APPLICATION OF DERIVATIVES

Important Questions

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Multiple Choice questions-

1. The rate of change of the area of a circle with respect to its radius r at r = 6 cm is:

(a) 10π

(b) 12π

(c) 8π

(d) 11π

2. The total revenue received from the sale of x units of a product is given by R(x)=3x²+36x+5. The marginal revenue, when x = 15 is:

(a) 116

(b) 96

(c) 90

(d) 126.

3. The interval in which y = x² e^-x is increasing with respect to x is:

(a) (-∞, ∞)

(b) (-2,0)

(c) (2, ∞)

(d) (0, 2).

4. The slope of the normal to the curve y = 2x² + 3 sin x at x = 0 is

(a) 3

(b) $\frac{1}{3}$

(c) -3

(d) –$\frac{1}{3}$

5. The line y = x + 1 is a tangent to the curve y² = 4x at the point:

(a) (1, 2)

(b) (2, 1)

(c) (1, -2)

(d) (-1, 2).

6. If f(x) = 3x² + 15x + 5, then the approximate value of f(3.02) is:

(a) 47.66

(b) 57.66

(c) 67.66

(d) 77.66.

7. The approximate change in the volume of a cube of side x meters caused by increasing the side by 3% is:

(a) 0.06 x³ m³

(b) 0.6 x³ m³

(c) 0.09 x³m³

(d) 0.9 x³ m³

8. The point on the curve x² = 2y, which is nearest to the point (0, 5), is:

(a) (2 √2, 4)

(b) (2 √2, 0)

(c) (0, 0)

(d) (2, 2).

9. For all real values of x, the minimum value of $\frac{1-x+x^2}{1+x+x^2}$ is

(a) 0

(b) 1

(c) 3

(d) $\frac{1}{3}$

10. The maximum value of [x (x – 1) + 1]^1/3, 0 ≤ x ≤ 1 is

(a) ($\frac{1}{3}$)^1/3

(b) $\frac{1}{2}$

(c) 1

(d) 0

  Very Short Questions:

Long Questions:

Assertion and Reason Questions:

1. Two statements are given-one labelled Assertion (A) and the other labelled Reason (R). Select the correct answer to these questions from the codes (a), (b), (c) and (d) as given below.

Assertion(A):For each real ‘t’, then exist a point C in [t,t+π]

such that f′(C) = 0

Reason (R):f(t)=f(t+2π) for each real t

2. Two statements are given-one labelled Assertion (A) and the other labelled Reason (R). Select the correct answer to these questions from the codes(a), (b), (c) and (d) as given below.

Assertion (A):One root of x³−2x²−1= 0 and lies between 2 and 3.

Reason(R):If f(x) is continuous function and f [a], f[b] have opposite signs then at least one or odd number of roots of f(x)=0 lies between a and b.

Case Study Questions:

1. An architecture design a auditorium for a school for its cultural activities. The floor of the auditorium is rectangular in shape and has a fixed perimeter P.

Based on the above information, answer the following questions.

2. Rohan, a student of class XII, visited his uncle’s flat with his father. He observe that the window of the house is in the form of a rectangle surmounted by a semicircular opening having perimeter 10m as shown in the figure.

Based on the above information, answer the following questions.

Answer Key-

Multiple Choice questions-

Very Short Answer:

Long Answer:

or Max ./Min. of A (x), A’ (x) = 0

20 – (2 + π)(2x) + πx = 0

20 + x(π – 4 – 2π) = 0

20 – x(4 + π) = 0

Case Study Answers:

1. Answer :

2. Answer :

Assertion and Reason Answers:

1. a) Both A and R are true and R is the correct explanation of A.

Solution:

Given that f(x)=2+cosx

 Clearly f(x) is continuous and differentiable everywhere Also f′(x) = −sinx⇒ f′(x=0)

⇒ −sinx = 0 ⇒ x = nπ

∴ These exists C ∈ [t, t+π] for t ∈ R

 such that f′ (C) = 0

∴ Statement-1 is true Also

f(x) being periodic function of period 2π

∴ Statement-2 is true, but Statement-2 is not a correct explanation of Statement -1.

2. (a)  Both A and R are true and R is the correct explanation of A.

Solution:

Given f(x)=x³−2x²−1=0

   Here, f(2)=(2)³−2(2)²−1=8−8−1=−1

   and f(3)=(3)³−2(3)²−1=27−18−1=8 

∴f(2)f(3)=(−1)8=−8<0 

⇒ One root of f(x) lies between 2 and 3 

∴ Given Assertion is true Also Reason R is true and valid reason 

∴ Both A and R are correct and R is correct explanation of A.

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