class xi math question 2011 the collegiate

MATHEMATICS FIRST YEAR CHAPTER WISE QUESTIONS

CHAPTER # 01 (SETS)

Q#1 Let U = {0, 1, 2, 3}, A = {0, 1}, B = {1, 2}, C = {2, 3} [1991]

Prove that: A × (BUC) = (A×B) U (A×B)

Q#2 If A = {2, 3}, B = {3, 4} and C = {4, 5} [1992]

Prove that: A× (B∩C) = (A×B) ∩ (A × C)

Q#3 IF U = {1,2……….6} , A = {1,2,3} , B = {1 , 3 , 5} and C = {2 , 4 , 6}, [1994]

Find (A – C) and verify that: A∩B A AUB

Q#4 Verify the property A × (B U C) = (A × B) U (A × C) in the following sets [1997]

A = { a , b} , B = {b , c} , C = { c , d}

Q#5 Let A = {0,2,4} , B = {1,2} and C = {3,4} then prove that: [1998]

A × (BUC) = (A × B) U (A × C)

Q#6 Let U = {1,2,3,4,5,6} , A = {1,2,3,4} , B = {1,3,4,5} [1999]

Show that (A ∩ B)′ = A′ U B′

Q#7 Let U = {2,3,4,5,6} , A = {2,3} , B = {3,4} , C = {5,6} [2000]

Show that (B – C)′ = B′

Q#8 If U = {1,2,3,4,5,6} and A = {2,3} , B = {3,4} , C = {4,5} [2001]

Find (A × B) ∩ (A × C)

Q#9 If A = {2,3} , B = {3,4} , C = {c,f} and U = {2,3,4,c,f} Find [2003]

(A × B) U (A × C) and B′ - A′

Q#10 If A = {1,2,4} , B = {2,3,4} , U = {1,2,3,4,5} [2004]

Find (A U B)′ and (A ∩ A′)′

Q#11 If A and B are subsets of the universal set U, then prove that: [2005]

A U B = A U (A′ ∩ B)

Q#12 If A and B are the subsets of a set U, then prove that: [2006]

A U B = A U (A′ ∩ B)

Q#13 If U = {a , b , c , d , e} , A = { a , b , c} , B = {b , c , d} [2007]

Find (A ∩ B′)′

Prepared by:

Prof S.M. Noaman


Q#14 If A = {2,3} , B = { 3 , 4} and C = {4,5} [2008]

Find A × (B U C)

CHAPTER # 02 (REAL & COMPLEX NUMBER SYSTEM)

Q#01 Express the following equation in terms of the conjugate coordinates [1991]

x2+ y2 = 9.

Q#02 Solve the complex equation (x + 3i)2 = 2yi [1991]

Q#03 Simplify (x,3y) , (2x , -y) [1991]

Q#04 Show that Z = 1 ± i, satisfies the equation: [1992]

Z2 – 2Z + 2 = 0

Q#05 If Z1 = 1 + i an Z2 = 3 + 2i, evaluate: [1994]

(i) (Z1)2 (ii) Z1Z2

Q#06 Show that (1 – i) 4 is a real number. [1995]

Q#07 What is the imaginary part of (2+i )2

3−4 i?

[1996]

Q#08 Prove that |Z1Z2|=|Z1||Z2| , where Z1, Z2 are the complex numbers. [1997]

Q#09 Solve the complex equation (x + 2yi)2 = xi. [1998]

Q#10 Find the additive inverse and the multiplicative inverse of (2, – 3). [1998]

Q#11 Is there a complex number whose additive and multiplicative inverse are equal?

[1998]

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Prof S.M. Noaman


Q#12 Divide 4 + i by 3 – 4i [1999]

Q#13 Prove that ( 325 ,−425 ) is the multiplicative inverse of (3,4).

[1999]

Q#14 Multiply (-3,5) by (2,-1) [1999]

Q#15 Separate into and imaginary parts: [2000]

1+2i2−i

and hence find |1+2i2−i | Q#16 By using definition of multiplicative inverse of two order pairs, [2000]

Find the multiplicative inverse of (5,2).

Q#17 Solve the equation (2,3)(x , y) = (- 4,7) [2000]

Q#18 Define modulus and the conjugate of complex number z = x + iy, [2001]

Where i=√−1 , If z= 1+i1−i

, then show that z z=|z|2

Q#19 Verify that1+3 i3−5 i

+ 217

=−417

+ 717

i

[2001]

Q#20 Find the multiplicative inverse of √3+ i√3+ i

, separating the

real and imaginary parts. [2002]

Q#21 Solve the following complex equation: [2002]

(x , y) (2,3) = (5 , 8)

Q#22 If Z1 = 1 + i and Z2 = 3 – 2i, then evaluate |5 Z1=4 z2|. [2003]

Q#23 Separate 7+5 i4+3i into real and imaginary parts.

[2003]

Q#24 Find the additive inverse and multiplicative inverse of (3,-4) [2003]

Prepared by:

Prof S.M. Noaman


Q#25 If Z1 and Z2 are complex numbers, verify |Z1 . Z2|=|Z1|,|Z2| [2004]

Q#26 Solve the following complex equation (x,y) . (2,3) = (5,8). [2004]

Q#27 Solve the complex equation (x + 2yi) 2 = xi. [2005]

Q#28 Show that (a ,b ) . [ a

a2+b2−b

a2+b2 ]=(1,0)

[2006]

Q#29 If z = (x , y) , then show that z . z=|z|2 [2006]

Q#30 If Z1 = 1 + i and Z2 = 3 – 2i, evaluate |5 Z1−4 Z2| . [2007]

Q#31 Solve the complex equations (x , y) . (2 , 3) = (-4 , 7) [2007]

Q#32 Separate the following into real and imaginary parts: [2007]

1+2 i3−4 i

+ 25

Q#33 Express x2 + y2 = 9 in terms of conjugate coordinates. [2008]

Q#34 If z1 = 1 + i , z2 = 3 – 2i , evaluate |5 z1−4 z2| [2008]

Q#35 Find the real and imaginary parts of i (3 + 2i). [2008]

Q#36 Find the multiplicative inverse of the complex number (3 , -5). [2008]

Q#37 Solve the complex equation: (x,y).(2,3)=(-4,7) [2010]

Q#38 Solve the complex equation: (x + 3i)2 = 2yi [2011]

Prepared by:

Prof S.M. Noaman


CHAPTER # 03 (QUADRATIC EQUATIONS)

Q#01 Solve the equation (2x + 5)4 + (2x + 1)4 = 82. [1991]

Q#02 For what values of p and q will both the roots of the equation. [1991]

y2 + (2p – 4)y = 31 + 5 vanish?

Q#03 If α, β are the roots of px2 + qx + r = 0 , p ≠ 0, form the equation whose roots are:

(2α+ 1α ) ,(2 β+ 1

β ) [1992]

Q#04 Form an equation whose roots are – 1 ± i. [1994]

Q#05 Solve the equation: [1994]

√ 1−xx

+√ x1−x

=136

Q#06 Find the condition that one root of the equation px2 + qx + r = 0 may be square of the other.

[1994]

Q#07 If α , β are the roots of the equation lm2 + mx + n = 0 , 1 ≠ 0 form the equation whose

are α+ 1α

,β+ 1β2

[1995]

Q#08 If 4x+1 = 64, what is the value of x? [1996]

*2 *3 *4 *5

Q#09 If α , β are the roots of x2 – 3x + 1 = 0, find the equation whose roots are [1996]

1 + α + α2 and 1 + β + β2

Q#10 Find the value of ‘k’ by synthetic division method so that x – 1 is a factor of P(x)

where P(x) = 2kx4 + 7x3 + kx2 + 7x – 1. [1997]

Q#11 Find the solution set of the following equation. [1997]

1 + 3x + 4x2 + 3x3 + x4 = 0

Q#12 If α, β are the roots of the equation px2 + qx + r = 0 , p ≠ 0, form the equation whose

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Prof S.M. Noaman


roots are (α2 + β2) and ( 1α 2+ 1β2 )

[1998]

Q#13 Solve the equation √2x+7+√x+3=1 [1999]

Q#14 If α, β are the roots of the equation x2 – 3x + 2 = 0, form the equation whose roots

are (α + β)2 and (α – β)2. [2000]

Q#15 Show that ω155 + ω247 + i 360 = 0 where ω is the cube root of unity

and i = √−1 [2000]

Q#16 Show that all cube roots of –27 are –3, –3ω and –3ω2 where ω and ω2 are the

complex cube roots of unity. [2001]

