previous_years_questions_combined.pdf

Brilliant Education Centre

Previous years Board Papers

2008-2014

MOHAMMED ASHRAF

MATHEMATICS

CLASS XII

BRILLIANT EDUCATION CENTRE- DOHA QATAR

CLASS XII PREVIOUS YEARS BOARD QUESTIONS 2008 -2014 MATHEMATICS

Brilliant Education Centre AL Maadeed Street , Near Cambridge School - Doha Qatar Ph: 44936292, Mob; 55245385: Email : [email protected]

Chapter I Relations, Functions and Binary Operations

1.MARK QUESTIONS

1. Let A = {1, 2, 3}, B = {4, 5, 6, 7} and let (1,4),(2,5),(3,6)f be a function from A to B. State whether f is

one-one or not (All India 2010)

2. If :f R R is defined by 3 2,f x x define f f x (Foreign 2011, Delhi 2010)

3. Write : :fog if f R R and :g R R is given by f x x and 5 2g x x

(Foreign 2011)

4. Write : :fog if f R R and :g R R are given by 3:8g x x and 1/3:g x x

5. State the reason for the relation R in the set {1, 2, 3} given by R = {(1, 2), (2, 1)} not to be transitive

(Delhi 2011)

6. What is the range of the function 1

, 1?1

xf x x

x

(Delhi 2010)

7. If :f R R be defined by 1/3

33f x x , find fof x (All India 2010)

8. If f is an invertible function, defined as 3 4

5

xf x

write 1f x (Foreign 2010)

9. If :f R R and :g R R are given by 2sin 5 ,f x x and g x x find gof x

(Foreign 2010)

10. If 327f x x and 1/3g x x , find gof x (Foreign 2010)

11. If the function :f R R , defined by 3 4f x x is invertible, find 1f (All India 2010 C)

12. If :f R R defined by 3 5

2

xf x

is an invertible function, find 1f (All India 2009 C)

13. State whether the function :f N N given by 5f x x is injective, surjective or both

(All India 2008 C)

14. If :f R R defined by 2 7

4

xf x

is an invertible function, find 1f (Delhi 2008 C)




4 Marks Questions

15. Let A = R {3} and B = R {II}. Consider the function :f A B defined by 2

3

xf x

x

. Show that f is

one-one and onto and hence find 1f (Delhi 2012)

16. Show that :f N N given by 1,

1,

x if x is oddf x

x if x is even

is both one-one and onto

(All India 2012)

17. Let :f R R be defined as ( ) 10 7f x x . Find the function :g R R , such that Rgof fog I

(All India 2011)

18. Consider : 4,f R given by 2 4f x x . Show that f is invertible with the inverse 1f of f given

by 1 4,f y y where R+ is the set of all non-negative real numbers (Foreign 2011)

19. Show that the function :f W W defined by 1,

1,

n if n is evenf n

n if n is odd

is a bijective function

(All India 2011 C)

20. If :f R R be the function defined by 34 7,f x x show that f is a bijection

(Delhi 2011 C)

21. Let Z be the set of all integers and R is the relation on Z defined as R = {(a, b) : a, b Z and a b is divisible by

5}. Prove that R is an equivalence relation (Delhi 2010)

22. Show that the relation S in the set R of real numbers defined on, S = {(a, b) : a, b R and a b3} is neither

reflexive nor symmetric nor transitive (Delhi 2010)

23. Show that the relation S in set : 0 12A x Z x given by ( , ) : , ,S a b a b Z a b is divisible

by 4 is an equivalence relation. Find the set of all elements related to 1 (All India 2010)

24. Show that the relation S defined on set N N by (a, b) S (c, d) a + d = b + c is an equivalence relation

(All India 2010)

25. Consider : 5,f R given by 29 6 5f x x x , show that f is invertible with

16 1

3

yf y

(Foreign 2010)

26. Let :f X Y be a function. Define a relation R on X given by ( , ) : ( ) ( )R a b f a f b . Show that R is an

equivalence relation on X (All India 2010 C)




27. Show that a function :f R R given by , , , 0f x ax b a b R a is a bijection

(Delhi 2010 C)

28. Prove that the relation R in set A = {1, 2, 3, 4, 5} given by ( , ) :R a b a b is even is an equivalence relation

(Delhi 2009)

29. Let :f N N be defined by

1,

2

,2

nif n is odd

f nn

if n is even

for all ,n N find whether f is bijective

(All India 2009)

30. Show that relation R in the set of real numbers, defined as 2, :R a b a b is neither reflexive, nor

symmetric nor transitive (Foreign 2009)

31. If the function :f R R is given by 2 3 1f x x x and :g R R is given by 2 3,g x x find (i)

fog (ii) gof (All India 2009, 2008 C)

32. If the function :f R R is given by 3

3

xf x

and :g R R is given by 2 3,g x x find (i) fog (ii)

gof. Is 1 ?f g

Binary Operations (Delhi 2009 C)

1.Mark Questions :

1. Let * be a binary operation on N given by a * b = LCM (a, b) for all a, b N. Find 5 * 7

(Delhi 2012)

2. Let : R R R is defined as a * b = 2a + b. Find (2 * 3) * 4 (All India 2012)

3. If * is a binary operation on set of integers I defined by a * b = 3a + 4b 2. Find value of 4 * 5

(All India 2011 C)

4. Let * is a binary operation on set of integers I, defined by a * b = 2a + b 3. Find value of 3 * 4

(Delhi 2011 C)

5. If the binary operation * on set of integers z is defined by a * b = a + 3b2. Then, find the value of 2 * 4

(Delhi 2009)

6. Let * is the binary operation N given by a * b = HCF of a and b, where, a, b N. Write the value of 22 * 4

(All India 2009)

7. Let * be a binary operation on set Q of rational numbers defined as 5

aba b . Write the identity for *, if any

(All India 2009 C)




4 MARKS QUESTIONS

8. Consider the binary operations * : R R R and :o R R R defined as a b a b and .aob a For all

a, b R. Show that * is commutative but not associative, o is associative but not commutative

(All India 2012)

9. Consider the binary operation * on the set {1, 2, 3, 4, 5} defined by a * b = min {a, b}. Write operation table of

operation * (Delhi 2011)

10. Let * be a binary operation on Q, defined by 3

*5

aba b . Show that * is commutative as well as associative.

Also, find its identity, if it exists (Delhi 2010)

11. Let A N N and * be a binary operation on A defined by (a, b) * (c, d) = (a + c, b + d). Show that * is

commutative. Also, find identity element for * on A, if any (Foreign 2010)

12. Let * be the binary operation on N given by a * b = LCM of a and b. Find 20 * 16. Is * (i) commutative (ii)

associative (All India 2008 C)

13. Let * be a binary operation on set Q of rational numbers such that a * b = (2a b)2, a, b Q. Find

3 * 5, 5 * 3. Is 3 * 5 = 5 * 3 ? (Delhi 2008 C)

Chapter II Inverse Trigonometric Functions

1 Mark Questions

1. Write the value of 1 11 1cos 2sin

2 2

(Delhi 2012)

2. Find principal value of 1 1tan 3 sec 2 (All India 2012)

3. Write the value of 1 1sin sin

3 2

(Delhi 2011)

4. Write the value of 1 3tan tan

4

(Delhi 2011)

5. Write the value of 1 7cos cos

6

(Delhi 2011, 2009, All India 2009)

6. What is the principal value of 1 12 2cos cos sin sin

3 3

(All India 2011, 2009 C)

7. What is the principal value of tan 1

(1) ? (Foreign 2011, 2008 C)

8. Using principal values, write the value of 1 3sin

2

(All India 2011 C)




9. Write the principal value of 1 1sin

2

(Delhi 2011 C)

10. Write the principal value of 1 3sin

2

(Delhi 2010)

11. What is the principal value of sec 1

(2) ? (All India 2010)

12. Using principal values, find the value of 1 13cos cos

6

(All India 2010 C)

13. If 1 1tan 3 cot2

x , find x (All India 2010 C)

14. Write the principal value of 1 3sin sin

5

(Delhi 2009)

15. Write principal value of 1 7cos cos

6

(All India 2009)

16. Using principal values, evaluate 1 11

tan 1 sin2

(Delhi 2009 C)

17. Find principal value of 1 3cos

2

(All India 2008 C)

4 Marks Questions

18. Prove that 1 1 18 3 36sin sin cos

17 5 85

(Delhi 2012)

19. Prove that 1 costan , ,

1 sin 4 2 2 2

x xx

x

(Delhi 2012)

20. Prove that 1 1 14 12 33cos cos cos

5 13 65

(All India 2012)

21. Prove the following 1 1 sin 1 sincot ; 0,

2 41 sin 1 sin

x x xx

x x

(Delhi 2011, All India 2009)

