mathematics for class xi by maths n...

M A T H S N M E T H O D S

S E C T O R 1 0 , D W A R K A , N E W

D E L H I 1 1 0 0 7 5

9 8 1 1 1 6 0 4 4 2

M A T H S . K M R @ G M A I L . C O M

1 / 5 / 2 0 1 4

Mr. Mukesh Kumar,Msc.(Maths),B.Sc.Maths(H),University Of Delhi He has hands on experience of more than 20 years in teaching and

guiding students and teachers for class IX, X, XI, XII, engineering

entrance exam etc. Because of wide experience exposure with all

categories of students, he understands the psychological aspects of

students apart from intricacies of math.

MATHEMATICS FOR

CLASS XI BY

MATHS N METHODS

Class XI (Mathematics) www.mathsNmethods.in

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Ch1 Sets

For 1 mark 1. If A and B are two sets such that n(A∩B)=10,n(A)=28 and n(B)=32,find n(A∪B).

1. If A and B are two sets such that n(A∪B)=50,n(A)=28 and n(B)=32,find n(A∩B).

2. How many subsets will be of a set containing 3 elements?

3. Find no. of elements in power set of A, if A= {1, 2, 4} ?

4. Find no. of elements in P(A) if A = {1,2,3}

5. Find P (A) if A= {2}.

6. Draw Venn diagram of B-A.

7. Find no. of elements in P(A) if A = {1,2,3}

8. Write in set builder form for the set A = {a,e,i,o,u}

9. Write in set builder form for the{�� , �

� , �� , �

� , � ,

}.

10. If A = {ø}, then find � ��, �� .

11. Write the set {x: x2<4, x is an integer} in roster form.

12. If A = {ø}, then find �� .

For 4 marks

1. A college awarded 38 medals in football, 15 in basketball and 20 in cricket. If these medals went to

a total of 58 men and only 3 men got medals in all the 3 sports, how many received medals in

exactly 2 0f the 3 sports ?

2. Write all subsets of the set {ø,1,2}.

3. Write all subsets of the set {ø,1,0}.

4. In a committee, 60 people speak French, 20 speak Spanish and 10 speak both Spanish and French.

How many speak at least one of the 2 languages?

5. In a committee, 50 people speak French, 20 speak Spanish and 10 speak both Spanish and French.

How many speak at least one of the 2 languages?

6. A and B are 2 sets If A∩X = B∩X =ø and AUX =BUX for some set X, prove that A=B.

7. For A = {1,3,5,7} , B= {1,2,3,4}, U={1,2,3,4,5,6,7,8,9}. Find (a) A-B (c) A-ø (c) A’ B

’

8. For A ={3,4,6,7} , B= {2,4,6,8} verify that (A B)’=A’U B’ for U={1,2,3,4,5,6,7,8}.

9. Verify De Morgan’s law by Venn diagram.

10. If U = {1,2,3, . . . , 10 } , A = {1,2,3} and B = { 3,4,5} then find (a) A – B (b) (A U B)’

(c) (A ∩ B)’ (d) B - A

11. For any sets A and B, show that

P(A ∩ B) = P(A) ∩ P(B)

For 6 marks


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1. In a survey of 100 people, it was found that 28 people read newspaper H, 30 people read

newspaper T, 42 people read newspaper I, 8 read both H and T, 10 read both H and I, 5 read both T

and I, 3 read all three newspapers. Find the no. of people who read (a) at least one of the

newspapers and (b) newspaper I only.

2. In a survey of 60 people, it was found that 25 people read newspaper H, 26 people read newspaper

T, 26 people read newspaper I, 9 read both H and I, 11 read both H and T, 8 read both T and I, 3

read all three newspapers. Find the no. of people who read at least one of the newspapers.

3. A college awarded 38 medals in football, 15 in basketball and 20 in cricket. If these medals went to

a total of 58 men and 0nly 3 men got medals in all the three sports, how many received medals in

exactly (a)1 of 3 sports (b)2 of 3 sports .


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Ch2 Relation and Functions For 1 mark

1.Find the domain of the function defined by f(x) = ��√��.

2. Find the domain of the function f(x)=� !��!� � ��!�

3. If A={1,2}, B={2,4,5}, then find A X B

4. If A={1,2}, B={3,4}, then find A X (B X ø ).

5. Let f=((1,1),(2,3),(0,-1),(-1,-3)} be a linear function from Z into Z. Find f(x).

6. Define modulus function .

7. If A = {1,2 } and B = { x,y } Write A X B.

8. If A×B ={(a,x),(a,y),(b,x),(b,y)} find the set A.

9. Let A= {1,2} and B= {3,4}. Find the no. of relations from A to B.

For 4 marks

1. Find the domain and range of the real function √"� − 9

2. Find the domain and range √9 − "�

3. Find the domain and range of the function √49 − "�

4. Let A={1,2,3,4,8}. Let R be the relation on set A defined by {(a,b): a, b ЄA, b is exactly

divisible by a} a) write R in roster form b)Find the range of R.

5. Let f=((1,1),(2,3),(0,-1),(-1,-3)} be a linear function from Z into Z. Find f(x).

6. If A = {1, 2, 3, . . .,18}. Let R be the relation on A defined by R={(a,b): a,b ЄA, 3a-

b=0} then (a) Write R in roster form (b) Find the domain of R (c) Find the range of

R.