Q#17 If α and β are the roots of the equation px2 + qx + r = 0 (p ≠ 0), form an

equation whose roots are α2 and β2. [2001]

Q#18 For what value of ‘k’ will the equation. k3x2 +(2k2 – 1) + 4k = 0, have equal

roots? [2002]

Q#19 If α , β are the roots of px2 + qx + r = 0, form the equation whose roots are

1/α2,1/β2 [2002]

Q#20 Prove that ω49 + ω101 + ω150 = 0, where ω is a complex cube root of unity. [2003]

Q#21 If α , β are the roots of 3x2 – 7x + 8 = 0, find the value of: [2003]

i) 1

α2+ 1α2

ii) [α+ 1α ] [ β+ 1

β ]Q#22 For what value of k will x + 5 be a factor of 2x3 + kx2 – 2x + 15? [2004]

Prepared by:

Prof S.M. Noaman


Q#23 Solve the equation (x – 9) (x – 7) (x + 1) – 384 = 0. [2004]

Q#24 If α and β are the roots of αx2 + bx + c = 0, a ≠ 0, find the equation whose

roots are α3 and β3. [2004]

Q#25 Solve the equation: [ x−1x ]2

+3[ x+ 1x ]=0

[2005]

Q#26 Find the condition that one root of px2 + qx + r = 0, p ≠ 0 may be double

the other root. [2005]

Q#27 Solve y6 – 26y3 – 27 = 0. [2006]

Q#28 Find all the cube roots of 27. Also show that their sum is zero. [2006]

Q#29 Find ′k′ if one root of 4y2 – 7ky + k + 4 = 0 is zero. [2006]

Q#30 Prove that the roots of, the equation x2 – 2x (m + 1/m) + 3 = 0 are real. [2007]

Q#31 If α , β are the roots of the equation px2 – qx – r = 0 , p ≠ 0, find the

equation whose roots are α3 and β3. [2007]

Q#32 Solve the following equation: [2008]

x−3x+3

+ x+3x−3

=2 415

Q#33 Find the equation whose roots are reciprocal of the roots of x2 – 6x + 8 = 0. [2008]

Determine the value of k for which the roots of the following equation are equal:

(k + 1)x2 + 2(k + 3)x + 2k + 3 = 0

Q#34 If α , β are the roots of the equation ax2 + bx + c = 0, from the equation

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whose roots are .α+1α

,β+1β

[2009]

Q#35 If ∝ and β are the roots of the equatoion px2+qx + r = 0, (p≠0), Find the value of ∝2+ β2.

[2010]

Q#36 If {1, ω , ω2}are the cube roots of unity, Prove that

(2+ω2 )= 32+ω

[2010]

Q#37 Solve α , β are the roots of the equation x2 – 3x + 2 = 0 from an equation whose roots

are (α+β)2 and (α – β)2. [2011]

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Prof S.M. Noaman


CHAPTER # 04 (SYSTEMS OF TWO EQUATIONS INVOLVING TWO

VARIABLES)

Q#01 The sum of the circumferences of two circles is 24π

meters and the sum of their

areas is 80π square meters. What are the radii of the circles?

[1991]

Q#02 Solve and check:

[1992]

2x + 3y = 2

2x2– 3y2 = 20

Q#03 Solve for x:

[1996]

xa−x

–a−xx

=a


[1998]

3x + 2y = 7

3x2 – 2y2 = 25


[1999]

2x2 + y2 = 13

5x2 – 2y2 = - 8

Q#06 Solve the following system of Equations.

[2000]

x – y – 5 = 0 x2 + 2xy + y2 = 9

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Prof S.M. Noaman


Q#07 The sum of the circumferences of two circles is 42πcm

and the sum of their

areas is 261π square cm. What are the radii of the circles?

[2001]

Q#08 Solve the following system of Equations:

[2002]

x2 + y2 = 169

x – y – 13 = 0

Q#09 Solve the following system of equations:

[2003]

i) 12x2 – 25xy + 12y2 = 0 ii) x2 + y2 = 25

Q#10 Solve the equations: x + y = 5

[2005]

2x+ 3

y=2

Q#11 Solve the system of equations:

[2007]

xy = 6

x2 + y2 =13


[2008]

x+ y=5

3x+ 2

y=2


[2009]

4 x+3 y=25 ,

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Prof S.M. Noaman


4x+ 3

y=2

Q#14 Solve: x + y = 5 [2010]

3x+ 2

y=2

Q#15 Solve the following system of equations:[2011]

x2 + y2 = 169

x – y = 13

CHAPTER # 05 (MATRICES)

Q#01 Verify that (AX)t = xtAt, where: [1991]

A=[1 −1 22 2 0]∧X=[2 1

0 21 −1]

Q#02 If A=[ 4 −2 05 6 −7

−3 1 9 ] ,C=[+2 +3−4 0−1 +3] [1992]

Then compute Ct At.

Q#03 What is a null matrix? Prove that:[1994]

{[ ω ω2 1ω2 1 ωω ω2 1 ]+[ 1 ω ω2

ω ω2 1ω2 1 ω ]}[ 1ωω2]=[000 ]

Prepared by:

Prof S.M. Noaman


Where ω is a complex cube root of unit.

Q#04 If A=[ 1 2 34 6 8

−1 2 −3 ] ,B=[ 2 3 4−4 6 −81 2 5 ], then verify [1996]

That (AB)t = BtAt.

Q#05 Define a Singular Matrix. Find the value of λ for which the following matrix

becomes singular.[1997]

A=[ λ 1 01 λ 20 2 λ]

Q#06 Solve the system of equations by Cramer’s rule:[1999]

2x + y + z = 1

x – 2y – 3z = 1

3x + 2y + 4z = 5

Q#07 Solve for Z : [−2 34 −1][1 z 5

2 4 z ] [−310 ]=[ 2−14] [2001]

Q#08 Define the following terms: [2001]

i) Transpose of a matrix ii) Diagonal matrix

iii) Scalar matrix iv) Unit matrix

Q#09 Perform the Matrix multiplication. [2002]

[a b c ] [ x f gf y hg h z ] [

abc ]

Q#10 If A=[ i 00 −i ] ,B=[0 i

i 0] where i = √−1 , verify that

A2 = B2 = -I2

[2003]

Q#11 Solve for x .[−2 34 −1] [1 x 5

2 4 x ][−310 ]=[2 −14 ]t [2004]

Prepared by:

Prof S.M. Noaman


Q#12 If possible, find the matrix X such that: [2 −30 1 ] . X=[−2 5

8 −7][2006]

Q#13 Solve the following for x: [−2 34 −1] [1 x 5

2 4 x ][−310 ]=[ 2−14 ][2007]

Q#14 Prove. {[ 1 ω ω2

ω ω2 1ω2 1 ω ]}+[ ω ω2 1

ω2 1 ωω ω2 1 ][ 1ωω2]=[000 ] [2008]

Q#15 Solve for x: [−2 34 −1] [1 x 5

2 4 x ][−310 ]=[2 −14 ] [2009]

Q#16 Verify that: [ Sinθ −CosθCosθ Sinθ ][ Sinθ Cosθ

−Cosθ Sinθ ]=[1 00 1]

[2010]

Q#17 Solve for x:[−2 34 −1] [1 x 5

2 4 x ][−310 ]=[2 −14 ] t [2011]

CHAPTER # 06 (DETERMINANTS & INVERSE MATRICES)

Prepared by:

Prof S.M. Noaman


Q#01 Solve by matrix method.[1991]

9x + 7y + 3t = 6

5x – y + 4t = 1

6x + 8y + 2t = 4

Q#02 Solve: [ x2−a2 x2 x

b2−a2b2bc2−a2c2 c ]=0

[1992]

Q#03 Solve the Equations by Cramer’s Rule.[1992]

x + 2y + z = 8

2x – y + z = 3

x + y – z = 0

Q#04 Making use of the properties of the determinants, solve

for x the equation: [ x2+a2 x2 x

b2+a2b2 xc2+a2 c2c ]=0

[1994]

Q#05 Use matrix method to solve the system of equations: [1995]

2x1 – x2 + 2x3 = 4

x1 + 10x2 – 3x3 = 10

-x1 + x2 + x3 = -6

Q#06 If A=[ 1 2 34 6 8

−1 2 −1] find kA and k׀ A׀

[1996]

Q#07 By suing the properties of determinant, express the following determinant in

factors: [ 1 1 1a b ca3 b3 c3] [1997]

Q#08 Solve the following equation by using Cramer’s Rule:[1997]

x + y = 5

y + z = 7

z + x = 6

Q#09 Using the properties of determinants, prove that:[1998]