22. Find the value of 1 1tan tan

x x y

y x y

(Delhi 2011)

23. Prove that 1 11 1 1 1tan cos , 1

4 21 1 2

x xx x

x x

(All India 2011, 2010 C)




24. Prove the following 1 1 11 1 312tan tan tan

2 7 17

(All India 2011, Delhi 2009 C, 2008 C)

25. Prove that 1 1 11 1 1tan tan tan

2 5 8 4

(All India 2011, 2008 C)

26. Prove that 1 19 9 1 9 2 2sin sin

8 4 3 4 3

(Foreign 2011)

27. Solve following equation for x 1 11 1tan tan , 0

1 2

xx x

x

(Foreign 2011 C, 2008 C, All India 2010, 2009 C)

28. Prove that 1 1 11 2 1 3tan tan tan tan

4 9 2 4

(All India 2011 C)

29. Solve for x, 11

cos 2sin , 09

x x (All India 2011 C)

30. Prove that 1 13 172tan tan

4 31 4

(Delhi 2011 C)

31. Solve for x,

21 1

2

2 1tan cot , 1 1

1 2 3

x xx

x x

(Delhi 2011 C)

32. Prove that, 1 11 1

tan cos , 0,12 1

xx x

x

(Delhi 2010)

33. Prove the following 1 1 112 3 56cos sin sin

13 5 65

(Delhi 2010)

34. Prove that,

31 1 1

2 2

2 3tan tan tan

1 1 3

x x xx

x x

(All India 2010)

35. Prove that 2

1 1

2

1cos tan sin cot

2

xx

x

(All India 2010)

36. Solve for x, 1 1cos sin

2 6

xx

(All India 2010 C)

37. Prove that, 1 11 12tan tan3 7 4

(All India 2010 C)

38. Solve for x, 1 1tan tan ; 6 0

2 3 4

x xx

(Delhi 2010 C)




39. Prove that, 1 1 13 8 36sin sin cos5 17 85

(Delhi 2010 C)

40. Prove that, 1 1 14 12 33cos cos cos

5 13 65

(Delhi 2010 C, 2009)

41. Solve for x, 1 1 18

tan 2 tan 2 tan , 079

x x x

(Delhi 2010 C)

42. Solve for x, 2 tan 1

(cos x) = tan 1

(2 cosec x) (All India 2009)

43. Prove that, 1 1 14 5 16sin sin sin

5 13 65 2

(Delhi 2009)

44. Solve for x, tan 1

3x + tan 1

2x = 4

(Delhi 2009, All India 2009 C, 2008)

45. Prove that, 1 1 13 3 8tan tan tan

4 5 19 4

(All India 2009 C)

46. Prove that, 1 1 1 11 1 1 1tan tan tan tan

3 5 7 8 4

(All India 2009 C, Delhi 2008 C)

47. Solve for x, tan 1

x + 2 cot 1

x = 2

3

(Delhi 2009 C)

48. Solve for x, 1 11 1tan tan

2 2 4

x x

x x

(Delhi 2009 C)

49. Solve for x, 1 11tan tan ,0 1

1 4

xx x

x

Chapter III Matrices

One Mark Questions

1. Simplify cos sin sin cos

cos sinsin cos cos sin

(Delhi 2012)

2. Find the value of y x from following equation 5 3 4 7 6

27 3 1 2 15 14

x

y

(All India 2012)

3. Write the value of x y + z from following equation

9

5

7

x y z

x z

y z

(Foreign 2011)




4. Write the order of product matrix

1

2 2 3 4

3

(Foreign 2011)

5. If a matrix has 5 elements, write all possible orders it can have (All India 2011)

6. For a 2 2 matrix, A = [aij] whose elements are given by aij = i/j, write the value of a12 (Delhi 2011)

7. From the following matrix equation, find the value of x 4 3 4

5 3 5 6

x y

y

(Foreign 2010)

8. Find x from the matrix equation 1 3 5

4 5 2 6

x

(Foreign 2010)

9. If 3 4 19

4 1 15

x

x

, find value of x (Foreign 2010)

10. If cos sin

sin cosA

, then for what value of , A is an identity matrix ? (Delhi 2010)

11. If 1 2 3 1 7 11

3 4 2 5 23k

, then write the value of k (Delhi 2010)

12. If A is a matrix of order 3 4 and B is matrix of order 4 3, find order of matrix (AB) (Delhi 2010 C)

13. If 1 7 1

2 5 4 5

x y

y

, then find x (Delhi 2010 C)

14. If 2 3 6 0

0 4 6 4

x y y

, then find x (All India 2010 C)

15. If 3 2 5 2

3 7 3 7

y x x

, then find y (All India 2010 C)

16. If 2 1 4 1

5 2 5 0

x

x y

, find the value of y (All India 2009 C)

17. Find the value of x, if 3 1 2

2 3 5 3

x y y

y x

(All India 2009)

18. Find the value of y, if 2 2 2

5 3 5

x y

x

(All India 2009)

19. Find the value of x, if 2 5 6 5

3 3 2

x y

y

(All India 2009)




20. If 2 4

3 2A

and 1 3

2 5B

, find A B (All India 2008 C)

21. If 2 3 0 3

4 2 8 2

x y y

x

, find x and y (All India 2008 C)

22. Find x and y, if 1 3 0 5 6

20 1 2 1 8

y

x

(Delhi 2008)

Transpose of Matrices

1 MARK QUESTIONS

1. If 1 2

3 4A

, find A A (All India 2010 C)

2. If 3 3

2 4A

, find A A where A transpose of A (All India 2009 C)

3. If matrix A = [1 2 3], write AA (Delhi 2009)

4 MARKS QUESTIONS

4. For the following matrices A and B, verify that

1

; 4 , 1 2 1

3

AB B A A B

(All India 2010)

5. If

2 0 1

2 1 3

1 1 0

A

, find value of A2 3A + 2I (All India 2010)

6. Express the following matrix as a sum of symmetric and a skew-symmetric matrix and verify your result;

3 2 4

3 2 5

1 1 2

(All India 2010)

7. Using Elementary Row Transformation (ERT), find inverse of matrix 6 5

5 4A

(Foreign 2010)

8. Find A 1

by using ERT for matrix 3 2

7 5A

(Foreign 2010)

9. Using elementary transformation, find inverse of matrix 2 5

1 3A

(Delhi 2010)




6 MARKS QUESTIONS

Using elementary transformation, find inverse of following matrices

10. (a)

1 1 2

1 2 3

3 1 1

(Delhi 2012) (b)

2 0 1

5 1 0

0 1 3

(Foreign 2011)

1 3 2

3 0 1

2 1 0

(Delhi 2010) (d)

1 2 2

1 3 0

0 2 1

(All India 2010)

(e)

3 0 1

2 3 0

0 4 1

(All India 2009) (e)

1 2 3

2 5 7

2 4 5

(All India 2008)

Chapter IV- Determinants

1 Mark Questions

1. Let A be a square matrix of order 3 3. Write the value of |2A|where, |A|= 4. [All India 2012]

2. For what value of x, 2( 1) 2

2

x xA

x x

is a singular matrix ? [All India 2011C]

3. For what value of x, the matrix 2 4 4

5 3

x

x

is a singular matrix ? [All India 2011C]

4. For what value of x, the matrix 2 4

2 3

x

x

is a singular matrix ? [Delhi 2011C]

5. For what value of x, matrix 6 4

3 1

x

x

is a singular matrix ? [Delhi 2011C]

6. For what value of x, the matrix 5 1

2 4

x x

is singular ? [Delhi 2011]

7. Evaluate

0 0

0 0

cos15 sin15

sin 75 cos75 [All India 2011]

8. If 3 4

1 1 2

x x

x , write positive value of x. [Foreign 2011; All India 2008C]

9. If 3 1

2 3A

, then find |adj A|. [Delhi 2010C]

10. If |A| = 2, where A is a 2 2 matrix, and |adj A|. [All India 2010C]




11. What is the value of

4

4 ?

