BINOMIAL THEOREM

(Parabola, Ellipse & Hyperbola)

MATHEMATICS

Quest

Question bank on Compound angles, Trigonometric eqⁿ and ineqⁿ, Solutions of Triangle & Binomial There are 142 questions in this question bank.

Select the correct alternative : (Only one is correct) Q.1 If x + y = 3 – cos4θ and x – y = 4 sin2θ then

(A) x⁴ + y⁴ = 9 (B) x+ y=16

(C) x³ + y³ = 2(x² + y²) (D) x+ y=2 Q.2 If in a triangle ABC, b cos^2A

2 + a cos^{2 B} 2 = ³

2 c then a, b, c are :

(A) in A.P. (B) in G.P. (C) in H.P. (D) None

Q.3 If tanB =

A cos n 1

A cos A sin n

− 2 then tan(A + B) equals (A) (1 n)cosA

A sin

− ^(B) _sinA

A cos ) 1 n ( −

(C) (n 1)cosA A sin

− ^(D) (n 1)cosA A sin +

Q.4 Given a² + 2a + cosec² π

2 (a x+ )

F HG I

KJ

= 0 then, which of the following holds good?

(A) a = 1 ; x

2 ∈I (B) a = –1 ; x

2 ∈I

(C) a ∈ R ; x ∈φ (D) a , x are finite but not possible to find Q.5 If A is the area and 2s the sum of the 3 sides of a triangle, then :

(A) A ≤ ^s

2

3 3 (B) A = ^s

2

2 (C) A > ^s

2

3 (D) None

Q.6 The exact value of cos2 cos cos cos cos cos

28

3 28

6 28

9 28

18 28

27 28

π π π π π π

ec + ec + ec is equal to

(A) – 1/2 (B) 1/2 (C) 1 (D) 0

Q.7 In any triangle ABC, (a + b)² sin^2C

2 + (a − b)² cos^2C 2 =

(A) c (a + b) (B) b (c + a) (C) a (b + c) (D) c²

Q.8

( ) ( ) ( )

( ) ( )

tan . cos sin

cos . tan

x x x

x x

− + − −

− +

π π π

π π

2

3 2

7 2

2 3

2 3

when simplified reduces to :

(A) sinx cosx (B) −sin² x (C) −sinx cosx (D) sin²x Q.9 If in a ∆ ABC, sin³A + sin³B + sin³C = 3 sinA · sinB · sinC then

(A) ∆ ABC may be a scalene triangle (B) ∆ ABC is a right triangle (C) ∆ ABC is an obtuse angled triangle (D) ∆ ABC is an equilateral triangle

Q.10 In a triangle ABC, CH and CM are the lengths of the altitude and median to the base AB. If a = 10, b = 26, c = 32 then length (HM)

[3]

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Quest

θ for all permissible vlaues of θ

(A) is less than – 1 (B) is greater than 1

Q.15 If the median of a triangle ABC through A is perpendicular to AB then ^tan tan

Q.18 ^{sin cos}

sin cos

Q.20 In a triangle ABC, angle B < angle C and the values of B & C satisfy the equation 2 tanx - k (1 + tan² x) = 0 where (0 < k < 1) . Then the measure of angle A is :

(A) π/3 (B) 2π/3 (C) π/2 (D) 3π/4

Quest

Q.22 In a ∆ ABC, if the median, bisector and altitude drawn from the vertex A divide the angle at the vertex into four equal parts then the angles of the ∆ ABC are :

(A) ²

[5]

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Quest

Q.30 With usual notation in a ∆ ABC, if R = k

(

^{r r}

) (

^{r r}

) (

^{r r}

)

r r r r r r

1 2 2 3 3 1

1 2 2 3 3 1

+ + +

+ + where k has the value equal to:

(A) 1 (B) 2 (C) 1/4 (D) 4

Q.31 If a cos³ α + 3a cosα sin² α = m and a sin³ α + 3a cos² α sinα = n . Then (m + n)^2/3 + (m − n)^2/3 is equal to :

(A) 2a² (B) 2 a^1/3 (C) 2a^2/3 (D) 2a³

Q.32 In a triangle ABC , AD is the altitude from A . Given b > c , angle C = 23° & AD = ^{a b c}

b² − c²

then angle B = [JEE ’94, 2]