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Prof S.M. Noaman


[1 a a2

1 b b2

1 c c2]=(a−b ) (b−c )(c−a)

Q#10 use matrix method to solve the following equations: [1998]

x + y + z = 22x – y – z = 1x – 2y – 3z = -3

Q#11 Find the inverse of the Matrix by adjoint method:[1999]

A=[ 2 2 31 −1 0

−1 2 1]Q#12 Express the following determinant factors form by using properties of determinants.

|1 1 1a b ca3 b3 c3| [2000]

Q#13 Apply matrix method to solve the following system of Equations: [2000]

9x + 7y + 30 = 6

5x – y + 40 = 1

6x + 8y + 20 = 4

Q#14 Using the properties of determinants, show that:[2001]

|1 x yz1 y zx1 z xy|=( x− y ) ( y−z )(z−x )

Q#15 Use by Adjoint method to solve the equations.[2002]

2x – y + 2z = 4

x + 10y – 3z = 10

x – y – z = 6

Q#16 Use properties of determinants Solve for x.[2002]

|1 a b1 a x1 x c|=54

Q#17 Using the properties of determinants, show that:[2003]

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Prof S.M. Noaman


|1 x x2

1 y y2

1 z z2|=(x− y )( y−z )(z−x )

Q#18 Find the inverse of matrix A=[1 2 12 −1 11 1 −1] by adjoint

method. [2003]

Q#19 Find the adjoint of the matrix. A=[ 1 2 34 6 8

−1 2 −3 ][2004]

Q#20 Solve the determinant for x ,|1 a b1 x b1 c x|=0

[2004]Q#21 Evaluate, using the properties of the determinants.

[2005]¿

Q#22 Use the matrix method to solve the system of linear equations: [2005]

3x + 2y + 4z = 5

2x + y + z = 1

x – 2y – 3z = 1

Q#23 Use the matrix method to solve the following system of linear Equations: [2006]

2x – y + 2z = 4

x + 10y – 3z = 10

-x + y + z = -6

Q#24 Prove the following by using the properties of determinants: [2007]

| 1 1 1x y zyz zx xy|=( x− y ) ( y−z )(z−x )

Q#25 Find the inverse of a matrix A by the adjoint method if:[2007]

A=[0 1 11 3 02 1 −1]

Q#26 Prove[2008]

|a+x a aa a+x aa a a+x|=x3(3 a+x) by using properties of determinants.

Prepared by:

Prof S.M. Noaman


Q#27 Find the inverse of A=[ 2 2 31 −1 0

−1 2 1] by Adjoint method.

[2008]

Q#29 Solve the following system of equations by using the matrix method: [2009]

x + y = 5

y + z = 7

z + x = 6

Q#30 Using properties of determinants, show that:[2010]

|1 1 1a b ca2 b2 c2|=( a−b ) (b−c )(c−a)

Q#31 Use the Adjoint method to solve the given Equations:[2010]

2x – y + 2z = 4

x + 10y – 2z = 10

x – y – z = 6

Q32 Using the properties of determinant show that:[2011]

| 1 1 1α β γβγ γγ αβ|=(α−β ) (β−γ )(γ−α)

Q#33 Using Cramer’s rule, solve the following system of Equations: [2011]

x + y – z = 2

x + 2y + z = 7

3x – y + 2z = 12

Q#34 Find A-1 by Adjoint method if A=[1 3 52 4 79 8 6 ]

[2011]

Prepared by:

Prof S.M. Noaman


CHAPTER # 07 (GROUPS)

Q#01 Let A = {1, ω ,ω2}, where ω is a complex cube root of unity. Check whether the given set is closed, commutative and associative with respect to the operation of multiplication. [1991]

Q#02 Let S = {1ω,ω 2}, ω being a complex cube root of unity.[1992]

Construct a composition table with respect to multiplication in C and show that(S, .) is a Group.

Prepared by:

Prof S.M. Noaman


Q#03 Let S = {1, ω, ω2} ω being a complex cube root of unity. Construct a composite table with respect to multiplication on “C” and show that (S.,) is a group. [1994]

Q#04 State whether the following are True or False. Given reasons. [1995]

Is(z) a group where is defined by a * b = 0, for all a, b Є Z.

Q#05 i) If p and q are two integers p ≠ 0 , p = -q which of the following must be true:

* p < q * p < q[1996]

* p – q < 0 * pq < 0 * p + q < 0

ii) A groupoid (s.*) is a semi-group if_____?

iii) Is ordered pair (3,5) equal to the ordered pair (5,3)?

iv) The multiplicative identity in C is______.

Q#06 Let A = {0,1,2,4} Define a* b = a, ∀a ,b ϵ A . Construct the Composition table for * in A.

[1996]

Q#07 Define binary operation, a * b = 4a b ∀ a , b , ϵ Q. If ′.′ Represents the ordinary multiplication, then show that: [1997]

i) ¼ is the identity element w.r.t.*

ii)112

is the inverse of 34 under*.

Q#08 Show that the set of all the matrices of the form.

[cosθ −sinθsinθ cosθ ][1997]

Where θ is a real number, form an Abelian group w.r.t. multiplication matrices

Q#09 0 is a real number from an Abelian group w.r.t. multiplication. [1997]

Q#10 Let A = {1, ω, ω2}, where ω is a complex cube root of unity. Check whether the given set is closed, cumulative and associative, with respect to the

operation of multiplication. [1998]

Q#11 Show that (R,*) is a group if ‘*’ is defined in R by a * b = 7ab, ∀a, b ϵR.[1999]

Q#12 Let us define a binary operation * in the set of rational numbers Q by a * b = 5ab for a, b ϵ Q. Prove that * is associative and commutative.

Also, verify that 15is identity w.r.t. * element and

750

is the

inverse element of 27w.r.t.

[2000]

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Prof S.M. Noaman


Q#13 Let * be defined in Z by p * q = p + q + 4, thence show that: [2001]

i) * is associative ii) Identity with respect to * exists in Z.iii) Every element of Z has an inverse under * in Z;

where Z is the set of integers.

Q#14 Define a binary operation (*) in Q, by a * b = 4a.b, where*Represents ordinary multiplication. Also show that: [2002]

a) (*) is commutative.b) (*) is associative.c) ½ is the identity element under (*).

d)112

is the inverse element of ¾ under (*).

Q#15 Construct the multiplication table to show that multiplication is a binary operation on S = {1,-1,i,-i}, where i=√−1. [2003]

Q#16 Let S = {1, ω, ω2}, where ω is a complex cube root of unity. Check whether the given set ‘S’ is closed, commutative and associative with respect to the operation

of multiplication. [2004]

Q#17 Show that S = {1,-1, i , -i} forms a finite abelian group under usual multiplication of complex numbers.

[2005]

Q#18 Using the multiplication table, find whether ‘+’ is a binary operation on S = {1,ω,ω2}

[2006]

Q#19 Let S = {1 , -1 , i , -i }, construct the composition table w.r.t. to Multiplication and show that (.) is commutative and associative in S. [2007]

Q#20 T is the set of real number of the form a + b√2, where a , b ϵ Q and a , b are not simultaneously zero. Show that (T, ●) is a group where ‘●’is an ordinary multiplication.

[2008]

Q#21 Let * be defined in Z, the set of all integers, as a*b = a + b + 3. Show that: [2009]

a) *is commutative and associative.b) Identity w.r.t.* exists in Z.c) Every element of Z has an inverse w.r.t.*.

Prepared by:

Prof S.M. Noaman


Q#22 Using the multiplication table show that multiplication is a binary operation on S = {1,-1, i , -i }. Also show that (•) is commutative. [2010, 2011]

CHAPTER # 08 (SEQUENCES & SERIES)

Q#01 Find the sum to 20 terms of an A.P whose 4th term is 7 and the 7th term is 13. [1991]Q#02 If a rubber ball is dropped on the floor from a height of 27

meters, it always rebounds to two-third of the distance of the previous fall; find the distance it will have traveled before hitting the ground for the seventh time. [1991]

Q#03 If A, G, H be respectively the A.M., the G.M. and H.M. between any two numbers

and Ax = Gy = Hz, prove that x , y , z are in G.P. [1991]

Q#04 The sum of four numbers in A.P. is 20. The ratio of the product of the first and the

last numbers and the product of 2nd and 3rd numbers is 2:3; find the numbers.

[1992]

Q#05 The sum of infinite terms of a G.P. is 4 and the sum of their cubes is 192;

find the G.P. [1992]

Q#06 Find the three numbers in A.P. whose sum is 12 and whose product is 28. [1994]

Q#07 Sum the series 1+32+ 54+ 78+… to infinity.

