2 +bxy +3y2 –5x + 2y –3= 0represents a circle also find
the radius and centre of the circle.
9. If x2 + y2 + 2gx + 2 fy –12 = 0 represents a circle with a centre (2,3) find g,f and the radius of
the
circle
10. If x2 + y2 + 2gx + 2 fy = 0 represents a circle with centre (-4,-3) then find g,f and radius of the
circle
11. If x2 + y2 – 4x + 6 y + c = 0 represents a circle with radius 6 then find the value of c.

12. Find the equation of a circle which is concentric with x2 + y2 – 6x – 4 y –12 = 0 and
passing through (-2, 14)
13. Does the point (-2.5, 3.5) lie inside, outside or on the circle x2 + y2 = 25 ?

MULTIPLE CHOICE QUESTIONS

1. The diameters of a circle are along 2x + y – 7 = 0 and x +3y –11= 0. Then the equation of
this
circle, which also passes through (5, 7) is
2 2
1) x + y – 4x – 6 y –16 2 2
2) x + y – 4x – 6 y – 20 = 0
=0
2 2
2 2 4) x + y + 4x + 6 y –12 = 0
3) x + y – 4x – 6 y –12
=0

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2.
2 2
Consider the circle x + y – 4x – 2 y + c = 0 whose centre is A(2, 1). If the point P(10, 7)
is such that the line segment PA meets the circle in Q with PQ = 5 then c =
1) +15 2) 20 3) 30 4) -20

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3. The abscissae of two points A and B are the roots of the equation 2 2
x + 2ax – b = 0 and their

ordinates are the roots of the equation y2 + 2 py – q2 = 0 . The radius of the circle with AB as
diameter is

1) a2 + b2 + p2 + q2 2) a2 + p2

3) b2 + p2 4) a2 + b2 – p2 + q2

4. The equation of the circle concentric with the circle

2 2
x + y – 6x +12 y +15 = 0 and of double its
area is
2 2 2 2
1) x + y – 6x +12y –15 = 0 2) x + y – 6x +12y – 30 = 0

2 2 2 2
3) x + y – 6x +12y – 25 = 0 4) x + y – 6x +12y – 20 = 0

5. A square is inscribed in the circle 2

+ y – 2x + 4y + 3 = 0 its sides are parallel to the
Co-ordinate axes then one vertex of the square is
1) 2, –
(
2) 1– 2,–2 ( )
1+ 2 )
(
3) 1, –2 + 2 4) None

)
6. The equation of circle with centre (cosα, sinα) and radius 1 is

2 2 2 2
1) x + y = 1 2) x + y – 2x cosα– 2 y sinα= 0
2 2
3) x + y + 2x cosα+ 2 y = 0 2 2
4) x + y – 2x cosα–1 = 0
7. If r and r are radii of the circles 2 2 2
1 2 + y – 4x – 8y – 44 = 0 and x + y + 6x – 8y – 96 = 0
respectively then r : r is
1 2
1) 8:11 2) 9:11 3) 16:21 4) 64:123
8. The radius of the circle touching the lines 3x – 4 y + 5 = 0, 6x – 8 y – 9 = 0
is
23
1) 1 2)
15 20 19
3) 19 4) 20

9. The perimeter of the circle 3 ( x2 + y2 ) = 16 is

4
1) 16 8 1
2) π 3) 4) π
3

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10. The radius of circle 3 (x2 + y2 )– 3x + 9 y + 10 (Model Paper-II)

=0
15
15 5 3
1) 2 2) 2 3) 4) Not defined

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1 MARK QUESTIONS
11. Find the equation of a circle with
i) center (0, 2), radius '2' ii) centre (-2,3) and radius '4'
(1 1 1
iii) centre , and radius
iv) centre (1,1) and radius
  2
  2 4 12

v) centre (–a, –b) and radius a 2 – b2 vi) centre at (0,0) and radius r.

vii) centre (-3, 2) and radius 4.

10.2.PARABOLA
8 MARK QUESTIONS
***1. Find the equation of the parabola, whose axis is parallel to the x-axis and which passes through
the points (-2,1), (1,2) and (-1,3)
***2. Find the equation of the parabola whose axis is parallel to Y-axis and which passes through
the points (4,5), (-2,11) & (-4,21)
***3. Derive the equation of parabola y2 = 4ax in standard form.
**4. Find the equation of the parabola whose focus is (-2,3) & directrix is the line 2x+3y-4=0 also
find the length of the latus rectum and the equation of axis of the parabola.
*5. Find the area of the triangle formed by the lines joining the vertex of the parabola

x2=12y to the ends of its latus rectum.

*6. An equilateral triangle is inscribed in the parabola y2 = 4ax, where one vertex is at
the vertex of the parabola. Find the length of the side of the triangle.

4 MARK QUESTIONS
1. Find the equation of the parabola whose focus is S (3,5) and vertex A (1,3)
2. Find the equation of the parabola which is symmetric about the y-axis, and passes through
the point (2, -3).
3 The focus of a parabola mirror as shown in the diagram is at a distance of 5cm from its
vertex. If the mirror is 45cm deep, find the distance AB (from diagram)

4. A beam is supported at its ends by supports which are 12 metres apart. Since the load is
concentrated at its centre, there is a deflection of 3 cm at the centre and the deflected beam is
in the shape of a parabola. How far from the centre is the defection 1 cm?
5. If a parabolic reflector is 20 cm in diameter and 5 cm deep, find the focus.
An arch is in the form of a parabola with its axis vertical. The arch is 10 m high and
5 m wide at the base. How wide is it 2 m from the vertex of the parabola?

7. The cable of a uniformly loaded suspension bridge hangs in the form of a parabola. The
roadway which is horizontal and 100 m long is supported by vertical wires attached to

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the cable, the longest wire being 30 m and the shortest being 6 m. Find the length of a

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supporting wire attached to the roadway 18 m from the middle.
2 MARK QUESTIONS
1. Find the coordinates of the focus, axis, the equation of the directrix and latus rectum of the
following parabola
2
i) y = 8x ii) y2 = 12x iii) x2 = 6y iv) y2 = – 8x

v) x2=-16y (Model Paper-I) vi) y2 = 10x v i i ) x2 = –9y

2. Find the equation of the parabola that satisfies the following conditions:
i) Focus (6, 0); directrix is x = -6 i i ) Focus (0, –3); directrix y = 3
iii) Vertex (0,0); focus (3, 0) i v ) Vertex (0, 0) focus (– 2, 0)

v ) Vertex (0, 0) passing through (2, 3) and axis is along x-axis

v i ) Vertex (0, 0), passing through (5, 2) and symmetric with respect to y-axis

3. Find the equation of the parabola with vertex at (0, 0) and focus at (0, 2).
4. Find the equation of the parabola with focus (2, 0) and directrix x = - 2
MULTIPLE CHOICE QUESTIONS
1. Parabola has the origin as its focus and the line x=2 as the directrix. Then the vertex of the
parabola is at
1) (1,0) 2) (0, 1) 3) (2, 0) 4) (0, 2)
2. If the focus of a parabola is (0, -3) and its directrix is y = 3 then its equation is
2 2
1) x = –12 y 2) x = 12 y
2 2
3) y = –12x 4) y = 12x