7. If A = {1, 2, 3, 4, 6}. Let R be the relation on A defined by {(a,b): a,b ЄA, b is exactly

divisible by a} then (a) Write R in roster form (b) Find the domain of R (c) Find the

range of R.

8. The Cartesian product A X A has 9 elements among which are found (-1,0) and (0,1). Find

the set A and the remaining elements of A X A .


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Ch3 Trigonometry For 1 mark

1. Evaluate tan ��)�

2. Evaluate tan �)�

3. Evaluate cos�−17100�

4. Evaluate cos�− )� �

5. 1�� ℎ� 3456� �� 789��:�!;<= ��:

6. 1�� cosec ?��@ �� A��" = ��

�� 4�� " 5�� 3�� D64��4��. 7. Find cot ?− ��F

� @. 8. Find the value of cosec�− ��)

� �

9. If cot " = − �� , x lies in 2

nd quadrant then find sin " .

10. Find the principal solution(s) of tan " =��√�

11. Evaluate sin 500

- sin 700+ sin 10

0

For 4 marks

1. Prove that HIJ ��!HIJ ��!HIJ ��JK= ��!JK= ��!JK= �� = cot 3"

2. ��3� �ℎ4� �4�4" = �789��;<= �� ;<= �!;<=L �.

3. Prove that �MN9 � ! MN9 �� ! �JK= O� ! JK= ��

�HIJ � ! HIJ �� ! �HIJ O� ! HIJ �� = tan 6"

4. MN9�QRSM�Q�MN9QRSM�QMN9�QMN9Q!RSMQRSMQ = �4�2�

5. P��3� �ℎ4� MN9��MN9��!MN9�RSM��RSM� = tanx

6. Find the value of tan �)U�

7. 1�� sin ?��@ , A�� ?�

�@ 4�� 4� ?��@ �� A��" = − �

� , " �� 2�� D6��4��. 8. V�53� �� " ∶ 2 cos� " + 3��" = 0. 9. Solve for " ∶ sin 2" − sin 4" + sin 6" = 0 10. Solve for θ the equation cosθ+ cos3θ- 2cos2θ=0.

11. Solve for " 2cos� " + 3 sin " = 0


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12. V�53� √3 A��" + ��"=√2

13. V�53� ��2" + ��4" + ��6" = 0

14. 1�� ℎ� Y��45 ��56�� cot� " + �MN9� + 3 = 0

15. V�53� ∶ A��A " = cot " + √3

16. V�53� ∶ A��A " + cot " = √3

17. V�53� ∶ Z√3 − 1[ cos " + �√3 + 1� sin " = 2 (2nπ± )� + )

��)

18. Solve: sinU " + cosU " = �� (

)U)

19. Solve : tan " + tan 2" + tan 3" = tan " tan 2" tan 3"

20. Prove that tan " + 2 tan 2" + 4 tan 4" + 8 cot 8" = cot "

21. Prove that sin20osin40

osin60

osin80

o= �

� . 22. ��3� �ℎ4� ��100��300��500��700 = �

�. 23. ��3� �ℎ4� A��200A��300A��400A��800 = √�

� 24. Prove that A��120A��240A��480A��960 = − �

� 25. ��3� �ℎ4� A��200A��400A��600A��800 = �

�. 26. ��3� �ℎ4� �4�200�4�400�4�600�4�800 = 3

27. ��3� �ℎ4� ��180 = √�� .

28. ��3� �ℎ4� A��7 ��

0 = √2 + √3 + √4 + √6

29. Pr�3� �ℎ4� 2A��θ = _2 + `2 + √2 + 2cos8θ 30. Prove that �4�3"�4�2"�4" = �4�3" − �4�2" − �4�"

31. Prove that A��3". A��2" − A��3". A��5" − A��5". A��2" = 1. 32. ��3� �ℎ4� A��"A��2" − A��2"A��3" − A��3"A��" = 1

33. ��3� �ℎ4� ��" + ��3" + ��5" + ��7" = 4A��"A��2"��4"

34. ��3� �ℎ4� �4�500 = �4�400 + 2�4�100

35. RSM�:�MN9�:RSM�:!MN9�: = A��620

36. �

MN9�0: − √�RMS�0: = 4

37. ��3� JK=�a�b�JK= �a!b� = c �R

8

38. If in ∆�ef, ∠� = 600, �ℎ�� 3� �ℎ4�: �8!c + �

8!R = �8!c!R

39. In ∆�ef, ��3� �ℎ4�: 8 !c 8 �c = �!HIJ�Q�a� HIJ b

�!HIJ�Q�b� HIJ a

40. In ∆�ef, ��3� �ℎ4�: 4� cos�e − f� + i� cos�f − �� + A� cos�f − e� = 34iA

For 6 marks

1. Prove that: 1 sin cos

tan .1 sin cos 2

+ θ − θ θ = + θ + θ

2. Prove that: ( ) ( )3 34 cos 20 cos 40 3 cos 20 cos 40 .° + ° = ° + °


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3. ��3� �ℎ4� ��" + ��2" + ��4" + ��5" = 4 cos ?��@ cos ?��

� @ ��3" 4. ��3� �ℎ4� ��360 = `�0��√�

� . 5. ��3� �ℎ4� A�� 360 = √�!�

�

6. ��3� �ℎ4� cos� " + cos��" + �)� � + cos��" − �)

� � = ��

7. j34564�� cos� " + cos��" + )�� + cos��" − )

��

8. If tan � = kk�� , tan e = �

�k�� , �ℎ�� ℎ4� � − e = )�

9. If tan()� + l

�) = tan3()� + m

�), prove that ��n = �MN9m!JK=o m�!� JK= m

10. If cos " + cos p = �� , sin " + sin p = �

� , ��3� �ℎ4� cos ��q� � = ±�

��


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Ch4 PMI For 4 marks

1. Using principle of mathematical induction prove that n(n+1)(n+5) is divisible by 3 for

all natural n.

