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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.
Solutions of Test: Application of Derivatives- Assertion & Reason Type Questions questions in English are available as part of our Mathematics (Maths) Class 12 for JEE & Test: Application of Derivatives- Assertion & Reason Type Questions solutions in Hindi for Mathematics (Maths) Class 12 course. Download more important topics, notes, lectures and mock test series for JEE Exam by signing up for free. Attempt Test: Application of Derivatives- Assertion & Reason Type Questions | 15 questions in 30 minutes | Mock test for JEE preparation | Free important questions MCQ to study Mathematics (Maths) Class 12 for JEE Exam | Download free PDF with solutions
Test: Application of Derivatives- Assertion & Reason Type Questions - Question 1
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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
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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.

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