3.
2
If the parabola y = 4ax passes through the point (3, 2) then the length of its latus rectum is
2
1) 4 1
2) 3) 4) 4
3
3 3
4. If the vertex of the parabola is the point (-3, 0) and the directrix is the line x + 5=0 then its
equation is
1) y2 = 8( x + 3) 2) x2 = 8( y + 3)

3) y2 = –8( x + 3) 4) y2 = 8( x + 5)

10.3. ELLIPSE
4 MARK QUESTIONS
***1. Find the coordinates of the foci, the vertices, the length of major axis, the minor axis,
the eccentricity and the length of the latus rectum of the f o l l o w i n g ellipse
2 2
i) x + y = 1. ii) 2
x y2 1 iii) x2 y2 1iv) x2 y2 1
+ = = + + =
16 9 25 9 36 16 4 25

viii) 36x2 + 4y2 = 144

2 2
v) x + y 2
y2 x2 y2
1 vi) x vii)
=+ = +
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25 100 49 36 1 100 400 1

i x ) 16x2 + y2 = 16 x ) 4x2 + 9y2 = 36

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2
***2. Find the equation of ellipse with focus at (1,-1), e = and directrix
x+y+2=0
as
3
***3. Find the equation of ellipse in the standard form whose distance between foci is 2 and the
15
length of latus rectum is
2
***4. Find the equation of ellipse in the standard form such that distance between foci is 8 and
distance between directrices is 32.
***5. Find the coordinates of the foci, the vertices, the lengths of major and minor axes and the
eccentricity of the ellipse 9 x2 + 4 y 2 = 36
**6. A man running a race course notes that the sum of the distances from the two flag posts
form him is always 10 m and the distance between the flag posts is 8 m. find the equation
of the posts traced by the man.
**7. An arch is in the form of a semi-ellipse. It is 8 m wide and 2 m high at the centre.
Find the height of the arch at a point 1.5 m from one end.
**8. A rod AB of length 15cm rests in between two coordinate axes in such a way that the end point
A lie on x-axis and end point B lies on y-axis. A point P(x,y) is taken on the rod in such a way
that AP=6 cm. Show that the locus of P is an ellipse.

2 MARK QUESTIONS
1. Find the equation of the ellipse, with major axis along the x-axis and passing through the
points (4, 3) and (-1, 4) (Model paper-I)
2. Find the equation of the ellipse whose vertices are (±13,0) and foci are (±5,0)
3. Find the equation for the ellipse that satisfies the given conditions:

i ) Vertices (±5, 0), foci (±4, 0) i i ) Vertices (0, ±13), foci (0,±5)
i i i ) Vertices (±6, 0), foci(±4, 0)
i v ) Ends of major axis (±3, 0), ends of minor axis (0, ±2)

v ) Ends of major axis (0, ± 5 ) ends of minor axis (±1, 0)

v i ) Length of major axis 26, foci (±5, 0)

v i i ) Length of minor axis 16, foci (0, ±6) v i i i ) Foci (±3, 0), a = 4
i x ) b = 3, c = 4, centre at the origin; foci on the x axis.
x ) Centre at (0, 0), major axis on the y-axis and passes through the points (3, 2) and
(1, 6).
x i ) Major axis on the x-axis and passes through the points (4, 3) and (6, 2).
4. Find the equation of the ellipse, whose length of the major axis is 20 and foci are (0, ± 5) .
MULTIPLE CHOICE QUESTIONS
1 x=4, then the
1. The eccentricity of an ellipse, when its centre at the origin is equation of the ellipse
2
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is
. If one of the directrices is

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1) 3x2 + 4y2 =1 2) 4x2 + 3y2 =1 3) 4x2 + 3y2 =12 4) 3x2 + 4y2 =12
4
2. The equation of the ellipse whose focus is at (4, 0) and whose eccentricity is
5
x2 y2 x2 y2 x2 y2 x2 y2
=+ =+ =+ + =
1) 2 2
1 2) 2 2
1 3) 2 2
1 4) 2 2
1
3 5 5 3 5 4 4 5
3. The ellipse x2 + 4y2 = 4 is inscribed is a rectangular (rectangle) aligned with the co-ordinate
axes which in turn is inscribed in another ellipse that passes through the point (4,0). Then the
equation of the ellipse is
1) x2 +12y2 =16 2) 4x2 –8y2 = 48 3) 4x2 + 64y2 = 48 4) x2 +16y2 =16
1
4. If focus of an ellipse is at the origin the directrix is the line x=4 and the eccentricity is .
2
Then the length of the semi-major axis is
1) 2 2)
4
3)
5
4)
8
3 3 3 3
5. The major axis of an ellipse is three times minor axis, then the eccentricity is
1) 2 2 2) 2 2 1
3) 4)
3 3 3 3
6. The eccentricity of ellipse x2 + 3y2 = 6 is
2 3
2 2
1) 2) 4)
3 3 2 3
1 MARK QUESTIONS

x2 y2
7. +
Find the eccentricity of ellipse = 1
9 16

8. Find the length of latus rectum of ellipse x2 +16y2 =16(Model paper-II)

10.4.HYPERBOLA
4 MARKS QUESTIONS
***1. Find the coordinates of the foci and the vertices, the eccentricity, the length of the latus
rectum of the hyperbolas:
2 2
i) x – y 1 ii) y 2 –2 = 16 iii) x2 y2 1
16x – =
9 16 16 9
2 2 2 2
y2 x2
=–
iv) 1 v) 9y -4x = 36 v i ) 16x – = 576
9 27 9y
vii)5y2 – 9x2 = 36 v i i i ) 49y2 – 16x2 = 784 p
a
2. Find the equation of the hyperbola satisfying the given conditions: Foci 0, ( s
si
±
n
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g through (2, 3)
10 ),
3. Find the equation of the hyperbola of given length of transverse axis 6 whose vertex bisects
the distance between the centre and the focus.

