a)

b)

c)

d)

If f(x) = x³ + bx² + cx + d and 0 < b² < c then in (−∞, ∞)

a) f(x) is a strictly increasing function

b) f(x) has a local maximum

c) f(x) is a strictly decreasing function

d) f(x) is bounded

The maximum value of  in the interval  is attained at

a)

b)

c)

d)

The value of  is

a) 2asina + a²cosa

b) 2asina − a²cosa

c) 2acosa + a²sina

d) None of these

Statement 1 : If e^xy +ln(xy) + cos(xy) + 5 = 0 then

Statement 2 :  then

a) Statement 1 is false, Statement 2 is true

b) Statement 1 is true, Statement 2 is true. Statement 2 is a correct explanation of statement 1.

c) Statement 1 is true, Statement 2 is true. Statement 2 is not a correct explanation of statement 1.

d) Statement 1 is true, Statement 2 is false

If f(x) = sign(x³), then

a) f(x) is continuous but not differentiable at x = 0

b)

c)

d) f is not differentiable at x = 0

If  then  at x = 1 is

a) 6 sin(ln5)

b) 5 sin(ln6)

c) 12 sin(ln5)

d) 5 sin[ln(12)]

The function defined by
   is

a) Continuous at x = 1

b) Continuous at x = 3

c) Differentiable at x = 1

d) All of the above

If  then f is

a) An increasing function

b) A decreasing function

c) Both increasing and decreasing function

d) None of the above

Let g(x) be the inverse of the function f(x) and   then,  is equal to

a)

b)

c)

d)

Let  and  

where a, b are non–negative real numbers. The value of a, if (gof)x is continuous for all values of x, is

a) – 1

b) 0

c) 1

d) 2

If the function f(x) = ax³ + bx² + 11x – 6 satisfies the conditions of Rolle’s theorem in [1, 3] and  then the value of a, b are respectively

a) − 1, 6

b) −2, 1

c) 1, − 6

d) − 1,

Let f(x) satisfies the requirement of Lagranges’ mean value theorem in [0, 2]. If f(0) = 0 and  for all values of x in [0, 2] then

a) f(x) ≤ 2

b) |f(x)| ≤ 1

c) f(x) = 2x

d) f(x) = 3 for at least one value of x in [0, 2]

Let f(x) = e^x, g(x) = sin⁻¹x, h(x) = f(g(x)) then

a)

b)

c)

d)

If 2a + 3b + 6c = 0 then at least one root of the polynomial

ax² +bx + c =0 lies in the interval

a) (1, 2)

b) (0, 1)

c) (2, 3)

d) None of these

If the function  

be continuous at , then k is equal to

a) 3

b) 6

c) 12

d) None of these

The longest distance of the point (a, 0) from the curve 2x² + y² – 2x = 0 is given by

a)

b)

c)

d)

The shortest distance between the line y – x = 1 and the curve y² = x is

a)  

b)

c)

d)

Of all rectangles having a given perimeter p, find the one with the maximum area.

Show that the semi–vertical angle of a cone of maximum volume and given slant height is .

The parcel post regulations restrict parcels to be such that the length plus girth must not exceed 6 feet and the length must not exceed 3.5 feet. Determine the parcel of the greatest volume that can be sent by post if the form of the parcel be a right circular cylinder.

A window frame has three straight sides and the fourth (top) semi circular. If the total perimeter is restricted to 40 feet find the dimensions so that the greatest amount of light may be admitted.

If sum of lengths of hypotenuse and another side of a right angle is given show that the area of the triangle is maximum when the angle between the sides is .