[1994]

Q#08 If A, G , H are respectively the arithmetic, geometric and harmonic means

between any two numbers, prove that G2 = AH. [1994]

Prepared by:

Prof S.M. Noaman


Q#09 Sum the series: 0.6 + 0.66 + 0.666 + … to n terms. [1995]

Q#10 Prove that a , b , c are in either A. P. or G.P. or [1995]

H.P according as a(b−c )a−b

a or b or c.

Q#11 If the sum of first n natural numbers is 15 and the sum of the first n + 1 natural

numbers is 21, then ‘n’ is: [1996]

*3 *4 *5 *6 *10

Q#12 The sum of first 14 terms of 1,1.4,1.8….. is equal to the sum of first n terms of

5.6 , 5.8, 6.0, ……, find the value of n. [1996]

Q#13 The first three terms of a G.P. are x + 10 , x – 2, x – 0 respectively. Calculate the

value of x and hence find the sum to infinity of this series. [1996]

Q#14 Which term of the H.P. 6,2,6/5,……… is equal to 2/33? [1996]

Q#15 How many terms of the square -9,-6,-3….. must be taken for the sum of the

terms to be 66? [1997]

Q#16 The product of three number in G.P is 216 and the sum of their product in

pairs is 156, find the numbers. [1997]

Q#17 If x− yy−z

= xz

then prove that x,y,z are in H.P.

[1997]

Q#18 The sum of four terms in A.P is 4, the sum of the product of the first and the last

term and the two middle terms is – 38 ; find the numbers. [1998]

Q#19 If a rubber ball that is dropped on the floor from a height of 27 meters, always

rebounds one-third of the distance of the previous fall, find the distance it will have traveled before hitting the ground for the ninth time. [1998]

Prepared by:

Prof S.M. Noaman


Q#20 Find the sum of 15 terms of an A.P whose middle term is – 42. [1999]

Q#21 Find the sum of an infinite series in G.P if sum of the first and the fourth

terms is equal to 54 and the sum of the second and the third term is 36 find the series also.

[1999]

Q#22 Find the 12th term of an H.P. if given that its second term is 3 and 4th

terms is 2. [1999]

Q#23 The sum of the first Ŋ terms of two A.P′s are as 3n + 1:13 – 7n. Find ratio of

their second terms. [2000]

Q#24 Find the G.P whose second term is 2 and the sum to infinity is 8. [2000]

Q#25 Find the sum of all natural numbers between 250 and 1000 which are

exactly divisible by 7. [2001]

Q#26 Which term of the geometric sequence 27,18,12…..is 512729

?

[2001]

Q#27 The 12th term of an H.P. is 15 and the 19th term is

322

; find

the 10th term of the H.P.

[2001]

Q#28 Which term of H.P 12 ,413

,29,………is

142

?

[2002]Q#29 The base of a right –angled triangle is 10cm, and the sides of the triangle are in

A.P.; find the hypotenuse. [2002]

Q#30 The sum of infinite term of a G.P. is 3, and the sum of their cubes is 81;

find the G.P. [2002]

Q#31 Find the sum of 20 terms of an A.P. whose 4th terms is 7 and 7th term is 13. [2003]

Prepared by:

Prof S.M. Noaman


Q#32 Express the recurring decimal 0.237 as rational number. [2003]

Q#33 Simplify x ( y−z )( x− y )

when x, y, z are in

[2003]

i) A.P. ii) G.P and iii) H.P.

Q#34 Find the sum of an A.P. of 17 terms whose middle term is 5. [2004]

Q#35 Find the value of n such that an+1+bn+ 1

an+bn may become the

arithmetic mean between

a and b. [2004]

Q#36 Show that the product of n geometric means between a and b is (ab) n/2. [2004]

Q#37 Find the sum of all natural numbers between 1 and 100 which are not

exactly divisible by 2 or 3. [2005]

Q#38 Find three numbers in G.P whose sum is 19 and whose product is 216. [2005]

Q#39 The ratio of the sum of the first four terms of a G.P. to the sum of the first eight terms of the same G.P is 81:97; find the common ratio of the G.P. [2006]

Q#40 If the 3rd , 6th and the last terms of an H.P are respectively 1/3, 1/5 and 3/203, find the number of terms of the H.P.

[2006]

Q#41 The sum of four terms of an A.P. is 4 and the sum of the product of first and the last and the two middle terms is – 38, find the A.P? [2007]

Q#42 Find the G.P whose third term is 9/4 and whose sixth term

is24332

. [2007]

Q#43 The pth term of an H.P. is q and the qth term is p: find (p+q) th term and (p.q) th term.

[2007]

Q#44 The sum of p terms of an A.P is q the sum of q terms of the same. A.P. is p; find the sum of (p+q) terms.

[2008]

Prepared by:

Prof S.M. Noaman


Q#45 Find the 17th term of an H.P. whose first two terms are 6 and 8. [2008]

Q#46 Find the sum to n terms of the A.G. series: [2008]

1 + 3r + 5r + 7r + ……………….

Q#47 Express the value of the recurring decimal 0.423 as a common fraction. [2009]

Q#48 Show that the sum of the (p + q) term and (p – q)th term of an A.P is equal to twice the pth term.

[2009]

Q#49 If the pth term of an H.P. is q, the qth term is p; prove that

the (p+q)th term is pqp+q

.

[2009]

Q#50 If the sum of 8 terms of an A.P is 64 and the sum of its 19 terms is 361, find the sum of 31 terms of the A.P.

[2010]

Q#51 insert 4 harmonic means between 12 and 48/5. [2010]

Q#52 Find the sum of 20 terms of an A.P whose 4th term is 7 and 7th terms is 13. [2011]

Q#53 Which term of the sequence 27, 18 , 12 … is 512729

?

[2011]

Q#54 The 12th term of an H.P is 15and 19th term is

322

find the 4th term.

[2011]

CHAPTER # 09 (PERMUTATIONS & COMBINATIONS)

Q# 01 In how many ways can 4 gentlemen and 2 ladies be seated at a round-table so that the Ladies are not together? In how many of these ways will three particular.

gentlemen be next to each other? [1991]

Prepared by:

Prof S.M. Noaman


Q#02 If 6300=[ 104 , n] , then find n.

[1997]

Q#03 Find n and r if npr = 240 and nCr = 120. [2004]

Q#04 How many natural numbers may be formed by using 4 out of 1,2,3,4,5: [2005]

i) If the digits are not repeated?ii) If the digits may be repeated?iii) How many of them are even if the digits are not

repeated?Q#05 Find the total number of ways in which birds can perch on 4 trees when there is no restriction to the choice of a tree. In how many of these ways will one particular

bird be alone on a tree? [1992]

Q#06 In a class there are 8 boys and 5 girls. Two class representatives are to be chosen

from them. In how many ways can they be selected if: [2006]i) Both are chosen from all?ii) One is to be a boy and the other a girl?iii) The first is to be a boy and the second either a boy or

a girl?

Q#07 How many different digits may be formed using 1, 2, 3, 4, 5, 6, 7, if each number is

to be (i) odd(ii)even? [2008]

Q#08 In how many ways can 3 English,3 Sindhi and 2 Urdu books be arranged on a shelf

so as to have all the books in the same language together? [2009]

Q#09 In how many distinct ways can the letter of the word INTELLIGENCE be arranged?

[2010]

Q#10 In nP3 = 12 n/2P3, find n: [2011]

Q#11 A father has “8” children. He take them, there at a time to a zoo without taking the same 3 children more than once, how often will he go and how often does each child get the chance to go? [2011]

Q#12 The letters of the word MISSISSIPPI are scrambled and then arranged in some

Prepared by:

Prof S.M. Noaman


order. What is the probability that the four I’s are consecutive letters in the resulting arrangement?

[1994]

Q#13 In how many ways can a hockey eleven be chosen out of 15 players? [1994]

i) Include a particular player?ii) Exclude a particular player?

Q#14 How many different words can be formed from the letters of the word

INSTITUTION, all taken together? [1995]

Q#15 A party of 7 members is to be chosen from a group of 6 gents and 5 ladies. In how

many ways can the party be formed if it is to contain (i) exactly 4 ladies, (ii) at least 4 ladies, (iii) at the most 4 ladies?

[1995]

Q#16 The number of arrangements of the word SUCCESS are:[1996]*420 *5040 *120 *210

Q#17 A father has 8 children. He takes them, three at a time, to a zoo, as often as he can

without taking the same three children more than once. How often will he go, and how often will each child go?