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2 MARKS QUESTIONS
1. ±3) the equation of the hyperbola with foci (0,
Find (
and vertices 0, ± 11

 
 2 
2. Find the equation of the hyperbola satisfying the given conditions.
i) Vertices (±2, 0), foci (±3,0) ii)Vertices (0, ±5), foci (0,±8)
iii) Vertices (0, ±3), foci (0,±5) iv) Foci (±5, 0), the transverse axis is of length 8.
vii) Foci (0, ±13), the conjugate axis is of length 24
(
viii) Foci ±3 5, 0 ) , the latus rectum is of length 8

ix) Foci (±4, 0), the latus rectum is of length 12 x) Vertices (±7, 0), 4
=
3
3. Find the equation of the hyperbola where foci are (0, ±12)and the length of the latus rectum
is 36.
MULTIPLE CHOICE QUESTIONS
1. The equation of the conic with focus at (1, -1) directrix along x – y + 1 = 0 and
with eccentricity 2 is
1) x – y =
2 2
2) xy = 3) 2xy – 4x + 4 y + 1 = 4) 2xy + 4x – 4 y + 1 = 0
1 1 0
2. If one of the foci of the hyperbola is origin the corresponding directrix is 3x + 4 y + 1 = 0
and the eccentricity of the hyperbola is 5 , then equation of the hyperbola is
1) 4x + 11y + 24 xy + 6 x + 8 y + 1 = 0 2) 8x 2 + 9 y 2 + 24 xy + 6x + 6 y + 1 = 0
2 2

3) 8x 2 + 9 y2 + 24xy + 6x + 8 y + 1 = 0 4) 8x 2 + 9 y 2 – 24xy + 6x + 8 y + 1 = 0
3. The vertices of the hyperbola are (2, 0)(–2,0)the foci are (3,0)(–3,0) the equation of
the hyperbola is
2 2
1) x – y
2 2 2 2 2 2
1 2) x – y 1 3) x – y 1 4) x – y 1
5 4 4 5 5 2 2 5
4. The foci of the hyperbola 9 y2 – 4x 2 = 36 are the points

(
1) 0, ± ) 2) (±3,0) 3) (0, ±3) 4) (±13,0)

5. The distance between the foci of the hyperbola3 y 2 – 4 x – 6 y – 11 = 0 is

1) 4 2) 6 3) 8 4) 10

6. The equation 2
x + y = 1 represents a hyperbola if c
2

5–
9–
c c
1) lies between 5 and 9 2) does not lies between 5 and 9
3) lies between 0 and 5 4) lies between 0 and 9
7. If e and e ' are the eccentricities of the ellipse 5x + 9 y2 = 45 and the hyperbola
2

5x2 – 4 y 2 = 45 respectively then ee '

1) 9 2) 5 3) 4 4) 1
2 2
8. If the foci of the ellipse x
2
2 x – y = 1 coincide then b2 =
+y 1 and the hyperbola
25 16 4 b2
1) 4 2) 5 3) 8 4) 9
x y2
2 – =
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9. For the hyperbola 1 which of the following remains constant when α
cos2α sin2α
varies?
1) Eccentricity 2) Directrix
3) Abscissae of Vertices 4) Abscissae of foci

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1 MARK QUSTIONS
y2 x2
10. Find the eccentricity of hyperbola – =
2 1 2
11. b a
Find the eccentricity of rectangular hyperbola (or) equilateral hyperbola (Model Paper-I)
12. Find the length of conjugate axis of hyperbola 3x2 – 4 y2 = 12

13. Find the length of transverse axis of hyperbola

4 y2 = 4

11. INTRODUCTION TO THREE DIMENSIONAL GEOMETRY

2 MARKS QUESTIONS
1. Find ‘x’ if the distance between (5,-1,7) and (x,5,1) is 9 units [MODEL PAPER-1]
1) 8 or 2 2) 8 or 3 3) 9 or 3 4) 9 or 2o
2. Show that the points P(-2,3,5) , Q(1,2,3) and R(7,0,-1) are collinear
3. Show that the points O (0, 0, 0) , A ( 2,-3, 3) and B (-2, 3,-3) are collinear. Find the
ratio in which each point divides the segment joining the other two points.
4. Are the points A (3.6.9), B(10,20,30), C (25,-41,5), the vertices of a right angled triangle ?
5. Verify the points (0,7,10) , (-1,6,6) and (-4,9,6) are the vertices of a right angled triangle.
6. Verify the points (-1, 2,1), (1,-2, 5,) , (4,-7, 8) and (2,–3, 4) are the vertices of a parallelogram.

7. Show that the points A(1,2,3), B(-1,-2,-1), C(2,3,2) and D(4,7,6) are the vertices of a
parallelogram ABCD, but it is not a rectangle.
8. Show that the Points A(1,1,1), B(4,1,1), C(4,5,1) and D(1,5,1) are the vertices of rectangle
9. If A and B be the points (3, 4, 5) and (-1, 3,-7) ,respectively find the equation of the set of

points P such that PA2 - PB2 = K 2 where k is a constant

10. Find the equation of set of points P such that PA2 + PB2 = 2k 2 , where A and B are the points
(3,4,5) and (-1,3,-7) respectively?
11. Verify the points A(0,7,-10),B (1,6,-6) and C(4,9,-6) are the vertices of an isosceles triangle
12. The centroid of a triangle ABC is at the point (1,1,1). If the coordinates of A and B are ( 3,-5, 7)

and (-1, 7,-6) , respectively, find the coordinates of the point C. (MODEL PAPER-2)

13. Find the equation of the set of points which are equidistant from the points (1,2,3) and (3,2,-1

14. Find the equation of the set of the points P such that its distances from the points
4,–5)
and B (–2,1, 4) are equal.
15. Find the equation of the set of points P, the sum of whose distances from A(4,0,0) and
B (-4,0,0) is equal to 10.
16. If a variable point which move such that 3PA = 2PB if A = (-2,2,3) and B = (13,-3,13) Prove

that P satisfies the equation y2 + z2 + 28x –12 y +10z – 247 = 0

17. Find the lengths of the medians of the triangle with vertices A (0, 0, 6 ) , B(0, 4, 0) and C(6, 0, 0)
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18. A (5,4,6) B (1,-1,3) and C(4,3,2) are three points Find the coordinates of the point in which the

bisector BAC meets the side BC .

