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Test: Application of Derivatives- Assertion & Reason Type Questions - JEE MCQ


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15 Questions MCQ Test Mathematics (Maths) Class 12 - Test: Application of Derivatives- Assertion & Reason Type Questions

Test: Application of Derivatives- Assertion & Reason Type Questions for JEE 2024 is part of Mathematics (Maths) Class 12 preparation. The Test: Application of Derivatives- Assertion & Reason Type Questions questions and answers have been prepared according to the JEE exam syllabus.The Test: Application of Derivatives- Assertion & Reason Type Questions MCQs are made for JEE 2024 Exam. Find important definitions, questions, notes, meanings, examples, exercises, MCQs and online tests for Test: Application of Derivatives- Assertion & Reason Type Questions below.
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Test: Application of Derivatives- Assertion & Reason Type Questions - Question 1

Directions : In the following questions, A statement of Assertion (A) is followed by a statement of Reason (R). Mark the correct choice as.

Assertion (A): At x = π/6, the curve y = 2cos2 (3x) has a vertical tangent.

Reason (R): The slope of tangent to the curve y = 2cos2 (3x) at x = π/6 is zero.

Detailed Solution for Test: Application of Derivatives- Assertion & Reason Type Questions - Question 1

Given y = 2cos2(3x)

dy/dx = 2 x 2 x cos(3x) x (- sin 3x) x 3

dy/dx = -6 sin 6x dx

∴ R is true.

Since the slope of tangent is zero, the tangent is parallel to the X-axis. That is the curve has a horizontal tangent at x = π/6. Hence A is false.

Test: Application of Derivatives- Assertion & Reason Type Questions - Question 2

Directions : In the following questions, A statement of Assertion (A) is followed by a statement of Reason (R). Mark the correct choice as.

Assertion (A): The function y = [x(x – 2)]2 is increasing in (0, 1) ∪ (2, ∞)

Reason (R): dy/dx = 0, when x = 0, 1, 2

Detailed Solution for Test: Application of Derivatives- Assertion & Reason Type Questions - Question 2

y = [x(x-2]2

= [x2 - 2x]2

∴ dy/dx = 2(x2 - 2x) (2x - 2)

or dy/dx = 4x(x-1)(x-2)

On equating dy/dx = 0

4x(x - 1)(x - 2) = 0 ⇒ x = 0, x = 1, x = 2

∴ Intervals are (-∞=, 0), (0,1), (1,2), (2,∞)

Since, dy/dx > 0 in (0,1) or (2, ∞)

∴ f(x) is increasing in (0,1) ∪ (2, ∞)

Both A and R are true. But R is not the correct explanation of A.

Test: Application of Derivatives- Assertion & Reason Type Questions - Question 3

Directions : In the following questions, A statement of Assertion (A) is followed by a statement of Reason (R). Mark the correct choice as.

The equation of tangent at (2, 3) on the curve y2 = ax3 + b is y = 4x – 5.

Assertion (A): The value of a is ±2

Reason (R): The value of b is ±7

Detailed Solution for Test: Application of Derivatives- Assertion & Reason Type Questions - Question 3

y2 = ax3 + b

Differentiate with respect to x,

Since (2, 3) lies on the curve

y2 = ax3 + b

Or 9 = 8a + b …. (i)

Also from equation of tangent

y = 4x – 5

slope of the tangent = 4

either, a = 2 or a = - 2

For a = 2,

9 = 8(2) + b

or b = - 7

∴ a = 2 and b = - 7 and

for a = - 2,

9 = 8(- 2 )+b or b = 25

or a = - 2 and b = 25

Hence A is true and R is false.

Test: Application of Derivatives- Assertion & Reason Type Questions - Question 4

Directions : In the following questions, A statement of Assertion (A) is followed by a statement of Reason (R). Mark the correct choice as.

The total revenue received from the sale of x units of a product is given by R(x) = 3x2 + 36x + 5 in rupees.

Assertion (A): The marginal revenue when x = 5 is 66.

Reason (R): Marginal revenue is the rate of change of total revenue with respect to the number of items sold at an instance.

Detailed Solution for Test: Application of Derivatives- Assertion & Reason Type Questions - Question 4

Marginal revenue is the rate of change of total revenue with respect to the number of items sold at an instance. Therefore R is true.