[1997]

Q#18 A department in a college consists of 6 professors and 8 students. A study tour is to

be arranged. In how many ways can a party of seven members be chosen so as to include:

[1998]i) Exactly 4 professors? ii) At least 4

professors?

Q#19 The letters of the word MISSISSIPPI are scramble and then arranged in some

order. What is the probability that four “Is” are consecutive letters in the resulting arrangements?

[1998]

Q#20 A team of 14 players is to be formed out of 20 players. In how many ways can the

selection be made if a particular players is to be: [1999]

i) Included ii) Excluded?

Prepared by:

Prof S.M. Noaman


Q#21 A coin was tossed 3 times; find the probability of getting: [1999]

i) at least one head ii) no tailiii) exactly two tails iv) at the most two

heads

Q#22 In how many ways can 5 essential prayers be offered, each in a different mosque,

out of 7 mosques in the area? [2000]

Q#23 In how many ways can a troop of 5 soldiers be formed out of 7 soldiers? [2000]

Q#24 There are seven mosque in a locality. In how many ways can Aslam offer five

obligatory prayers in different mosques such that:[2001]i) Zohar prayer is offered in the mosque A? ii) Zohar prayer is not offered in the mosque A?

Q# 25 In how many ways can 3 books on mathematics, 2 books on Physics and 2 books on

the same subject always remain together? [2002]

Q#26 How many different natural numbers of 3 different digits each can be formed from

the digits 1,2,3,4,5,7,9 if each number is to be (i) odd , or (ii) even? [2003]

Q#27 In how many ways can 3 English, 2 Urdu and 2 Sindhi books be arranged on a shelf

so as to keep all the books in each language together?[2004]

Q#28 How many natural numbers of four digits can be formed with four digits 2,3,5,7

when no digits is being used more than once in each number? What will be their numbers if the given digits are 2,3,5,0?

[2007]

Q#29 In how many ways can three books on Chemistry, 2 books on Physics and 2 books

on Mathematics be placed on a shelf such that the books on the same subject always remain together?

[2007]

Prepared by:

Prof S.M. Noaman


CHAPTER # 10 (INTRODUCTION TO PROBABILITY) Q#01 The dice are rolled once. Find the probability of getting a multiple of 3 or a sum of 10.

[1991]

Q#02 A committee of 4 members is to be chosen by lot from a group of 7 men and [1992] women. Find the probability that the committee will consist of 2 men and 2 women.

Q#03 A committee of 4 members is to be chosen by lot from a group of 7 men and [1993]

women. Find the probability that the committee will consist of 2 men and 2 women.

Prepared by:

Prof S.M. Noaman


Q#04 Out of 12 eggs, 2 are bad. From these eggs 3 are chosen at random to make a cake.

What is the probability that the cake contains. [1995]

i) exactly one bad egg ii) at least one bad egg

Q#03 A prize of any 4 books is to be given out of 7 Arabic and 5 French books. Find the

probability that the prize will consist of 2 Arabic and 2 French books. [2000]

Q#07 From a group of 6 men and 5 women a committee of 4 members is to be formed.

What is the probability that the committee will consists of 2 men and 2 women?[2001]

Q#08 A room has 3 lamps, from a collection of 12 light bulbs of which 8 are no good, a

person selects 3 at random and puts then sockets. What is the probability that he will have light?

[2002]

Q#09 Nine tennis teams play friendly matches, each team has to play against each other.

How many matches have to be played in all?[1996]

Q#10 In a fruit basket there are one dozen organs and half a dozen apples. If a person has

to select a fruit at random what is the chance of its being an orange? [1996]

Q#11 A committee of 4 members is to be chosen by lot from a group of 7 teachers and 5

students. Find the probability that the committee will consist of 2 teachers and two students.

[1997]

Q#12 If a die is rolled twice, what is the probability that: [2003]i) The sum of the points on it is? ii) There is at least one 5?

Q#13 A coin is tossed twice. Find the probability of (i) at least one head, exactly one tail.

[2004]

Prepared by:

Prof S.M. Noaman


Q#14 The probability that the principal of a college has a television is 0.6, a refrigerator

0.2 and both television and refrigerator is 0.06; find the probability that the

principal has at least a television or a refrigerator. [2005]

Q#15 Of the students attending a lecture 50% could not see what was being written on the

blackboard, 40% of them could not hear what the lecturer was saying: a particular unfortunate 30% of them fell into both of these categories. What is the probability that a students picked at random was able to see and hear satisfactorily?[2006]

Q#16 A fair coin is tossed 3 times. Find the probability of getting (i) at least one head

and (ii) at most two heads.[2007]

Q#17 A drawer contains 50 bolts and 150 nuts. Half of the bolts and half of the nuts are

rusted. If one item is chosen, what is the probability that it is either rusted or is a bolt?

[2008]

Q#18 A fair coin is tossed 3 times, find he probability of getting:[2008]i) At least one head and ii) at most two heads

Q#19 The probability that a man has a T.V. is 0.6, a V.C.R. is 0.2, and a T.V. and V.C.R.

both is 0.04. What is the probability that the man has either a T.V. or

a V.C.R? [2009]

Q#20 A word consists of 5 consonants and 4 vowels. Three letters are chosen at random. What is the probability that more than one vowel will be selected? [2011]

Prepared by:

Prof S.M. Noaman


CHAPTER # 11 (MATHEMATICAL INDUCTION) Q#01 Prove that 2n > n2, for an integral values of n ≥ 5.

[1991]

Q#02 Find the sum of the following series: [1992]

162 + 172 + 182 . . . + 402.

Q#03 Prove that a + ar2 + ar3 + . . . + arn-1

[1994]

¿a(1−rn)1− y

, Where r # 1

Q#04 Prove that: 11.3

+ 43.5

+ 95.7

+. . .n2

(2n−1 )(2n+1)=

n(n+1)2(2n+1)

[1995]

Q#05 Prove the following preposition by mathematical induction: [1997]

2! > 2n, ∀n ϵN, n ≥ 4

Q#06 By the principle of mathematical induction show that (23n+2 – 28n – 4) is

divisible by 49, ∀n ϵ N. [1998]

Q#07 Use mathematical induction to prove that: [1999]

Prepared by:

Prof S.M. Noaman


22 + 42 + 62 + 82 + . . . + (2n)2 = 23 n(n + 1)(2n + 1)

Q#08 Prove the following proposition by using principle of Mathematical induction:

2 + 6 + 12 + . . . + n (n + 1) = 13 n(n +1)(n+2)

[2000]

Q#09 Prove by mathematical induction that: [2001]

2 + 5 + 8 + ……………. + (3n – 1) = n2 (3n + 1)

Q#10 Prove the following proposition by the method of Mathematical Induction: [2002]

(32n+2 – 28 – 4) is divisible by 49 ,∀ n ∈N.

Q#11 Prove by mathematical induction that 10n + 3.4n+2 + 5 is divisible

by 9, ∀n ϵN. [2003]


1.3 + 2.4 + 3.5 + . . . + n(n + 2) = 16 n (n +1)(2n+7)

Q#13 Without using calculator, find the sum: [2005]

112 + 122 + 132 + . . . + 302

Q#14 Prove by mathematical induction: [2006]

12 + 32 + 52 + . . . + (2n – 1)2 = 13 n(2n – 1) (2n + 1)

Q#15 Use mathematical induction to prove that: [2007]

12 + 22 + 32 + . . . + n2 + n (n+1 )(2n+1)

6

Q#16 Prove that 2 + 4 + 6 + . . . + 2n = n(n + 1), for all natural numbers “n”. [2008]

Prepared by:

Prof S.M. Noaman


Q#17 Prove the following proposition by the principle of Mathematical Induction: [2009]

2+6+12+ .. .+n (n+1 )=13n(n+1)(n+2)

Q#18 Prove by mathematical induction that 10n + 3.4n+2 + 5 is divisible by “9” for all n ∈N.

[2010]


2 + 6 + 12 + . . . + n (n + 1) = 13n(n+1)(n+2),∀ n ∈N.

CHAPTER # 12 (THE BINOMIAL THEOREM) Q#01 If y=

34–3.54.8

+ 3.5 .74.8.12

+❑ , . .. [1991]

Prove that y2 + 2y – 7 = 0

Prepared by:

Prof S.M. Noaman


Q#02 Find the middle term in the expansion of [ x−1x ]10

∗¿

[1994]

Q#03 If i be a quantity so small that i3 may be neglected in comparison to c3, [1994]

prove that: √ c1+c

+√ cc−1

=2+ 3 i2

4c2

Q#04 Sum the series: [1995]

1−32.12+ 2.53.6

.1

22−2.5 .83.6 .9

.1

23+ .. .

