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QUESTION: 1

The two curves x^{3} - 3xy^{2} + 2 = 0 and 3x^{2} y - y^{3} - 2 = 0

Solution:

m_{1}m_{2} = -1 ⇒ Curves are orthogonal

QUESTION: 2

The greatest value of

Solution:

QUESTION: 3

The function f (x) = cot^{-1} x+ x increases in the interval

Solution:

QUESTION: 4

The real number x (x > 0) when added to its reciprocal gives the minimum sum at x equals

Solution:

Minimum occurs at x = 1

QUESTION: 5

If the function f (x) = 2x^{3} - 9ax^{2} + 12a^{2}x + 1, where a > 0 , attains its maximum and minimum at p and q respectively such that p² = q, then ‘a’ equals

Solution:

QUESTION: 6

If 2a + 3b + 6c = 0, then at least one root of the equation ax^{2} + bx + c = 0 lies in the interval

Solution:

By Roll’s theorem, at least one root of f ' (x) =0 lies in (0, 1)

QUESTION: 7

A point on the parabola y^{2} = 18x at which the ordinate increases at twice the rate of the abscissa, is

Solution:

y² = 18 x .....(i)

Differentiating w.r.t t

QUESTION: 8

The normal to the curve x = a (1 + cos θ) , y = a sin θ at 'θ' always passes through the fixed point

Solution:

Equation of normal at point (a(1 + cos θ) ,a sin θ) is

It is clear that normal passes through fixed point (a, 0)

QUESTION: 9

Angle between the tangents to the curve y = x^{2} - 5x + 6 at the points (2, 0) and (3, 0) is

Solution:

Product of slopes = -1 ⇒ angle between tangents is π/2

QUESTION: 10

A function is matched below against an interval where it is supposed to be increasing, Which of the following pairs is incorrectly matched?

Solution:

QUESTION: 11

A spherical iron ball 10 cm in radius is coated with a layer of ice of uniform thickness that melts at a rate of 50 cm³/min. When the thickness of ice is 15 cm, then the rate at which the thickness of ice decreases, is

Solution:

QUESTION: 12

Area of the greatest rectangle that can be inscribed in the ellipse

Solution:

Any point on ellipse is P (x, y) = (a cosθ, b sinθ)

Area of rectangle = 4ab cosθ sinθ

= 2ab sinθ

∴ Maximum area = 2ab.

QUESTION: 13

Let f (x) be differentiable for all x. If f (1) =-2 and f '(x) __>__ 2 for x ∈ (1, 6) then

Solution:

QUESTION: 14

The normal to the curve x = a (cos θ + θ sin θ), y = a (sin θ - θ cos θ) at any point 'θ' is such that

Solution:

Equation of normal is

⇒ Angle made with x – axis is

QUESTION: 15

The function has a local minimum at

Solution:

QUESTION: 16

A value of c for which the conclusion of Mean value Theorem holds for the function f (x) = log_{e} x on the interval [1, 3] is

Solution:

QUESTION: 17

The function f (x) = tan^{-1} (sin x cos x) is an increasing function in

Solution:

QUESTION: 18

Suppose the cubic x^{3} - px + q = 0 has three distinct real roots where p > 0 and q > 0. Then, which one of the following holds?

Solution:

Graph of y = x^{2} - px + q cuts x-axis at three distinct pints

QUESTION: 19

The shortest distance between the line y - x= 1 and the curve x =y^{2 }is

Solution:

Condition for shortest distance is slope of tangent to x = y² must be same as slope of line y = x + 1

QUESTION: 20

Given p (x) = x^{4} + ax^{3} + bx^{2} + cx + d such that x = 0 is the only real root of p' (x) = 0 . If p (-1) < p (1) , then in the interval [-1,1]

Solution:

p ' (x) has only one change of sign. ⇒ x = 0 is a point of minima.

P(-1) =1- a + b + d P(0) = d

P(1) =1+ a + b + d

⇒ P (-1) < P (1) , P (0) < P (1) , P (-1) > P ( 0)

⇒ P (-1) is not minimum but P(1) is maximum

QUESTION: 21

The equation of the tangent to the curve that is parallel to the x-axis is

Solution:

(2, 3) is point of contact thus y = 3 is tangent

QUESTION: 22

Let f : R → R be defined by If f has a local minimum at x = - 1, then a possible value of k is

Solution:

∴ f has a local minimum at x = - 1

Possible value of k is – 1

QUESTION: 23

Let f : R → R be a continuous function defined by

Statement-1 :

Statement -2 :

Solution:

Hence correction option is (d)

QUESTION: 24

The normal to the curve

Solution:

Slope of normal at (1, 1) = - 1

∴ Equation of normal y -1 = -1(x -1)

y +x= 2

Solving we get point as

So in forth quadrant

QUESTION: 25

Consider the function, f (x) =| x - 2 | + | x - 5 | .x ∈ R

Statement -1 : f '(4) = 0

Statement -2 : f is continous in [2, 5], differentiable in (2, 5) and f (2) = f (5)

Solution:

f (x) is constant is [2, 5]

QUESTION: 26

A spherical balloon is filled with 4500 π cubic meters of helium gas. If a leak in the balloon causes the gas to escape at the rate of 75 π cubic meters per minute, then the rate (in metres per minute) at which the radius of the balloon decreases 49 minutes after the leakage began is

Solution:

QUESTION: 27

Let a, b ∈ R be such that the function f given by f (x) = In | x | +bx^{2} + ax, x ≠ 0 has extreme values at x =-1 and x = 2.

Statement – 1: f has local maximum at x = - 1 and at x = 2.

Statement-2 :

Solution:

QUESTION: 28

The real number k for which the equation, 2x^{3} + 3x + k = 0 has two distinct real roots in [0, 1]

Solution:

f ' (x) = 6x^{2}+ 3

⇒ so f(x) is increasingequation cannot have 2 distinct roots.

QUESTION: 29

The function f (x) = sin^{4} x+ cos^{4} x Increasing if

Solution:

f '(x) = 4 sin^{3} x cos x - 4 cos^{3} x sin x = - sin 4 x for increasing f '(x) >0

QUESTION: 30

dt then f decreasing in the interval

Solution:

f ' (x) < 0

e^{x} (x -1) ( x - 2) < 0

1 < x<2

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