4

a b c

b c a

c a b

[Foreign 2010]

12. What positive value of x makes following pair of determinants equal ?

2 3 16 3,

5 5 2

x

x. [All India 2010]

13. If A is a non-singular matrix of order 3 and |adj A| = |A|k, then what is value of k ? [All India 2009C]

14. Evaluate 7 2

210 5

. [Delhi 2009C]

15. Find x from equation 4

0.2 2

x

x [ All India 2009]

16. Write value of

2 3 4

5 6 8

6 9 12x x x

[ Delhi 2009]

17. If 1 2

,4 2

A

then find k, if |2A| = k .|A| [Foreign 2009]

18. Write value of

a b b c c a

b c c a a b

c a a b b c

[ All India 2009]

19. Evaluate2cos 2sin

sin cos

[ Delhi 2008C]

20. Evaluate a ib c id

c id a ib

[ Delhi 2008]

4 Marks questions

1. Using properties of determinants, prove that =2

[ Delhi 2012]

2. Prove 4

b c a a

b c a b abc

c c a b

[All India 2012]




3. Using properties of determinants, prove that

2

2 2 2 2

2

4

a ab ac

ba b bc a b c

ca cb c

[Delhi 2011; All India 2011C]

4. Using properties of determinants, prove that following:

2 2 2

3 3 3

( )( )( )

x y z

x y z xyz x y y z z x

x y z

[Delhi 2011, 2010C]

5. Using properties of determinants, prove the following:

2

4 2 2

2 4 2 (5 4)(4 )

2 2 4

x x x

x x x x x

x x x

[ Delhi 2011, 2009]

6. Using properties of determinants, solve the following for x:

2 2 3 3 4

4 2 9 3 16 0

8 2 27 3 64

x x x

x x x

x x x

[All India 2011]


0

a x a x a x

a x a x a x

a x a x a x

[All India 2011]


0

x a x x

x x a x

x x x a

[All India 2011]

9. Prove, using properties of determinants

2 (3 )

y k y y

y y k y k y k

y y y k

[Foreign 2011]


3

2 2

2 2 ( )

2 2

a b c a a

b b c a b a b c

c c c a b

[Foreign 2011]





3

2

2 2( )

2

x y z x y

z y z x y x y z

z x z x y

[Foreign 2011; All India 2009C; 2008]

12. Prove using properties of determinants

2

2 2 2 2

2

1

1 1

1

a ab ac

ab b bc a b c

ca cb c

[All India 2011C; Foreign 2009]

13. Prove that 2 2 2 ( )( )( )( )

a b c

a b c a b b c c a ab bc ca

bc ca ab

[Delhi 2011C]

14. Prove that 2

b c c a a b a b c

q r r p p q p q r

y z z x x y x y z

[All India 2010C; Delhi 2009C, 2008C]

15. Prove that

2 2 2

2 2 2 3

2 2 2

( )

( ) 2 ( )

( )

b c a a

b c a b abc a b c

c c a b

[All India 2010C]

16. Prove that 2 2 2 ( )

[Delhi 2010C, 2008C]

17. Using properties of determinants, prove the following

3 3 3

1 1 1

( )( )( )( )a b c a b b c c a a b c

a b c

[ Delhi 2009C]


1 1 1

1 1 1

1 1 1

x

y xyz xy yz zx

z

[All India 2009, 2008]

19. Using properties of determinants, prove that 35 4 4 2

10 8 8 3

x y x x

x y x x x

x y x x

[All India 2009]




20. Prove that

1 1 1

2 3 2 1 3 2 1

3 6 3 1 6 3

p p q

p p q

p p q

[All India 2009]


2 2

2 2 2 2 3

2 2

1 2 2

2 1 2 (1 )

2 2 1

a b ab b

ab a b a a b

b a a b

[Delhi 2009, 2008]

22. Prove that 3 3 3 3

a b c

a b b a c a a b c abc

b c c a a b

[Delhi 2009]

23. Prove that

2

2 3 2

2

1

1 (1 )

1

x x

x x x

x x

[Foreign 2009]

24. Show that if x y z and

2 3

2 3

2 3

1

1 0,

1

x x x

y y y

z z z

then 1 + xyz = 0. [Delhi 2008C]

6 Marks Questions

1. Using matrices, solve the following system of equations

x y + 2z = 7

3x + 4y 5z = 5

2x y + 3z = 12 [Delhi 2012]

2. Using matrices, solve the following system of linear equations

x + y z = 3

2x + 3y + z = 10

3x y 7z = 1 [All India 2012]

3. Using matrix method, solve the following system of equations

2 3 10 4 6 54, 1,

x y z x y z

6 9 202, , , 0x y z

x y z [Delhi 2011]

4. Using matrices, solve the following system of equations:

4x + 3y + 2z = 60;




x + 2y + 3z = 45;

6x + 2y + 3z = 70 [All India 2011]


x + 2y + z = 7;

x + 3z = 11;

2x 3y = 1 [All India 2011; Delhi 2008C]


x + 2y 3z = 4;

2x + 3y + 2z = 2;

3x 3y 4z = 11 [All India 2011, 2008]

7. Use product

1 1 2 2 0 1

0 2 3 9 2 3

3 2 4 6 1 2

to solve the system of equations

x y + 2z = 1;

2y 3z = 1;

3x 2y + 4z = 2 [ Foreign 2011]

8. If

2 1 1

3 0 1

2 6 0

A

, find A 1

using A 1

solve the following of equations

2x y + z = 3, 3x z = 0 and 2x + 6y 2 = 0 [All India 2011C]

9. If

1 2 0

0 1 1

2 0 3

A

find A- 1

and hence solve the following system of equations

x 2y + z = 0, y + z = 2, 2x 3z = 10 [All India 2011C]

10. If

1 2 0

2 1 3

0 2 1

A

and

7 2 6

2 1 3 ,

4 2 5

B

find AB, hence solve system of equations

x 2y = 10; 2x + y + 3z = 8 and 2y + z = 7 [Delhi 2011C]

11. If

3 4 2

2 3 5 ,

1 0 1

A

find A 1


3x 4y + 2z = 1, 2x + 3y + 5z = 7 and x + z = 2. [ Delhi 2011C]




12. If

8 4 1

10 0 6 ,

8 1 6

A

find A 1


8x 4y + z = 5, 10x + 6z = 4, 8x + y + 6z = 5

2. [All India 2010C]

13. If

1 1 0

2 3 4

0 1 2

A

and

2 2 4

4 2 4 ,

2 1 5

B

find AB, use this to solve the system of equation

x y = 3, 2x + 3y + 4z = 17, y + 2z = 7. [All India 2010C]

14. If

3 2 1

4 1 2

7 3 3

A

find A 1

. Hence solve the following system of equations

3x + 2y + z = 6; 4x y + 2z = 5; 7x + 3y 3z = 7. [Delhi 2010C]


x + y + z = 1; x 2y + 3z = 2; x 3y + 5z = 3 [ All India 2009C]


8x + 4y + 3z = 18; 2x + y + z = 5; x + 2y + z = 5; [All India 2009C]


x + y z = 3; 2x + 3y + z = 10; 3x y 7z = 1 [Delhi 2009C]

18. Using matrices, solve

2x + 8y + 5z = 5; x + y + z = -2; x + 2y z = 2 [Delhi 2009C]

19. Using matrices, solve system of linear equations

x + y + z = 6; x + 2z = 7; 3x + y + z = 12 [All India 2009]

20. Solve the using matrix method

3x 2y + 3z = 8; 2x + y z = 1; 4x 3y + 2z = 4 [Foreign 2009]


x y + 2z = 1; 2y 3z = 1; 3x 2y + 4z = 2 [All India 2008C]


2x + y + z = 7; x y z = 4; 3x + 2y + z = 10 [All India 2008C]

Chapter V- Continuity and Differentiability

4 Marks Questions




1. Find the value of a, if the function ( )f x defined by

2 1, 2

( ) , 2

1, 2

x x

f x a x

x x

is continuous at x = 2.

[All India 2011C]

2. Discuss the continuity of the function ( )f x at 1

2x , when ( )f x is defined as follows:

11 0

, 22

1( ) 1,

2

3 1, 1

2 2

xx

f x x

x x

[Delhi 2011C]

3. Find the relationship between a and b so that the function f defined by

1, 3( )

3 3

ax if xf x

bx if x

is continuous at x = 3. [All India 2011]

4. Find the value of a for which the function f defined as

3

sin ( 1), 02

( )tan sin

, 0

a x x

f xx x

xx

is continuous at x = 0. [Delhi 2011]

5. If the function ( )f x given by

3 , 1

( ) 11, 1

5 2 , 1

ax b if x

f x if x

ax b if x

is continuous at x = 1, find the value of

a and b. [Delhi 2011; All India 2010]

6. Find the values of a and b such that the following function f(x) is a continuous function :

5, 2

( ) , 2 10

21, 10

x

f x ax b x

x

[Delhi 2011]

7. Find the value of k so that the function f defined by

cos,

2 2( )

3,2

k xif x

xf x

if x

is continuous at 2

x

[Foreign 2011]




8. Find the value of k so that the function f defined by 1,

( )cos ,

kx if xf x

x if x

is continuous at x = .

[Foreign 2011]

9. For what values of , is the function 2 0( 2 ),

( )04 1,

if xx xf x

if xx

continuous at x = 0 ?

[Foreign 2011]

10. If the function

2

1 cos 4, 0

( ) , 0

, 016 4

xx

x

f x a x

xx

x

is continuous at x = 0, find the value of a.