(A) 157° (B) 113° (C) 147° (D) none

Q.33 The value of cotx + cot(60º + x) + cot(120º + x) is equal to :

(A) cot3x (B) tan3x (C) 3 tan3x (D) ^{3 9}

3

2 3

−

− tan tan tan

x

x x

Q.34 In a ∆ ABC, cos 3A + cos 3B + cos 3C = 1 then : (A) ∆ ABC is right angled

(B) ∆ ABC is acute angled (C) ∆ ABC is obtuse angled

(D) nothing definite can be said about the nature of the ∆.

Q.35 The value of ^{3 76 16}

76 16

+ ° °

° + °

cot cot

cot cot is :

(A) cot44º (B) tan44º (C) tan2º (D) cot46º

Q.36 If the incircle of the ∆ ABC touches its sides respectively at L, M and N and if x, y, z be the circumradii of the triangles MIN, NIL and LIM where I is the incentre then the product xyz is equal to :

(A) Rr² (B) rR² (C) ¹

2 Rr² (D) ¹

2 rR² Q.37 The number of solutions of tan (5π cosθ) = cot (5π sinθ) for θ in (0, 2π) is :

(A) 28 (B) 14 (C) 4 (D) 2

Q.38 If A = 340⁰ then 2 sinA2

is identical to

(A) 1+sinA + 1−sinA (B) − 1+sinA − 1−sinA (C) 1+sinA − 1−sinA (D) − 1+sinA + 1−sinA

Q.39 AD, BE and CF are the perpendiculars from the angular points of a ∆ ABC upon the opposite sides.

The perimeters of the ∆ DEF and ∆ ABC are in the ratio : (A) ^{2 r}

R (B) ^r

R

2 (C) ^r

R (D) ^r

R 3 where r is the in radius and R is the circum radius of the ∆ ABC

Quest

Q.40 The value of cosec π

18 – 3 ^sec π 18 is a

(A) surd (B) rational which is not integral

(C) negative natural number (D) natural number Q.41 In a ∆ ABC if b + c = 3a then cot^B

2 · cot^C

2 has the value equal to :

(A) 4 (B) 3 (C) 2 (D) 1

Q.42 The set of values of ‘a’ for which the equation, cos 2x + a sin x = 2a − 7 possess a solution is :

(A) (− ∞, 2) (B) [2, 6] (C) (6, ∞) (D) (− ∞, ∞)

Q.43 In a right angled triangle the hypotenuse is 2 2 times the perpendicular drawn from the opposite vertex.

Then the other acute angles of the triangle are (A) π

3 ^&

π

6 ^(B)

π 8 ^&

3 8

π (C) π

4 ^&

π

4 ^(D)

π 5 ^&

3 10

π

Q.44 Let f, g, h be the lengths of the perpendiculars from the circumcentre of the ∆ ABC on the sides a, b and c respectively . If ^a

f b g

c

+ + h = λ ^{a b c}

f g h then the value of λ is :

(A) 1/4 (B) 1/2 (C) 1 (D) 2

Q.45 In ∆ ABC, the minimum value of

∏

∑

2 cot A

2 cot B 2. cot A

2 2 2

is

(A) 1 (B) 2 (C) 3 (D) non existent

Q.46 If the orthocentre and circumcentre of a triangle ABC be at equal distances from the side BC and lie on the same side of BC then tanB tanC has the value equal to :

(A) 3 (B)

3

1 (C) – 3 (D) –

3 1

Q.47 The general solution of sin x + sin 5x = sin 2x + sin 4x is :

(A) 2nπ (B) nπ (C) nπ/3 (D) 2 nπ/3

where n ∈ I

Q.48 The product of the distances of the incentre from the angular points of a ∆ ABC is : (A) 4 R² r (B) 4 Rr² (C)

(

^{a b c R}

)

s (D)

( )