2. Using principle of mathematical induction prove that n(n+1)(2n+1) is divisible by 6 for

all natural n.

3. Using principal of mathematical induction, prove that x2n –y2n is divisible by x-y.

For 6 marks

1. Prove �

�.� + �

�. + �

.O + . . . +�

��9!��.��9!�� = 9

��9!�� by using the principle of mathematical

induction for all n є N.

2. Using principal of mathematical induction, prove that 13+2

3 +3

3 +4

3 +…n

3 = ?9�9!��

� @�.

3. Using principal of mathematical induction, prove that 12

+22 +3

2 +4

2 +…n

2 = 9�9!��9!��

.

4. Prove �

�.� + �

�.U + �

U.�� + . . . +�

��9��.��9!�� = 9

�9!�� by using the principle of mathematical

induction for all n є N.

5. Using principle of mathematical induction prove that

�

�.�.� + ��.�.� + �

�.�.� + … + �9�9!��9!�� = 9�9!��

��9!��9!�� for all natural n.


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Ch5 Complex Numbers For 1 mark

1. Find the least positive integral value of n for which ?�!N��N@

9=1

2. Write the value of Ns: o!Ns: L!Ns: t!Ns: u!Ns: v

Ns o!Ns L!Ns t!Ns u!Ns v

3. If ��N��!N� = 4 + �i, �ℎ�� ℎ� 3456� �� 4� + i�

4. Find the amplitude of �N

5. Write the value of �!�N!�N ��N!�N

6. Find tann �� n �� ℎ� 4w�5��6�� 8!Nc8�Nc

7. If z= �

��HIJ l�NMN9 l, then find Re(z)

8. Write in polar form of (� 25)

3

9. If (1+ �) (1+2�) (1+3�). . . (1+n�)=a+ �b, then express 2.5.10.17. . .(1+n2)

10. If Z=1 − cos n + �� n, then find |Z|

For 4 marks

1. Find real θ for which � ! �KJK=θ��KJK=θ is purely imaginary.

2. 1�� 4�p 2 A�w�5�" ��. Z1 and Z2, prove that

Re(Z1Z2) =Re(Z1)Re(Z2)-Im(Z1)Im(Z2)

3. Find the modulus of �!N��N -

��N�!N .

4. Find the Square root of (a) 5+12� (b) √−1 (c)-8-6 � (d)- √−1

For 6 marks

1. Convert the complex number ��!N√� into polar form.

2. Convert the complex number z = N��

HIJx o !N JK= )/� in the polar form.

3. If (x+ � y)3 =u + � v, then prove that

z� + {

q = 4(x2 – y

2)

4. Convert the complex number ��U�!N√� into polar form.

5. Write the complex number Z= �!�N��K in polar form.


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Ch6 Linear Inequalities For 1 mark

1. Solve for real x, 4-4x > 15-3x.

2. Solve 2-3x > 1-x.

3. Solve 12x < 55, x

4. Represent the solution on number line

-3<4- �� ≤ 18

For 4 marks 1. A manufacturer has 640 liters of 8% solution of acid. How many liters of 2% acid solution must be added so that

acid content in the resulting mixture will be more than 4% and less than 6%?

2. A manufacturer has 600 liters of 12% solution of acid. How many liters of 30% acid solution must be added so

that acid content in the resulting mixture will be more than 15% and less than 18%?

3. I.Q. of a person is given by formula I.Q.=}.Q.b.Q. ~100, where M.A. stands for mental age and C.A., stands for

chronological age. If 75≤ �. �. ≤ 135 for a group of a 9 years children. Find the range of their mental age.

4. A manufacturer has 600 liters of 12% solution of acid. How many liters of 30% acid solution must be added so

that acid content in the resulting mixture will be more than 15% and less than 18%.

5. Solve and show the solution on number line,

-5 ≤ ��

� ≤ 9 .

For 6 marks

1. Solve the following systems of linear inequalities graphically:

" + 2p ≤ 10, " + p ≥ 1, " − p ≤ 0, " ≥ 0, p ≥ 0 . 2. Solve the following systems of linear inequalities graphically:


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3" + 2p ≤ 150, " + 4p ≤ 80, " ≤ 15, " ≥ 0, p ≥ 0 . 3. Solve the following system of linear inequations graphically:

X+2y≤40, 3x+y≥30, 4x+3y≥60, x≥0, y≥o.

4. Solve the following system of linear inequations graphically:

X+2y≤40, 3x+y≥30, 4x+3y≥60, x≥0, y≥o.

5. Solve the following systems of linear inequalities graphically:

2" + p ≥ 4, " + p ≤ 3, 2" − 3p ≤ 6 .