[All India 2011C]

11. Find the values of a and b such that the function defined as follows is continuous :

2; 2

( ) ; 2 5

3 2; 5

x x

f x ax b x

x x

[Delhi 2010]

12. For what value of k is the function defined by

2( 2), 0( )

3 1, 0

k x if xf x

x if x

continuous at

x = 0 ? Also write whether the function is continuous at x = 1. [Delhi 2010]

13. Find all points of discontinuity of f, where f is defined as follows :

| | 3, 3

( ) 2 , 3 3

6 2, 3

x x

f x x x

x x

[Delhi 2010]

14. Show that the function defined as follows, is continuous at x = 1, x = 2 but not differentiable these at

2

3 2, 0 1

( ) 1 22 ,

5 4, 2

x x

f x xx x

x x

[Delhi 2010]




15. Show that the function f(x) defined by

sincos , 0

2,( )0

4(1 1 )0,

xx x

x

f xx

xx

x

is continuous at x = 0.

[All India 2009C]

16. If the function defined by

2 1, 2

( ) , 2

1, 2

x x

f x a x

x x

is continuous at x = 2, x = 2, find the value of a. Also discuss

the continuity of f(x) at x = 3. [Delhi 2009C]

17. Show that 3

5 4, 0 1( )

4 3 , 1 2

x when xf x

x x when x

is continuous at x = 1. [Delhi 2008C]

18. If the following function f(x) is continuous at x = 0, find the value of k : 21 cos 2

, 0( ) 2

, 0

xx

f x x

k x

[All India 2008C]

19. If f(x) defined by the following, is continuous at x = 0, find the values of a, b and c.

2

3/ 2

sin( 1) sin, 0

( ) 0

0

a x xif x

x

f x c if x

x bx xif x

bx

[All India 2008]

20. For what value of k is the following function continuous at x = 2 ?

2 1 ; 2

( ) ; 2

3 1 ; 2

x x

f x k x

x x

[Delhi 2008]

Topic -2 Differentiability

Previous years Examination Questions

4 Marks Questions

1. If (cos x)y = (cos y)

x find

dy

dx. [Delhi 2012]

2. If sin y = x sin (a + y), prove that

2sin ( )

sin

dy a y

dx a

[Delhi 2012]




3. If 1 1sin , cos, ,t tx a y a

show that dy/dx= -y/x [All India 2012]

4. Differentiate

21 1tan 1

x

x

wrt x. [All India 2012]

5. If y = (tan 1

x)2, show that

22 2 2

2( 1) 2 ( 1) 2

d y dyx x x

dx dx [Delhi 2012]

6. If xy = e

x y , show that

2

log

log( )

dy x

dx xe

[All India 2011, 2010]

7. If 1

tan log ,x ya

show that

22

2(1 ) (2 ) 0.

d y dyx x a

dx dx [All India 2011]

8. Differentiate

2cos

2

1

1

x x xxx

wrt x. [Delhi 2011]

9. If x = a ( - sin ), y = a ( 1 + cos ), find 2

2

d y

dx. [Delhi 2011]

10. Prove that

22 2 1 2 2sin .

2 2

d x a xa x a x

dx a

[Foreign 2011]

11. If 2log 1 ,y x x

prove that

22

2( 1) 0

d y dyx x

dx dx [Foreign 2011]

12. If 2 2log 1 1 ,x x y x show that 2(1 ) 1 0.dyx xydx

[All India 2011C]

13. If x = a ( +sin ) and y = a (1 cos ), find 2

2

d y

dx. [All India 2011C]

14. If y = a sin x + b cos x, prove that

2

2 2 2.dy

y a bdx

[ All India 2011C]

15. If x = a (cos + sin ) and y = a (sin - cos ), find 2

2

d y

dx [All India 2011C, 2008]

16. If x = a ( - sin ) and y = a (1 + cos ), find dy

dx at

3

. [All India 2011C, 2008]

17. If cos log tan2

tx a t

and y = a sin t, find

dy

dx. [All India 2008; Delhi 2011C]

18. If 1 1 0,( ),x y y x x y then prove that 2

1

(1 )

dy

dx x

[Delhi 2011C]




19. If y = (sinx cos x)(sinx cosx)

, 3

,4 4

x then find

dy


20. If

21 2 3 1cos ,

13

x xy

find dy


21. If y = (cot 1

x)2, then show that (x

2 + 1)

2

22

22 ( 1) 2.

d y dyx x

dx dx [Delhi 2010C]

22. If 1cos , 1,y ec x x then show that

22 2

2( 1) (2 1) 0

d y dyx x x


23. If

21 3 4 1cos ,

5

x xy

find ,dy

dx [All India 2010]

24. If 1cos , 1 1,a xy e x

then show that 2

2 2

2(1 ) 0

d y dyx x a y


25. Find ,dy

dxif y = (cos x)

x + (sin x)

1/x. [Delhi 2010]

26. Find ,dy

dxif

1 2sin 1 1y x x x x

[Delhi 2010]

27. If y = ex sin x, prove that

2

22 2 0

d y dyy

dx dx [All India 2010C]

28. If y = (x)x + (sin x)

x, find

dy


29. If y = (sin x)x + sin

1 x , find

dy

dx. [All India 2009C; Delhi 2009]

30. If y = Aemx

+ Benx

, show that

2

2( ) 0

d y dym n mny

dx dx [All India 2009C]

31. If y = 3 cos (log x) + 4 sin (log x), show that

22

20.

d y dyx x y

dx dx [Delhi 2009, 2009C]

32. If y = (x)sinx

+ (logx)x, find

dy

dx. [Delhi 2009]

33. If (cos log tan )2

x a

and y = a sin , find the value of 2

2

d y

dxat

4

. [Delhi 2009C]

34. If y = (log x)x + (x)

cosx, find

dy

dx. [Delhi 2009C]




35. If y = ex (sin x + cos x), then show that

2

22 2 0.

d y dyy


36. If

1

2

sin,

1

xy

x

show that

22

2(1 ) 3 0

d y dyx x y

dx dx . [All India 2009]

37. Differentiate the following function wrt x, (x)cosx

+ (sinx )tan x

[Delhi 2009]

38. Differentiate the following function wrt x, xsinx

+ (sinx)cos x

[ Delhi 2009]

39. If y = xcot x

+ (sin x)x, find

dy


40. If xy + y2 = tan x + y, find

dy


41. If

2cos

2

1(log ) ,

1

x xy xx

find

dy

dx. [Delhi 2008C]

42. If

21 5 12 1sin ,

13

x xy

find dy

dx. [All India 2008]

43. If 1 1 sin 1 sincot ,

1 sin 1 sin

x xy

x x

find dy

dx. [Delhi 2008]

44. If 2

2

1 11 log 1y x

x x

, find

dy

dx. [Delhi 2008]

45. Differentiate the following function wrt x 1 1 1tan

1 1

x x

x x

[ Delhi 2008]

Chapter VI Application of Derivatives

Topic I - Errors and Approximation, Increasing and Decreasing functions

4 MARKS QUESTIONS

1. Using differentials, find approximate value of 49.5 (Delhi 2012)

2. A ladder 5 m long is leaning against a wall. Bottom of ladder is pulled along the ground away from wall at the

rate of 2 m/s. How fast is the height on the wall decreasing when the foot of ladder is 4 m away from the wall ?

(All India 2012)

3. Sound is pouring from the pipe at the rate of 12 cm3/s. The falling sand forms a cone on a ground in such a way

that the height of cone is always one-sixth of radius of the base. How fast is the height of sand cone increasing

when the height is 4 cm ? (Delhi 2011)

4. Prove that 4sin

2 cosy

is an increasing function in 0,

2

(All India 2011)




5. If the radius of sphere is measured as 9 cm with an error of 0.03 cm, then find the approximate error in calculating

its surface area (All India 2011)

6. Find the intervals in which the function sin cos ,0 2f x x x x is strictly increasing or strictly

decreasing (Foreign 2011, All India 2009)

7. Show that the function 3 23 3 ,f x x x x x R (All India 2011 C)

8. Find the intervals in which the function 3 2

1 2f x x x is (a)Increasing (b)Decreasing

(All India 2011 C)

9. Find the intervals in which the function 3 22 9 12 20f x x x x is (a)Increasing (b) Decreasing

(Delhi 2011 C)

10. Find the intervals in which function 3 22 9 12 15f x x x x is (a)Increasing (b) Decreasing

(Delhi 2011 C)

11. Find the intervals in which the function 3 22 15 36 17f x x x x is increasing or decreasing

(All India 2010 C)

12. Find the intervals in which the following function is (a)Increasing (b)Decreasing

3 22 9 12 15f x x x x (All India 2010 C, 2008, 2008 C)

