VBQs Binomial Theorem Class 11 Mathematics with solutions has been provided below for standard students. We have provided chapter wise VBQ for Class 11 Mathematics with solutions. The following Binomial Theorem Class 11 Mathematics value based questions with answers will come in your exams. Students should understand the concepts and learn the solved cased based VBQs provided below. This will help you to get better marks in class 11 examinations.

**Binomial Theorem VBQs Class 11 Mathematics**

**Question. Let (x + 10) ^{50} + (x – 10)^{50} = a_{0} + a_{1}x + a_{2}x^{2} + …. + a_{50}x^{50}, for all x ∈ R; then a_{2}/a_{0} is equal to :**

(a) 12.50

(b) 12.00

(c) 12.25

(d) 12.75

## Answer

C

**Question. If the third term in the binomial expansion of (1 + x ^{log2x})^{5 }equals 2560, then a possible value of x is:**

(a) 1/4

(b) 4√2

(c) 1/8

(d) 2√2

## Answer

A

**Question. The number of terms in the expansion of (1 + x) ^{101} (1 + x^{2} – x)100 in powers of x is:**

(a) 302

(b) 301

(c) 202

(d) 101

## Answer

C

**Question. If for positive integers r > 1, n > 2, the coefficients of the (3r) ^{th} and (r + 2)^{th} powers of x in the expansion of (1 + x)^{2n} are equal, then n is equal to : **

(a) 2r + 1

(b) 2r –1

(c) 3r

(d) r + 1

## Answer

A

**Question. If the fourth term in the Binomial expansion of (2/x + x ^{log8x})^{6} (x > 0) is 20 × 8^{7}, then a value of x is:**

(a) 8

^{3}

(b) 8

^{2}

(c) 8

(d) 8

^{–2}

## Answer

B

**Question. If the fourth term in the binomial expansion of **

** is equal to 200, and x > 1, then the value of x is:**

(a) 100

(b) 10

(c) 10^{3}

(d) 10^{4}

## Answer

B

**Question. The positive value of l for which the co-efficient of x2 in the expression x ^{2}(√x + λ/x^{2})^{10} is 720, is:**

(a) 4

(b) 2√2

(c) √5

(d) 3

## Answer

A

**Question. If the fractional part of the number 2 ^{403}/15 is k/15 , then k is equal to: **

(a) 6

(b) 8

(c) 4

(d) 14

## Answer

B

**Question. The natural number m, for which the coefficient of x in the binomial expansion of (x ^{m} + 1/x^{2})^{22} is 1540, is ______.**

## Answer

13

**Question. If n is the degree of the polynomial, **

** and m is the coefficient of xn in it, then the ordered pair (n, m) is equal to **

(a) (12 , (20)^{4})

(b) (8, 5 (10)^{4})

(c) (24 , (10)^{8})

(d) (12, 8 (10)^{4})

## Answer

D

**Question. If α and β be the coefficients of x ^{4} and x^{2} respectively in the expansion of (x + √(x^{2} – 1))^{6} + (x – √(x^{2} – 1))^{6} , then:**

(a) α + β = 60

(b) α + β = –30

(c) α – β= 60

(d) α – β = –132

## Answer

D

**Question. If {4n – 3n – 1 : n ∈ N } and Y = {9(n -1) : nε N}, where N is the set of natural numbers, then X ∪ Y is equal to:**

(a) X

(b) Y

(c) N

(d) Y – X

## Answer

B

**Question. The coefficient of x ^{2} in the expansion of the product (2 – x^{2}). ((1 + 2x + 3x^{2})6 + (1 – 4x^{2})6) is**

(a) 106

(b) 107

(c) 155

(d) 108

## Answer

A

**Question. If (27) ^{999} is divided by 7, then the remainder is :**

(a) 1

(b) 2

(c) 3

(d) 6

## Answer

D

**Question. Statement – 1 :** For each natural number n, (n + 1)^{7}–1 is divisible by 7.**Statement – 2 :** For each natural number n, n^{7} – n is divisible by 7.

(a) Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.