Ch7 Permutations & Combinations For 1 mark

1. How many even numbers of 3-digit can be formed by using the digits 1 to 9 if no digit is repeated.

2. Find x, if ��! +

�! =

�!

3. How many chords can be drawn through 21 points on a circle?

4. A polygon has 44 diagonals. Find the no. of its sides.

5. If P(n,t)=840; C(n,t)=35, then find t.

6. If C(n,12)= C(n,8) then find n.

7. Find C(n,3) if C(n,4)+ C(n,5)=C(10,5).

8. Find C(n,2) if C(n,4)+ C(n,5)=C(10,5).

9. In how many ways can 5 letters be posted in 4 letter boxes?

10. Find the position of the word ‘SACHIN’ if all the letters of the word ‘SACHIN’ are written in all possible

orders as written in a dictionary.

11. Find the word at 24th position if all the letters of the word ‘PRAGATI’ are written in all

possible orders as written in a dictionary

12. Find the word at 23rd

position if all the letters of the word ‘PRAGATI’ are written in all

possible orders as written in a dictionary


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For 4 marks

1. 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?

2. From a class of 20 students, 8 are to be chosen for an excursion party. There are 4 student

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?

3. How many words with or without meaning each of 3 vowels and 2 consonants can be formed from the

letters of the word “DAUGHTER”.

4. How many ways can a cricket team be selected from a group of 25 players containing

10batsmen, 8bowlers, 5allrounder and 2wicketkeepers assume that the team of 11 players

requires 5batsmen, 3allrounder , 1wicketkeeper and 2 bowlers.

5. Find the number of permutation of the letters of the word “MATHEMATICS”. In how many of these

arrangements

(a) Do all the vowels occur together?

(b) Do the vowels never occur together?

6. Find the no. of arrangement 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 all the vowels never occur together?

(iv)do the words begin with I and end in P?

7. In how many ways can 3 prizes be distributed among 4 boys, when

(i)no boy gets more than 1 prize?

(ii)a boy may get any no. of prizes?

(iii)no boy gets all the prizes?

8. Find the sum of all the numbers that can be formed with the digits 2,3,4,5 taken all at a time.

9. Find the sum of all the numbers that can be formed with the digits 1,2,3,4 taken all at a time.

10. How many even numbers are there with 3 digits such that if 5 is one of the digits, then 7 is next digit?

11. Determine the no. of natural numbers smaller than 104, in the decimal notation of which all the digits are

distinct.

12. Prove that 35! Is divisible by 216

. What is the largest integer n such that 35! Is divisible by 2n?

13. There are 10 points in a plane. No three of which are in the same st. line, excepting 4 points, which are

collinear. Find the (i) no. of st. lines obtained from the pairs of these points; (ii) no. of triangles that can be

formed with the vertices as these points.

14. Find the number of ways in which 5 boys and 5 girls be seated in a row so that

(i) no 2 girls may sit together.


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(ii) all the girls sit together and all the boys sit together.

(iii) all the girls are never together.

15. Find the position of the word ‘RANDOM’ if all the letters of the word ‘RANDOM’ are written in all possible

orders as written in a dictionary.

16. How many words can be formed by taking 4 letters at a time out of the letters of the word

‘MATHEMATICS’.

17. A committee of 4 persons is to be formed from 5 men and 3 women. In how many ways the

committee can be formed if it includes (a) at least 3 men ( b) at least 2 women ?

Ch8 Binomial Theorem For 1 mark


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1. Find the fifth term in the expansion of (2x-y)

10

2. Find the 28th

term in the expansion of (1+2x+x2)

15.

3. Find the 88th term in the expansion of (1-6x+9x2)45.

4. Find the 13th

term in the expansion of (9x-��)

18

5. Find the 4th

term from the end in the expansion of (�

� - �o )

17

6. Find the coefficient of 5th

term in the expansion of (1-2x)12

.

7. Find the coefficient of �� ℎ� �"�4�� 1 + "�9 ?1 + �

�@9. 8. Write the number of terms in the expansion of (1-3x+3x

2-x

3)

8

9. Find the sum of the coefficients in the expansion of (1-3x+x2)

111

10. Find a positive value of m for which the coefficient of x2 in the expansion (1+x)

m is 6.

11. Which is larger (1.01)1000000

or 10,000?

12. Find the middle term in the expansion of (x2+y

2)

6.

13. Find the middle term in the expansion of (x+�

��)12

.

14. Find the number of irrational terms in the expansion of ?4st + 7 s

s:@��

15. The number of terms with integral coefficient in the expansion of ?17so + 35s

@00

For 4 marks

1. Find the coefficient of x

5 in the expansion of the product (1+2x)

6(1-x).

2. Find the coefficient of x5 in the expansion of (1+x)

21+(1+x)

22+…(1+x)

30

3. If the sum of the binomial coefficient of the expansion (3x+��)

n is equal to 256, then find the term independent

of x.

4. Find the term independent of x in the expansion of (9x – �

�√� )18

.

5. Find the term independent of x in the expansion of (1+x)m

(x+��)

n

6. Find (x+y)4 - (x-y)

4 using binomial and hence evaluate (√5 + √3)

4 - (√5 - √3)

4.

