CBSE Class 12 Mathematics Application Of Derivatives MCQs Set 03

Question. If sum of two numbers is 3, the maximum value of the product of first and the square of second is
(a) 4
(b) 3
(c) 2
(d) 1

Answer : A

Question. If f(x) = x³ – 7x² + 15, then the approximate value of f(5.001) is
(a) 34.995
(b) – 30.995
(c) 24.875
(d) None of these

Answer : B

Question. If the error committed in measuring the radius of sphere, then … will be the percentage error in the surface area.
(a) 1%
(b) 2%
(c) 3%
(d) 4%

Answer : D

Question. If y = (4x – 5) is a tangent to the curve y² = px³ + q at (2, 3), then
(a) p = – 2, q = – 7
(b) p = – 2, q = 7
(c) p = 2, q = – 7
(d) p = 2, q = 7

Answer : C

Question. The radius of a sphere initially at zero increases at the rate of 5 cm/sec. Then its volume after 1 sec is increasing at the rate of :
(a) 50 π
(b) 5 π
(c) 500 π
(d) None of these

Answer : C

Question. The interval in which the function f(x) = 4×2 + 1/x is decreasing is :
(a) (-1/2 , 1/2)
(b) [-1/2 , 1/2]
(c) (–1, 1)
(d) [–1, 1]

Answer : A

Question. The function f (x) = x² –2x is strictly increasing in the interval:
(a) (–2, –1)
(b) (–1, 0)
(c) (0, 1)
(d) (1, 2)

Answer : D

Question. If for a function f (x), f ‘(a) = 0, f “(a) = 0, f ”'(a) > 0, then at x = a, f (x) is
(a) Minimum
(b) Maximum
(c) Not an extreme point
(d) Extreme point

Answer : C

Question. The normal to a given curve is parallel to x-axis if
(a) dy/dx = 0
(b) dy/dx = 1
(c) dy/dx = 0
(d) dy/dx = 1

Answer : A

Question. The slope of the tangent to the curve x = 3t² + 1, y= t³ –1 at x = 1 is:
(a) 1/2
(b) 0
(c) –2
(d) ∞

Answer : B

Question. The volume V and depth x of water in a vessel are connected by the relation V = 5x – x²/6 and the volume of water is increasing , at the rate of 5 cm3/sec, when x = 2 cm. The rate at which the depth of water is increasing, is
(a) 5/18 cm / sec
(b) 1/4 cm / sec
(c) 5/16 cm / sec
(d) None of these

Answer : D

Question. The angle of intersection to the curve y = x², 6y = 7 – x³ at (1, 1) is :
(a) π/2
(b) π/4
(c) π/3
(d) π

Answer : A

Question. The curve given by x + y = exy has a tangent parallel to the Y-axis at the point
(a) (0, 1)
(b) (1, 0)
(c) (1, 1)
(d) None of these

Answer : B

Question. What is the x-coordinate of the point on the curve f (x) = √x (7x – 6), where the tangent is parallel to x-axis?
(a) -1/3
(b) 2/7
(c) 6/7
(d) 1/2

Answer : B

Question. What is the interval in which the function f (x) = √9-x² is increasing? (f (x)>0)
(a) 0 < x < 3
(b) – 3 < x < 0
(c) 0 < x < 9
(d) – 3 < x < 3

Answer : B
 

Case Based Questions

Shobhit′s father wants to construct a rectangular garden using a brick will on one side of the garden and wire fencing for the other three sides as shown in figure. He has 200 ft. of wire fencing.

Based on the above information, answer the following questions:

Question. If x denotes the length of side of garden perpendicular to brick wall and y denotes the length of side parallel to brick wall, then find the relation representing total amount of fencing wire.
(a) x + 2y = 150
(b) x + 2y = 50
(c) y + 2x = 200
(d) y + 2x = 100

Answer : C

Question. To construct a garden using 200 ft. of fencing, we need to maximise its
(a) volume
(b) area
(c) perimeter
(d) length of the side

Answer : B

Question. Maximum value of A(x) occurs at x equals
(a) 50 ft.
(b) 30 ft.
(c) 26 ft.
(d) 31 ft.