R’(x) = 6x + 36

R’(5) = 66

∴ A is true.

R is the correct explanation of A.

Test: Application of Derivatives- Assertion & Reason Type Questions - Question 5

Directions : In the following questions, A statement of Assertion (A) is followed by a statement of Reason (R). Mark the correct choice as.

Assertion (A): The function f(x) = x3 – 3x2 + 6x – 100 is strictly increasing on the set of real numbers.

Reason (R): A strictly increasing function is an injective function.

Detailed Solution for Test: Application of Derivatives- Assertion & Reason Type Questions - Question 5

f(x) = x3 - 3x2+ 6x - 100

f(x) = 3x2 - 6x + 6

= 3[x2 - 2x + 2]

= 3[(x-1)2+1]

Since f’(x) > 0; x ∊ R

f(x) is strictly increasing on R.

Hence A is true.

For a strictly increasing function,

x1 > x2

⇒ f(x1) > f(x2)

i.e.) x1 = x2

⇒ f(x1) — f(x2)

Hence, a strictly increasing function is always an injective function.

So R is true.

But R is not the correct explanation of A.

Test: Application of Derivatives- Assertion & Reason Type Questions - Question 6

Directions : In the following questions, A statement of Assertion (A) is followed by a statement of Reason (R). Mark the correct choice as.

Assertion (A): For the curve y = 5x – 2x3, if x increases at the rate of 2 units/sec, then at x = 3 the slope of curve is decreasing at 36 units/sec.

Reason (R): The slope of the curve is dy/dx

Detailed Solution for Test: Application of Derivatives- Assertion & Reason Type Questions - Question 6

The slope of the curve y = f(x) is dy/dx. R is true

Given

Curve is y = 5x – 2x3

Rate of Change of the slope is decreasing by 72 units/s.

A is false.

Test: Application of Derivatives- Assertion & Reason Type Questions - Question 7

Directions : In the following questions, A statement of Assertion (A) is followed by a statement of Reason (R). Mark the correct choice as.

Assertion (A): The function is a decreasing function of x throughout its domain.

Reason (R): The domain of the function

Detailed Solution for Test: Application of Derivatives- Assertion & Reason Type Questions - Question 7

log (1 + x) is defined only when x + 1 > 0 or x > -1

Hence R is true.

For increasing function,

dy/dx is always greater than zero.

Is always increasing throughout its domain.

Hence A is false.

Test: Application of Derivatives- Assertion & Reason Type Questions - Question 8

Directions : In the following questions, A statement of Assertion (A) is followed by a statement of Reason (R). Mark the correct choice as.

Assertion (A): The slope of normal to the curve x2 + 2y + y2 = 0 at (–1, 2) is –3.

Reason (R): The slope of tangent to the curve x2 + 2y + y2 = 0 at (–1, 2) is 1/3

Detailed Solution for Test: Application of Derivatives- Assertion & Reason Type Questions - Question 8

Slope of tangent at (-1,2)

Hence R is true.

Slope of normal at (-1,2)

Hence A is true.

R is the correct explanation for A.

Test: Application of Derivatives- Assertion & Reason Type Questions - Question 9

Directions : In the following questions, A statement of Assertion (A) is followed by a statement of Reason (R). Mark the correct choice as.

Assertion (A): The altitude of the cone of maximum volume that can be inscribed in a sphere of radius r is 4r/3.

Reason (R): The maximum volume of the cone is 8/27 of the volume of the sphere.

Detailed Solution for Test: Application of Derivatives- Assertion & Reason Type Questions - Question 9

Let radius of cone be x and its height be h.

∴ OD = (h – r)

Volume of cone

∴ at h = 4r/3, Volume is maximum

Maximum volume

Hence both A and R are true.

R is not the correct explanation of A.

Test: Application of Derivatives- Assertion & Reason Type Questions - Question 10

Directions : In the following questions, A statement of Assertion (A) is followed by a statement of Reason (R). Mark the correct choice as.

A particle moves along the curve 6y = x3 + 2.

Assertion (A): The curve meets the Y axis at three points.