Q#05 Find the first five terms of the expansion of (3 – 2x)2. [1996]

Q#06 Write the formula for the nth term of sequence 12,9,6 . . . [1996]

Q#07 If |x|<1 , prove that (1+ x )12+(1−x )

23

(1+x )+(1−x )12

=1−5 x6

[1997]

Q#08 Identity the series: 1+23.12+ 2.53.6

.122

+ 2.5 .83.6 .9

.123

+. . .

[1997]

as the binomial expansion and find its sum.

Q#09 Using Binomial Theorem, write the term independent of x

in the expansion of ( 4 x23 − 32x )

9

and simplify it.

[1999]

Q#10 Busing using general term in the expansion of (a + b)n ;

find the term independent of x in [ x− 12 x ]

8

. Point out the

middle term also. [2000]

Q#11 When x is so small that its square and higher power may

be neglected, find the value of [1+ 23 x ]52+√4+2 x

√(4+x )3

[2002]

Q#12 Using the general term of Binomial Theorem, determine

the term independent of x in the expansion of [32 x2− 13 x ]

6

[2002]

Prepared by:

Prof S.M. Noaman


Q#13 Find the first negative term in the expansion of (1+2 x )72 ,

using the formula for general term. [2003]

Q#14 Identity the following series as binomial expansion and find the sum. [2003]

1+ 34+ 3.54.8

+ 3.5 .74.8 .12

+ .. .

Q#15 Find the term independent of x in the expansion of [ x−2x ]10

by using the general term formula. [2004]

Q#16 Evaluate to four decimal places the expansion of (1.03 )13

[2004]

Q#17 If x = 3y + 6y2 + 10y3 + . . . prove that: [2005]

y=13x− 1.4

32 .2!x2+ 1.4 .7

33.3 !x3+ .. .

Q#18 If x=13+ 1.33.6

+ 1.3 .53.6 .9

+ .. .prove that x2 + 2x – 2 = 0

[2006]

Q#19 If |x|<1 , prove √1+x+3√(1−x )2

(1−x )+√1+x=1−5

6x approximately.

[2007]

Q#20 Using the general term of the binomial theorem, determine the term independent of x in the expansion of (x – ½ x)10. [2007]

Q#21 Find the first negative in the expression of (1+2 x )52

[2008]

Q#22 If y = 2x + 3x2 + 4x3 + . . . Then show that: [2008]

x= y2− 1.3

222 !y2+ 1.3.5

23.3 !y3+1.3 .5 .7

24 .4 !y4+. . .

Q#23 If x be a quantity so small that x3 may be neglected in comparison with y3,

Prove that: √ yy+x

+√ yy−x

=2+ 3 x2

4 y2

[2009]

Q#24 Write in the simplified form the term independent of x in the expansion of

Prepared by:

Prof S.M. Noaman


(x− 1x2 )

15

[2009]

Q#25 If Y34+ 3.54.8

+ 3.5 .74.8 .12

+ .. . Prove that y2 + 2y – 7 = 0.

[2009]

Q#26 Find the 1st negative term in the expansion of [1+ 3 x2 ]92

[2010]

Q#27 Show that 3√4=1+ 14+ 1.34.6

+ 1.3 .54.6 .8

+. . .

[2010]

Q#28 Find the term independent of x in the finomial expansion

of (x− 1x2 )

15

[2011]

Q#29 Show that √31+ 13+ 1.3

32.2 !+1.3 .533.3 !

+. . .

[2011]

CHAPTER # 13 (FUNDAMENTALS OF TNGONOMETRY) Q#01 How far does boy on a bicycle travel in 10 revolutions if the diameters of the wheels of his bicycle are equal to 56 cm each? [1991]

Q#02 A belt 24.75 meters long passes around a 3.5cm diameter pulley. As the belt

makes 3 complete revolutions in a minute, how many radians does the wheel turn in one second?

[1992]

Q#03 If tanθ3

√5and ρ(θ) is in the third quadrant, draw unit circle

to show ρ (θ) and find theremaining trigonometric functions. [1994]

Prepared by:

Prof S.M. Noaman


Q#04 If a point on the rim of a 21cm diameter wheel travels 5040 cm in a minute, through

how many degrees does the wheel turn is a second? [1995]

Q#05 If cosθ=2

√5and sinθ < θ then find the value of sinθ.

[1997]

Q#06 A belt 24.75 meters long passes around a 2.5cm diameter pulley. As the belt makes 4

complete revolutions in a minute, find the number of radians through which the wheel turns is one second.

[1998]

Q#07 How does a boy travel on a bicycle in 15 revolutions if the diameter of each of the

wheels is 14cm? [1999]

Q#08 If sinθ=−√32

; ρ(θ) lies in the third quadrant, find all the

remaining Trigonometric functions. [1999]

Q#09 If sinθ=45 and ρ(θ) is not in the first quadrant, then find the

values of all the remaining trigonometric functions. [2000]

Q#10 If tanθ=34∧ρ (θ ) is in the 3rd quadrant, find the remaining

trigonometric functions, using the definition of radian function with x2 + y2 =

1. [2001],[2011]

Q#11 Find the arc length of a circle whose diameter is 28 cm and whose central angle is

of measure 41o. [2001]

Q#12 If tanθ=512

∧ρ(θ) is not in the third quadrant, find of

remaining [2002] trigonometric functions, using the definition of radian function

with x2 + y2 + 1.

Prepared by:

Prof S.M. Noaman


Q#13 A belt 24.75 meters long passage around a 3.5 cm diameter pulley. As the belt

makes 3 complete revolutions in a minute, how many radians does the wheel turn in one second?

[2002]

Q#14 A belt 23.85 meters long passes around 1.58 cm diameter pulley. If the belt makes

two complete revolutions in a minute how many radians does the wheel turn in one second?

[2003]

Q#15 If cotθ = - 2 and ρ(θ) is in the second quadrant, find the values of all the trigonometric functions by using the definition of radian function. [2003]

Q#16 How far does a boy on a bicycle travel in 12 revolutions if the diameter of the wheel of his bicycle is each equal to 56 cm? [2004]

Q#17 If tanθ=−13

∧sin θ<θ , find the values of all trigonometric

functions by using the definition of radian function. [2004]

Q#18 If tanθ=−43

∧ρ(θ) is not in the second quadrant, use radian

function to find the remaining trigonometric functions. [2005]

Q#19 Find the remaining trigonometric function if tan θ is

negative and sinθ=35 [2006]

Q#20 If tanθ=−13

∧ρ(θ) is not in the 2nd quadrant, find the

remaining trigonometric function using the definition of radian function x2 + y2 = 1. [2007]

Q#21 A belt 24.75 meters passes around a 3.5cm diameter pulley as the belt makes

3 complete revolution in a minute. How many radians does wheel turn in a

second? [2007]

Q#22 Find the remaining trigonometric functions if cotθ = 3 and sin θ is positive. [2008]

Q#23 If a point on the rim of a 21 cm diameter flywheel travels 5040 meters in a minute,

Prepared by:

Prof S.M. Noaman


through how many radians does the flywheel turn in one second? [2009],[2011]

Q#24 If tanθ = ½ , find the remaining trigonometric functions when “θ” lies in the 3rd quadrant.

