CBSE Class 12 Mathematics Applications Of Derivatives Assignment Set C

1. A balloon, which always remains spherical, has a variable diameter 3/2(2x+ 1) .Find the rateof change of its volume with respect to x.

2 .The side of a square sheet is increasing at the rate of 4 cm per minute. At what rate is the area increasing when the side is 8 cm long ?

3. The radius of a circle is increasing at the rate of 0.7 cm/sec. what is the rate of increase of its circumference ?

 

1. Find the point on the curve y² = 8x for which the abscissa and ordinate change at the same rate?

2. A man 2 metre high walks at a uniform speed of 6km /h away from a lamp post 6 metre high. Find the rate at which the length of his shadow increases. Also find the rate at which the tip of the shadow is moving away from the lamp post.

3. The length of a rectangle is increasing at the rate of 3.5 cm/sec and its breadth is decreasing at the rate of 3cm/sec. find the rate of change of the area of the rectangle when length is 12 cm and breadth is 8 cm

 

 

CASE STUDY QUESTIONS

1. Yash wants to prepare a handmade gift box for his friend’s birthday at his home. For making lower part of the box, he took a square piece of paper of each side equal to 10 cm.
Based on the above information answer the following questions.

Question. If x cm be the size of square piece cut from each corner of the paper of size 10 cm, then possible value of x will be given by interval
(a) (0,10)
(b) ( 5,10)
(c) (0,5),
(d) (10,15)
Answer : A

Question. Yash is interested to maximise the volume of the box, So what will be the side of the square to be cut to maximise the volume
(a) 5cm
(b) 5/3cm
(c) 3cm
(d) 4cm
Answer : B

Question. Volume of the open box formed by folding up the cutting corner can be expressed as
(a) V=2x(10-2x)(10-2x)
(b) V=x(10-2x)(10-2x)
(c) V=x(10-x)(10-2x)
(d) V= x(10-x)(10-x)
Answer : B

Question. The value of x for which 𝑑𝑉/𝑑𝑥 = 0 𝑖𝑠
(a) 0,5
(b) 5/3 ,0
(c) 5/3,5
(d) 3,4
Answer : C

Question. The maximum volume is
(a) 1000cm3/27
(b) 3000cm3/27
(c) 2000cm3/27
(d) 1000cm3/27
Answer : C

2. A tank with rectangular base of length x metre, breath y metre and rectangular side, open at the top is to be constructed so that the depth is 1 m and volume is 9𝑚3.If building of tank is Rs 70 per square metre for the base and Rs 45 per square metre for the sides?

Based on above information answer the following questions.

Question. What is the cost of the base?
(a) 9xy
(b) 70xy
(c) xy
(d) 50xy
Answer : B

Question. For what value of x, C is minimum?
(a) 2
(b) 1
(c) 3
(d) 5
Answer : A

Question. What is the least cost of construction?
(a) Rs 1000
(b) Rs 1170
(c) Rs 1270
(d) 1570
Answer : C

Question. AWhat is the cost of making all the sides?
(a) 90(x+y)
(b) 90xy
(c) 9(x+y)
(d) 40(x +y)
Answer : A

ASSERTION AND REASON

Question. Assertion (A) : The tangent to the curve 𝒚 = 𝒙^𝟑 − 𝒙^𝟐 − 𝒙 + 𝟐 at (1,1) is parallel to the x- axis .
Reason (R ): The slope of the tangent to the curve at (1,1) is zero.
A. Both A and R are true and R is the correct explanation of A
B. Both A and R are true but R is the correct explanation of A
C. A is true but R is false
D. A is false but R is true
Answer : A

Question. Assertion(A): 𝒚 = 𝐥𝐨𝐠(𝟏 + 𝒙) − 𝟐𝒙/𝟐+𝒙 ,x>-1 is a decreasing function of x throughout its domain
Reason (R ): 𝒅𝒚/𝒅𝒙 > 0 for all x>-1
A. Both A and R are true and R is the correct explanation of A
B. Both A and R are true but R is the correct explanation of A
C. A is true but R is false
D. A is false but R is true
Answer : A

Question. Assertion(A): Function f(x)== 𝐱^𝟑 − 𝟑𝐱^𝟐 + 𝟑𝐱 + 𝟐 is always increasing. Reason(R):Derivative f’(x) is always negative.
A. Both A and R are true and R is the correct explanation of A
B. Both A and R are true but R is the correct explanation of A
C. A is true but R is false
D. A is false but R is true
Answer : C

Question. Function f(x) = logcosx is strictly increasing on (𝟎, 𝝅/2)
Reason( R): Slope of tangent on the above curve is negative in the given interval.
A. Both A and R are true and R is the correct explanation of A
B. Both A and R are true but R is the correct explanation of A
C. A is true but R is false
D. A is false but R is true
Answer : D