13. Find the intervals in which the function 2

1 2f x x x is increasing or decreasing

(All India 2009 C)

14. Find the intervals in which 3 212 36 17f x x x x is increasing or decreasing function

(Delhi 2009 C)

15. The length x of a rectangle is decreasing at the rate of 5 cm/min and the width y is increasing at the rate of 4

cm/min. When x = 8 cm and y = 6 cm, find the rate of change of (a) the perimeter (b) area of rectangle

(All India 2009 C)

16. Find intervals in which the function f given by 3 31

, 0f x x xx

is (a)Increasing (b)Decreasing

(Delhi 2009 C)

17. If 23 15 5,f x x x find the approximate value of 3.02f using differentials (Delhi 2008 C)

6. Marks Questions

18. Find the intervals in which the following function 2 320 9 6f x x x x is

(a)Strictly increasing (b) Strictly decreasing (All India 2010)




19. Find the intervals in which the function f given by sin cos ,0 2f x x x x is strictly increasing or

strictly decreasing (Delhi 2010)

20. Find values of x for which 2

2f x x x is an increasing function. Also, find the points on the curve

where the tangent is parallel to X-axis (Delhi 2010)

Tangents and Normals

1Marks Questions

1. Find the slope of tangent to the curve y = 3x2 6 at the point on it whose x coordinate is 2

(All India 2009 C)

2. Find the slope of tangent to the curve y = 3x2 4x at point whose x coordinate is 2 (Delhi 2009 C)

3. Find the slope of the tangent to the curve y = 3x4 4x at x = 1 (All India 2008 C)

4. For the curve y = 3x2 + 4x, find the slope of tangent to the curve at point where x coordinate is 2

(Delhi 2008 C)

4 Marks Questions

5. Find the points on curve y = x3 11x + 5 at which equation of tangent is y = x 11 (Delhi 2012 C)

6. Find he points on the curve x2 + y

2 2x 3 = 0 at which tangent is parallel to X-axis (Delhi 2011)

7. Find the points on the curve y = x3 at which the slope of the tangent is equal to y coordinate of the point

(Foreign 2011)

8. Find the equation of tangent to curve x = sin 3t, y = cos 2t at 4

t

(All India 2011 C, 2008)

9. Find the equations of tangents to the curve y = (x2 1) (x 2) at the points where the curve cuts the X-axis

(All India 2011 C)

10. Find the equation of tangent to the curve 4x2 + 9y

2 = 36 at the point 3cos ,2sin (Delhi 2011 C)

11. Find the equation of tangent to the curve y = x4 6x

3 + 13x

2 10x + 5 at point x = 1, y = 0

(Delhi 2011 C)

12. Find the equation of tangent to the curve x = a sin3 t and y = a cos

3 t at the point where

4t

(All India 2010 C, 2009 C)

13. Find the equation of tangent to the curve 2

7

5 6

xy

x x

at the point where it cuts the X-axis

(All India 2010 C, 2010)

14. Find the equations of the normal to the curve y = x3 + 2x + 6 which are parallel to line x + 14y + 4 = 0

(Delhi 2010)

15. Find the equation of tangent to the curve x2 + 3y = 3, which is parallel to line y 4x + 5 = 0




(Delhi 2009 C)

16. Find the equation of tangent to the curve 3 2y x which is parallel to the line 4x 2y + 5 = 0

(Delhi 2009)

17. At what points will the tangent to the curve y = 2x3 15x

2 + 36x 21 be parallel to X-axis ? Also, find the

equations of tangent to the curve (All India 2008)

6Marks Questions

18. Find the equations of tangent and normal to the curve 1 cos , sinx y at 4

(All India 2010)

Maxima and Minima

6 Marks Questions

1. Show that the height of a closed right circular cylinder of given surface and maximum volume is equal to

diameter of base (Delhi 2012)

2. Prove that radius of right circular cylinder of greatest curved surface area which can be inscribed in a given cone

is half of that of the cone (All India 2012)

3. An open box with a square base is to be made out of a given quantity of cardboard of area C2 square units. Show

that the maximum volume of box is

3

6 3

C (All India 2012)

4. Show that the right circular cone of least curved surface and given volume has an altitude equal to 2 times the

radius of the base (Delhi 2011)

5. A window has the shape of a rectangle surmounted by an equilateral triangle. If the perimeter of the window is 12

m, find the dimensions of the rectangle that will produce the largest area of the window

(Delhi 2011)

6. A window is in the form of a rectangle surmounted by a semi-circular opening. The total perimeter of the window

is 10m. Find the dimensions of the rectangle so as to admit maximum light through the whole opening

(All India 2011)

7. Show that of all the rectangles inscribed in a given fixed circle, the square has the maximum area

(All India 2011)

8. Show that of all the rectangles with a given perimeter, the square has the largest area (Delhi 2011)

9. Show that of all the rectangles of given area, the square has the smallest perimeter (Delhi 2011)

10. Show that the semi-vertical angle of a right circular cone of maximum volume and given slant height is 1tan 2

(All India 2011, 2008)




11. Find the volume of the largest right circular cylinder that can be inscribed in a sphere of radius r cm

(All India 2011)

12. Find the point on the curve y2 = 2x which is at a minimum distance from the point (1, 4)

(All India 2011, 2009)

13. Show that the right circular cylinder of given volume, open at the top, has minimum total surface area, if its height

is equal to the radius of the base (Delhi 2011, 2009)

14. A wire of length 28 m is to be cut into two pieces. One of the two pieces is to be made into a square and the other

into a circle. What should be the lengths of two pieces so that the combined area of circle and square is minimum

? (All India 2010)

15. An open tank with a square base and vertical sides is to be constructed from a metal sheet so as to hold a given

quantity of water. Show that the total surface area is least when depth of the tank is half its width

(All India 2010)

16. Show that the volume of the largest cone that can be inscribed in a sphere of radius R is 8

27 of the volume of the

sphere (Delhi 2010)

17. Find the maximum area of an isosceles triangle inscribed in the ellipse

2 2

125 16

x y , with its vertex at one end of

the major axis (Delhi 2010)

18. If the length of three sides of a trapezium other than the base is 10 cm each, find the area of the trapezium, when it

is maximum (All India 2010)

19. Show that the right circular cylinder, open at the top and of given surface area and maximum volume is such that

its height is equal to the radius of the base (Delhi 2010)

20. A manufacturer can sell x items at a price of Rs. 5100

x

each. The cost price of x items is Rs. 500

5

x

.

Find the number of items he should sell to each maximum profit (All India 2009)

21. If the sum of the lengths of the hypotenuse and a side of a right angled triangle is given, show that the area of the

triangle is maximum when the angle between them is 3

(All India 2009)

22. A tank with rectangular base and rectangular sides, open at the top is to be constructed so that its depth is 2m and

volume is 8 m3. If building of tank cost Rs.70 per sq.m for the base and Rs.45 per sq.m for sides. What is the cost

of least expensive tank ? (Delhi 2009)

23. Given the sum of the perimeter of a square and a circle, show that the sum of their area is least when the side of

the square is equal to diameter of their circle (All India 2008)




24. Show that the height of the closed right circular cylinder, of given volume and minimum total surface area, is

equal to the diameter (All India 2008)

25. Show that the volume of the greatest cylinder can be inscribed in a cone of height h and semi-vertical angle is

3 24 tan27

h (All India 2008)

26. Show that the height of the cylinder of maximum volume that can be inscribed in a cone of height h is 1

3h

(Delhi 2008)

Chapter VII- Integrals

1 mark Questions

1. Evaluate (1 )x xdx [Delhi 2012]

2. Evaluate 3

2

1dx

x [Delhi 2012]

3. Evaluate 2

2

04 x dx [All India 2012]

4. Given (tan 1)sec ( )x xe x xdx e f x C write ( )f x satisfying above [All India 2012]

5. Write the value of 2 16

dx

x [Delhi 2011]

6. Write the value

2

2

sec

cos

xdx

ec x [Delhi 2011]

7. Write the value of 2

2 3sin

cos

xdx

x

[Delhi 2011]

8. Evaluate 21

dx

x [All India 2011]

9. Evaluate

2(log )xdx

x [All India 2011]

10. Evaluate

1tan

21

xedx

x

[All India 2011]

11. Evaluate 3

21 1

dx

x [Foreign 2011]

12. Evaluate

2(1 log )xdx

x

[Foreign 2011; Delhi 2009]




13. Evaluate

2 2

2 2

x x

x x

e edx

e e

[ Foreign 2011]

14. Evaluatecos x

dxx

[ Foreign 2011]

15. Evaluate 2

2cos

sin

xdx

x [ All India 2008, 2009, 2010C]

16. Evaluate 1

20

2

1

xdx

x [ All India 2008, 2011C]