7. Simplify Z" + √" − 1[+ Z" − √" − 1[

8. Find the 5th term from the end in the expansion of ?�o� − �

� @O.


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9. If the 21st

and 22nd

terms in the expansion of (1+x)44

are equal, then find the value of x.

10. In the binomial expansion of (a-b)n,n≥ 5, the sum of 5

th and 6

th terms is zero. Then find

8c.

11. If a and b denotes the sum of the coefficients in the expansion of (1-3x+10x2)

n and (1+x)

n respectively then find

the relation between a and b.

12. If a and b denotes the coefficients of xn in the expansion of (1+x)

2n and (1+x)

2n-1 respectively then find the

relation between a and b.

13. If n be a positive integer, then using binomial theorem prove that 62n

-35n-1 is divisible by 1225 n∈ �.

14. If n be a positive integer, then using binomial theorem prove that 24n+4

-15n-16 is divisible by 225 n∈ �.

For 6 marks

1. The first 3 terms in the binomial expansion (a + b)

n are 729 , 7290 and 30375 respectively. Find b,

a and n.

2. The second, third and fourth terms in the binomial expansion (x + a) n

are 240, 720 and 1080

respectively. Find x, a and n.

3. The 3rd

, 4th

and 5th

terms in the expansion of (x+a)n are respectively 84,280 and 560, find the

values of x, a and n.

4. Find the term independent of x in the expansion of (9x – �

�√� )18

.

5. If three consecutive coefficient in the expansion of (1+x)n be 165,330 and 462, find n.

6. If three consecutive coefficient in the expansion of (1+x)n are in the ratio 6:33:110, find n.

7. If in the expansion of (1+x)n the coefficient of 14

th , 15

th and 16

th are in A.P. find n.

8. Find the coefficients of a4 in the product (1+2a)

4 (2-a)

5 using binomial theorem.

9. Find n, if the ratio of the fifth term from the beginning to the fifth term from the end in the expansion of

(√2 + �

√�)n

is √6: 1.

10. Find n, if the ratio of the 7th

term from the beginning to the 7th

term from the end in the expansion

of (√2o + �√�o )n is

� .


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Ch9 Sequence and Series


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For 1 mark

1. Find the sum of the series �� + �

� + �U + . . .

2. Find " if − � , " , −

� are in G.P

3. If the fifth term of a GP is 2, then write the product of its 9 terms.

4. The 3rd

term in a G.P is 4. Find the product of first 5 terms.

5. The sum of an infinite GP is 8, its second term is 2, find the first term.

6. If the sum of n terms of an AP is 2n2+5n, then write its n

th term.

7. If AM and GM of roots of a quadratic equation are 4 and 7, respectively, then obtain the quadratic

equation.

8. The income of a person is ₹ 3,00,000, in the first year and he receives an increase of ₹ 10,000 to his

income per year for the next 19 years. Find the total amount, he received in 20 years.

For 4 marks

1. Find the sum to n terms of the sequence 7, 77, 777, 7777 . . .

2. Find the sum to n terms of the sequence 0.4, 0.44, 0.444, 0.4444 . . .

3. Find the sum to infinite terms of the sequence 0.5, 0.55, 0.555, 0.5555 . . .

4. The sum of an infinite geometric series is 15 and the sum of the squares of these terms is 45. Find

the series.

5. Convert 2.4444… in fraction using GP.

6. Find the least value of n for which the sum 1+3+32+… to n terms is greater than 7000.

7. Find the sum to n terms of the sequence 3 x 12 + 5 x 2

2 + 7 x 3

2

8. Find the sum to n terms of the series: 5 +11 + 19 +29 +41 . . .

9. Find the sum to n terms of the series: 1 +5 + 12 +22 +35 . . . (A 9 �9!��

� )

10. Find the sum to n terms of the series: 5 +7 + 13 +31 +85 . . . [A �� 3�� + 8� − 1� ]

11. Find the sum to n terms of the series : 1X2 + 2X3 + 3X4 + 4X5 + . . .

12. Find the sum of 1+�

�!� + ��!�!� + … �� w�.

13. Find the sum of �� + �

�!� !�L + ��!� !�L … �� w�. (A

9 !9��9 !9!��)

14. The sum of n terms of 2 AP’s are in the ratio 5n+4:9n+6. Find the ratio of their 18th

term.

15. If 8�! c�

8��s! c��s is the AM between a and b , then find the value of n.

16. Find two positive numbers whose difference is 12 and whose AM exceeds the GM by 2. (A 16,4)

numbers.

17. If a is the AM of b and c and the GMs are G1 and G2, then prove that G13+G2

3=2abc

18. If one GM G and 2 AMs A1 and A2 be inserted between 2 given quantities, prove that

G2=(2A1-A2)(2A2-A1).

19. If AM and GM of two positive numbers are 10 and 8 respectively, find the numbers.


18 | P a g e

20. If one AM, A and 2 GMs " 4�� p �� between any 2 positive numbers, show that � q + q

� = 2A

21. If x be the AM and y,z be 2 GM between any 2 positive then prove that qo!�o

�q� = 2.

22. If the sum of n terms of an A.P is 3n2+5n and m

th term is 164, find the value of m.