R s c b a

Q.49 Number of roots of the equation cos² 3 1sin 2

3

4 1 0

x+ + x

− − = which lie in the interval [−π, π] is

(A) 2 (B) 4 (C) 6 (D) 8

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Quest

Q.50 sec sec

8 1

4 1

θ θ

−

− is equal to

(A) tan 2θ cot 8θ (B) tan 8θ tan 2θ (C) cot 8θ cot 2θ (D) tan 8θ cot 2θ Q.51 In a ∆ABC if b = a

(

³− and ∠C = 30¹

)

⁰ then the measure of the angle A is

(A) 15⁰ (B) 45⁰ (C) 75⁰ (D) 105⁰

Q.52 Number of values of θ ∈[ ,0 2π] satisfying the equation cotx – cosx = 1 – cotx. cosx

(A) 1 (B) 2 (C) 3 (D) 4

Q.53 The exact value of cos²73º + cos²47º + (cos73º . cos47º)is

(A) 1/4 (B) 1/2 (C)3/4 (D) 1

Q.54 ^{In a} ∆ABC, a = a₁ = 2 , b = a₂ , c = a₃ such that a_p+1 = 



 



 −

− −

− p p

p 2 p p

2 p

5 a 2 p 2 4

3 a 5 where p = 1,2 then

(A) r₁ = r₂ (B) r₃ = 2r₁ (C) r₂ = 2r₁ (D) r₂ = 3r₁

Q.55 The expression, ^tan

( ) (

^cos

)

cos( )

3 2

3 2

2

π π

α α

π α

− −

− + cos^_^{α− π}_^

2 sin(π − α) + cos(π + α) sin^_^{α − π}_^ 2 when simplified reduces to :

(A) zero (B) 1 (C) −1 (D) none

Q.56 The expression [1 − sin(3π − α) + cos(3π + α)] 1 3 2

5

−  − 2

 

 +  −

 





 



sin π cos

α π

α when simplified reduces to :

(A) sin 2α (B) − sin 2α (C) 1 − sin 2α (D) 1 + sin 2α

Q.57 If ‘O’ is the circumcentre of the ∆ ABC and R₁, R₂ and R₃ are the radii of the circumcircles of triangles OBC, OCA and OAB respectively then ^a

R b R

c

1 2 R3

+ + has the value equal to:

(A) ^{a b c} R

2 ³ (B) ^R

a b c

3

(C) ^4∆₂

R (D) ₂

R 4

∆

Q.58 The maximum value of ( 7 cosθ + 24 sinθ ) × ( 7 sinθ – 24 cosθ ) for every θ∈R.

(A) 25 (B) 625 (C)

2

625 (D)

4 625

Q.59 4 sin5⁰ sin55⁰ sin65⁰ has the values equal to (A) 3 1

2 2

+ (B) 3 1

2 2

− (C) 3 1

2

− (D) 3 3 1

2 2

d

−

i

Quest

Q.60 If x, y and z are the distances of incentre from the vertices of the triangle ABC respectively then z

y x

c b a

is equal to

(A)

∏

^tanA₂ ^(B)

∑

2

cotA (C)

∑

2

tanA (D)

∑

2 sinA

Q.61 The medians of a ∆ ABC are 9 cm, 12 cm and 15 cm respectively . Then the area of the triangle is

(A) 96 sq cm (B) 84 sq cm (C) 72 sq cm (D) 60 sq cm

Q.62 If x = ^nπ

2 , satisfies the equation sin^x

2 − cos^x

2 = 1 − sinx & the inequality ^x

2 2

3

− π ≤ 4π , then:

(A) n = −1, 0, 3, 5 (B) n = 1, 2, 4, 5

(C) n = 0, 2, 4 (D) n = −1, 1, 3, 5

Q.63 The value of 1

9 1 3

9 1 5

9 1 7

+ 9

F HG I

KJ F

⁺

HG I

KJ F

⁺

HG I

KJ F

⁺

HG I

KJ

cosπ cos π cos π cos π

is

(A) 9

16 (B) 10

16 (C) 12

16 (D) 5

16

Q.64 The number of all possible triplets (a₁ , a₂ , a₃) such that a₁+ a₂ cos2x + a₃ sin² x = 0 for all x is

(A) 0 (B) 1 (C) 3 (D) infinite

Q.65 In a ∆ABC, a semicircle is inscribed, whose diameter lies on the side c. Then the radius of the semicircle is

(A) a b 2

+

∆ (B)

c b a

2

− +

∆ (C)

s

2∆ (D)

2 c Where ∆ is the area of the triangle ABC.