(b) Statement-1 is true, Statement-2 is true; Statement-2 is NOT a correct explanation for Statement-1

(c) Statement-1 is true, Statement-2 is false

(d) Statement-1 is false, Statement-2 is true

## Answer

A

**Question. In the expansion of (x/cosθ + 1/sinθ) ^{16}, if l_{1} is the least value of the term independent of x when π/8 ≤ 0 ≤ π/4 and l_{2} is the least value of the term independent of x when π/16 ≤ 0 ≤ π/8 , then the ratio l_{2} : l_{1} is equal to :**

(a) 1 : 8

(b) 16 : 1

(c) 8 : 1

(d) 1 : 16

## Answer

B

**Question. The number of integral terms in the expansion of (√3 + ^{8}√5)^{256} is**

(a) 35

(b) 32

(c) 33

(d) 34

## Answer

C

**Question. If the constant term in the binomial expansion of **

** is 405, then |k| equals:**

(a) 9

(b) 1

(c) 3

(d) 2

## Answer

C

**Question. If for some positive integer n, the coefficients of three consecutive terms in the binomial expansion of (1 + x) ^{n+5} are in the ratio 5 : 10 : 14, then the largest coefficient in this expansion is :**

(a) 462

(b) 330

(c) 792

(d) 252

## Answer

A

**Question. In the binomial expansion of (a – b) ^{n}, n ≥ 5, the sum of 5th and 6th terms is ero, then a/b equals**

(a) (n – 5) / 6

(b) (n – 4) / 5

(c) 5/ (n – 4)

(d) 6/ (n – 5)

## Answer

B

**Question. For natural numbers m, n if (1 – y) ^{m} (1 + y)^{n }= 1+ a_{1}y + a_{2}y^{2} + ……. and a_{1} = a_{2} = 10, then (m,n) is**

(a) (20, 45)

(b) (35, 20)

(c) (45, 35)

(d) (35, 45)

## Answer

D

**Question. If the number of integral terms in the expansion of (3 ^{1/2} + 5^{1/8})^{n} is exactly 33, then the least value of n is :**

(a) 264

(b) 128

(c) 256

(d) 248

## Answer

C

**Question. The total number is irrational terms in the binomial expansion of (7 ^{1/5} – 3^{1/10})^{60} is :**

(a) 55

(b) 49

(c) 48

(d) 54

## Answer

D

**Question. The coefficient of x ^{18} in the product (1+x)(1–x)^{10} (1+x+x^{2})^{9} is :**

(a) 84

(b) –126

(c) –84

(d) 126

## Answer

A

**Question. The ratio of the coefficient of x ^{15} to the term independent of x in the expansion of (x^{2} + 2/x)^{15} is :**

(a) 7 : 16

(b) 7 : 64

(c) 1 : 4

(d) 1 : 32

## Answer

D

**Question. The middle term in the expansion of (1 – 1/x)n (1 – x)n in power of x is**

(a) ^{–2n}C_{n–1}

(b) ^{– 2}nCn

(c) ^{2n}C_{n–1}

(d) 2nC_{n}

## Answer

D

**Question. If x is so small that x ^{3} and higher powers of x may be neglected, then **

** may be approximated as**

(a) 1 – 3/8 x^{2}

(b) 3x + 3/8 x^{2}

(c) – 3/8 x^{2}

(d) x/2 – 3/8 x^{2}

## Answer

C

**Question. If x is positive, the first negative term in the expansion of (1+ x) ^{27/5} is**

(a) 6th term

(b) 7th term

(c) 5th term

(d) 8th term

## Answer

D

**Question. 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) 10/3

(c) -3/10

(d) -5/3

## Answer

C

**Question. A ratio of the 5th term from the begining to the 5th term from the end in the binomial expansion of (2 ^{1/3} + 1/2(3)^{1/3})^{10} is:**

(a) 1 : 2(6)

^{1/3}

(b) 1/1 : 4(16)

^{1/3}

(c) 4(36)

^{1/3}: 1

(d) 2(36)

^{1/3}: 1

## Answer

C

**Question. If ^{20}C_{1} + (2^{2}) ^{20}C_{2} +(3^{2}) ^{20}C_{3}+ ………. + (20^{2}) ^{20}C_{20} = A(2^{β}), then the ordered pair (A, β) is equal to :**

(a) (420, 19)

(b) (420, 18)

(c) (380, 18)

(d) (380, 19)

## Answer

B

**Question. **

**Statement – 1 :** S_{3} = 55 × 2^{9}.**Statement – 2 :** S_{1} = 90 × 2^{8} and S_{2} = 10 × 2^{8}.

(a) Statement -1 is true, Statement -2 is true ; Statement – 2 is not a correct explanation for Statement -1.

