A stone tablet 7 feet high is placed on a wall with base 9 feet above the level of an observer’s eye. How far from the wall should the observer stand that the angle of vision subtended by the tablet at the observer’s eye may be maximum?

The value of  is

a) 2asina + a²cosa

b) 2asina - a²cosa

c) 2acosa + a²sina

d) None of  these

If f ( x ) = sgn (x³) then

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

b) f ʹ(0⁺) = 2

c) f ʹ(0^-) = 1

d)  f is not derivable at x = 0

If a + b + c = 0, then the equation 3ax² + 2bx + c = 0 has

a) At least one root in [ 0, 1 ]

b) One positive and one negative root

c) No real root

d) None of the above

  is equal to

a) 1

b) 1

c) 0

d) None of these

If the function f(x) = ax³ + bx² + 11x – 6 satisfies the conditions of Rolle’s theorem in [1, 3] and  = 0 then the values of a and b are respectively

a) -1, 6

b) -2, 1

c) 1, -6

d) -1,

If f(x) = cos(lnx) then
f  f     

is equal to

a) cos (x – y)

b) log (x – y)

c) cos (x + y)

d) None of these

The value of f(0), so that

f(x) =

is continuous everywhere is

a) 3(ln4)³

b) 4(ln4)³

c) 12(ln4)³

d) 15 (ln4)³

If  where
. Given F(5) = 5, then f(10) is equal to

a) 5

b) 10

c) 0

d) 15

Multiple choices with one or more than one correct answers
  then

a) x = f(y)

b) f(1) = 3

c) y increases with x for x < 1

d) f is a rational function of x

Given  and f(x) = cosx – x(x + 1). Find the range of f (A).

Subjective Problems
Let f (x + y) = f (x) . f (y) for all x, y. Suppose f (5) = 2 and  = 3. Find f (5).

The values of  lies in the interval .  .  .

Let  , 0 < x < 2 are integers m ≠ 0, n > 0 and let p be the left hand derivative of |x − 1| at x = 1. If , then

a) n = −1, m = 1

b) n = 1, m = −1

c) n = 2, m = 2

d) n > 2, n = m

Let f(x) be a non constant differentiable function defined on (−∞, ∞) such that f(x) = f(1 – x) and  then

a)  vanishes at twice an (0, 1)

b)

c)

d)

The domain of the derivative of the function
f (x) =

a) R  { 0 }

b) R

c) R

d) R

Using the relation , or otherwise prove that

a) True

b) False

a) True

b) False

For all ,

a) True

b) False

Cosine of angle of intersection of curve y = 3^{x – 1}lnx and y = x^x – 1 is

If

then the value of  is

a) 0

b) 1

c) −1

d) 2

Let  then for all x

a) f is not continuous

b) f is not differentiable for all x

c) f ʹ is continuous

d) None of the above

If  

Then f(x) is continuous and differentiable at x = 1, if

a)

b)

c)

d)