17. Evaluate

3 2 1

1

x x xdx

x

[ Delhi 2011C]

18. Evaluate 1

20

1

1dx

x [Delhi 2011C]

19. Write value of 2

1 sin

cos

xdx

x

[ All India 2011C]

20. Evaluate

2

2

1xdx

x

[ Delhi 2010C]

21. Evaluate / 4

3

/ 4sin xdx

[Delhi 2010C]

22. Evaluate 2sec (7 4 )x dx [All India 2010]

23. Evaluate log x

dxx

[ All India 2009C]

24. Evaluate log(sin )

tan

xdx

x [All India 2009C]

25. Evaluate 2(cos cot ) xec x x e dx [ Delhi 2009C]

26. Evaluate

2sec xdx

x [ All India 2009]

27. Evaluate sin x

dxx

[ All India 2009]

28. Evaluate 1/ 2

20

1

1dx

x [ All India 2009]

29. Evaluate

2

2

sec

3 tan

xdx

x [Delhi 2009]




30. If 1

2

0(3 2 ) 0,x x k dx find value of k. [Delhi 2009]

31. Evaluate

2

31

xdx

x [Delhi 2008]

4 Marks Questions

32. Evaluate sin sin 2 .sin3x x xdx [Delhi 2012]

33. Evaluate 2

2

(1 )(1 )dx

x x [Delhi 2012]

34. Evaluate sinxe xdx [Foreign 2011]

35. Evaluate 2

3 5

8 7

xdx

x x

[Foreign 2011]

36. Evaluate

2

4

4

16

xdx

x

[ All India 2011C]

37. Evaluate

41

20

1

1

xdx

x

[All India 2011C]

38. Evaluate

2

4

1

1

xdx

x

[Delhi 2011C]

39. Evaluate sin cos

sin 2

x xdx

x

[Delhi 2011C]

40. Evaluate / 2

0

sin

1 cos

x xdx

x

[All India 2011]

41. Evaluate 2 2

2

( 1)( 3)

xdx

x x [Delhi 2011]

42. Evaluate 2

5 3

4 10

xdx

x x

[Delhi 2011; All India 2010]

43. Evaluate 2 1 sin 2

1 cos 2

x xe dxx

[All India 2010C]

44. Evaluate 2 2( 1)( 2)

dx

x x [ Delhi 2010C]

45. Evaluate 2

1log(log )

(log )x dx

x

[Delhi 2010C]




46. Evaluate the following 2

( 2)( 3)

xdx

x x

[All India 2010]

47. Evaluate the following

22

21

5

4 3

xdx

x x [All India 2010]

48. /3

/ 6

sin cos

sin 2

x xdx

x

[ Delhi 2010]

49. 21

(1 2 )

xdx

x x

[Delhi 2010]

50. Evaluate sin 4 4

1 cos 4

x xe dxx

[Delhi 2010]

51. Evaluate (sin cos )

sin 2

x xdx

x

[ All India 2009C]

52. Evaluate 2

2 5

7 6

xdx

x x

[Delhi 2009C]

53. Evaluate 25 4 2

dx

x x [All India 2008C]

54. Evaluate 25 4

x

x x

edx

e e [Delhi 2009]

55. Evaluate 2

3

2 5

xdx

x x

[All India 2008C]

56. Evaluate 1sinx xdx [All India 2008C]

57. Evaluate .log( 1)x x dx [Delhi 2008C]

6 Marks Questions

58. Evaluate

1

2

sin

1

x xdx

x

[Delhi 2012]

59. Evaluate

2

2

1

( 1) ( 3)

xdx

x x

[Delhi 2012]

60. Evaluate 2

6 7

( 1) ( 4)

xdx

x x

[All India 2011]

61. Evaluate tan cotx x dx [Delhi 2010C]




62. Evaluate 2

2

1

( 2)( 1)

x xdx

x x

[All India 2009C]

63. Evaluate

3

3

tan tan

1 tan

x xdx

x

[Delhi 2009C]

64. Evaluate

2

4 2 1

xdx

x x [Delhi 2008C]

Topic 2 : Properties of Definite integrals and Limit of Sums

Previous Years Examination Questions

4 Marks Questions

1. Evaluate 20

sin

1 cos

x xdx

x

[All India 2012; Delhi 2011C, 2009C, 2008]

2. Evaluate 2

3

1x xdx

[All India 2012]

3. Evaluate / 4

0log(1 tan )x dx

[All India 2011; 2008C]

4. Evaluate 1

0

1log 1 dx

x

[All India 2011]

5. Evaluate 1

20

log(1 )

1

xdx

x

[All India 2011C]

6. Evaluate 0 1 sin

xdx

x

[Delhi 2010]

7. Evaluate 0

tan

sec cos

x xdx

x ecx

[All India 2009C]

8. Evaluate 5

52x dx

[All India 2008C]

9. Using properties of definite integrals, evaluate the following 0

a xdx

x a x [Delhi 2008C]

10. Evaluate 0

tan

sec tan

x xdx

x x

[All India 2008; Delhi 2008]

11. Evaluate / 2

0logsin xdx

[All India 2008]

12. Evaluate 1

1 2

0cot 1 x x dx [Delhi 2008]

6 Marks Questions




13. Evaluate 3

2

1(2 5 )x x dx as a limit of a sum [Delhi 2012]

14. Evaluate / 2

1

02sin cos tan (sin )x x x dx

[Delhi 2011]

15. Evaluate / 2

4 40

sin cos

sin cos

x x xdx

x x

[Delhi 2011; All India 2010C]

16. Evaluate /3

/ 6 1 tan

dx

x

[All India 2011]

17. Evaluate 4

2

1( )x x dx as a limit of a sum. [Delhi 2011, 2010]

18. Evaluate / 4

0

sin cos

9 16sin 2

x xdx

x

[Delhi 2011]

19. Evaluate 2

2

0(3 2)x dx as a limit of a sum [All India 2011C]

20. Evaluate 2

3

1x xdx

[All India 2011C]

21. Evaluate 0

tan

sec cos

x xdx

x ecx

[Delhi 2011C, 2008]

22. Evaluate 4

11 2 4x x x dx [Delhi 2011C]

23. Evaluate 2

2

0( )x x dx as limit of a sum [Delhi 2011C]

24. Evaluate 3

2

1(2 3)x dx as limit of a sum [Delhi 2010, 2009C]

25. Evaluate 2

2

1( 5 )x x dx as limit of a sum [Delhi 2010C]

26. Evaluate 3

2

1(3 2 )x x dx as limit of sum [Delhi 2010]

27. Evaluate 0

tan

sec tan

x xdx

x x

[Delhi 2010; All India 2008]

28. Evaluate 5

2

2(3 5)x dx as limit of sum [All India 2009C]

29. Evaluate 2 2 2 20 cos sin

xdx

a x b x

[ All India 2009]

30. Evaluate

cos

cos cos0

x

x x

edx

e e

[ Delhi 2009]

31. Evaluate / 2

0(2logsin logsin 2 )x x dx

[Delhi 2009]




32. Evaluate 3

2

1( )x x dx as limit of sum [All India 2008C]

33. Evaluate / 2

0 sin cos

xdx

x x

[All India 2008]

34. Evaluate 1

0sin

a xdx

a x

[All India 2008]

35. Evaluate 1

1 2

02 tan x dx [All India 2008]

36. Evaluate a

a

a xdx

a x

[Delhi 2008]

37. Evaluate 3

2

1( 5 )x x dx as the limit of a sum [Delhi 2008C]

38. Evaluate 1

1 2

0cot (1 )x x dx [Delhi 2008]

Chapter 8- Applications of Integration

6 Marks Questions

1. Using method of integrals find the area of region bounded by lines 3x 2y + 1 = 0, 2x + 3y 21 = 0 and

x 5y + 9 = 0. [ Delhi 2012]

2. Find area of the region 2 2, : 4, 2x y x y x y [ All India 2012]

3. Using integration, find the area of the triangular region whose sides have the equations y = 2x + 1,

y = 3x + 1 and x = 4. [ All India 2011C; Delhi 2011]

4. Find the area of the region 2 2, : 1x y x y x y [ All India 2011; Delhi 2010C]

5. Find the area of the region given by 2, :x y x y x . [ Delhi 2011C; All India 2009; 2008C]

6. Using integration, find the area of ,ABC the coordinates of whose vertices are A (2, 5), B (4, 7) and

C (6, 2). [ Delhi 2011; All India 2010C]

7. Sketch the graph of 3y x and evaluate the area under the curve 3y x above X-axis and between x = -

6 to x = 0. [ All India 2011]

8. Using method of integration, find the area of the region bounded by the lines 2x + y = 4; 3x 2y = 6;

x 3y + 5 = 0 [ Foreign 2011; Delhi 2009]




9. Find the area of the region enclosed between the two circles x2 + y

2 = 4 and (x 2)

2 + y

2 = 4.