Q.66 For each natural number k , let C_k denotes the circle with radius k centimeters and centre at the origin.

On the circle C_k , a particle moves k centimeters in the counter- clockwise direction. After completing its motion on C_k , the particle moves to C_k+1 in the radial direction. The motion of the particle continues in this manner .The particle starts at (1, 0).If the particle crosses the positive direction of the x- axis for the first time on the circle C_n then n equal to

(A) 6 (B) 7 (C) 8 (D) 9

Q.67 If in a ∆ ABC, ^{cosA cos cos} a

B b

C

= = c then the triangle is

(A) right angled (B) isosceles (C) equilateral (D) obtuse Q.68 If cos A + cosB + 2cosC = 2 then the sides of the ∆ ABC are in

(A) A.P. (B) G.P (C) H.P. (D) none

Q.69 If A and B are complimentary angles, then : (A) ¹

2 1

+ 2



 

  +

 

 tanA tanB

= 2 (B) 1

2 1

+ 2



 

  +

 

 cotA cotB

= 2

[9]

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Quest

(C) 1

2 1

+ 2



 

  +

 

 secA cos

ecB

= 2 (D) 1

2 1

− 2



 

  −

 

 tanA tanB

= 2

Q.70 The value of , 3 cosec20° − sec20° is :

(A) 2 (B) ^{2 20}

40 sin sin

°

° (C) 4 (D) ^{4 20}

40 sin sin

°

°

Q.71 If in a ∆ ABC, cosA·cosB + sinA sinB sin2C = 1 then, the statement which is incorrect, is (A) ∆ ABC is isosceles but not right angled (B) ∆ ABC is acute angled

(C) ∆ ABC is right angled (D) least angle of the triangle is ^π 4

Q.72 The set of values of x satisfying the equation, ^tan

( )

^x ₄

2

−π

− 2

( ) ( )

x 2 cos

4 sin2x

25 . 0

−π

+ 1 = 0, is :

(A) an empty set (B) a singleton

(C) a set containing two values (D) an infinite set

Q.73 The product of the arithmetic mean of the lengths of the sides of a triangle and harmonic mean of the lengths of the altitudes of the triangle is equal to :

(A) ∆ (B) 2 ∆ (C) 3 ∆ (D) 4 ∆

[ where ∆ is the area of the triangle ABC ]

Q.74 If in a triangle sin A : sin C = sin (A − B) : sin (B − C) then a² : b² : c²

(A) are in A.P. (B) are in G.P.

(C) are in H.P. (D) none of these

[ Y G ‘99 Tier - I ] Q.75 The number of solution of the equation,

∑

= 5

1 r

) x r

cos( = 0 lying in (0, p) is :

(A) 2 (B) 3 (C) 5 (D) more than 5

Q.76 If θ = 3 α and sin θ = ^a a² +b²

. The value of the expression, a cosec α − b sec α is

(A) ¹

2 2

a +b (B) 2 a² +b² (C) a + b (D) none

Q.78 The value of cot 71 2

0

+ tan 671 2

0

– cot 671 2

0

– tan71 2

0

is :

(A) a rational number (B) irrational number (C) 2(3 + 2 3) (D) 2 (3 – 3⁾

Q.79 If in a triangle ABC ^{2cosA cos 2cos} a

B b

C c

a b c

b

+ + = + ca then the value of the angle A is : (A) 8

π (B)

4

π (C)

3

π (D)

2 π

Quest

where ∆ is the area of the triangle

Q.87 If θ is eliminated from the equations x = a cos(θ – α) and y = b cos (θ – β) then Q.88 The general solution of the trigonometric equation

tan x + tan 2x + tan 3x = tan x · tan 2x · tan 3x is

[11]

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Quest

Q.89 If log_ab + log_bc + log_ca vanishes where a, b and c are positive reals different than unity then the value of (log_ab)³ + (log_bc)³ + (log_ca)³ is