[2010]

Q#25 A belt 24.75 meter long passes over a 1.5 cm diameter pulley. As the belt makes two complete revolutions in a minute, how many vadians does the wheel turn in one second? [2010]

CHAPTER # 14 (TRIGONOMETRIC IDENTITIES)

Q#01 Prove that tan (α−β )= tanα−tanβ1+tanα . tanβ

[1991]

Q#02cot θcosesθsin θ+ tanθ

=coses θ . cotθ [1991]

Q#03 cos4θ = 8 cos4 θ – 8. cos2 θ + 1 [1991] & [2011]

Q#04sin 7θ−sin5θcos7θ+cos 5θ

=tan θ [1991] &

[2011]

Q#05 Prove that sinu+sin v=2sin( u+v2 )cos( u−v

2 ) [1992]

Q#06 Sin6 θ + cos6 θ = 1 – 3sin2 θ cos2 θ[1992]

Q#07 tan (α+β )= tanα+ tan β7−tanα . tan β

[1992]

Prepared by:

Prof S.M. Noaman


Q#08 1+cos2θ= 2

1+ta n2θ[1992]

Q#09 sin 5θ – sin 3θ = 4 sin θ cos 3θ2cos

5θ2

[1994]

Q#10tan θ+sinθsin θ− tan θ

=sec θ1+cosθ1−secθ

(sec θ≠1) [1994]

Q#11 If tanα= √34−√3

∧tanβ= √34+√3

,Find tan(α+β) [1995]

Q#12 Prove that tan( π4 + θ2 )=√ 1+sinθ

1−sinθ=sec θ+tanθ [1995]

Q#13 sin 5θ−sin3θ+sin 2θ=4 sinθ cos 3θ2cos

5θ2

[1996]

Q#14tan θ+sinθsin θ−tan θ

=sec θ .( 1+cosθ1−sec θ );Sec θ≠1 [1996]

Q#15 Express cosθ in terms of cotθ.[1997]

Q#16 Prove that the points A(6 , 13), B(5,7) and C(4,1) are collinear. [1997]

Q#17cosec A

cosec A−1+ cosec Acosec A+1

=2 se c2 A

[1997]

Q#18 √ 1+cosθ1−cosθ

=cot θ2

[1997]

Q#19 cosα .cos (α – β )+sinα sin (α−β )=cosβ [1997]

Q#20cosθ−sinθcosθ+sinθ

= cotθ−1cot θ+1

[1998]

Q#21 tan (α+β )= tanα+ tan β1−tanα tan β

[1998]

Prepared by:

Prof S.M. Noaman


Q#22sin 2θsinθ

− cos 2θcosθ

=sec θ

[1998]

Q#23 sinu+sin v=sin( u+v2 )cos( u−v

2 ) [1998]

Q#24sinθ+sin 3θ+sin 5θcosθ+cos3θ+cos5θ

= tan 3θ [1998]

Q#25cot θ+cosec θsinθ+ tan θ

=cosecθ cot θ [1999] & [2011]

Q#26tan A+B /2tan A−B /2

= sin A+sinBsin A−sinB

[1999]

Q#27sin θ1−cot θ

+ cos1−tan θ

=cosθ+sin θ [1999]

Q#28 Show that (0,0) (a,0) , (b , c) and (b – a , c) taken in order form the vertices of a parallelogram.

[2000]

Q#29 cos 4θ = 8cos4 θ – 8 cos2 θ + 1[2000]

Q#30 ¿ [2000]

Q#31 cos α cos(α – β) + sin α sin (α – β) = cos β[2000]

Q#32cosθ1−tanθ

+ sin θ1−cot θ

=sinθ+cosθ [2001]

Q#33sinα−sin βsinα+sin β

=tan(α−β

2 )tan(α+ β

2 )[2001]

Q#34 sin (α + β). sin (α – β) = sin2α – sin2β[2001]

Q#35 sin 7θ−sin 5θ+sin 2θ=4 sinθ 7θ2cos

5θ2cos [2002]

Prepared by:

Prof S.M. Noaman


Q#36sin 3θsinθ

−cos 3θcosθ

=2 [2002] &

[2010]

Q#37 √ 1−cosθ1+cosθ=cosecθ−cot θ [2003]

Q#38 tanθ2= sin θ1+cosθ

=1−cosθsinθ

[2003]

Q#39 sin6θ – sin 4θ + sin 2θ = 4 sin θ cos 2θ cos 3 θ[2003]

Q#40 cos(α + β). Cos(α – β) = cos2 α – sin2 β[2004]

Q#41 cos 4θ = 8 cos4 θ – 8cos2 θ + 1[2004]

Q#42tanθ+sinθ

cosec θ+cot θ=tan θ .sin θ [2004]

Q#43 Cos6θ + Sin6θ = 1 – 3sin2θCos2θ[2005]

Q#44 tanθ2=1−cos θ

sin θ [2005] &

[2010]

Q#45sin 3θsinθ

−cos 3θcosθ

=2

[2005]

Q#46 Prove that Cos3θ = 4 cos3θ – 3cosθ[2010]

Q#47 Show that the points (2,1),(5,a) and (2,6) are the vertices of a right – angle triangle.

[2010]Q#48 Express all the trigonometric functions in terms of cosecθ.

[2006]

Prepared by:

Prof S.M. Noaman


Q#48 sin6 θ – cos6 θ = 1 – 3 sin2 θcos2 θ.[2006]

Q#49 cos(θ+ф) cos (θ – ф) = cos2 θ – sin2 ф[2006]

Q#50 cos4 θ = 8 cos4 θ – 8cos2 θ + 1.[2006]

Q#52 cos4 θ = 8cos4 θ – 8 cos2 θ + 1[2007]

Q#53sin 3θcosθ

+ cos3θsin θ

=2cosec θ

[2007]

Q#54sinθ1+cosθ

+1+cosθsin θ

=2cosec θ [2007]

Q#55 tan3θ=3 tan θ−ta n3θ1−3 ta n2θ

[2008]

Q#56sin 2θsinθ

− cos 2θcosθ

=sec θ

[2008]

Q#57sin 7θ−sin5θcos7θ+cos 5θ

=tan θ

[2008]

Q#58 If α=√32

∧β= 1√2

∧both ρ (α )∧ρ ( β ) are in the first quadrant find

the value of sin(α+β )[2008]

Q#591

1−sinθ+ 1sinθ

=2 se c2θ

[2009]

Q#60 cos(α + β) cos (α – β) = cos2 α – sin2 β.[2009]

Q#61 cos4 θ = 8cos4 θ – 8cos2 θ+1.[2009]

Prepared by:

Prof S.M. Noaman


CHAPTER # 15 (GRAPHS OF TNGONOMETRIC FUNCTIONS) Q#01 Draw the grapy of cos 2θ, where -π ≤ θ ≤ π.

[1994] Q#02 Draw the grapy of cos 2 θ, where π ≤ θ ≤ π.

[1996]

Q#03 Draw the graph of the function y = sin(-θ), -180o ≤ 0 ≤ 180o and find the value of sin (– 150o) from the graph.

[1997]

Q#04 Draw the graph of y = cos θ2 for – 2 π ≤ θ ≤ 2 π.

[1998]

Q#05 Draw the graph of cos θ when – π ≤ 0 ≤ π.[1999]

Q#06 Draw the graph of y = sin 2 θ , where π2≤θ≤

π2.

[2001]

Q#07 Draw the graph of y = tan x for – 90o < x < 90o. [2002] & [1992]

Q#08 Draw the graph of y = sin 2x , where – π ≤ x ≤ π. From the graph find the value of

sin 140o. [2003]

Prepared by:

Prof S.M. Noaman


Q#09 Draw the graph of y = cos2x, where π2≤×≤

π2

[2005]

Q#10 Draw the graph of ½ cos 2 θ if – π ≤ θ ≤ π.[2007]

Q#11 Sketch the graph of sin θ, where 0 ≤ θ ≤ 2π.[2008]

Q#12 Draw the graph of sin θ when -π ≤ θ ≤ π/2.[2009]

Q#13 Draw the graph of cos2x, where –π ≤ x ≤ π.

Q#14 Draw the graph of sin x, when -180o ≤ x ≤ 180o.

CHAPTER # 16 (SOLUTIONS OF TRIANGLES) Q#01 An aeroplane is flying at a height of 9000 meters. If the angle of depression to a field

marker is 23o, find the aerial distance. [1991]

Q#02 A plane is heading due east at 675 km/h. A wind is blowing in from the north – east

at 75 km/h; find the direction of the plane and it ground speed. [1991]

Q#03 If the measure of the two sides of a triangle are 4 and 5 units find the third side so that the area of the triangle is 6 Sqr. units. [1991]

Prepared by:

Prof S.M. Noaman


Q#04 A man observe that the angle of elevation of the top of a mountain is 50o from a

point on the ground. On walking 150 meters away from the point, the angle of elevation becomes 4.45o. Find the height of the mountain. [1992]

Q#05 The three sides of the triangular lot have lengths 9 , 12 and 16cm respectively. Find

the measure of its largest angle and the area of the lot. [1992]

Q#06 Show that in any triangle ABC [1994]

1r2

+ 1r12+1r22 +1r 32=

a2+b2+c2

∆

Q#07 A man standing on one side of a road observes that the measure of the angle

Subtended by the top of a five-storey building on the opposite side of the road is 35o.

When he crosses the road of width 35 meters to approach the building, he finds the

Measure of the angle to be 65o; find the height of the building and distance of the

initial position of the man from the building. [1994]

Q#08 Solve the triangle ABC in which ∟B = 60 o, ∟ A = 49o and C = 39 inches. [1994]

Q#09 Show that r1= (s−c ) cot β2=4 R sin

α2cos

β2cos

γ2

[1995]

Q#10 Show that in any triangle ABC, [1996]

1r2

+ 1r12+1r22 +1r 32=

a2+b2+c2

∆2

Q#11 A man standing on one side of a road observes that the measure of the angle

Subtended by the top of a five-storey building on the opposite side of the road is 35o.

