Application of Derivatives Class 12 Mcq Test (120601)

An Apache helicopter of the enemy is flying along the path given by the curve \(f(x) = x^2 + 7\). A soldier, placed at the point (1, 2), wants to shoot the helicopter when it is nearest to him. What is the nearest distance?
A) \(\sqrt{10}\) units
B) \(\sqrt{5}\) units
C) 5 units
D) 10 units

Find two positive numbers x and y such that \(x + y = 60\) and \(xy^3\) is maximum.
A) x = 15, y = 45
B) x = 20, y = 40
C) x = 25, y = 35
D) x = 30, y = 30

Find the maximum and minimum values of the function \(x + sin 2x\) on the interval \([0, 2\pi]\).
A) Maximum value: 3, Minimum value: -1
B) Maximum value: 2, Minimum value: -2
C) Maximum value: 4, Minimum value: 0
D) Maximum value: 1, Minimum value: -3

A rectangular sheet of tin 45 cm by 24 cm is to be made into a box without a top. What should be the side of the square to be cut off so that the volume of the box is maximum?
A) 6 cm
B) 8 cm
C) 10 cm
D) 12 cm

What is the value of 'a' in the function \(x^4 – 62x^2 + ax + 9\), if the function attains its maximum value at \(x = 1\) on the interval [0, 2]?
A) 33
B) 45
C) 28
D) 51

Find two positive numbers whose sum is 16 and the sum of whose cubes is minimum.
A) 4 and 4
B) 3 and 5
C) 2 and 6
D) 5 and 3

For what values of a, the function f given by \(f(x) = x^2 + ax + 1\) is increasing on [1, 2]?
A) All values of a
B) No values of a
C) Some values of a
D) None of these

A ball, thrown into the air from a building 60 meters high, travels along a path given by \(h(x) = 60 - x^2\). What is the maximum height the ball will reach?
A) 60 meters
B) 30 meters
C) 45 meters
D) 15 meters

Find two positive numbers x and y such that their sum is 35 and the product \(x^2 y^5\) is a maximum.
A) x = 5, y = 30
B) x = 10, y = 25
C) x = 15, y = 20
D) x = 20, y = 15

A square piece of tin of side 18 cm is to be made into a box without a top. What should be the side of the square to be cut off so that the volume of the box is the maximum possible?
A) 3 cm
B) 4 cm
C) 5 cm
D) 6 cm