(A) an odd prime (B) an even prime

(C) an odd composite (D) an irrational number

Q.90 If the arcs of the same length in two circles S₁ and S₂ subtend angles 75° and 120° respectively at the centre. The ratio

2 1

S

S is equal to

(A) 5

1 (B)

16

81 (C)

25

64 (D)

64 25

Q.91 Number of principal solution of the equation tan 3x – tan 2x – tan x = 0, is

(A) 3 (B) 5 (C) 7 (D) more than 7

Q.92 The expression

°

°

°

−

°

20 sin

· 20 tan

20 sin 20 tan

2 2

2 2

simplifies to

(A) a rational which is not integral (B) a surd

(C) a natural which is prime (D) a natural which is not composite Q.93 The value of x that satisfies the relation

x = 1 – x + x² – x³ + x⁴ – x⁵ + ... ∞

(A) 2 cos36° (B) 2 cos144° (C) 2 sin18° (D) none

Select the correct alternatives : (More than one are correct) Q.94 If sinθ = sinα then sin ^θ

3 = (A) sin α

3 ^{(B) sin}

π α 3− 3



 

 (C) sin ^{π α} 3+ 3



 

 (D) −sin ^{π α} 3+ 3



 



Q.95 Choose the INCORRECT statement(s).

(A sin 82¹ 2

°

. cos 37¹ 2

°

and sin 127¹ 2

°

. sin 97¹ 2

°

have the same value.

(B) If tanA = ³

4− 3 & tanB = ³

4+ 3 then tan(A − B) must be irrational.

(C) The sign of the product sin2 . sin3 . sin5 is positive.

(D) There exists a value of θ between 0 & 2π which satisfies the equation ; sin⁴ θ – sin² θ – 1 = 0.

Q.96 Which of the following functions have the maximum value unity ?

(A) sin² x − cos² x (B) ^{sin2 cos2}

2

x− x

(C) − ^{sin2 cos2} 2

x− x

(D) ⁶ 5

1 2

1 3 sinx+ cosx



 



Quest

Q.97 If the sides of a right angled triangle are {cos2α + cos2β + 2cos(α + β)} and {sin2α + sin2β + 2sin(α + β)}, then the length of the hypotenuse is :

(A) 2[1+cos(α − β)] (B) 2[1 − cos(α + β)]

(C) 4 cos^2α−β

2 (D) 4sin^2α+β

2 Q.98 An extreme value of 1 + 4 sinθ + 3 cosθ is :

(A) −3 (B) −4 (C) 5 (D) 6

Q.99 The sines of two angles of a triangle are equal to ⁵ 13 & ⁹⁹

101 . The cosine of the third angle is :

(A) 245/1313 (B) 255/1313 (C) 735/1313 (D) 765/1313

Q.100 It is known that sinβ = ⁴

5 & 0 < β < π then the value of

3 ²

6

sin ( ) cos ( )

sin

α β cos α β

α

+ − π +

is:

(A) independent of α for all β in (0, π/2) (B) ⁵

3 for tan β > 0 (C) ^{3 7 24}

15 ( + cot )α

for tanβ < 0 (D) none

Q.101 If x = secφ − tanφ & y = cosecφ + cotφ then : (A) x = ^y

y +

− 1

1 (B) y = ¹

1 +

− x

x (C) x = ^y

y

− + 1

1 (D) xy + x − y + 1 = 0 Q.102 If 2 cosθ + sinθ = 1, then the value of 4 cosθ + 3sinθ is equal to

(A) 3 (B) –5 (C) 7

5 ^{(D) –4}

Q.103 If sint + cost = ¹

5 then tan ^t

2 is equal to :

(A) −1 (*B) –¹

3 (C) 2 (D) − ¹

6 BINOMIAL

There are 39 questions in this question bank.

Q.104 Given that the term of the expansion (x^1/3 − x^−1/2)¹⁵ which does not contain x is 5m where m ∈ N, then m =

(A) 1100 (B) 1010 (C) 1001 (D) none

Q.105 In the binomial (2^1/3 +3^−1/3)ⁿ, if the ratio of the seventh term from the beginning of the expansion to the seventh term from its end is 1/6, then n =

(A) 6 (B) 9 (C) 12 (D) 15

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Quest

Q.106 If the coefficients of x⁷ & x⁸ in the expansion of 2 + 3



 

 x ⁿ

are equal, then the value of n is :

(A) 15 (B) 45 (C) 55 (D) 56

Q.107 The coefficient of x⁴⁹ in the expansion of (x – 1) 



 



 − 2

x 1 



 



 − ₂ 2

x 1 ... 



