JEE Mains 2017

has 7 friends, 4 of them are ladies and 3 are men. His wife

A man

X

Y

also has 7 friends, 3 of them are ladies and 4 are men. Assume

X

have no common friends. Then, the total number of ways in which

Y

X

together can throw a party inviting 3 ladies and 3 men, so that 3 friends of each of

Y

X

are in this party, is (a) 485 (b) 468 (c) 469 (d) 484

Y

Topic:

Permutations and Combinations

Exam:

JEE Mains 2017

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Question 2

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}) + \dots +

(^{21} C_{10} -^{10} C_{10})

is (a)

2^{21} - 2^{11}

2^{21} - 2^{10}

2^{20} - 2^{9}

2^{20} - 2^{10}

Topic:

Binomial Theorem

Exam:

JEE Mains 2017

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Question 3

For any three positive real numbers

a, b

c

, if

9 (25 a^{2} + b^{2}) + 25 (c^{2} - 3 a c) = 15 b (3 a + c)

$b, c$ and a are in GP
$b, c$ and a are in AP
$a, b$ and $c$ are in AP
$a, b$ and $c$ are in GP

Topic:

Sequences and Series

Exam:

JEE Mains 2017

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Question 4

at the point, where the curve intersects the

The normal to the curve

y (x - 2) (x - 3) = x + 6

Y

-axis passes through the point (a)

(- \frac{1}{2}, - \frac{1}{2})

(\frac{1}{2}, \frac{1}{2})

(\frac{1}{2}, - \frac{1}{3})

(\frac{1}{2}, \frac{1}{3})

Topic:

Application of Derivatives

Exam:

JEE Mains 2017

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Question 5

is a constant of integration, then the ordered pair

I_{n} = \int tan^{n} x d x, (n > 1)

. If

I_{4} + I_{6} = a tan^{5} x + b x^{5} + C

, where

C

(a, b)

is equal to (a)

(- \frac{1}{5}, 1)

(\frac{1}{5}, 0)

(\frac{1}{5}, - 1)

(- \frac{1}{5}, 0)

Topic:

Integrals

Exam:

JEE Mains 2017

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Question 6

\int_{π /4}^{3 π /4} \frac{d x}{1 + c o s x}

is equal to (a)

- 2

(b) 2 (c) 4 (d)

- 1

Topic:

Integrals

Exam:

JEE Mains 2017

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Question 7

(2 + sin x) \frac{d y}{d x} + (y + 1) cos x = 0

y (0) = 1

y (\frac{π}{2})

is equal to (a)

\frac{1}{3}

- \frac{2}{3}

- \frac{1}{3}

\frac{4}{3}

Topic:

Differential Equations

Exam:

JEE Mains 2017

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Question 8

The eccentricity of an ellipse whose centre is at the origin is

1/2

. If one of its directrices is

x = - 4

, then the equation of the normal to it at

(1, \frac{3}{2})

$2 y - x = 2$
$4 x - 2 y = 1$
$4 x + 2 y = 7$
$x + 2 y = 4$

Topic:

Conic Sections

Exam:

JEE Mains 2017

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Question 9

If a hyperbola passes through the point

P (2, 3)

and has foci at

(\pm 2, 0)

, then the tangent to this hyperbola at

P

also passes through the point

$(32, 23)$
$(22, 33)$
$(3, 2)$
$(- 2, - 3)$

Topic:

Conic Sections

Exam:

JEE Mains 2017

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Question 10

The distance of the point

(1, 3, - 7)

from the plane passing through the point

(1, - 1, - 1)

having normal perpendicular to both the lines

\frac{x - 1}{1} = \frac{y + 2}{- 2} = \frac{z - 4}{3}

\frac{x - 2}{2} = \frac{y + 1}{- 1} =

\frac{z + 7}{- 1}

, is (a)

\frac{20}{74}

units (b)

\frac{10}{83}

units (c)

\frac{5}{83}

units (d)

\frac{10}{74}

units

Topic:

Three Dimensional Geometry

Exam:

JEE Mains 2017

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Question 11

a = 2 \hat{i} + \hat{j} - 2 \hat{k}, b = \hat{i} + \hat{j}

c

be a vector such that

∣ c - a ∣ = 3, ∣ (a \times b) \times c ∣ = 3

and the angle between

c

a \times b

3 0^{\circ}

. Then,

a \cdot c

is equal to

$\frac{25}{8}$
$2$
$5$
$\frac{1}{8}$

Topic:

Vector Algebra

Exam:

JEE Mains 2017

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Question 12

For three events

A, B

C

, if

P

(exactly one of

A

B

occurs

) = P (

exactly one of

B

C

occurs

) = P

(exactly one of

C

A

occurs

) = \frac{1}{4}

(all the three events occur simultaneously)

P

= \frac{1}{16}

, then the probability that atleast one of the events occurs, is (a)

\frac{7}{32}

\frac{7}{16}

\frac{7}{64}

\frac{3}{16}

Topic:

Probability

Exam:

JEE Mains 2017

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Question 13

If two different numbers are taken from the set

{0, 1, 2, 3, \dots, 10}

, then the probability that their sum as well as absolute difference are both multiple of 4 , is (a)

\frac{6}{55}

\frac{12}{55}

\frac{14}{45}

\frac{7}{55}

Topic:

Probability

Exam:

JEE Mains 2017

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Question 14

A box contains 15 green and 10 yellow balls. If 10 balls are randomly drawn one-by-one with replacement, then the variance of the number of green balls drawn is (a)

\frac{12}{5}

(b) 6 (c) 4 (d)

\frac{6}{25}

Topic:

Probability

Exam:

JEE Mains 2017

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Question 15

5 (tan^{2} x - cos^{2} x) = 2 cos 2 x + 9

, then the value of

cos 4 x

is (a)

- \frac{3}{5}

\frac{1}{3}

\frac{2}{9}

- \frac{7}{9}

Topic:

Trigonometric Functions

Exam:

JEE Mains 2017

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Question 16

For

x \in (0, \frac{1}{4})

, if the derivative of

tan^{- 1} (\frac{6 x x}{1 - 9 x ^{3}})

x \cdot g (x)

g (x)

equals (a)

\frac{9}{1 + 9 x ^{3}}

\frac{3 x x}{1 - 9 x ^{3}}

\frac{3 x}{1 - 9 x ^{3}}

\frac{3}{1 + 9 x ^{3}}

Topic:

Inverse Trigonometric Functions

Exam:

JEE Mains 2017

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Question 17

is (a) invertible (b) injective but not surjective (c) surjective but not injective (d) neither injective nor surjective

The function

f : R \to [- \frac{1}{2}, \frac{1}{2}]

defined as

f (x) = \frac{x}{1 + x ^{2}}

Topic:

Relations and Functions

Exam:

JEE Mains 2017

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Question 18

a, b, c \in R

. If

f (x) = a x^{2} + b x + c

be such that

a + b + c = 3

f (x + y) = f (x) + f (y) + x y, \forall x, y \in R

is equal to (a) 330 (b) 165 (c) 190 (d) 255

\sum_{n = 1}^{10} f (n)

Topic:

Relations and Functions

Exam:

JEE Mains 2017

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Question 19

be a complex number such that

ω

2 ω + 1 = z

, where

z = - 3

. If

∣ ∣ 111 1 - ω^{2} - 1 ω^{2} 1 ω^{2} ω^{7} ∣ ∣ = 3 k

, then k is equal to (a)

- z

z

- 1

(d) 1

Topic:

Complex Number and Quadratic Equations

Exam:

JEE Mains 2017

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Question 20

be an integer such that the triangle with vertices

k

(k, - 3 k), (5, k)

units. Then, the orthocentre of this triangle is at the point
(a)

(- k, 2)

has area

28 sq.

(2, - \frac{1}{2})

(1, \frac{3}{4})

(1, - \frac{3}{4})