23. The sum of first 3 terms of a GP is �� and their product is -1. Find the terms.

24. If ax=b

y=c

z and x,y,z are in G.P, then Prove that log a ,log b, log c are in G.P

25. Three numbers whose sum is 15 are in A.P if 1,4,19 be added to them respectively, the resulting

numbers are in G.P find the numbers.

26. Find 3 numbers whose sum is 52 and the sum of whose product in pairs is 624.

27. If a,b,c are in AP and a,x,b and b,y,c are in GP show that x2,b

2,y

2 are in AP.

28. If �

�!q , �q , �

q!� are in A.P then show that x,y,z are in G.P

29. If S1, S2,S3 are sums to n terms, 2n terms and 3n terms respectively of a G.P, then prove that

S1(S3-S2)=(S2-S1)2.

For 6 marks

1. If a,b,c are in AP ; b,c,d are in GP and �R , �

� , �� 4�� in AP. Prove that a,c,e are in GP.

2.If � , D, � 4�� 4�� ℎ� �D64��, �"� + 2D" + � = 0 4�� "� + 2�" + � = 0 ℎ43� 4 A�ww�� , �ℎ�� ℎ�� ℎ4� �

� , �� , �

� 4�� .

3. The sum of 2 numbers is 6 times their geometric means, show that the numbers are in the ratio

(3 + 2√2): (3 - 2√2).

4. If the sum of m terms of an AP is equal to the sum of either the next n terms or the

next p terms, then prove that �w + �� ? �k − �

�@ = �w + �� ? �k − �

9@. 5.


19 | P a g e

Ch 10 Straight Lines

For 1 mark

1. Reduce the equation x-y =4 in normal form.

2. Find y intercept for the equation 2" + 3p − 5 = 0. 3. Find y intercept for the equation 12" + 3p − 5 = 0. 4. Find the slope of the line which makes an angle of 135

0 with x axis in clockwise direction.

5. Find the distance between the lines 5x+3y-7=0 and 15x+9y+14=0. 6. Find the angle between the lines 2x-y+3=0 and x+2y+3=0. 7. Find the value of λ for which the lines 3x+4y=5, 5x+4y=4 and λx+4y=6 meet at a point.

8. Find the equation of the line with slope �� and which is concurrent with the lines 4x+3y-7=0 and

8x+5y-1=0. 9. Find the value of q if the lines x+q=0,y-2=0 and 3x+2y+5=0 are concurrent.

For 4 marks

1. Find the equation of the line whose perpendicular distance from the origin is 4 and the

angle which the normal makes with the positive direction of x axis is 150.

2. Find the equation of the right bisector of the line segment joining the points (3,4) and (-1,2).

3. 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.

4. Find the equation of the line passing through the intersection of the lines x+y+3=0 and x-

y+2=0 and having y-intercept equal to 4.

5. Find the coordinates of the foot of perpendicular from the point (3,8) to the line x+3y=7. 6. Find the coordinates of the foot of perpendicular from the point (-1,3) to the line 3x-4y=16.

For 6 marks 1. Find the equation of the straight line which passes through the

point (3,4) and the sum of its intercepts on the axes is 14.

2. A line is such that its segment between the lines 5x-y +4 =0 and 3x+4y-4 =0 is bisected

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

3. Find the image of the point (3,8) with respect to the line x+3y=7 assuming the line to

be plane mirror.


20 | P a g e

Ch11 Conic Section

For 1 mark

1. Find the equation of the parabola whose focus(6,0); directrix x=-6.

2. Find the eccentricity of the hyperbola 16x2 -9y

2 =576.

3. Find the equation of the parabola whose focus(0,6); directrix y= -6.

4. Find the length of latus rectum of the hyperbola 16x2 -9y

2 =576.

5. Find the eccentricity of the hyperbola 9x2 -16y

2 =144.

6. Find the length of latus rectum of the hyperbola 16x2 -9y

2 =576.

7. Find the equation of the parabola which is symmetric about the y- axis , and passes through the

points (2,-3).


21 | P a g e

For 4 marks

1. Find the equation of the circle passing through the points (2,-2) and (3,4) and whose centre

is on the line x + y =2.

2. 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.

3. 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.

4. Find the equation of the hyperbola whose foci are (0,∓12) and the length of the latus

rectum is 36.

5. Find the equation of the hyperbola whose foci are (∓4,0) and the length of the latus rectum

is 12.

6. Find the vertices, foci, eccentricity and the length of the latus rectum of the ellipse

5x2+12y

2=20.

7. Find the vertices, foci, eccentricity and the length of the latus rectum of the ellipse

5x2+15y

2=40.

For 6 marks

1. Find the equation of the circle which passes through (-1,1) and centre of the circle x

2+y

2-4x-6y-

5=0 and whose centre lies on the line x-3y-11=0.

2. Find the equation for the ellipse whose major axis on the x-axis and passes through the points

(4,3) and (6,2).

3. Find the equation for the ellipse whose major axis on the x-axis and passes through the points

(4,3) and (-1,4).


22 | P a g e

Ch12 3-D

For 1 mark

1. Write the octant of the point (2,-3,5)

2. Find the point on the x- axis which is at a distance of 4 from the point (1,2,3)

For 4 marks

1. 4 students in traditional dresses represent 4 states of India, standing at the points represented by

O(0,0,0), A(a,0,0), B(0,b,0) and C(0,0,c). Find the place, in terms of coordinates, where a girl representing

‘BHARATMATA’ be placed so that ‘BHARATMATA’ is equidistant from the 4 students. What message

does it convey?