Answer : A

Question. Area of the garden as a function of x, say A(x), can be represented as
(a) 200 + 2x²
(b) x – 2x²
(c) 200x – 2x²
(d) 200 – x²

Answer : C

Question. Maximum area of garden will be
(a) 2500 sq. ft
(b) 4000 sq. ft
(c) 5000 sq. ft
(d) 6000 sq. ft

Answer : C
 

The shape of a toy is given as f(x) = 6(2x⁴ – x²). To make the toy beautiful 2 sticks which are perpendicular to each other were placed at a point (2, 3), above the toy.

Question. Find the slope of the normal based on the position of the stick.
(a) 360
(b) –360
(c) 1/360
(d) −1/360

Answer : D

Question. Which value from the following may be abscissa of critical point?

Answer : B

Question. Find the second order derivative of the function at x = 5.
(a) 598
(b) 1,176
(c) 3,588
(d) 3312

Answer : C

Question. What will be the equation of the tangent at the critical point if it passes though (2, 3)?
(a) x + 360y = 1082
(b) y = 360x – 717
(c) x = 717y + 360
(d) none

Answer : B

Question. At which of the following intervals will f(x) be increasing?
(a) (–∞, –1/2) ∪ (1/2, ∞)
(b) (–1/2, 0) ∪ (1/2, ∞)
(c) (0, 1/2) ∪ (1/2, ∞)
(d) (–∞, –1/2) ∪ (0, ∞)

Answer : B
 

The government declare that farmers can get ₹300 per quintal for their onions on 1st July and after that the price will be dropped by ₹3 per quintal per extra day. Shyam’s father has 80 quintal of onions in the field on 1st July and he estimates that crop is increasing at the rate of 1 quintal per day.

Based on the given information, answer the following questions:

Question. Revenue R as a function of x can be represented as
(a) R(x) = 3x² – 60x – 24000
(b) R(x) = –3x² + 60x + 24000
(c) R(x) = 3x² + 40x + 16000
(d) R(x) = 3x² – 60x – 1400

Answer : B

Question. If x is the number of days after 1st July, then price and quantity of onion respectively can be expressed as
(a) ₹(300 – 3x), (80 + x) quintals
(b) ₹(300 – 3x), (80 – x) quintals
(c) ₹(300 + x), 80 quintals
(d) None of these

Answer : A

Question. On which day should Shyam’s father harvest the onions to maximise his revenue?
(a) 11^th July
(b) 20^th July
(c) 12^th July
(d) 22^nd July

Answer : A

Question. Find the number of days after 1^st July, when Shyam’s father attain maximum revenue.
(a) 10
(b) 20
(c) 12
(d) 22

Answer : A

Question. Maximum revenue is equal to
(a) ₹20,000
(b) ₹24,000
(c) ₹24,300
(d) ₹24,700

Answer : C
 

The Relation between the height of the plant (y in cm) with respect to exposure to sunlight is governed by the following equation y = 4x – (1/2)x² where x is the number of days exposed to sunlight.

Question. What is the number of days it will take for the plant to grow to the maximum height?
(a) 4
(b) 6
(c) 7
(b) 10

Answer : A

Question. The rate of growth of the plant with respect to sunlight is _

Answer : B

Question. What will be the height of the plant after 2 days?
(a) 4 cm
(b) 6 cm
(c) 8 cm
(d) 10 cm

Answer : B

Question. What is the maximum height of the plant?
(a) 12 cm
(b) 10 cm
(c) 8 cm
(d) 6 cm

Answer : C

Question. If the height of the plant is 7/2 cm, the number of days it has been exposed to the sunlight is
(a) 2
(b) 3
(c) 4
(d) 1

Answer : D