(b) Statement -1 is true, Statement -2 is false.

(c) Statement -1 is false, Statement -2 is true .

(d) Statement – 1 is true, Statement 2 is true ; Statement -2 is a correct explanation for Statement -1.

## Answer

B

**Question. In a shop there are five types of ice-creams available. A child buys six ice-creams.****Statement – 1 :** The number of different ways the child can buy the six ice-creams is ^{10}C_{5}.**Statement – 2 :** The number of different ways the child can buy the six ice-creams is equal to the number of different ways of arranging 6 A’s and 4 B’s in a row.

(a) Statement -1 is false, Statement-2 is true

(b) Statement -1 is true, Statement-2 is true; Statement -2 is a correct explanation for Statement-1

(c) Statement -1 is true, Statement-2 is true; Statement -2 is not a correct explanation for Statement-1

(d) Statement -1 is true, Statement-2 is false

## Answer

A

**Question. The value of **

** is equal to :**

(a) ^{51}C_{7} – ^{30}C_{7}

(b) ^{50}C_{7} – ^{30}C_{7}

(c) ^{50}C_{6} – ^{30}C_{6}

(d) ^{51}C_{7} + ^{30}C_{7}

## Answer

A

**Question. The term independent of x in the expansion of (1/60 – x ^{8}/81) · (2x^{2} – 3/x^{2})^{6} is equal to :**

(a) –72

(b) 36

(c) –36

(d) –108

## Answer

D

**Question. The coefficient of x7 in the expansion of (1– x – x ^{2} + x^{3} )^{6} is**

(a) –132

(b) –144

(c) 132

(d) 144

## Answer

B

**Question. If the coefficients of x ^{–2} and x^{–4} in the expansion of **

** (x > 0), are m and n respectively, then m/n is equal to : **

(a) 27

(b) 182

(c) 5/4

(d) 4/5

## Answer

B

**Question. The remainder left out when 8 ^{2n} – (62)^{2n+1} is divided by 9 is:**

(a) 2

(b) 7

(c) 8

(d) 0

## Answer

A

**Question. The coefficient of x10 in the expansion of (1 + x) ^{2} (1 + x2)^{3} (1 + x3)4 is equal to**

(a) 52

(b) 44

(c) 50

(d) 56

## Answer

A

**Question. If {p} denotes the fractional part of the number p, then {3 ^{200}/8}, is equal to :**

(a) 5/8

(b) 7/8

(c) 3/8

(d) 1/8

## Answer

D

**Question. **

(a) Statement -1 is false, Statement-2 is true

(b) Statement -1 is true, Statement-2 is true; Statement -2 is a correct explanation for Statement-1

(c) Statement -1 is true, Statement-2 is true; Statement -2 is not a correct explanation for Statement-1

(d) Statement -1 is true, Statement-2 is false

## Answer

B

**Question. If the coefficients of the three successive terms in the binomial expansion of (1 + x)n are in the ratio 1 : 7 : 42, then the first of these terms in the expansion is:**

(a) 8th

(b) 6th

(c) 7th

(d) 9th

## Answer

C

**Question. If some three consecutive coefficients in the binomial expansion of (x + 1) ^{n} in powers of x are in the ratio 2:15:70, then the average of these three coefficients is:**

(a) 964

(b) 232

(c) 227

(d) 625

## Answer

B

**Question. The sum of the co-efficients of all even degree terms in x in the expansion of (x + √(x ^{3}-1))^{6} + (x – √(x^{3}-1))^{6} , (x > 1) is equal to :**

(a) 29

(b) 32

(c) 26

(d) 24

## Answer

D

**Question. If the coefficents of x3 and x4 in the expansion of (1+ ax + bx ^{2}) (1- 2x)^{18} in powers of x are both ero, then**

(a, b) is equal to:

(a) (14 , 272/3)

(b) (16 , 272/3)

(c) (16 , 251/3)

(d) (14 , 251/3)