When he crosses the road of width 35 meters to approach the building, he finds the

Measure of the angle to be 65o; find the height of the building and distance of the

initial position of the man from the building. [1996]

Prepared by:

Prof S.M. Noaman


Q#12 Solve the triangle ABC in which ∟B = 60 o, ∟ A = 49o and C = 39 inches. [1996]

Q#13 Find the length of the third side of a triangular building that faces 13.6 meters along

one street and 13.0 meters along another street. The angle of intersection of the two streets is 270o.

[1997]

Q#14 Prove that the area of a triangle. [1997]

∆=12

a2 sinα sin βSinα

, where ∆is the area of the given triangle

ABC.

Q#15 Prove that r1 , r2 , r3 = rs2, where r , r1 , r2 , r3 and s have usual meanings. [1997]

Q#16 Prove that in any equilateral triangle r: R : r1 = 1 : 2 : 3 , r. R and r1 have

usual meanings. [1998]

Q#17 Solve the triangle in which a = 15.2cm , b = 20.9 cm and c = 34.7cm and also find the

area of the triangle. [1998]

Q#18 A man is standing on the bank of a river. He observes that the measure of the angle

subtended by a tree on the opposite bank is 60o, when the retreats of 40 meters from the bank, he finds the measure of the angle to be 30o; find the height of the tree and the width of the river. [1998]

Q#19 Prove the Law of Cosine, cos α=b2+c2−a2

2bc(Show all the

working) [1999]

Q#20 A piece of a plastic strip 1 m is bent to form an isosceles triangle with 96o as its

largest angle; find the length of the sides. [1999]

Q#21 Prove that r1=4 R sinα2cos

β2cos

γ2

where R, r1 have their usual

meanings. [1999]

Prepared by:

Prof S.M. Noaman


Q#22 Solve the following triangle: [2000]

a = 54o 58 , b = 70 cms , c = 58 cms

Q#23 Prove that: rr1 r2 r3=∆2. State also the meaning of the notations. [2000]

Q#24 Deduce the complete formula: [2000]

sinα2=√ (s−b )(s−c)

bcWhere all the letters have their usual meanings

Q#25 A hiker walks due east at 4 km per hour and a second hiker starting from the same

point walks 55o north – east at the rate of 6km per hour. How far apart will they be after 2 Hours?

[2001]

Q#26 Show that in any triangle ABC. [2001]

1r2

+ 1r12+1r22 +1r 32=

a2+b2+c2

∆2

Q#27 Three sides of a triangular lot have lengths 14cm, 15 cm and 16cm ; find the

measure of the largest angle and the area of the lot. [2001]

Q#28 Derive the cosine law: a2 = b2 + c2 – 2bc cos α for a triangle ABC. [2002]

Q#29 Due to the wind blowing from 37o north – east, the pilot of an aeroplane whose

airspeed is 500km/hr has to head his plane at 325o in order to follow a courses of 310o; find the ground speed.

[2002]

Q#30 Prove: ∆=4 Rr cosα2cos

β2cos

γ2

where all the symbols have

their usual meanings. [2002] &

[2011]

Q#31 Prove that cos α2=√ S(S−α)

bc; where all symbols have their

usual meanings

Prepared by:

Prof S.M. Noaman


in ∆ ABC. [2003]

Q#32 Solve the triangle is α = 25o, a = 40cm, b = 20cm, [2003]

Q#33 Show that in any ∆ ABC ,1r2

+ 1r12+1r22+1r32=

a2+b2+c2

∆2

[2003]Where all the symbols have their usual meanings.

Q#34 An aeroplane is flying at a height of 9000 meters. If the angle of depression of a field market is 23o, find the aerial distance. [2004]

Q#35 Three points A,B,C form a triangle such that the ratio of the measures of their angles is 1:2:3. Find the ratio of the lengths of the sides. [2004]

Q#36 Find the area of a ∆ABC in which α = 30o , β = 63o and c = 7.3 cm. [2004]

Q#37 Prove that tanβ2=√ ( s−a )(s−c )

s (s−b) [2005]

Where s, a , b , c have their usual meanings.

Q#38 If three sides of a triangle are 11 cm, 13cm , 16cm, find the measure of the largest angle and the area of the triangle. [2005]

Q#39 If a = b = c , then prove that r : R : r1 = 1:2:3, where a , b , c , r , R1 , r1 have their usual meanings.

[2005]

Q#40 A plan is heading due east at 675km/h. The wind is blowing in from the north east at 75km/h, find the direction of the plane and its ground speed. [2006]

Q#41 Solve the triangle ABC in which a = b = c. [2006]

Q#42 Show that r1=4 r sinα2cos

β2cos

γ2

[2006]

Q#43 Prove that in any equilateral triangle r : R : r1 = 1 : 2 : 3, where r , R , r1 have their usual meanings.

[2007]

Prepared by:

Prof S.M. Noaman


Q#44 A piece of plastic strip 1 meter long is bent so form an isosceles triangle with 95o as the measure of its largest angle; find the lengths of the sides of the triangle. [2007]

Q#45 The measures of the two sides of a triangle are 4cm and 5cm, find the third side such that the area of the triangle is 6 square centimeters. [2007]

Q#46 A man is standing on the bank of a river. He observes that the measures of the angle

elevation subtended by the tree on the opposite bank is 65o, when the retreats of 35 meters from the bank, he finds the measure of the angle to be 35o; find the height of the tree and the width of the river. [2008]

Q#47 Find the area of the triangle ABC, if α = 30o β = 63o c = 7.3 cm. [2008]

Q#48 Prove that ∆=r2 cotα2cot

β2cot

γ2

where all notations have

their usual meanings. [2008]

Q#49 r1 r2 r3 = rs2, where the symbols have their usual meanings. [2008]

Q#50 Find the area of the triangle ABC when α = 51o, γ = 61o, b =√3cm [2009]

Q#51 Derive the Law of Cosine. [2009]

Q#52 Solve the triangle ABC when α = 49o , β = 60o , c = 39cm. [2009]

Q#53 If a = b = c, prove that r1: R: r = 3: 2: 1, where r1, R and r have their usual meanings.

[2009]

Q#54 Draw Law of Sines. [2011]

Q#55 Prove that r2=∆

s−b , where all the letters have their


Q#56 Show that in any ∆ ABC .1r1

+ 1r 2

+ 1r3

=1r , the letters have their


Prepared by:

Prof S.M. Noaman


Q#57 Solve the ΔABC in which a = 5cm , b = 10cm and c = 13cm [2010]

Q#58 Draw the Law of Cosine. [2010]

CHAPTER # 17(INVERSE TRIGONOMETRIC FUNCTIONS &

TRIGONOMETRIC EQUATIONS)

Q#01 Show that Tan-1 a + Cot-1 a = π /2; a ≥ 0 [1992]

Q#02 Find the value of: [1994]

sin(arc sin √32

+arc cos12 )+ tan(arc sin 12+arc tan−1(−1))

Q#03 Solve the equation tan2 x – 1 = 1cos x

.

[1994]

Q#04 Prove that tan−1113

+tan−1 17=ta n−1 2

9.

[1995]

Q#05 Draw the graph of y = arc cos x , where – 1 where -1 ≤ × ≤ 1. [1995]

Q#06 Solve √3cosθ+sin θ=2. [1995],[2007]

Q#07 Find the value of: [1996]

sin(arc sin √32

+arc cos12 )+ tan(arc sin 12+arc tan−1(−1))

Q#08 Solve the equation tan2 x – 1 = 1

cos× [1996]

Q#09 Solve 2sin2θ + 2√2 sinθ−3=0 [2006]

Prepared by:

Prof S.M. Noaman


Q#10 ta n−1( 13 )+ 12 tan−1( 17 )= π8 [2006],

[2007]

Q#11 Solve Cosθ + Cos2θ + 1 = 0 [2008]

Q#12 Prove that tan−1( 113 )+ta n1( 14 )=ta n−1( 13 )

[2008]

Q#13 Solve the trigonometric equation √1+cosθ−√1−sinθ=1 [2009]

Q#14 Solve 2sin2θ – 3sinθ – 2 = 0 [2010]

Q#15 Solve Cosθ + Cos2θ + 1 = 0 [2011]

Prepared by:

Prof S.M. Noaman

class xi math question 2011 the collegiate

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