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QUESTION: 1

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: 2

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

Solution:

Let a point on the ellipse be P (a cosθ, b sinθ)

So, In the rectangle formed as this point as one of the vertices,

(Length)/2 = a cosθ

(Width)/2 = b sinθ

Area of the rectangle = (2a cosθ)(2b sinθ)

= 4ab(sinθ)(cosθ)

= 2ab sin(2θ)

Max (Area of the rectangle) = 2ab

QUESTION: 3

Let f(x) be differentiable for all x. If f (1) =-2 and f '(x)³ ≥2 x ∈ (1/6) then

Solution:

By LMVT

QUESTION: 4

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

(y -θ sinq- cosθ )

QUESTION: 5

The function f (x) = x/2+2/x has a local minimum at

Solution:

Signs of f ' (x )

f ' (x ) changes sign as x crosses 2.

f (x ) has minima at x = 2.

QUESTION: 6

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: 7

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

Solution:

QUESTION: 8

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

Maximum at x = -√p/3 , minima at x = √p/3

QUESTION: 9

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

shortest distance =

QUESTION: 10

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: 11

The equation of the tangent to the curve y= x+4/x^{2} that is parallel to the x-axis is

Solution:

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

QUESTION: 12

Let 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: 13

Solution:

As f(x) is continuous and 1/3 belongs to range

QUESTION: 14

The normal to the curve x^{2} + 2xy - 3y^{2} = 0 at 1,1

Solution:

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

∴ Equation of normal y -1 = -1(x -1)

y +x= 2

Solving we get point as (1, 1) and (1/-1/3)

So in forth quadrant

QUESTION: 15

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: 16

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: 17

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 : a=1/2 and =-1/4

Solution:

QUESTION: 18

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

Solution:

f ' ( x ) = 6 x2+ 3

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

QUESTION: 19

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

Solution:

QUESTION: 20

Let dt then f decreasing in the interval

Solution:

QUESTION: 21

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

Solution:

mm = -1 ⇒ Curves are orthogonal

QUESTION: 22

The greatest value of f (x) = (x + 1)^{1/3} - (x - 1)^{1/3} on [0, 1] is

Solution:

is decreasing on [0, 1].

Greatest value of f (x ) is f ( 0) = 1 - ( -1) = 2

QUESTION: 23

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

Solution:

QUESTION: 24

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

Solution:

Signs of f ' (x )

Minimum occurs at x = 1

QUESTION: 25

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: 26

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

Solution:

f ( 0) = df (1) (∵ 2a + 3b + 6x = 0)

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

QUESTION: 27

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

Solution:

QUESTION: 28

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

Solution:

x = a (1 + cosθ ) , ya = sinθ

Equation of normal at point ( a (1 + cosθ ) ,a sinθ ) is

y - a sinθ = sinθ/cosθ (x-a(1+cosθ))

y cosθ = x- a ) sinθ

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

QUESTION: 29

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: 30

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

Solution:

Let f (x) = x^{2} + 6x +6 ⇒ f (x) = 3x (x +4)

Increasing in (-∞,-4) Let ( ) 2 2

f_{2} (x) = 3x^{2} - 2x + 1

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