 



 − ₄₉ 2

x 1 is equal to

(A) – 2 



 



 − ₅₀ 2

1 1 (B) + ve coefficient of x

(C) – ve coefficient of x (D) – 2 



 



 − ₄₉ 2 1 1

Q.108 The last digit of (3^P +2) is :

(A) 1 (B) 2 (C) 4 (D) 5

where P = 3⁴ⁿ and n ∈ N

Q.109 The sum of the binomial coefficients of 2 1 x x

n

 +

 

 is equal to 256 . The constant term in the expansion is

(A) 1120 (B) 2110 (C) 1210 (D) none

Q.110 The coefficient of x⁴ in ^x 2 x

3

2 10

 −

 

 is : (A) ⁴⁰⁵

256 (B) ⁵⁰⁴

259 (C) ⁴⁵⁰

263 (D) ⁴⁰⁵

512 Q.111 The remainder, when (15²³ + 23²³) is divided by 19, is

(A) 4 (B) 15 (C) 0 (D) 18

Q.112 Let (7+4 3)ⁿ = p + β when n and p are positive integers and β ∈ (0, 1) then (1 – β) (p + β) is (A) rational which is not an integer (B) a prime

(C) a composite (D) none of these

Q.113 If (11)²⁷ + (21)²⁷ when divided by 16 leaves the remainder

(A) 0 (B) 1 (C) 2 (D) 14

Q.114 Last three digits of the number N = 7¹⁰⁰ – 3¹⁰⁰ are

(A) 100 (B) 300 (C) 500 (D) 000

Q.115 The last two digits of the number 3⁴⁰⁰ are :

(A) 81 (B) 43 (C) 29 (D) 01

Q.116 If (1+x+x²)²⁵ = a₀+ a₁x + a₂x² +...+ a₅₀ .x⁵⁰ then a₀ + a₂ + a₄ + ... + a₅₀ is :

(A) even (B) odd & of the form 3n

(C) odd & of the form (3n−1) (D) odd & of the form (3n+1)

Quest

Q.117 The sum of the series (1²+1).1! + (2²+1).2! + (3²+1). 3! + ... + (n²+1). n! is : (A) (n+1). (n+2)! (B) n.(n+1)! (C) (n+1). (n+1)! (D) none of these Q.118 Let P_m stand for ⁿP_m . Then the expression 1.P₁+ 2.P₂ + 3.P₃ + ... + n.P_n =

(A) (n+1)! −1 (B) (n+1)! + 1 (C) (n+1)! (D) none of these Q.119 The expression ¹

4 1

1 4 1

2

1 4 1

2

7 7

x

x x

+

+ +











 −  − +





















 is a polynomial in x of degree

(A) 7 (B) 5 (C) 4 (D) 3

Q.120 If the second term of the expansion a a a

n 1 13

1

/ +













− is 14a^5/2 then the value of

n n

C C

3 2

is :

(A) 4 (B) 3 (C) 12 (D) 6

Q.121 If (1 + x) (1 + x + x²) (1 + x + x² + x³) ... (1 + x + x² + x³ + ... + xⁿ) ≡ a₀ + a₁x + a₂x² + a₃x³ + ... + a_mx^m then ^ar

r m

=

∑

0

has the value equal to

(A) n! (B) (n + 1)! (C) (n – 1)! (D) none

Q.122 The value of 4 {ⁿC₁ + 4 . ⁿC₂ + 4² . ⁿC₃ + ... + 4^{n − 1}} is :

(A) 0 (B) 5ⁿ + 1 (C) 5ⁿ (D) 5ⁿ − 1

Q.123 If n be a positive integer such that n ≥ 3, then the value of the sum to n terms of the series 1 . n −

(

ⁿ⁻¹

)

1! (n − 1) +

(

ⁿ⁻¹

) (

ⁿ⁻²

)