[ All India 2010C; Delhi 2008]

10. Using integration, find the area of the region [ Delhi 2010C]

11. Using integration, find the area of ,ABC coordinates of whose vertices are A (4, 1), B(6, 5) and C (8, 4).

[ All India 2010]

12. Find the area of circle 4x2 + 4y

2 = 9 which is interior to the parabola x

2 = 4y. [ All India 2010]

13. Using integration, find the area of the following region 2, : 1 5x y x y x [ Delhi 2010]

14. Using integration, find the area of the following region : 2 2

, : 19 4 3 2

x y x yx y

[ Delhi 2010]

15. Using integration, find the area of the region bounded by the curve x2 = 4y and the line x = 4y 2.

[ Delhi 2010]

16. Find the area of the region bounded by ellipse

2 2

2 21

x y

a b and the straight line 1.

x y

a b

[ All India 2009C]

17. Using integration, find the area of the region bounded by the triangle whose vertices are (1, 3), (2, 5) and (3, 4).

[ Delhi 2009C]

18. Find the area of the region bounded by the region enclosed by the curves

(x 6)2 + y

2 = 36 and x

2 + y

2 = 36. [ All India 2009C]

19. Find the area enclosed by the parabola 4y = 3x2 and the line 3x 2y + 12 = 0. [ All India 2009]

20. Find the area of the region enclosed by the parabola y2 = x and the line x + y = 2. [ Delhi 2009]

21. Find the area of the region enclosed between two circles x2 + y

2= 9 and (x 3)

2 + y

2 = 9. [ Delhi 2009]

22. Prove that the curves y2 = 4x and x

2 = 4y divide the area of the square bounded by x = 0, x = 4, y = 4 and

y = 0 into three equal parts. [ Delhi 2009]

23. Find the area of the region 2 2 2, : 4 ,4 4 9x y y x x y [ All India 2008C; Delhi 2008C]

24. Find the area of the region 2 2 2, : 6 , 16x y y x x y [ Delhi 2008C]

25. Using integration, find the area of triangular region whose vertices are (1, 0), (2, 2) and (3, 1).




[ All India 2008]

26. Find the area of the region bounded by the parabolas y2

= 4ax and x2 = 4ay. [ All India 2008]

Chapter IX - Differential Equations

4 Marks Questions

1. Find the differential equation of all circles in the first quadrant which touch the coordinate axes.

[All India 2010C]

2. Find the differential equation of family of all orders touching Y-axis at the origin.

[Delhi 2010; All India 2009, 2008C]

3. Find the differential equation of family of all circles touching X-axis at the origin.

[Delhi 2010C; All India 2009C]

4. Form the differential equation representing family of ellipses having foci on X-axis and centre at the origin.

[Delhi 2009C]

5. Form the differential equation representing family of curves given by (x a)2 + 2y

2 = a

2, where a is an arbitrary

constant. [All India 2009]

Topic 2 : Solution of different types of Differential equations

4 Marks Questions

Direction (Q. Nos. 1 to 3) solve the following differential equations

1. 2 22 2 0

dyx xy y

dx [Delhi 2012]

2. 2 2 2 21

dyx y x y

dx given that y = 1 when x = 0 [Delhi 2012]

3. 2 1 1;dy

x xdx

y = 0 when x = 2 [All India 2012]

4. Find the particular solution of the differential equation2 2(1 ) (1 ) 0,x xe dy y e dx given that y = 1 when x = 0.

[Foreign 2011; All India 2008C]

5. Solve the following differential equation cot 4 cos ,dy

y x x ecxdx

given that y = 0 when 2

x

.

[Foreign 2011]

6. Solve the following differential equation 2 21

1 2 ,1

dyx xy

dx x

given y = 0 when x = 1.

[ Foreign 2011]

7. Solve the following differential equation 2 2xdy ydx x y dx . [All India 2011]




8. Solve the following differential equation 23 .dy

y x xdx

[All India 2011]

9. Solve the following differential equation 2( 2 ) 0xdy y x dx [All India 2011]

10. Solve the following differential equation 3( ) 0xdy y x dx [All India 2011]

11. Solve the following differential equation 2tan (1 )sec 0x xe ydx e y dy [Delhi 2011]

12. Solve the following differential equation 2cos tan

dyx y x

dx [Delhi 2011]

13. Solve the following differential equation 2(1 )(1 log ) 0y x dx xdy [Delhi 2011]

14. Solve the following differential equation 21 2 cot ,( 0)x dy xy dx x dx x

[All India 2011]

15. Solve the following differential equation cos 0;dy y y

ecdx x x

y = 0 when x = 1.

[All India 2011, Delhi 2009]

16. Solve the following differential equation 2sin 0

yx y dx xdy

x

[Delhi 2011C]

17. Solve the following differential equation cot 0, 0dy

x y x xy x xdx

[Delhi 2011C]

18. Show that the following differential equation is homogeneous and then solve it

log 2 0y

ydx dy xdyx

[All India 2011]

19. Solve the following differential equation cos sin sin cos 0y y y y dy

x y y y x xx x x x dx

[All India 2010C]

20. Solve the following differential equation2 2log log 0

y yxy dx y x dy

x x

[Delhi 2010C]

21. Solve the following differential equation 2 21 2 4dy

x xy xdx

[All India 2010, 2008]

22. Solve the following differential equation 3 2 21 2dy

x x x x xdx

[All India 2010]

23. Solve the following differential equation 2 21

1 2 ; 11

dyx xy x

dx x

[All India 2010]

24. Solve the following differential equation2 2 2 21 0

dyx y x y xy

dx [All India 2010]




25. Show that the differential equation 2dy

x y x ydx

is homogeneous and solve it. [All India 2010]

26. Find the particular solution of the differential equation satisfying the given condition

2 2( ) 0; (1) 1x dy xy y dx y [Delhi 2010]

27. Find the general solution of the following differential equation:

2log . log

dyx x y x

dx x [Delhi 2010]

28. Find the particular solution of the differential equation satisfying the given condition tan ,dy

y xdx

given that y =

1 when x = 0. [ Delhi 2010]

29. Solve the following differential equation sec sindy

x y xdx

[ All India 2009C]

30. Solve the following differential equation log 2logdy

x x y xdx

[Delhi 2009, 2009C]

31. Solve the following differential equation 2cos tan

dyx y x

dx [All India 2009; Delhi 2008, 2008C]

32. Solve the following differential equation tandy y

x y xdx x

[All India 2009]

33. Solve the following differential equation 2 11 tandy

x y xdx

[Delhi 2009]

34. Solve the following differential equation cos sindy

y x xdx

[Delhi 2009]

35. Solve the following differential equation sec tandy

y x xdx

[ Delhi 2008C]

36. Solve the following differential equation 2 tan sindy

y x xdx

[All India 2008]

37. Solve the following differential equation 2 2 2

dyx y xy

dx [All India 2008]

38. Solve the following differential equation 2 2 2 0x y dx xydy given that y = 1 when x = 1

[Delhi 2008]

39. Solve the following differential equation (2 )

,(2 )

dy x y x

dx x y x

if y = 1 when x = 1. [Delhi 2008]

Chapter X - Vector Algebra




1Mark Questions

1. Find sum of vectors 2a i j k and 2 4 5b i j k and 6 7c i j k . [Delhi 2012]

2. Find the scalar components of AB with initial point A (2, 1) and terminal point B ( - 5, 7).[All India 2012]

3. For what values of a the vectors 2 3 4i j k and 6 8ai j k are collinear ? [Delhi 2011]

4. Write the direction cosines of vector 2 5i j k . [ Delhi 2011]

5. Write the position vector of mid-point of the vector joining points P (2, 3, 4) and Q (4, 1, -2).

[Foreign 2011]

6. Write a unit vector in the direction of 2 2a i j k . [All India 2011]

7. Find the magnitude of the vector 3 2 6a i j k . [ All India 2011C]

8. Find a unit vector in the direction of vector 2 3 6a i j k . [ All India 2011C]

9. If A, B and C are the vertices of a triangle ABC, then what is the value of AB + BC + CA ?

[ Delhi 2011C]

10. Find a unit vector in the direction of 2 3 6a i j k . [ Delhi 2011C]

11. Find a vector in the direction of 2 2a i j k which has magnitude 6 units. [ Delhi 2010C]

12. Find a position vector of mid-point of line segment AB, where A is point (3, 4, -2) and B is point (1, 2, 4)

[ Delhi 2010C]

