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# Competition Level Test: Application Of Derivative- 1

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## 30 Questions MCQ Test Mathematics For JEE | Competition Level Test: Application Of Derivative- 1

Competition Level Test: Application Of Derivative- 1 for JEE 2023 is part of Mathematics For JEE preparation. The Competition Level Test: Application Of Derivative- 1 questions and answers have been prepared according to the JEE exam syllabus.The Competition Level Test: Application Of Derivative- 1 MCQs are made for JEE 2023 Exam. Find important definitions, questions, notes, meanings, examples, exercises, MCQs and online tests for Competition Level Test: Application Of Derivative- 1 below.
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Competition Level Test: Application Of Derivative- 1 - Question 1

### The two curves x3 - 3xy2 + 2 = 0 and 3x2 y - y3 - 2 = 0

Detailed Solution for Competition Level Test: Application Of Derivative- 1 - Question 1 m1m2 = -1 ⇒ Curves are orthogonal

Competition Level Test: Application Of Derivative- 1 - Question 2

### The greatest value of Detailed Solution for Competition Level Test: Application Of Derivative- 1 - Question 2  Competition Level Test: Application Of Derivative- 1 - Question 3

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

Detailed Solution for Competition Level Test: Application Of Derivative- 1 - Question 3 Competition Level Test: Application Of Derivative- 1 - Question 4

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

Detailed Solution for Competition Level Test: Application Of Derivative- 1 - Question 4 Minimum occurs at x = 1

Competition Level Test: Application Of Derivative- 1 - Question 5

If the function f (x) = 2x3 - 9ax2 + 12a2x + 1, where a > 0 , attains its maximum and minimum at p and q respectively such that p² = q, then ‘a’ equals

Detailed Solution for Competition Level Test: Application Of Derivative- 1 - Question 5  Competition Level Test: Application Of Derivative- 1 - Question 6

If 2a + 3b + 6c = 0, then at least one root of the equation ax2 + bx + c = 0 lies in the interval

Detailed Solution for Competition Level Test: Application Of Derivative- 1 - Question 6 By Roll’s theorem, at least one root of f ' (x) =0 lies in (0, 1)

Competition Level Test: Application Of Derivative- 1 - Question 7

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

Detailed Solution for Competition Level Test: Application Of Derivative- 1 - Question 7

y² = 18 x       .....(i)
Differentiating w.r.t t  Competition Level Test: Application Of Derivative- 1 - Question 8

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

Detailed Solution for Competition Level Test: Application Of Derivative- 1 - Question 8 Equation of normal at point (a(1 + cos θ) ,a sin θ) is It is clear that normal passes through fixed point (a, 0)

Competition Level Test: Application Of Derivative- 1 - Question 9

Angle between the tangents to the curve y = x2 - 5x + 6 at the points (2, 0) and (3, 0) is

Detailed Solution for Competition Level Test: Application Of Derivative- 1 - Question 9 Product of slopes = -1 ⇒ angle between tangents is  π/2

Competition Level Test: Application Of Derivative- 1 - 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? Detailed Solution for Competition Level Test: Application Of Derivative- 1 - Question 10     Competition Level Test: Application Of Derivative- 1 - 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

Detailed Solution for Competition Level Test: Application Of Derivative- 1 - Question 11  Competition Level Test: Application Of Derivative- 1 - Question 12

Area of the greatest rectangle that can be inscribed in the ellipse Detailed Solution for Competition Level Test: Application Of Derivative- 1 - Question 12

Any point on ellipse is P (x, y) = (a cosθ, b sinθ)
Area of rectangle = 4ab cosθ sinθ
= 2ab sinθ
∴ Maximum area = 2ab. Competition Level Test: Application Of Derivative- 1 - Question 13

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

Detailed Solution for Competition Level Test: Application Of Derivative- 1 - Question 13 Competition Level Test: Application Of Derivative- 1 - 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

Detailed Solution for Competition Level Test: Application Of Derivative- 1 - Question 14  Equation of normal is ⇒ Angle made with x – axis is Competition Level Test: Application Of Derivative- 1 - Question 15

The function has a local minimum at

Detailed Solution for Competition Level Test: Application Of Derivative- 1 - Question 15  Competition Level Test: Application Of Derivative- 1 - Question 16

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

Detailed Solution for Competition Level Test: Application Of Derivative- 1 - Question 16 Competition Level Test: Application Of Derivative- 1 - Question 17

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

Detailed Solution for Competition Level Test: Application Of Derivative- 1 - Question 17  Competition Level Test: Application Of Derivative- 1 - Question 18

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?

Detailed Solution for Competition Level Test: Application Of Derivative- 1 - Question 18

Graph of y = x2 - px + q cuts x-axis at three distinct pints  Competition Level Test: Application Of Derivative- 1 - Question 19

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

Detailed Solution for Competition Level Test: Application Of Derivative- 1 - Question 19

Condition for shortest distance is slope of tangent to x = y² must be same as slope of line y = x + 1  Competition Level Test: Application Of Derivative- 1 - Question 20

Given p (x) = x4 + ax3 + bx2 + 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]

Detailed Solution for Competition Level Test: Application Of Derivative- 1 - Question 20  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

Competition Level Test: Application Of Derivative- 1 - Question 21

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

Detailed Solution for Competition Level Test: Application Of Derivative- 1 - Question 21  (2, 3) is point of contact thus y = 3 is tangent

Competition Level Test: Application Of Derivative- 1 - 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

Detailed Solution for Competition Level Test: Application Of Derivative- 1 - Question 22 ∴ f has a local minimum at x = - 1 Possible value of k is – 1

Competition Level Test: Application Of Derivative- 1 - Question 23

Let f : R → R be a continuous function defined by Statement-1 : Statement -2 : Detailed Solution for Competition Level Test: Application Of Derivative- 1 - Question 23  Hence correction option is (d)

Competition Level Test: Application Of Derivative- 1 - Question 24

The normal to the curve Detailed Solution for Competition Level Test: Application Of Derivative- 1 - Question 24 Slope of normal at (1, 1) = - 1
∴ Equation of normal y -1 = -1(x -1)
y +x= 2
Solving we get point as Competition Level Test: Application Of Derivative- 1 - 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)

Detailed Solution for Competition Level Test: Application Of Derivative- 1 - Question 25

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

Competition Level Test: Application Of Derivative- 1 - 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

Detailed Solution for Competition Level Test: Application Of Derivative- 1 - Question 26  Competition Level Test: Application Of Derivative- 1 - Question 27

Let a, b ∈ R be such that the function f given by f (x) = In | x | +bx2 + 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 : Detailed Solution for Competition Level Test: Application Of Derivative- 1 - Question 27  Competition Level Test: Application Of Derivative- 1 - Question 28

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

Detailed Solution for Competition Level Test: Application Of Derivative- 1 - Question 28

f ' (x) = 6x2+ 3
⇒ so f(x) is increasing equation cannot have 2 distinct roots.

Competition Level Test: Application Of Derivative- 1 - Question 29

The function f (x) = sin4 x+ cos4 x Increasing if

Detailed Solution for Competition Level Test: Application Of Derivative- 1 - Question 29

f '(x) = 4 sin3 x cos x - 4 cos3 x sin x = - sin 4 x for increasing f '(x) >0 Competition Level Test: Application Of Derivative- 1 - Question 30 dt then f decreasing in the interval

Detailed Solution for Competition Level Test: Application Of Derivative- 1 - Question 30

f ' (x) < 0
ex (x -1) ( x - 2) < 0
1 < x<2

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