MULTIPLE CHOICE QUESTIONS

1. The co-ordinate of the foot of the perpendicular from a point P (6,7,8) on the y-axis are
1) (6,0,0) 2) (0,7,0) 3) (0,0,8) 4) (0,7,8)
2. The length of the perpendicular drawn from the point P (13,5,12) on the x-axis is
(MODEL PAPER – 1)

1) 13 2) 213 3) 194 4) 313

3. A parallelopiped is formed by planes drawn through the points (1,2,5) and (4,5,8) parallel to
the co-ordinate planes, then the length of the diagonal of the parallelopiped is

1) 2 2 2) 3 3) 3 4) 3 3

4. The Locus of the point for which y=0 is

1) xy-plane 2) yz-plane 3) zx-plane 4) None of these

5 If the distance between the points P(0,a,3) and Q(3,0,7) is 41 then a =

1) -4 2) 4 3) ±4 4) None of these
6. The ratio in which the line joining (2,4,5) and (3,5,-9) is divided by the yz-plane
1) 2 : 3 2) 3 : 2 3) -2 : 3 4) 4 : -3
7. What is the locus of a point for which y = 0 , z=0 ?
1) x-axis 2) y-axis 3) z-axis 4) yz-plane
8. XOZ-plane divides the join of (2,3,1) and (6,7,1) in the ratio
1) 3 : 7 2) 2 : 7 3) -3 : 7 4) -2 : 4
9. The co-ordinates of the foot of the perpendicular drawn from the point P(3,4,5) on the yz-
plane are
1) (3,4,0) 2) (0,4,5) 3) (3,0,5) 4) (3,0,0)

10. Statement-I (Assertion) : Point (3, a2 –1, a2 – 3a + 2) lies on the x-axis if a=1

Statement-II (Reason) : On x-axis , y and z co-ordinates of every point are each equal to zero
1) Statement-I and Statement-II are true, Statement-II is a correct explanation for Statement-I
2) Statement-I and Statement-II are true, Statement-II is not a correct explanation for Statement-I
3) Statement-I is true , Statement-II is false,
4) Statement-I is false , Statement-II is true,
11. Find the distance between the points (2,3,5) and (4,3,1)

1) 20 2)
21 3)
22 4) None of these
12. Find the distance between the points (-3, 7, 2) and (2, 4,-1)

1) 40
41 42 43
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2) 3) 4)

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13. Find the distance between the points (–1, 3, –4) and (1, –3, 4) .

1) 104 2) 3) 4)
109 110 100
14. Find the distance of P (3, –2, 4) from the origin. [MODEL PAPER – 2]

1) 28 2) 29 3) 30 4) 32
1 MARK QUESTIONS
15. The point (x, 0, 0) lies on axis

16. The point (0, y, 0) lies on axis

17. The point (0, 0, z ) lies on axis

18. The point (x, y, 0) lies on plane

19. The point (x, 0, z) lines on plane

20. Find the octant in which the points (-3,1, 2) and (-3,1,-2) lie.

21. A point is on the x-axis. What are its y-coordinate and z-coordinates?
22. A point is in the XZ-plane. What can you say about its y-coordinate?
23. Name the octants in which the following points lie (1, 2, 3) , ( 4,-2, 3) , ( 4,-2,-5) , (4, 2,-5) ,

(-4, 2,-5) (-4, 2, 5) , (-3,-1, 6) , (-2,-4,-7) .

24. The x-axis and y-axis taken together determine a plane known as
25. The coordinates of the points in the XY-plane are of the form
26. Coordinate planes divide the space into octants
27. Three vertices of a parallelogram ABCD are A ( 3,-1, 2) , B (1, 2,-4) and C (-1,1, 2) find
the coordinates of the fourth vertex
28. If the origin is the centroid of the triangle PQR with vertices P (2a, 2, 6 ) , Q (-4, 3b,-10) and

R (8,14, 2c) , then find the values of a, b and c

15. Find the coordinates of the point which divides the line segment joining the points (1,-2,3)
and (3,4,-5) in the ratio 2 : 3 i) internally , and ii) externally

12. LIMITS AND DERIVATIVES

LIMITS:
8 MARKS QUESTIONS
a + bx, x<1

***1. Suppose f ( x ) = 4, x = 1 and if limf ( x ) = f (1) what are possible values of a and b ?

 x→1

 b - ax, x>1

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mx2 + n, x<0
***2. If 
f ( x ) =  nx + m, 0  x  1. For what integers m and n does both limf (x) and limf (x)
x→ 0 x→1
nx3 + m, x>1

exist?
f (x) - 2
***3. If the function f (x) satisfies lim = π, evaluate limf ( x ) .
x→1 x2 - x→1

 x + 1, x< 1
0

***4. If ( )  0,
f x = x = 0 For what value (s) of a does limf ( x ) exists?
 x - 1, x>0
x→a


4 MARKS QUESTIONS
 2x + 3, x0
***1. Find limf ( x ) and limf ( x ) , where f ( x ) = 
x→0
x→1
 3 ( x + 1) , x>0

***2. Let a1 , a2 , ...., an be fixed real numbers and define a function f ( x ) = ( x - a1 )( x - a2 ) ( x - an ) .

What is lim f ( x ) ? for some a s a1 , a2 , ....an , compute lim f ( x ) .
x→a1 x→a
xsina - asinx
***3. Compute lim
x→a x-a

***4. Compute lim (

sin πcos2x )
x→0
2
x
***5. Compute lim 1 + x-31-x
3
x→0
x
***6. If f ( x ) = -25 - x2 f (x) - f
(1 )
then find lim
x→1 x-1
2 MARKS QUESTIONS
1. Evaluate the following limits

3x2 - x - 10 1
x + 21
x4 - 81
i) lim 2 ii) lim 2 iii) lim
x→2 x -4 x→3 2x - 5x - 3 x→-2 x+2
2. Find the limit of the following
 x3 - 4x2 + 4x 
lim  3 x -2 4   x3 - 2x2 
2

i) lim  2  ii)  iii) lim  2 

x
→2
L x -4  x→2
L x - 4x + 4x x→2
L x - 5x + 6 

x-2 1 
iv) lim -

L x - x x - 3x + 2x 
x→1 2 3 2

3. Find the limit

x15 - 1 logex
i) lim ii) lim 1 + x - 1 iii) lim
x→1
10
x -1 4. Co pute
m
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x→0 x x→1 x-1
ax + xcosx sinax + bx
i) lim ii) lim a, b, a + b s 0
x→0 bsinx x→0 ax + sinbx
5. Compute the following

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x
i) lim x e - ( ii) lim
loge (1 +
iii) lim
(
log 1 + x3
x→0
2x) x
1) 1 - ) sin x
x→0 x→0
3

cosx
cos2x - 1
6. Evaluate i) lim ii) lim (cosecx - cotx)
-1
x→0 x→0

 x2 –1, 1
7. Find lim f ( where f ( x) =  2
x),
→
 –x –1, x>1
x 1

x
8. Evaluate lim f ( x
 , xs0
→
where f ( x) =  x
),
x 0  0, x = 0

x
, xs
0

9. Find limf ( x ) , where f ( x ) = x
x→0

 0, x = 0

10. Find limf ( x ) , where f ( x ) = x - 5

x 5
→
11. Compute lim ([x] + x) and lim ([x] + x) where [.] (denotes) integral part
x + -
→2 x→2