2 ! (n − 2) –

(

ⁿ⁻¹

) (

ⁿ⁻²

) (

ⁿ⁻³

)

3 ! (n − 3) + ... is :

(A) 0 (B) 1 (C) – 1 (D) none of these

Q.124 In the expansion of (1 + x)⁴³ if the co−efficients of the (2r + 1)^th and the (r + 2)^th terms are equal, the value of r is :

(A) 12 (B) 13 (C) 14 (D) 15

Q.125 The positive value of a so that the co−efficient of x⁵ is equal to that of x¹⁵ in the expansion of x a x

2 3

10

 +

 

 is (A) ¹

2 3 (B) ¹

3 (C) 1 (D) 2 ₃

Q.126 In the expansion of ^x

x x

x x x +

− + − −

−



 

 1

1

1

2 3 1 3 1 2

10

/ / / , the term which does not contain x is :

(A) ¹⁰C₀ (B) ¹⁰C₇ (C) ¹⁰C₄ (D) none

Q.127 If the 6^th term in the expansion of the binomial ¹_{8 3} ² ₁₀

8

x x x

/ + log



 

 is 5600, then x equals to

(A) 5 (B) 8 (C) 10 (D) 100

[15]

Quest Tutorials

North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

Quest

Q.128 Co-efficient of α^t in the expansion of,

(α + p)^{m − 1} + (α + p)^{m − 2} (α + q) + (α + p)^{m − 3} (α + q)² + ... (α + q)^{m − 1} where α ≠ − q and p ≠ q is :

(A) ^mCt

(

^{pt qt}

)

p q

−

− (B) ^mCt

(

^{pm t qm t}

)

p q

− −

−

−

(C) ^mCt

(

^{pt qt}

)

p q +

− (D) ^mCt

(

^{pm t qm t}

)

p q

− + −

−

Q.129 (1 + x) (1 + x + x²) (1 + x + x² + x³) ... (1 + x + x² + ... + x¹⁰⁰) when written in the ascending power of x then the highest exponent of x is ______ .

(A) 4950 (B) 5050 (C) 5150 (D) none

Q.130 Let

(

^{5+2 6}

)

ⁿ = p + f where n ∈ N and p ∈ N and 0 < f < 1 then the value of, f² − f + pf − p is

(A) a natural number (B) a negative integer

(C) a prime number (D) are irrational number

Q.131 Number of rational terms in the expansion of

(

^{2 + 43}

)

^{100 is :}

(A) 25 (B) 26 (C) 27 (D) 28

Q.132 The greatest value of the term independent of x in the expansion of

10

x sin cos

x 



 



 θ

+

θ is

(A) ¹⁰C₅ (B) 2⁵ (C) 2⁵ · ¹⁰C₅ (D) ₅⁵

10

2 C

Q.133 If (1 + x – 3x²)²¹⁴⁵ = a₀ + a₁x + a₂x² + ... then a₀ – a₁ + a₂ – a₃ + ... ends with

(A) 1 (B) 3 (C) 7 (D) 9

Q.134 Coefficient of x⁶ in the binomial expansion

2 9

x 2

3 3 x 4











 − is

(A) 2438 (B) 2688 (C) 2868 (D) none

Q.135 The term independent of 'x' in the expansion of 9 1 3

18

x− x



 

 , x > 0 , is α times the corresponding binomial co-efficient . Then 'α' is :

(A) 3 (B) ¹

3 (C) − ¹

3 (D) 1

Q.136 The expression [x + (x³−1)^1/2]⁵ + [x − (x³−1)^1/2]⁵ is a polynomial of degree :

(A) 5 (B) 6 (C) 7 (D) 8 [JEE’92, 6 + 2]

Quest

Q.137 Given (1 – 2x + 5x² – 10x³) (1 + x)ⁿ = 1 + a₁x + a₂x² + .... and that a₁² = 2a₂ then the value of n is

(A) 6 (B) 2 (C) 5 (D) 3

Q.138 The sum of the series aC₀ + (a + b)C₁ + (a + 2b)C₂ + ... + (a + nb)C_n is where Cr's denotes combinatorial coefficient in the expansion of (1 + x)ⁿ, n ∈ N