23 | P a g e

For 6 marks

1. Using section formulae prove that points (-2,3,5), (1,2,3) and (7,0,-1) are collinear. Also find the ratio in which

3rd

point divide line segment joining the first 2.

2. Using section formulae prove that points (3,2,-4), (5,4,-6) and (9,8,-10) are collinear. Also find the ratio in

which 3rd point divide line segment joining the first 2 .

Ch13 Limits & Derivatives

For 1 mark

1. Evaluate lim�→0��!√�!�

�

2 Evaluate lim�→0 JK= ��;<= ��.

3. Evaluate lim�→0 JK= ��;<= �� .

4. Evaluate lim�→∞

� ��!��


24 | P a g e

5. Evaluate lim�→0 JK= ��;<= ��.

6. Evaluate lim�→0��

�

7. Evaluate lim�→0��

√�!��

8. Evaluate lim�→��I� ��

��

9. Evaluate lim�→0�I��!��I� ��

�

10. Evaluate lim�→0�I� ��!�o�

JK=o �

For 4 marks

Evaluate the following Algebraic limits:-

1. lim�→��o�� !�

�L�U� !� 2. lim�→�Z� ��[s�

��o� U� ! �� 3. lim ��→�

�� ! �� − �

�o� ��

4. lim�→√��L� �

� ! �√��U 5. lim�→�/��

�√�� 6. lim�→�√� ��!√��

√� �� , x>1

7. lim�→�� − �

�o�� !��) 8. lim�→�/�U�o��

��L�� 9. lim�→0√8 !� �√8 ��

�

10. lim�→8√8!��√��√�8!��√� 11. lim�→�

��√�!��√�� 12. lim�→�

��√��√��

13. lim�→0√�!� �√�!�√�!�o�√�!� 14. lim�→�

�s:��0��t�� 15. lim�→8

��!��to��8!��to��8

16. Find � �� lim�→��L�� =lim�→�

�o��o� �� 17. lim�→�∞

√"� − " + 1 + "

18. lim�→∞

��s!�� s!�� 19. lim�→∞

√� !8 �√� !c √� !R �√� !� 20. lim9→∞

�o!�o!⋯!9o9L

21. lim9→∞� �

9 + �9 + �

9 + ⋯ + 9��9 � 22. lim9→∞

�9!��!!�9!��!�9!��!��9!��!

23. ��"� = 8� !c� !� , lim�→0��"� = 1 4�� lim�→∞

��"� = 1 �ℎ�� 3� �ℎ4� ��−2� = ��2� = 1


25 | P a g e

Evaluate the following Trigonometric limits:-

1. lim�→0��HIJ k��HIJ 9� 2. lim�→0

;<= ��JK= �JK=o � 3. lim�→0

JK= ��! JK= �JK= ��JK= ��

4. limq→0��!q� J�H��!q�� J�H �

q 5.lim�→0HI; ��RSM�R ��

� 6. lim�→0��HIJ �√HIJ ��

�

7. lim�→0U

�� 1 − cos � � − cos �

� + cos � � . cos �

� � 8. lim�→0��HIJ � HIJ �� HIJ ��

JK= �

9. lim�→0�8!�� JK=�8!��8 JK= 8

� 10.lim�→0√��√�!HIJ �

� 11. lim�→0√�!JK= ��√��JK= �

�

12. lim�→)/��;<= �

��xL

13.lim�→x

��JK= �?x

��@ 14. lim�→x�

HI; ��HIJ ��)�U��o

15. lim�→xL

�√��HIJ �!JK= ��t��JK= �� 16.lim�→x

HIJ ��!� HIJ �?x

��@o 17. lim�→xL

JK= ��HIJ ��x

L

18. lim�→8MN9√��MN9√8

��8 19.lim�→xL

√HIJ ��√JK= ��x

L 20. lim�→x

√��√�!JK= �√� HIJ �

21. lim�→0√��√�!HIJ �

JK= � 22.lim�→0��;<= � 23. lim�→0

��!��

24. Prove that lim�→��!�o��

�o�� = ��

�

25. ��"� = �4" − 5, �� " ≤ 2" − �, �� " > 2 find �, �� lim�→� ��"� exists.

26. f(x) = �2x + 3, if x ≤ 2 x + k, x > 2 , ¤ind the value of k, so that lim�→� f(x) exists.

27. f(x)=© m + nx, if x < 1 4, if x = 1n − mx, if x > 1 and if lim�→� f(x) =4 find the possible values of m and n.

28. Find the derivative of √tan " from the first principle.

.

29. Find the derivative of �!�� from the first principle.

30. Find the derivative of ��!�� from the first principle.


26 | P a g e

31. Find the derivative of �

MN9� from the first principle.

i. 32. Find the derivative of cos√2" + 1 from the first principle.

ii. 33. Find the derivative of cosec(2x - )

��) from the first principle.

41. Find the derivative of x2 cosx from the first principle.

42. Differentiate the following

(a) JK= ��HIJ �JK= �!HIJ � (b)

JK= �� HIJ �� JK= �!HIJ � (c)

J�H �!;<= �J�H ��;<= � (d)

� 789�J�H �!;<= � (e)

�!«S¬��«S¬�

(f) ��!JK= ��!�I� � (g)

��JK=� � (h)

�8� !c�!R (i)

�� HI; �√� (j) x

nlogax e

x

(k) cot�{�2" + 3�so} (l) `��√" (m) ex sin(logx) (o) (ax+b)

n (cx+d)

m (p)

� HIJxLJK= �

iii.