MULTIPLE CHOICE QUESTIONS

2
x -1
1. lim [MODEL PAPER –I]
x →1 x-1

1) 0 2) -2 3) 2 4) Does not exist

1 (1 - x ) 2

2. lim cos-1
x→0 x 1 + x2
1) 0 2) 1 3) 2 4) Does not exist
1
3.
x→0
(
lim 1 + tan 2x 2x
2

1 ) –
1
1) e 2 2) e 3) e 2
4) 1
(cosα ) + (sinα
x
-π
4. If < α < lim x
π x→2 ) -1
2 2 x-2
1) co s 2 α lo g c o s α 2) sin 2 α log sin α

3) co s 2 α lo g cos α + sin 2 α lo g sin α

4) 0
e
5. If Δ (x ) = -1x Δ (x )
then lim =
sinx - 1 1 x→0 x

1) 0 2) 1 3) 2 4) -1
1

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( ax + bx  x
6. lim   =
x→0
 2 
1 3) 1
1) 2) ab 4) ab
ab ab
7. tanx - sinx
lim
x→0 =
x3

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1 1
1) – 2) 0 3) 2 4)
2 2

8. If a, b, d > 0 and s1 ax - b x
then lim
x→
0
dx - 1

1) log
(a 2) log ( b 3) log d 4) log d
 
 
d b
  ( a a
  d  a
 
 b b

9. ( 1+tanx sinx
lim 
x→0  1+sinx 
1) 1 2) e–1
3) e–2 4) e

10. 1+ 2x - 1- 2x
lim -1
sin 2x
1
1) -1 2) 0 3) 1 4)
2
sinx
11. Assertion (A): lim =1
x→0 x
Reason (R): If lim f ( x ) = l, lim g ( x ) = m then lim f (x ) g (x ) = lm
x→a x→a x→a

1) Both A and R are true and R is correct explanation of A

2) Both A and R are true and R is not correct explanation of A
3) A is true R is false
4) A is false R is true
12. Assertion (A): lim x = 1
x→0 x
Reason (R): f : R/ {0} → R defined by f ( x ) =x
x has range {
-1,1}
1) Both A and R are true and R is correct explanation of A
2) Both A and R are true and R is not correct explanation of A
3) A is true R is false
4) A is false R is true
(3
Let f ( x ) = x + [x], x ϵR then f
'
13. is equal to

2 
3
1) 5
2) 1 3) 0 4)
2
2
'(
14. If f ( x ) = xcosx then f
π   is equal to
2
 
1) –π π π
2) 3) 4) π

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2 2 4
15. If f ( x ) = 1 + x + x2 + x3 + x4 + ... + xn then f ' (1) =

1) n
2) n+ 3) n ( n + 4) n + n 2
2
1 1)

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1 MARK QUESTIONS
16. Evaluate the following limits
( 22
i) lim x - 4x + 3
ii) limπr 2 iii) lim iv) + x5 + 1
 
x→π 7 r →1 x→ 4 x-2 lim
x-1
  x → -1

( x + 1)
5
-
vi) lim ax + vii) lim Lx3 - x2 viii) lim Lx (x + 1)
1 b +1
v) lim
x→0
x x→0 cx + 1 →1 x→3

+ bx + c
ix) lim 2 ,a+b+cs0
x→1 cx + bx + a

17. Find the limit of lim L1 + x + x2 + ... +  (Model Paper –I)
x10 
x→-1

 x2 + 1 
18. Find the limit of lim  
x→1
L x + 100  (Model Paper –II)
19. Compute
e3x - 1
i) lim x 4x
ii) lim e - sinx - iii) lim e3+x - e3 iv) lim e - 1
1
x→0
x x→0 x x→0 x x→0 x
x 5
v) lim e2+x - e2 vi) lim e - e sinx ex - e3
vii) lim e - viii) lim
x→0
x
x→5 x-5 x→0 x→3 x-3
1x

20. Show that lim = -1

x→2- x-2

(a > 0, b > 0, b s 1)
x
21. Compute lim a -
1
x
x→0 b -1
1
z3-1
22. Find lim
z→1 1
z6-1
23. Find the limit of the following
i) sinax sinax cosx
lim(Model Paper –II) ii) lim , a, b s iii) lim
0
x→0 sinbx
x→0 bx x→0 π - x

iv) sin ( π - v) tan2x

lim vi) lim xsec x
x) lim
x→π
- x) ππ
x→ x→0
2
2
DERIVATIVES :
4 (or) 8 MARKS QUESTIONS

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***1. Find the derivatives of the following functions from the first principle.
(i) x + 1 (ii) sin2x (iii) cos ax (iv) sec 3x (v) xsinx (vi) cos2x (vii) cos x
(viii) sinx (ix) tanx (xii) ( π
(x + 1) (xiii) cos x - (xiv) x3 - 27
 
8
 
x + 2x + 3
(xv) (x - 1 )( x - 1
(xvi) 2 (xvii) (xviii) f (x) =
2) x 1 x-2
x-1
(xix) f (x) = x + (xx) sinx + cosx
1
x

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100 99 2
x x x
***2. For the function f ( x ) = + + ... + + x + 1 . Prove that f ' (1) = 100f' ( 0 ) .
100 99 2
***3. Find the derivative of xn + axn-1 + a2xn-2 + ... +an-1x + an for some fixed real number a
***4. Find the derivative of the following functions
cosx ii) sinx + iii) secx - 1 iv) a + bsinx
i) 1 + cosx sinx - secx + 1 c + dcosx
sinx cosx 2 (π
x cos
   4  viii) x
vii)
v) sin (x + a) vi) 4x + 5sinx
sinx 1+
cosx 3x + tanx
7cosx x + cosx
x 5
x - cosx
xi)
ix) n x)
sinx tanx
sin x
2. MARKS QUESTIONS
1. Find the derivative of the following functions
1
(r 1+
i) (px + q ) ax + x
 x  ii) (ax + b )(cx + d iii) iv)
1
 2 b cx 1-
) +d x
1 b 2
px + qx + r

2
(Model Paper – II) vi) 2 vii)
v) ax + bx + c px + qx + r ax + b
a b
viii) - n n m
4cosx2 ix)24x) (ax + b ) xi) (ax + b) (cx + d)
x
x x
2. Compute the derivative of f ( x ) = sin2x

i) cotnx ii) cosec4x iii) sinmx.cosnx iv) sinmx.cosnx v) log (tan5x)