Reason (R): At the points (2,5/3) and (–2, –1) the ordinate changes two times as fast as the abscissa.

Detailed Solution for Test: Application of Derivatives- Assertion & Reason Type Questions - Question 10

On Y axis, x = 0. The curve meets the Y axis at only one point, i.e., (0,⅓)

Hence A is false.

Put x = 2 and –2 in the given equation to get y

∴ The points are (2, 5/3), (-2,-1)

R is true.

Test: Application of Derivatives- Assertion & Reason Type Questions - Question 11

Directions : In the following questions, A statement of Assertion (A) is followed by a statement of Reason (R). Mark the correct choice as.

The sum of surface areas (S) of a sphere of radius ‘r’ and a cuboid with sides x/3, x and 2x is a constant.

Assertion (A): The sum of their volumes (V) is minimum when x equals three times the radius of the sphere.

Reason (R): V is minimum when

Detailed Solution for Test: Application of Derivatives- Assertion & Reason Type Questions - Question 11

Given

i.e., r2(54 + 4p) = S

or r2(54 + 4p) = 4?r2 + 6x2

or 6x2 = 54r2

or x2 = 9r2

or x = 3r

Hence both A and R are true.

R is the correct explanation of A.

Test: Application of Derivatives- Assertion & Reason Type Questions - Question 12

Directions : In the following questions, A statement of Assertion (A) is followed by a statement of Reason (R). Mark the correct choice as.

The radius r of a right circular cylinder is increasing at the rate of 5 cm/min and its height h, is decreasing at the rate of 4 cm/min.

Assertion (A): When r = 8 cm and h = 6 cm, the rate of change of volume of the cylinder is 224p cm3/min

Reason (R): The volume of a cylinder is

Detailed Solution for Test: Application of Derivatives- Assertion & Reason Type Questions - Question 12

The volume of a cylinder is V=?r2h.

So R is false.

∴ Volume is increasing at the rate of 224? cm3/min.

∴ A is true.

Test: Application of Derivatives- Assertion & Reason Type Questions - Question 13

Directions : In the following questions, A statement of Assertion (A) is followed by a statement of Reason (R). Mark the correct choice as.

Assertion (A): The equation of tangent to the curve y = sin x at the point (0, 0) is y = x.

Reason (R): If y = sin x, then dy/dx at x = 0 is 1

Detailed Solution for Test: Application of Derivatives- Assertion & Reason Type Questions - Question 13

Given y = sin x

dy/dx = cos x

∴ R is true.

Equation of tangent at (0, 0) is y – 0 = 1(x – 0)

⇒ y = x.

Hence A is true.

R is the correct explanation of A.

Test: Application of Derivatives- Assertion & Reason Type Questions - Question 14

Directions : In the following questions, A statement of Assertion (A) is followed by a statement of Reason (R). Mark the correct choice as.

Consider the function f(x) = sin4x + cos4x.

Assertion (A): f(x) is increasing in

Reason (R): f(x) is decreasing in

Detailed Solution for Test: Application of Derivatives- Assertion & Reason Type Questions - Question 14

f(x) = sin4 x + cos4 x

or

f ’(x) = 4sin3 xcos x – 4cos3 xsin x

= – 4sin xcos x [– sin2 x + cos2 x]

= – 2sin 2x cos 2x

= – sin 4x

On equating,

f ’(x) = 0

or – sin 4x = 0

or 4x = 0, ?, 2?, ...........

Both A and R are true. But R is not the correct explanation of A.

Test: Application of Derivatives- Assertion & Reason Type Questions - Question 15

Directions : In the following questions, A statement of Assertion (A) is followed by a statement of Reason (R). Mark the correct choice as.

AB is the diameter of a circle and C is any point on the circle.

Assertion (A): The area of DABC is maximum when it is isosceles.

Reason (R): DABC is a right-angled triangle.

Detailed Solution for Test: Application of Derivatives- Assertion & Reason Type Questions - Question 15

Let the sides of rt.? ABC be x and y

∴ x2 + y2 = 4r2

or Area is maximum, when ? is isosceles.

Hence A is true.

Angle in a semicircle is a right angle. ∠C = 90°

⇒ ?ABC is a right-angled triangle.

∴ R is true.

R is the correct explanation of A.

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