13. Write a vector of magnitude 9 units in the direction of vector 2 2i j k . [ All India 2010]

14. Write a vector of magnitude 15 units in the direction of vector 2 2i j k . [ Delhi 2010]

15. What is the cosine of angle which the vector 2i j k makes with Y-axis ? [ Delhi 2010]

16. Find a unit vector in the direction of vector 6 2 3b i j k . [ All India 2009C]

17. Find a unit vector in the direction of vector 2 2a i j k [Delhi 2009C]

18. Write a unit vector in the direction of vector 2 2b i j k [Delhi 2009]

19. Write a unit vector of the direction of vector 2 6 3a i j k [ Delhi 2009]




20. Find the magnitude of the vector 2 6 3a i j k . [All India 2008C]

21. Find a unit vector in the direction of 2a i j k [Delhi 2008C]

22. If 2a i j k and 3 5 ,b i j k find a unit vector in the direction a b . [ All India 2008]

23. Find a unit vector in the direction of 3 2 6a i j k . [ Delhi 2008]

4 Marks Questions

24. Find a vector of magnitude 5 units and parallel to the resultant of 2 3a i j k and 2b i j k .

[Delhi 2011]

25. Let ,a i j k 4 2 3b i j k and 2c i j k . Find a vector of magnitude 6 units which is parallel to

the vector 2 3 .a b c [ All India 2010]

26. Find the position vector of a point R which divides the line joining two points P and Q whose position vectors are

2a b and 3a b respectively, externally in the ratio 1 : 2. Also show that P in the mid-point of line segment

RQ. [ Delhi 2010]

1 Mark Questions

1. Find when projection of 4a i j k on 2 6 3b i j k is 4 units. [Delhi 2012]

2. Write value of . .k j i j k [All India 2012]

3. If . 0a a and . 0,a b then what can be concluded about the vector b ? [ Foreign 2011]

4. Write the projection of vector i j on the vector i j [ All India 2011]

5. Write the angle between vectors a and b with magnitude 3 and 2 respectively, having . 6a b .

[ All India 2011]

6. For what value of are the vectors 2i j k and 2 3i j k perpendicular ? [ All India 2011C]

7. If 3,a 2b and angle between a and b is 600, find .a b . [ Delhi 2011C]

8. Find the value of , if the vectors 2 3i j k and 3 2 4i j k are perpendicular to each other.

[ All India 2010C]

9. If 2, 3a b and . 3,a b find the projection of b on a . [ All India 2010C]




10. Vectors a and b are such that 3,a 2

3b and a b is a unit vector. Write the angle between

a and b [ All India 2010]

11. If a and b are two vectors such that . ,a b a b then find angle between a b . [ All India 2010]

12. Find , if 2 6 14 7i j k i j k = 0. [ All India 2010]

13. Find . ,a b if 2a i j k and 2 3b i j k . [ All India 2009C]

14. Find . ,a b find 3 2a i j k and 2 3 3b i j k . [ Delhi 2009C]

15. Find the value of P, if 2 6 27 3 0i j k i j Pk [ All India 2009]

16. If P is a unit vector and (x P). (x + P) = 80, then find x . [ All India 2009]

17. Find the angle between a and b with magnitudes 1 and 2 respectively and when 3a b .

[ Delhi 2009]

18. Write the value of P for which 3 2 9a i j k and 3b i Pj k are parallel vectors.

[ Delhi 2009]

19. Find the projection of a on b if . 8a b and 2 6 3b i j k . [ Delhi 2009]

20. Find value of following: .( ) .( ) .( )i j k j i k k i j [ All India 2008C]

21. Find a b if 7 7a i j k and 3 2 2 .b i j k [ Delhi 2008C]

22. If 3, 2a b and . 3,a b find angle between a and b . [ All India 2008]

23. Find angle between vectors a i j k and .b i j k [ Delhi 2008]

24. For what value of , 2a i j k and 2 3b i j k are perpendicular to each other ?

[ Delhi 2008]

4 Marks Questions




25. If , ,a b c are three vectors such that 5,a 12,b 13c and 0,a b c find the value of

. . .a b b c c a . [ Delhi 2012]

26. Let 4 2 , 3 2 7a i j k b i j k and 2 4 .c i j k Find a vector p which is perpendicular to both a

and b and . 18.p c [ All India 2012, 2010]

27. Find a unit vector perpendicular to ach of the vectors a b and a b , where 3 2 2a i j k and

2 2b i j k . [ Delhi 2011]

28. If two vectors a and b are such that 2a and 1b and . 1,a b then find (3 5 ).(2 7 )a b a b .

[ Delhi 2011]

29. If vectors 2 2 3 ,a i j k 2b i j k and 3c i j are such that a b is perpendicular to c , then

find value of . [ Foreign 2011; All India 2009C]

30. Using vectors find the area of triangle with vertices A (1, 1, 2) and B(2, 3, 5) and C(1, 5, 5).

[ All India 2011]

31. If ,a b and c are three vectors such that 3, 4a b and 5c and each one of these is perpendicular to the

sum of other two, find a b c . [ All India 2011C, 2010C]

32. If ,a b and c are three mutually perpendicular vectors of equal magnitude, show that a b c is equally inclined

to ,a b and c . Also find the angle [Delhi 2011C]

33. Using vectors, find the area of triangle with vertices A (2, 3, 5), B (3, 5, 8) and C (2, 7, 8).

[ Delhi 2010C]

34. The scalar product of vector i j k with the unit vector along the sum of vectors 2 4 5i j k and

2 3i j k is equal to one. Find the value of . [ All India 2009, 2008C]

35. If a b c d and a c b d , show that a d is parallel to b c , where a d and b c .

[ Delhi 2009]

36. Three vectors ,a b and c satisfy the condition 0,a b c Find the value of . . .a b b c c a , if 1, 4a b

and 2.c [ All India 2008C]

37. Find a vector of magnitude 5 units, perpendicular to each of the vectors ( )a b and ( )a b , where




a i j k and 2 3b i j k . [ Delhi 2008C]

38. If ,a b and c are such that 0,a b c and 3, 5a b and 7.c Find angle between a and b .

[ All India 2008; Delhi 2008]

39. If a i j k and ,b j k find a vector c such that a c b and . 3.a c [ Delhi 2008]

Chapter XI-Three Dimensional Geometry

Topic 1 : Straight Line

1 Mark Questions

1. If a line has direction ratios 2, 1, 2, then what are its direction cosines ? [Delhi 2012]

2. What are the direction cosines of a line which makes equal angles with the coordinate axes.

[Foreign 2011; All India 2009]

3. Write the vector equation of the line given by5 4 6

.3 7 2

x y z [Delhi 2011, 2010]

4. Equation of line is 4 3 2

2 2 1

x y z . Find the direction cosines of a line parallel to above line.

[All India 2011C]

5. If the equations of line AB are3 2 5

1 2 4

x y z

, write the direction ratios of line parallel to above line AB.

[Delhi 2011C]

6. Find the value of , such that the line 2 1 5

6 4

x y z

is perpendicular to the plane 3x y 2z = 7.

[All India 2010C]

7. Find the distance of point (2, 3, 4) from x-axis. [ Delhi 2010C]

8. Write the equation of line parallel to the line 2 3 5

3 2 6

x y z

and passing through point (1,2, 3).

[All India 2009C]

9. Write the direction cosines of a line parallel to the line 3 2 2

3 2 6

x y z

. [ Delhi 2009C]

10. The equation of line 2 5 4 6

4 3 6

x y z is .Find the direction cosines of the line parallel to this line.

[All India 2008]




11. The equation of line is given by4 3 2

2 5 6

x y z . Write the direction cosines of the line parallel to above

line. [Delhi 2008C]

12. If P = (1, 5, 4) and Q = (4, 1, -2), find the direction ratios of PQ. [All India 2008]

4 Marks Questions

13. Find the vector and Cartesian equations of line passing through point (1, 2, -4) and perpendicular to two lines.

8 19 10

3 16 7

x y z

and

15 29 5

3 8 5

x y z

[Delhi 2012]

14. Find the angle between following pair of lines 2 1 3

2 7 3

x y z

and

2 2 8 5

1 4 4

x y z

and check

whether the lines are parallel or perpendicular. [Delhi 2011]

15. Find the shortest distance between lines whose vector equations are

(1 ) ( 2) (3 2 )r t i t j t k

( 1) (2 1) (2 1)r s i s j s k [All India 2011]

16. Find the shortest distance between lines [Foreign 2011]

6 2 2 ( 2 2 )r i j k i j k

4 (3 2 2 )r i k i j k

17. Find shortest distance between the lines [Foreign 2011]

( 2 ) ( )r i j k i j k

(2 ) (2 2 )r i j k i j k

18. Find the equation of the perpendicular from point (3, -1, 11) to line2 3

2 3 4

x y z . Also, find the coordinates

of foot of perpendicular and the length of perpend

previous_years_questions_combined.pdf

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