## Answer

B

**Question. Let **

** Then a _{7} / a_{13} is equal to ___________.**

## Answer

800

**Question. **

(a) – 4

(b) 6

(c) – 8

(d) 10

## Answer

A

**Question. If the 7th term in the binomial expansion of **

** is equal to 729, then x can be :**

(a) e^{2}

(b) e

(c) e/2

(d) 2e

## Answer

B

**Question. If n is a positive integer, then (√3+1) ^{2n} – (√3 –1)^{2n} is :**

(a) an irrational number

(b) an odd positive integer

(c) an even positive integer

(d) a rational number other than positive integers

## Answer

A

**Question. If (2+x/3) ^{55} is expanded in the ascending powers of x and the coefficients of powers of x in two consecutive terms of the expansion are equal, then these terms are:**

(a) 7th and 8th

(b) 8th and 9th

(c) 28th and 29th

(d) 27th and 28th

## Answer

A

**Question. The sum of the rational terms in the binomial expansion of (2 ^{1/2} + 3^{1/5})^{10} is :**

(a) 25

(b) 32

(c) 9

(d) 41

## Answer

D

**Question. The coefficient of x ^{4} in the expansion of (1 + x + x^{2} + x^{3})^{6} in powers of x, is ____________.**

## Answer

120

**Question. The smallest natural number n, such that the coefficient of x in the expansion of (x ^{2} + 1/x^{3})^{n} is ^{n}C_{23}, is :**

(a) 38

(b) 58

(c) 23

(d) 35

## Answer

A

**Question. The number of terms in the expansion of (y ^{1/5} + x^{1/10})^{55} , in which powers of x and y are free from radical signs are**

(a) six

(b) twelve

(c) seven

(d) five

## Answer

A

**Question. If ƒ(y) = 1 – (y – 1) + (y – 1) ^{2} – (y – 1)^{3} + … – (y – 1)^{17}, then the coefficient of y^{2} in it is**

(a) 17C

_{2}

(b) 17C

_{3}

(c) 18C

_{2}

(d) 18C

_{3}

## Answer

D

**Question. The sum of the co-efficients of all odd degree terms in the expansion of (x+ √x ^{3}–1) + (x– √x^{3} -1) ,(x >1) is :**

(a) 0

(b) 1

(c) 2

(d) – 1

## Answer

C

**Question. The coefficient of x ^{–5} in the binomial expansion of **

(a) 1

(b) 4

(c) – 4

(d) – 1

## Answer

A

**Question. If the coefficients of x ^{2} and x^{3} are both ero, in the expansion of the expression (1 + ax + bx^{2}) (1–3x)^{15} in powers of x, then the ordered pair (a, b) is equal to:**

(a) (28, 861)

(b) (–54, 315)

(c) (28, 315)

(d) (–21, 714)

## Answer

C

**Question. If the coefficient of x ^{7} in [ ax^{2} + (1/bx) ]^{11} equals the coefficient of x^{7} in [ ax – (1/bx) ]^{11} , then a and b satisfy the relation**

(a) a – b = 1

(b) a + b = 1

(c) a/b = 1

(d) ab = 1

## Answer

D

**Question. For a positive integer n, (1 + 1/x) ^{n} is expanded in increasing powers of x. If three consecutive coefficients in this expansion are in the ratio, 2 : 5 : 12, then n is equal to __________.**

## Answer

118

**Question. The coefficient of x ^{n} in expansion of (1 + x)(1 – x)^{n} is**

(a) ( –1)

^{n–1}n

(b) ( – 1)

^{n}(1 – n)

(c) ( – 1)

^{n–1}(n – 1)

^{2}

(d) (n – 1)

## Answer

B

**Question. The sum of the series 2 · ^{20}C_{0} + 5 · ^{20}C_{1} + 8 · ^{20}C_{2} + 11 · ^{20}C_{3} + … + 62 · ^{20}C20 is equal to :**

(a) 2

^{26}

(b) 2

^{25}

(c) 2

^{23}

(d) 2

^{24}

## Answer

B

**Question. If the sum of the coefficients in the expansion of (a + b) ^{n} is 4096, then the greatest coefficient in the expansion is**

(a) 1594

(b) 792

(c) 924

(d) 2924

## Answer

C

**Question. The coefficient of t ^{4} in the expansion of ( 1 – t^{6} / 1 – t )^{3}**

(a) 14

(b) 15

(c) 10

(d) 12

## Answer

B

**Question. The sum of the real values of x for which the middle term in the binomial expansion of (x ^{3}/3 + 3/x)^{8} equals 5670 is :**

