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

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

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 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 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

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

What are the two numbers whose sum is 24 and whose product is as large as possible?
A) 12 and 12
B) 10 and 14
C) 8 and 16
D) 6 and 18

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

The profit from a grove of orange trees is given by \(P(x) = ax + bx^2\), where a and b are constants and x is the number of orange trees per acre. How many trees per acre will maximize the profit?
A)\( -\frac{b}{2a}\)
B) \(-\frac{a}{2b}\)
C) \(-\frac{2b}{a}\)
D) \(-\frac{2a}{b}\)

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