(x +x+2
2

vi) log  2  vii) cos (logx + ex ) (Model Paper – II) viii) cosecxcotx
 x - x + 2 
ix) sinnx x) x4 (5sinx -
xi) (x2 + 1)cosx xii) (ax 2
+ sinx)(p + qcosx)
3cosx)
xiii)
(x + cosx)(x - tanx) xiv)
(x +secx)(x - tanx)
1 MARK QUESTIONS
x 3 x
1. Find the derivative of 5 logx + x e
2. If f ( x ) = 1 + x + x2 + .... + x100 , then find f ' (1)

3. If f (x) = 2x2 + 3x - 5, then prove that f ' (0) + 3.f ' (-1) = 0

4. Find the derivative of x2 – 2 at x = 10

5. Find the derivative at x = 2 of the function f ( x) = 3x
6. Find the derivative of the following

i) f ( x ) = 3
1 ii) 2x - iii) (5x3 + 3x - 1 ) ( x - (5 + 3x)
iv)
4
x 1)

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v) x5
(3 - 6x -9
vi) x-4 (3 - 4x-5 ) vii) sinxcosx viii) 5secx + 4cosx
)
x
ix) 3cotx + 5cosecx xii) 5sinx - 6cosx + 7 xiii) 2tanx - 7secx xiv) 5sinx + e logx

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13. STATISTICS
8 MARKS QUESTIONS:
***1. Find the mean deviation about the mean for the following data
Marks obtained 10-20 20-30 30-40 40-50 50-60 60-70 70-80
Number of
2 3 8 14 8 3 2
students
***2. Find the mean deviation about the mean for the following data
Income
100- 200- 300- 400- 500- 600- 700-
per day in 0-100
200 300 400 500 600 700 800
₹
Number
4 8 9 10 7 5 4 3
of persons
***3. Find the mean deviation about the mean for the following data
Height in cms 95-105 105-115 115-125 125-135 135-145 145-155
Number of boys 9 13 26 30 12 10
***4. Calculate the mean deviation about median for the following data
Class 0-10 10-20 20-30 30-40 40-50 50-60
Frequency 6 7 15 16 4 2
***5. Find the mean deviation about median for the following data
Marks 0-10 10-20 20-30 30-40 40-50 50-60
Number of
6 8 14 16 4 2
girls
****6. The diameters of circles (in mm) drawn in a design are given below
Diameters 33-36 37-40 41-44 45-48 49-52
No of
15 17 21 22 25
children’s
Calculate the standard deviation and mean diameter of the circles. (MODEL PAPER-1)
***7. Find the mean, variance and standard deviation using short-cut method
(BOARD MODEL PAPER)
Height 100- 105- 110-
70-75 75-80 80-85 85-90 90-95 95-100
In cms 105 110 115
No of
3 4 7 7 15 9 6 6 3
children

***8. Calculate the mean, variance and standard deviation for the following distribution
Class 30-40 40-50 50-60 60-70 70-80 80-90 90-100
Frequency
3 7 12 15 8 3 2
fi
***9. Find the mean and variance for the following frequency distributions
Classes 0-10 10-20 20-30 30-40 40-50
Frequencies 5 8 15 16 6
***10. Find the mean and variance for the following frequency distributions

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Classes 0-30 30-60 60-90 90-120 120-150 150-180 180-210
Frequencies 2 3 5 10 3 5 2

***11. Find the variance and standard deviation for the following data

xi 4 8 11 17 20 24 32

fi 3 5 9 5 4 3 1
***12. Find the mean and variance for each of the following data
xi 6 10 14 18 24 28 30

fi 2 4 7 12 8 4 3
***13. Find the mean and variance for each of the following data
xi 92 93 97 98 102 104 109

fi 3 2 3 2 6 3 3
***14. Find the mean and standard deviation using short-cut method
xi 60 61 62 63 64 65 66 67 68

fi 2 1 12 29 25 12 10 4 5
**15. Find mean deviation about the mean for the following data
xi 2 5 6 8 10 12

fi 2 8 10 7 8 5
**16. Find the mean deviation about the mean for the following data
xi : 5 10 15 20 25

fi : 7 4 6 3 5
**17. Find the mean deviation about the mean for the following data
xi : 10 30 50 70 90
fi : 4 24 28 16 8
**18. Find the mean deviation about the median for the following data

xi 3 6 9 12 13 15 21 22

fi 3 4 5 2 4 5 4 3
**19. Find the mean deviation about the median for the following data
xi : 5 7 9 10 12 15

fi : 8 6 2 2 2 6
**20. Find the mean deviation about the median for the following data
xi : 15 21 27 30 35

fi : 3 5 6 7 8
*21. The mean and variance of eight observations are 9 and 9.25 respectively. If six of the
observations are 6,7,10,12,12, and 13, find the remaining two observations

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*22. The mean and variance of seven observation are 8 and 16 respectively. If five of the
observations are 2,4,10,12,14; find the remaining two observations
*23. The mean and standard deviation of six observations are 8 and 4 respectively. If each
observation is multiplied by 3, find the new mean and new standard deviation of the
resulting observations
*24. The mean of 5 observations is 4.4 and their variance is 8.24. If three of the observations
are 1,2 and 6, find the other two observations

2 MARKS QUESTIONS:
1. Find the mean deviation about the mean for the following data 4,7,8,9,10,12,13,17.
2. Find the mean deviation about the mean for the following data 38,70,48,40,42,55,63,46,54,44
3. Find the mean deviation about the mean for the following data: 6,7,10,12,13,4,8,12
4. Find the mean deviation about the mean for the following data
12,3,18,17,4,9,17,19,20,15,8,17,2,3,16,11,3,1,0,5
5. Find the mean deviation about the median for the following data 3,9,5,3,12,10,18,4,7,19,21

6. Find the mean deviation about the median for the following data
13,17,16,14,11,13,10,16,11,18,12,17
7. Find the mean deviation about the median for the following data
36,72,46,42,60,45,53,46,51,49
8. Find the mean and variance for each of the following data 6,7,10,12,13,4,8,12
9. Find the variance of the following data 6,8,10,12,14,16,18,20,22,24
10. Find the mean and variance for each of the following data First n natural numbers
11. Find the mean and variance for each of the following data First 10 multiples of 3