(a) 0

(b) 6

(c) 4

(d) 8

## Answer

A

**Question. If the term independent of x in the expansion of **

** is k, then 18k is equal to :**

(a) 5

(b) 9

(c) 7

(d) 11

## Answer

C

**Question. If the number of terms in the expansion of ( 1 – 2/x + 4/x ^{2} )^{n} , x ≠ 0, is 28, then the sum of the coefficients of all the terms in this expansion, is :**

(a) 243

(b) 729

(c) 64

(d) 2187

## Answer

B

**Question. The sum of coefficients of integral power of x in the binomial expansion (1–2√x) ^{50} is :**

(a) 1/2(3

^{50}– 1)

(b) 1/2(2

^{50}– 1)

(c) 1/2(3

^{50}+ 1)

(d) 1/2(3

^{50})

## Answer

C

**Question. Let α > 0, β > 0 be such that α ^{3} +β^{2} = 4. If the maximum value of the term independent of x in the binomial expansion of (αx^{1/9} + βx^{1/6})^{10} is 10k, then k is equal to :**

(a) 336

(b) 352

(c) 84

(d) 176

## Answer

A

**Question. The value of r for which ^{20}C_{r}^{20}C_{0} + ^{20}C_{r-1}^{20}C_{1} + ^{20}C_{r-2}^{20}C_{2} + … +^{20}C_{0}^{20}C_{r} is maximum, is :**

(a) 15

(b) 20

(c) 11

(d) 10

## Answer

B

**Question. If **

** then K is equal to:**

(a) (25)^{2}

(b) 2^{25} – 1

(c) 2^{24}

(d) 2^{25}

## Answer

D

**Question. The positive integer ust greater than (1 + 0.0001) ^{10000} is**

(a) 4

(b) 5

(c) 2

(d) 3

## Answer

D

**Question. The value of ( ^{21}C_{1} – ^{10}C_{1}) + (^{21}C_{2} – ^{10}C_{2}) + (^{21}C_{3} – ^{10}C_{3}) + (^{21}C_{4} – ^{10}C_{4}) + …. + (^{21}C_{10} – ^{10}C_{10}) is :**

(a) 2

^{20}– 2

^{10}

(b) 2

^{21}– 2

^{11}

(c) 2

^{21}– 2

^{10}

(d) 2

^{20}– 2

^{9}

## Answer

A

**Question. The coefficient of x ^{1012} in the expansion of (1 + x^{n} + x^{253})^{10}, (where n ≤ 22 is any positive integer), is**

(a) 1

(b)

^{10}C

_{4}

(c) 4n

(d)

^{253}C

_{4}

## Answer

B

**Question. The sum of the series ^{20}C_{0} – ^{20}C_{1} + ^{20}C_{2} – ^{20}C_{3} + …. – …. + ^{20}c_{10} is**

(a) 0

(b)

^{20}C

_{10}

(c) –

^{20}C

_{10}

(d) 1/2

^{20}C

_{10}

## Answer

D

**Question. r and n are positive integers r > 1, n > 2 and coefficient of (r+2) ^{th} term and 3r^{th} term in the expansion of (1 + x)^{2n} are equal, then n equals**

(a) 3r

(b) 3r + 1

(c) 2r

(d) 2r + 1

## Answer

C

**Question. The coefficients of x ^{p} and x^{q} in the expansion of (1+ x )^{p+q} are**

(a) equal

(b) equal with opposite signs

(c) reciprocals of each other

(d) none of these

## Answer

A

**Question. The coefficient of x4 in the expansion of (1 + x + x ^{2})^{10} is _______.**

## Answer

615

**Question. If the sum of the coefficients of all even powers of x in the product (1 + x + x ^{2} + … + x^{2n}) (1 – x + x^{2} – x^{3} + … + x^{2n}) is 61, then n is equal to_______.**

## Answer

30

**Question. The term independent of x in the binomial expansion of (1 – 1/x + 3x ^{5})(2x^{2} – 1/X)^{8} is :**

(a) 496

(b) –496

(c) 400

(d) –400

## Answer

C

**Question. The term independent of x in expansion of **

(a) 4

(b) 120

(c) 210

(d) 310

## Answer

C