Question. Assertion(A): 𝒚 = 𝒆𝒙 is always strictly increasing function.
Reason (R): 𝒅𝒚/𝒅𝒙 = 𝒆^𝒙 > 0 for all real values of x.
A. Both A and R are true and R is the correct explanation of A
B. Both A and R are true but R is the correct explanation of A
C. A is true but R is false
D. A is false but R is true
Answer : A

Question. Assertion (A): Slope of the tangent to the curve y = 𝟑𝒙𝟒 − 𝟒𝒙 at x=4 is 764 Reason (R): The value of 𝒅𝒚/𝒅𝒙 = 𝟏𝟐𝒙^𝟑 − 𝟒 is 764 at x= 4
A. Both A and R are true and R is the correct explanation of A
B. Both A and R are true but R is the correct explanation of A
C. A is true but R is false
D. A is false but R is true
Answer : A

Question. Assertion (A) Tangent to the curve 𝒚 = 𝟐𝒙^𝟑 + 𝒙^𝟐 + 𝟐 at the point (-1,0) is parallel to the line y = 4x + 3
Reason (R): Slope of the tangent at (-1,0) is 4 equal to the slope of the given line .
A. A is false but R is true
B. Both A and R are true but R is the correct explanation of A
C. A is true but R is false
D. Both A and R are true and R is the correct explanation of A
Answer : A

Question. Assertion(A): Function f(x) =𝒙 + 𝟏/x is strictly increasing in the interval (-1,1)
Reason(R) : Derivative f’(x) <0 in the interval
A. Both A and R are true and R is the correct explanation of A
B. Both A and R are true but R is the correct explanation of A
C. A is true but R is false
D. A is false but R is true
Answer : C

Question. Assertion(A): At the (3,27) on the curve 𝒚 = 𝒙^𝟑, slope of the tangent is equal to y coordinate of the point.
Reason (R): 𝒅𝒚/𝒅𝒙 = 𝟑𝒙^𝟐 = 𝟐𝟕 𝒂𝒕 𝒙 = 𝟑
A. Both A and R are true and R is the correct explanation of A
B. Both A and R are true but R is the correct explanation of A
C. A is true but R is false
D. A is false but R is true
Answer : A

Question. Assertion(A): Y = sinx is increasing in the interval (𝝅/𝟐, 𝝅)
Reason(R): 𝒅𝒚/𝒅𝒙 is negative in the given interval.
A. Both A and R are true and R is the correct explanation of A
B. Both A and R are true but R is the correct explanation of A
C. A is true but R is false
D. A is false but R is true
Answer : C

Question. Assertion(A): Tangents to the curve y = 𝟕𝒙^𝟑 + 𝟏𝟏 at the points where x = 2 and x = – 2 are parallel.
Reason(R): Slope of tangents at both the points are equal.
A. Both A and R are true and R is the correct explanation of A
B. Both A and R are true but R is the correct explanation of A
C. A is true but R is false
D. A is false but R is true
Answer : A

Question. Assertion (A): Maximum value of the function f(x) =(𝟐𝒙 − 𝟏)^𝟐 + 𝟑 is 3.
Reason(R): f(x)≥ 𝟑 for all real values of x.
A. Both A and R are true and R is the correct explanation of A
B. Both A and R are true but R is the correct explanation of A
C. A is true but R is false
D. A is false but R is true
Answer : D 

Question. Assertion(A): The line y = x + 1 is a tangent to the curve 𝒚𝟐= 4x at the point (1,2).
Reason (R) : Slope of tangent to the given curve at the given point is 1 and the point also satisfies equation of the line.
A. Both A and R are true and R is the correct explanation of A
B. Both A and R are true but R is the correct explanation of A
C. A is true but R is false
D. A is false but R is true
Answer : A

Question. Assertion f(x) = 𝒆^𝒙 do not have maxima and minima
Reason ( R) : f ’(x) =𝒆𝒙 ≠ 𝟎 for all real values of x.
A. Both A and R are true and R is the correct explanation of A
B. Both A and R are true but R is the correct explanation of A
C. A is true but R is false
D. A is false but R is true
Answer : A

Question. Assertion (A): x = 0 is the point of local maxima of the function f given by 𝒇 = 𝟑𝒙^𝟒 + 𝟒𝒙^𝟑 − 𝟏𝟐𝒙^𝟐 +12
Reason(R): 𝒇′(𝒙) = 𝟎 𝒂𝒕 𝒙 = 𝟎 𝒂𝒏𝒅 𝒂𝒍𝒔𝒐 𝒇′′(𝒙) < 0 𝑎𝑡 𝑥 = 0
A. Both A and R are true and R is the correct explanation of A
B. Both A and R are true but R is the correct explanation of A
C. A is true but R is false
D. A is false but R is true
Answer : A