12. The range of the following data (marks scored by 10 students in mathematics) is
15,20,31,62,13,6,41,86,21,74 is
13. The sum of 10 items is 12 and the sum of their squares is 18. The standard deviation is
4 3 2 1
1) 2) 3) 4)
5 5 5 5
14. PROBABILITY
8 MARKS QUESTIONS
***1. In an entrance test that is graded on the basis of two examinations, the probability of a
randomly chosen student passing the first examination is 0.8, and the probability of passing
the second examination is 0.7. The probability of passing at least one of them is 0.95. What is
the probability of passing both?
***2. The probability that a student will pass the final examination in both English and Hindi is
0.5, and the probability of passing neither is 0.1. If the probability of passing the English
examination is 0.75, what is the probability of passing the Hindi examination?
***3. The number lock of a suitcase has 4 wheels, each labelled with ten digits, i.e., from 0 to 9.
The lock opens with a sequence of four digits with no repeats. What is the probability of a
person getting the right sequence to open the suitcase?
***4. From the employees of a company, 5 persons are selected to represent them in the managing
committee of the company. Particulars of five persons are as follows:

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S.No. Name Sex Age in years

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1 Harish M 30
2 Rohan M 33
3 Sheetal F 46
4 Alis F 28
5 Salim M 41
A person is selected at random from this group to act as a spokesperson. What is the
probability that the spokesperson will be either male or over 35 years?
***5. Two dice are thrown. The events A, B and C are as follows:
A: getting an even number on the first die.
B: getting an odd number on the first die.
C: getting the sum of the numbers on the dice ≤ 5.
Describe the events
(i) A' (ii) not B (iii) A or B
(iv) A and B (v) A but not C (vi) B or C
(vii) B and C (viii) A n B' n C'
***6. Three coins are tossed once. Find the probability of getting
(i) 3 heads (ii) 2 heads (iii) at least 2 heads
(iv) at most 2 heads (v) no head (vi) 3 tails
(vii) Exactly two tails (viii) no tail (ix) atmost two tails
***7. A fair coin is tossed four times, and a person wins Rs 1 for each head and loses Rs 1.50 for
each tail that turns up. From the sample space, calculate how many different amounts of
money you can have after four tosses and the probability of having each of these amounts.
***8. If 4-digit numbers greater than 5,000 are randomly formed from the digits 0, 1, 3, 5, and 7,
what is the probability of forming a number divisible by 5 when, (i) the digits are repeated?
(ii) the repetition of digits is not allowed?
**9. A die is thrown, find the probability of the following events.
(i) A prime number will appear.
(ii) A number greater than or equal to 3 will appear.
(iii) A number less than or equal to one will appear.
(iv) A number more than 6 will appear.
(v) A number less than 6 will appear.
**10. Three coins are tossed. Describe
(i) Two events which are mutually exclusive.
(ii) Three events which are mutually exclusive and exhaustive.
(iii) Two events which are not mutually exclusive.
(iv) Two events which are mutually exclusive but not exhaustive.
(v) Three events which are mutually exclusive but not exhaustive.
**11. A die is thrown. Describe the following events:
(i) A: a number less than 7 (ii) B: a number greater than 7
(iii) C: a multiple of 3 (iv) D: a number less than 4
(v) E: an even number greater than 4 (vi) F: a number not less than 3
Also, find A u B , A n B , B u C , E n F , D n E , A – C D – E , E n F ' , F '

**12. Out of 100 students, two sections of 40 and 60 are formed. If you and your friend are among
the 100 students,
what is the probability that
(a) you both enter the same section?
(b) you both enter the different sections?
**13. In Class XI of a school, 40% of the students study Mathematics, and 30% study Biology.
10% of the class study both Mathematics and Biology. If a student is selected at random
from the class, find the probability that he will be studying Mathematics or Biology.
*14. Three coins are tossed once. Let A denote the event ‘three heads show”, B denote the event
“two heads and one tail show”, C denote the event” three tails show and D denote the
event ‘a head shows on the first coin”. Which events are

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(i) Mutually exclusive? (ii) Simple? (iii) Compound?
*15. Two dice are thrown. The events A, B and C are as follows:
A: getting an even number on the first die.
B: getting an odd number on the first die.
C: getting the sum of the numbers on the dice ≤ 5.
state true or false. (Give reasons for your answer)
(i) A and B are mutually exclusive
(ii) A and B are mutually exclusive and exhaustive
(iii) A = B'
(iv) A and C are mutually exclusive
(v) A andB' are mutually exclusive.
(vi) A' , B' , C are mutually exclusive and exhaustive.
4. MARKS QUESTIONS
***1. A and B are two events such that P(A) = 0.54, P(B) = 0.69 and P(A ∩ B) = 0.35. Find
I) P(A ∪ B) (ii) P( A 'n B ') (iii) P( A n B ') (iv) P(B n A ')
***2. A and B are events such that P(A) = 0.42, P(B) = 0.48 and P(A and B) = 0.16. Determine
i) P(not A), ii) P(not B) and iii) P(A or B)
1 1
***3. If E and F are events such a that P ( E ) = , P ( F ) and P ( E and F ) = ,
1 8
=
4 2
Find i) P ( E or F ) ii) P(not E and not F)
***4. A die has two faces, each with the number ‘1’, three faces, each with the number ‘2’ and one
face with the number ‘3’. If a die is rolled once, determine
i) P(2) ii) P(1 or 3) iii) P(not 3)
***5. 4 cards are drawn from a well-shuffled deck of 52 cards. What is the probability of
obtaining 3 diamonds and one spade?
***6. Find the probability that when a hand of 7 cards is drawn from a well shuffled deck of 52
cards, it contains
i) all Kings ii) 3 Kings iii) at-least 3 Kings.
***7. A card is selected at random from a pack of 52 cards.
a) How many points are there in the sample space?
b) Calculate the probability that the card is an ace of spades.
c) Calculate the probability that the card is (i) an ace (ii) black card
***8. One card is drawn from a well shuffled deck of 52 cards. If each outcome is equally likely,
calculate the probability that the card will be
i) a diamond ii) not an ace iii) a black card (i.e., a club or, a spade)
iv) not a diamond v) not a black card.
***9. A bag contains 9 discs of which 4 are red, 3 are blue and 2 are yellow. The discs are similar
in shape and size. A disc is drawn at random from the bag. Calculate the probability that it
will be
i) red, ii) yellow, iii) blue, iv) not blue, v) either red or blue.
***10. If A, B, C are three events associated with a random experiment, prove that
P (A u B u C) = P ( A) + P ( B ) + P (C ) – P ( A n B ) – P ( A n C ) – P ( B n C ) + P ( A
nBnC)
***11. Two students Anil and Ashima appeared in an examination. The probability that Anil will
qualify the examination is 0.05 and that Ashima will qualify the examination is 0.10. The
probability that both will qualify the examination is 0.02. Find the probability that
(a) Both Anil and Ashima will not qualify the examination.
(b) At-least one of them will not qualify the examination and
(c) Only one of them will qualify the examination
***12. A committee of two persons is selected from two men and two women. What is the
probability that the committee will have (a) no man? (b) one man? (c) two men?
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***13. In a lottery, a person chooses six different natural numbers at random, from 1 to 20, and if
these six numbers match with the six numbers already fixed by the lottery committee, he