(A) (a + 2nb)2ⁿ (B) (2a + nb)2ⁿ (C) (a +nb)2^{n – 1} (D) (2a + nb)2^{n – 1}

Q.139 The coefficient of the middle term in the binomial expansion in powers of x of (1 + αx)⁴ and of (1 – αx)⁶ is the same if α equals

(A) – 3

5 (B)

3

10 (C) –

10

3 (D)

5 3

Q.140 (2n+1) (2n+3) (2n+5) ... (4n−1) is equal to : (A) ^{( ) !}

. ( ) ! ( ) ! 4

2 2 2

n

n n

n (B) ^{( ) ! !}

. ( ) ! ( ) ! 4

2 2 2

n n

n n

n (C) ^{( ) ! !}

( ) ! ( ) ! 4

2 2

n n

n n (D) ^{( ) ! !}

! ( ) ! 4

2 2

n n

n n

Q.141 If S_n =

∑

= n

0

r r

nC 1

and T_n =

∑

= n

0

r r

nC r

then

n n

S T

is equal to

(A) 2

n (B) 1

2

n− (C) n – 1 (D)

2 1 n 2 −

Q.142 The coefficient of x^r (0 ≤ r ≤ n−1) in the expression :

(x+2)ⁿ⁻¹ + (x+2)ⁿ⁻². (x+1) + (x+2)ⁿ⁻³ . (x+1)² + ... + (x+1)ⁿ⁻¹ is :

(A) ⁿC_r (2^r −1) (B) ⁿC_r (2^n−r −1) (C) ⁿC_r (2^r +1) (D) ⁿC_r (2^n−r +1)

[17]

Quest Tutorials

North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

Quest

Answers

Select the correct alternative : (Only one is correct)

Q.1 D Q.2 D Q.3 A Q.4 B Q.5 A Q.6 D Q.7 D

Q.8 D Q.9 D Q.10 C Q.11 D Q.12 D Q.13 A Q.14 B

Q.15 C Q.16 A Q.17 A Q.18 B Q.19 D Q.20 C Q.21 D

Q.22 C Q.23 A Q.24 D Q.25 C Q.26 D Q.27 B Q.28 A

Q.29 D Q.30 C Q.31 C Q.32 B Q.33 D Q.34 C Q.35 A

Q.36 C Q.37 A Q.38 D Q.39 C Q.40 D Q.41 C Q.42 B

Q.43 B Q.44 A Q.45 A Q.46 A Q.47 C Q.48 B Q.49 B

Q.50 D Q.51 D Q.52 B Q.53 C Q.54 D Q.55 A Q.56 B

Q.57 C Q.58 C Q.59 B Q.60 B Q.61 C Q.62 B Q.63 A

Q.64 D Q.65 A Q.66 B Q.67 C Q.68 A Q.69 A Q.70 C

Q.71 C Q.72 A Q.73 B Q.74 A Q.75 C Q.76 B Q.78 B

Q.79 D Q.80 B Q.81 C Q.82 C Q.83 A Q.84 C Q.85 C

Q.86 B Q.87 B Q.88 D Q.89 A Q.90 C Q.91 C Q.92 D

Q.93 C

Select the correct alternatives : (More than one are correct)

Q.94 ABD Q.95 BCD Q.96 ABCD Q.97 AC Q.98 BD Q.99 BC

Q.100 ABC Q.101 BCD Q.102 AC Q.103 BC

BINOMIAL Select the correct alternative : (Only one is correct)

Q.104 C Q.105 B Q.106 C Q.107 A Q.108 D Q.109 A Q.110 A

Q.111 C Q.112 D Q.113 A Q.114 D Q.115 D Q.116 A Q.117 B

Q.118 A Q.119 D Q.120 A Q.121 B Q.122 D Q.123 A Q.124 C

Q.125 A Q.126 C Q.127 C Q.128 B Q.129 B Q.130 B Q.131 B

Q.132 D Q.133 B Q.134 B Q.135 D Q.136 C Q.137 A Q.138 D

Q.139 C Q.140 B Q.141 A Q.142 B

In document Maths+quest (Page 151-168)