JEE Mains 2019

A committee of 11 members is to be formed from 8 males and 5 females. If

m

is the number of ways the committee is formed with at least 6 males and

n

is the number of ways the committee is formed with at least 3 females, then (a)

m = n = 68

m + n = 68

m = n = 78

n = m - 8

Topic:

Permutations and Combinations

Exam:

JEE Mains 2019

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Question 2

The number of 6 digits numbers that can be formed using the digits

0, 1, 2, 5, 7

and 9 which are divisible by 11 and no digit is repeated, is (a) 60 (b) 72 (c) 48 (d) 36

Topic:

Permutations and Combinations

Exam:

JEE Mains 2019

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Question 3

Suppose that 20 pillars of the same height have been erected along the boundary of a circular stadium. If the top of each pillar has been connected by beams with the top of all its non-adjacent pillars, then the total number of beams is (a) 180 (b) 210 (c) 170 (d) 190

Topic:

Permutations and Combinations

Exam:

JEE Mains 2019

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Question 4

The number of ways of choosing 10 objects out of 31 objects of which 10 are identical and the remaining 21 are distinct, is (a)

2^{20} - 1

2^{21}

2^{20}

2^{20} + 1

Topic:

Permutations and Combinations

Exam:

JEE Mains 2019

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Question 5

A group of students comprises of 5 boys and

n

girls. If the number of ways, in which a team of 3 students can randomly be selected from this group such that there is at least one boy and at least one girl in each team, is 1750 , then

n

is equal to (a) 28 (b) 27 (c) 25 (d) 24

Topic:

Permutations and Combinations

Exam:

JEE Mains 2019

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Question 6

Consider a class of 5 girls and 7 boys. The number of different teams consisting of 2 girls and 3 boys that can be formed from this class, if there are two specific boys

A

, who refuse to be the members of the same team, is (a) 350 (b) 500 (c) 200 (d) 300

B

Topic:

Permutations and Combinations

Exam:

JEE Mains 2019

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Question 7

Consider three boxes, each containing 10 balls labelled

1, 2, \dots, 10

. Suppose one ball is randomly drawn from each of the boxes. Denote by

n_{i}

, the label of the ball drawn from the

i

th box,

(i = 1, 2, 3)

. Then, the number of ways in which the balls can be chosen such that

n_{1} < n_{2} < n_{3}

is (a) 82 (b) 120 (c) 240 (d) 164

Topic:

Permutations and Combinations

Exam:

JEE Mains 2019

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Question 8

men and two women participating in a chess tournament. Each participant plays two games with every other participant. If the number of games played by the men between themselves exceeds the number of games played between the men and the women by 84, then the value of

There are

m

m

is (a) 12 (b) 11 (c) 9 (d) 7

Topic:

Permutations and Combinations

Exam:

JEE Mains 2019

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Question 9

Medium

Consider the statement: "

P (n) : n^{2} - n + 41

is prime."' Then, which one of the following is true? (a) Both

P (3)

P (5)

are true. (b)

P (3)

is false but

P (5)

is true. (c) Both

P (3)

P (5)

are false. (d)

P (5)

is false but

P (3)

is true.

Topic:

Principle of Mathematical Induction

Exam:

JEE Mains 2019

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Question 10

If the fourth term in the binomial expansion of

(x^{(\frac{1}{1 + l o g _{10} x})} + x^{\frac{1}{12}})^{6}

is equal to 200 , and

x > 1

, then the value of

x

is (a) 100 (b)

1 0^{4}

1 0^{3}

Topic:

Binomial Theorem

Exam:

JEE Mains 2019

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Question 11

If the fourth term in the binomial expansion of

(\frac{2}{x} + x^{l o g_{8} x})^{6}, (x > 0)

20 \times 8^{7}

, then the value of

x

is (a)

8^{- 2}

8^{3}

8^{2}

Topic:

Binomial Theorem

Exam:

JEE Mains 2019

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Question 12

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) 227 (c) 232 (d) 625

Topic:

Binomial Theorem

Exam:

JEE Mains 2019

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Question 13

If the coefficients of

x^{2}

are both zero, in the expansion of the expression

x^{3}

(1 + a x + b x^{2}) (1 - 3 x)^{15}

in powers of

x

, then the ordered pair

(a, b)

is equal to (a)

(28, 315)

(- 21, 714)

(28, 861)

(- 54, 315)

Topic:

Binomial Theorem

Exam:

JEE Mains 2019

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Question 14

The smallest natural number

n

, such that the coefficient of

x

, is (a) 35 (b) 23 (c) 58 (d) 38

(x^{2} + \frac{1}{x ^{3}})^{n}

^{n} C_{23}

Topic:

Binomial Theorem

Exam:

JEE Mains 2019

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Question 15

The coefficient of

x^{18}

in the product

(1 + x) (1 - x)^{10} (1 + x + x^{2})^{9}

is (a) 84 (b)

- 126

- 84

(d) 126

Topic:

Binomial Theorem

Exam:

JEE Mains 2019

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Question 16

The term independent of

x

(\frac{1}{60} - \frac{x ^{8}}{81}) \cdot (2 x^{2} - \frac{3}{x ^{2}})^{6}

is equal to (a)

- 72

(b) 36 (c)

- 36

- 108

Topic:

Binomial Theorem

Exam:

JEE Mains 2019

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Question 17

The term independent of

x

(\frac{1}{60} - \frac{x ^{8}}{81}) \cdot (2 x^{2} - \frac{3}{x ^{2}})^{6}

is equal to (a)

- 72

(b) 36 (c)

- 36

- 108

Topic:

Binomial Theorem

Exam:

JEE Mains 2019

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Question 18

If the fractional part of the number

\frac{2 ^{403}}{15}

\frac{k}{15}

, then

k

is equal to (a) 14 (b) 6 (c) 4 (d) 8

Topic:

Binomial Theorem

Exam:

JEE Mains 2019

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Question 19

The coefficient of

t^{4}

is (a) 12 (b) 10 (c) 15 (d) 14

(\frac{1 - t ^{6}}{1 - t})^{3}

Topic:

Binomial Theorem

Exam:

JEE Mains 2019

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Question 20

If the third term in the binomial expansion of

(1 + x^{l o g_{2} x})^{5}

equals 2560 , then a possible value of

x

is (a)

42

\frac{1}{4}

\frac{1}{8}