JEE Mains 2016

If all the words (with or without meaning) having five letters, formed using the letters of the word SMALL and arranged as in a dictionary, then the position of the word SMALL is (a) 46 th (b) 59 th (c) 52 nd (d) 58 th

Topic:

Permutations and Combinations

Exam:

JEE Mains 2016

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Question 2

If the number of terms in the expansion of

(1 - \frac{2}{x} + \frac{4}{x ^{2}})^{n}, x \neq = 0

, is 28 , then the sum of the coefficients of all the terms in this expansion, is (a) 64 (b) 2187 (c) 243 (d) 729

Topic:

Binomial Theorem

Exam:

JEE Mains 2016

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Question 3

th and 9 th terms of a nonconstant

If the

2 nd, 5

AP

are in

GP

, then the common ratio of this GP is

$\frac{8}{5}$
$\frac{4}{3}$
1
$\frac{7}{4}$

Topic:

Sequences and Series

Exam:

JEE Mains 2016

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Question 4

If the sum of the first ten terms of the series

(1 \frac{3}{5})^{2} + (2 \frac{2}{5})^{2} + (3 \frac{1}{5})^{2} + 4^{2} + (4 \frac{4}{5})^{2} + \dots

, is

\frac{16}{5} m

m

is equal to

Topic:

Sequences and Series

Exam:

JEE Mains 2016

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Question 5

Medium

Let

p = lim_{x \to 0^{+}} (1 + tan^{2} x)^{1/2 x}

is equal to (a) 2 (b) 1 (c)

lo g p

\frac{1}{2}

\frac{1}{4}

Topic:

Limits and Derivatives

Exam:

JEE Mains 2016

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Question 6

For

x \in R, f (x) = ∣ lo g 2 - sin x ∣

g (x) = f (f (x))

, then (a)

g

is not differentiable at

x = 0

g^{'} (0) = cos (lo g 2)

g^{'} (0) = - cos (lo g 2)

g

is differentiable at

x = 0

g^{'} (0) = - sin (lo g 2)

Topic:

Continuity and Differentiability

Exam:

JEE Mains 2016

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Question 7

A wire of length 2 units is cut into two parts which are bent respectively to form a square of side

= x

units and a circle of radius

= r

units. If the sum of the areas of the square and the circle so formed is minimum, then (a)

2 x = (π + 4) r

(4 - π) x = π r

x = 2 r

2 x = r

Topic:

Application of Derivatives

Exam:

JEE Mains 2016

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Question 8

lim_{n \to \infty} (\frac{( n + 1 ) ( n + 2 ) \dots 3 n}{n ^{2 n}})^{1/ n}

is equal to (a)

\frac{18}{e ^{4}}

\frac{27}{e ^{2}}

\frac{9}{e ^{2}}

(d) 3

lo g 3 - 2

Topic:

Integrals

Exam:

JEE Mains 2016

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Question 9

is an arbitrary constant). (a)

The integral

\int \frac{2 x ^{12} + 5 x ^{9}}{( x ^{5} + x ^{3} + 1 ) ^{3}} d x

is equal to, (where,

C

\frac{- x ^{5}}{( x ^{5} + x ^{3} + 1 ) ^{2}} + C

\frac{x ^{10}}{2 ( x ^{5} + x ^{3} + 1 ) ^{2}} + C

\frac{x ^{5}}{2 ( x ^{5} + x ^{3} + 1 ) ^{2}} + C

\frac{- x ^{10}}{2 ( x ^{5} + x ^{3} + 1 ) ^{2}} + C

Topic:

Integrals

Exam:

JEE Mains 2016

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Question 10

and satisfies the differential equation,

If a curve

y = f (x)

passes through the point

(1, - 1)

y (1 + x y) d x = x d y

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

is equal to (a)

- \frac{2}{5}

- \frac{4}{5}

\frac{2}{5}

\frac{4}{5}

Topic:

Differential Equations

Exam:

JEE Mains 2016

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Question 11

The centres of those circles which touch the circle,

x^{2} + y^{2} - 8 x - 8 y - 4 = 0

, externally and also touch the

X

-axis, lie on (a) a circle (b) an ellipse which is not a circle (c) a hyperbola (d) a parabola

Topic:

Conic Sections

Exam:

JEE Mains 2016

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Question 12

be the point on the parabola,

Let

P

y^{2} = 8 x

, which is at a minimum distance from the centre

C

of the circle,

x^{2} + (y + 6)^{2} = 1

. Then, the equation of the circle, passing through

C

and having its centre at

P

$x^{2} + y^{2} - 4 x + 8 y + 12 = 0$
$x^{2} + y^{2} - x + 4 y - 12 = 0$
$x^{2} + y^{2} - \frac{x}{4} + 2 y - 24 = 0$
$x^{2} + y^{2} - 4 x + 9 y + 18 = 0$

Topic:

Conic Sections

Exam:

JEE Mains 2016

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Question 13

Medium

The eccentricity of the hyperbola whose length of the latusrectum is equal to 8 and the length of its conjugate axis is equal to half of the distance between its foci, is

$\frac{4}{3}$
$\frac{4}{3}$
$\frac{2}{3}$
$3$

Topic:

Conic Sections

Exam:

JEE Mains 2016

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Question 14

The distance of the point

(1, - 5, 9)

from the plane

x - y + z = 5

measured along the line

x = y = z

is (a)

310

103

\frac{10}{3}

\frac{20}{3}

Topic:

Three Dimensional Geometry

Exam:

JEE Mains 2016

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Question 15

If the line,

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

lies in the plane,

l x + m y - z = 9

is equal to (a) 26 (b) 18 (c) 5 (d) 2

l^{2} + m^{2}

Topic:

Three Dimensional Geometry

Exam:

JEE Mains 2016

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Question 16

Let

a, b

be three unit vectors such that

c

a \times (b \times c) = \frac{3}{2} (b + c)

. If

b

is not parallel to

c

, then the angle between

a

b

$\frac{3 π}{4}$
$\frac{π}{2}$
$\frac{2 π}{3}$
$\frac{5 π}{6}$

Topic:

Vector Algebra

Exam:

JEE Mains 2016

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Question 17

If the standard deviation of the numbers 2 ,

3, a

and 11 is

3.5

, then which of the following is true? (a)

3 a^{2} - 26 a + 55 = 0

3 a^{2} - 32 a + 84 = 0

3 a^{2} - 34 a + 91 = 0

3 a^{2} - 23 a + 44 = 0

Topic:

Statistics

Exam:

JEE Mains 2016

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Question 18

Let two fair six-faced dice

A

be thrown simultaneously. If

B

E_{1}

is the event that die

A

shows up four,

E_{2}

is the event that die

B

shows up two and

E_{3}

is the event that the sum of numbers on both dice is odd, then which of the following statements is not true? (a)

E_{1}

E_{2}

are independent (b)

E_{2}

E_{3}

are independent (c)

E_{1}

E_{3}

are independent (d)

E_{1}, E_{2}

E_{3}

are not independent

Topic:

Probability

Exam:

JEE Mains 2016

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Question 19

, then the number of real values of

0 \leq x < 2 π

x

, which satisfy the equation

cos x + cos 2 x + cos 3 x + cos 4 x = 0

, is (a) 3 (b) 5 (c) 7 (d) 9

Topic:

Trigonometric Functions

Exam:

JEE Mains 2016

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Question 20

The Boolean expression

(p \land \sim q) \lor q \lor (\sim p \land q)

is equivalent to (a)

\sim p \land q

p \land q

p \lor q