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wins the prize. What is the probability of winning the prize in the game? [Hint order of the
numbers is not important.]
***14. Check whether the following probabilities P(A) and P(B) are consistently defined
i) P(A) = 0.5, P(B) = 0.7, P(A ∩ B) = 0.6
ii) P(A) = 0.5, P(B) = 0.4, P(A ∪ B) = 0.8
***15. A box contains 10 red marbles, 20 blue marbles and 30 green marbles. 5 marbles are drawn
at random from the box, what is the probability that
i) all will be blue? ii) at least one will be green?
***16. In a certain lottery, 10,000 tickets are sold, and ten equal prizes are awarded. What is the
probability of not getting a prize if you buy (a) one ticket (b) two tickets (c) 10 tickets?
***17. Three letters are dictated to three persons, and an envelope is addressed to each of them, the
letters are inserted into the envelopes at random so that each envelope contains exactly one
letter. Find the probability that at least one letter is in its proper envelope.
**18. In a relay race there are five teams A, B, C, D and E. (a) What is the probability that A, B
and C finish first, second and third, respectively. (b) What is the probability that A, B and
C are first three to finish (in any order) (Assume that all finishing orders are equally likely)
**19. On her vacations Veena visits four cities (A, B, C and D) in a random order. What is the
probability that she visits
i) A before B? ii) A before B and B before C?
iii) A first and B last? iv) A either first or second?
v) A just before B?
**20. Fill in the blanks in the following table.
P(A) P(B) P(A ∩ B) P(A ∪ B)
(i) 1/3 1/5 1/15 …
(ii) 0.35 … 0.25 0.6
(iii) 0.5 0.35 … 0.7
**21. A fair coin with 1 marked on one face and 6 on the other and a fair die are both tossed. Find
the probability that the sum of numbers that turn up is (i) 3 (ii) 12
**22. There are four men and six women on the city council. If one council member is selected for
a committee at random, how likely is it that it is a woman?
**23. Events E and F are such that P(not E or not F) = 0.25 state whether E and F are mutually
exclusive.
*24. Let a sample space be S = {ω1 , ω2 ,..., ω6 }.Which of the following assignments of probabilities
to each outcome are valid?
Outcomes x x2 x3 x4 x5 x6
1
(a) 1 / 6 1 / 6 1 / 6 1 / 6 1 / 6 1 / 6
(b) 1 0 0 0 0 0
(c) 1 / 8 2 / 3 1 / 3 1 / 3 –1 / 4 –1 / 3
(d 1 / 12 1 / 12 1 / 6 1 / 6 1/6 3/2
) 0.1 0.2 0.3 0.4 0.6 0.6
(e)
*25. An experiment involves rolling a pair of dice and recording the numbers that come up.
Describe the following events:
A: the sum is greater than 8,
B: 2 occurs on either die,
C: the sum is at least 7 and a multiple of 3. Which pairs of these events are mutually
exclusive?
*26. A coin is tossed three times, consider the following events.
A: ‘No head appears’
B: ‘Exactly one head appears’ and

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C: ‘Atleast two heads appear’. Do they form a set of mutually exclusive and exhaustive
events?
*27. Two dice are thrown and the sum of the numbers which come up on the dice is noted. Let us
consider the following events associated with this experiment
A: ‘the sum is even’.
B: ‘the sum is a multiple of 3’.
C: ‘the sum is less than 4’.
D: ‘the sum is greater than 11’. Which pairs of these events are mutually exclusive?
MULTIPLE CHOICE QUESTIONS
1. The probability that a leap year selected at random will contain 53 Sundays is
1 2 6 5
1) 2) 3) 4)
7 7 7 7
2. If two dice are thrown, then the probability of getting the same number on both the faces is
2 3 5 1
1) 6 2) 6 3) 6 4) 6

3. If two dice are thrown, then the probability of getting a total score of seven is
1 2 3 5
1) 2) 3) 4)
6 6 6 6
4. If A and B are two events of a sample space with P ( A u B ) = P ( A) + P ( B ) , Then they are
1) Mutually exclusive 2) Exhaustive
3) 1 and 2 4) None
5. A letter is chosen at random from the word ‘ASSASSINATION’. Find the probability that
the letter is (i) a vowel (ii) a consonant
6. If P ( A) = 0.5 P ( B ) = 0.3. (given that A and B are mutually exclusive) Then ( A' n B' ) =
1) 0.6 2) 0.5 3) 0.7 4) 0.2
7. One card is drawn at random from a well shuffled deck of 52 cards. Then the probability
than the card be not a red card
2 1 1 1
1) 2) 3) 4)
5 5 2 3
8. Which of the following is false
1) P ( E ) = 0 e E is an impossible event 2) 0  P ( E ) < 1
3) P ( E ) = 1 e E is a certain event 4) P( E ) + P ( E ') = 1
9. Which of the following statement is not correct? P is a probability function of the sample
space S.
1) P ( E ) > 06E ϵ P (S ) 2) P ( S ) = 1
3) P (ф) = 0
4)
P ( E1 n E2 ) = P ( E1 ) + P ( E2E)2 are mutually exclusive
where
10. In the experiment of throwing a die, consider the following events
A= {1, 3, 5} , B= {2, 4, 6} & C= {1, 2,3}. Are these events equally likely?
11. In the experiment of throwing a die, consider the following events
A = {1, 3,5} , B = {2, 4} and C = {6}. Are these events mutually exclusive?
12. In the experiment of throwing a die, consider the following events
A = {2, 4, 6} , B = {3, 6}& C = {1, 5, 6}.Are these events exhaustive?
13. A coin is tossed twice, what is the probability that at least one tail occurs?
14. If 2/11 is the probability of an event A, what is the probability of the event ‘not A’.
15. Given P(A) = 3/5 and P(B) = 1/5 . Find P(A or B), if A and B are mutually exclusive events.

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16. If E and F are events such that P(E) = ¼ , P(F) = ½ and P(E and F) = 1/8 , find
(i) P(E or F), (ii) P(not E and not F)
17. Events E and F are such that P(not E or not F) = 0.25, state whether E and F are mutually
exclusive
18. A letter is chosen at random from the word ‘ASSASSINATION’. Find the probability that
the letter is (i) a vowel (ii) a consonant

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