Ch14 Mathematical Logics For 1 mark

For 4 marks

For 6 marks

Ch15 Statistics


27 | P a g e

For 1 mark

For 4 marks

1. Find the mean and variance for the following marks of class 11 of a school:

Marks 0 – 20 20 – 40 40 – 60 60 – 80 80 – 100

No. of students 1 2 8 9 5

2. Find mean deviation from median of ages of teachers of a school:

Ages(in yrs) 25 – 29 30 – 34 35 – 39 40 – 44 45 – 49 50-55

No. of teachers 5 8 7 5 11 9


Marks 0 – 10 10 – 20 20 – 30 30 – 40 40 – 50 50-60

No. of students 6 6 14 16 4 2

For 6 marks

1. Find the coefficient of variation for the following marks of class 11 of a school:

Marks 0 – 20 20 – 40 40 – 60 60 – 80 80 – 100



Marks 0 – 20 20 – 40 40 – 60 60 – 80 80 – 100



Ages(in yrs) 25 – 29 30 – 34 35 – 39 40 – 44 45 – 49 50 – 54 55 – 59

No. of teachers 5 8 7 50 11 12 9


Ages(in yrs) 25 – 29 30 – 34 35 – 39 40 – 44 45 – 49 50 – 54 55 – 59

No. of teachers 5 8 7 50 11 12 9


28 | P a g e

Ch16 Probability For 1 mark

1) Find probability of A or B, if A and B are mutually exclusive events & P(A)=1/3 P(B)=1/4.

2) 3 coins are tossed. Describe 3 events which are mutually exclusive.

3) Given P(A)= �� 4�� P�B� = �

�. Find P(A or B), if A and B are independent events.

4) 3 coins are tossed. Describe 3 events which are mutually exclusive and exhaustive.

5) From a well shuffled pack of 52 cards a card is drawn if it is club then find the probability that

2nd

card drawn is king.

6) Given P(A)= �� 4�� P�B� = �

�. Find P(A or B), if A and B are mutually exclusive events.

7) From a well shuffled pack of 52 cards a card is drawn if it is club then find the probability that

2nd card drawn is queen.

8) Find the probability that an ordinary year has 52 Sundays.

For 4 marks

1. Two dice are thrown find the probability that either the sum of the numbers appearing is

observed to be 6 or first die shows an even no .

2. Two dice are thrown together. Find the probability of getting an even number on the first

die or a total of 8.

3. Two dice are thrown together. What is the probability that the sum of the numbers on the

two faces is divisible by 3 or 4.

4. What are odds in favour of throwing at least 8 in a single throw of two dice?

5. Two dice are thrown, find (i) the odds in favour of getting a sum of 5 (ii) the odds against

getting a sum 6.

6. If a number of two digits is formed with the digits 2,3,5,7,9 without repetition of digits, what

is the probability that the number formed is 35?

7. Two cards are drawn at random from a pack of 52 playing cards. Find the probability of

getting

(i) a red card or a diamond

(ii) a black card or a king

8. Three cards are drawn at random from a pack of 52 playing cards. Find the probability that

it includes (i) at least 2 kings (ii) at most 2 kings (iii) exactly 2 kings

9. Four cards are drawn at random from a pack of 52 playing cards. Find the probability of

getting

(i) all the four cards of same colour

(ii) all the four cards of same suit

(iii) all the four cards of the same number

10. A committee of 2 persons is selected from 2 men and 2 women. What is the probability that

the committee will have a) 2 men b) no man? 11. Find probability that the all the vowels will never occur together in the arrangement of all

the letters of the word MATHEMATICS.


29 | P a g e

12. Find probability that the all the vowels will never occur together in the arrangement of all

the letters of the word ASSASSINATION.

13. A die is thrown twice and the sum of the numbers appearing is observed to be 6. What is

the conditional probability that the number 4 has appeared on first die?

14. There are 4 envelopes corresponding to 4 letters. If the letters are placed in the envelopes

at random, what is the probability that all the letters are not placed in the right envelopes?

15. What is the probability that in a group of 3 people at least two will have the same birthday?

Assume that there are 365 days in a year.

16. A bag contains 3 red, 6 white and 7 black balls. Two balls are drawn at random; find the

probability that both are black.

17. Six boys and six girls sit in a row randomly, find the probability that all the six girl sit

together.

18. A drawer contains 50 bolts and 150 nuts. Half of the bolts and half of the nuts are rusted. If

one item is chosen at random, what is the probability that it is rusted or a bolt.

19. The probability that a student will pass the final examination in both SST and Math is 0.05

and the probability of passing neither is 0.1. If the probability of passing the SST

examination is 0.75, what is the probability of passing the Math examination?

20. A box contains 4 red, 5 white and 6 black balls. A person draws 4 balls from the box at

random. Find the probability of selecting at least one ball of each colour.

21. A natural number is chosen at random from among the first 500. What is the probability

that the number so chosen is divisible by 3 or 5?

mathematics for class xi by maths n...

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