JEE Advanced Level Test: Applications of Derivatives - JEE MCQ

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

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

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

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

Suppose the cubic x³ - px + q = 0 has three distinct real roots where p > 0 and q > 0. Then, which one of the following holds?

The shortest distance between the line y - x= 1 and the curve x =y² is

Given p ( x ) = x ⁴ + ax³ + 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]

The equation of the tangent to the curve y= x+4/x² that is parallel to the x-axis is

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

The normal to the curve  x² + 2xy - 3y² = 0 at 1,1

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)

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

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

The real number k for which the equation, 2x³ + 3x + k = 0 has two distinct real roots in [0, 1]

The function f ( x ) = sin⁴ x+ cos⁴ x is increasing if 

​Let  dt then f decreasing in the interval

The two curves x³ - 3xy² + 2 = 0 and 3x ² y - y³ - 2 = 0

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

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

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

If the function f ( x ) = 2x³ - 9ax² + 12a ²x + 1, where a >0 , attains its maximum and minimum at p and q respectively such that p² = q, then 'a' equals

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

A point on the parabola y² = 18x at which the ordinate increases at twice the rate of the abscissa, is

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

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

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