Question(s) from Search: IIT

Multiple choices

If f (x) = then

a) f (x) is continuous ∀ x  ℝ

b) f (x) > 0 ∀ x > 1

c) f (x) is continuous but not differentiable ∀ x  ℝ

d) f (x) is not differentiable at two points

Eight chairs are numbered one to eight. Two women and three men wish to occupy one chair each. First the women choose the chairs from amongst the chairs marked 1 to 4 then the men select the chairs from amongst the remaining. The number of possible arrangements is

a)

b)

c)

d) None of these

Find  if f(x) =

a) 0

b)

c)

d)

An n digit number is a positive number with exactly n–digits. Nine hundred distinct n–digit numbers are to be formed with only the three digits 2, 5 and 7. The smallest value of n for which this is possible is

a) 6

b) 7

c) 8

d) 9

Let f be a twice differentiable function such that  and , . Find h (10) if

h (5) = 1.

a) 0

b) 1

c) 2

d) 4

The number of arrangements of two letters of the word BANANA in which two N’s do not appear adjacently is

a) 40

b) 60

c) 80

d) 100

If , then find the values of n and r

Let

Determine a and b so that f is continuous at x = 0.

a)

b)

c)

d)

A box contains 2 white balls, 3 black balls and 4 red balls. In how many ways can three balls be drawn from a box if at least one black ball is to be included in the draw?

A committee of 12 is to be formed from 9 women and 8 men. In how many ways this can be if at least five women have to be in the committee? In how many ways in these committees (i) The women are in majority, (ii)The men are in majority

If a function f : is an odd function such that  for x ε [a, 2a] and the left hand derivative at

x = a is 0 then find the left hand derivative at x =  

a) 0

b) 1

c) a

d) 2a

A country produces 90% of its food diet. The population grows continuously at a rate of 3% per year. Its annual food production every year is 4% more than that of last year. Assuming that the average food requirement per person remains constant, prove that the country will become self sufficient in food after n years, where n is the smallest integer bigger than or equal to

f(x) is a function such that  and the tangent at any point passes through (1, 2). Find the equation of the tangent.

a) x = 2

b) y = 2

c) x + y = 2

d) x – y = 2

Let f(x) =

If f is continuous for all x, then k is equal to

a) 3

b) 5

c) 7

d) 9

 

 

Then

a) 0

b) 1

c) 2

d) 4

The complex numbers  satisfying  are the vertices of the triangle which is

a) of zero area

b) right angle isosceles

c) equilateral

d) obtuse angled isosceles

Let f(x) = x|x|. The set of points where f(x) is twice differentiable is .  .  .  .

a) ℝ

b) 0

c) ℝ − {0, 1}

a) – 1

b) 0

c) 1

d) 2

If  is purely real where ω = α + iβ, β ≠ 0 and z ≠ 1 then the set of real values of z is

a)  

b)  

c)  

d)  

If a continuous function f defined on the real line ℝ, assumes positive and negative values in ℝ then the equation f(x) = 0 has a root in ℝ. For example, it is known that if a continuous function f on ℝ is positive at some points and its minimum value is negative then the equation f(x) = 0 has a root in ℝ. Consider the function f(x) =  for all real x where k is a real constant.

The line y = x meets y =  for k ≤ 0 at

a) No point

b) One point

c) Two points

d) More than two points

(One or more than one correct answer)
Let  and  be complex numbers such that  and . If has positive real part and  has negative imaginary part, then  may be

a) Zero

b) Real and positive

c) Real and negative

d) None of these

Find the values of x and y for which the following equation is satisfied

a) x = y = −1

b) x = y = 3

c) x = 1, y = 3

d) x = 3, y = −1

If = x + iy then

a) x = 3, y = 1

b) x = 1, y = 3

c) x = 0, y = 3

d) x = 0, y = 0

It is given that n is an odd integer greater than 3 and not a multiple of 3. Prove that  is a factor of  

If A =  and B = then the value of α for which A² = B is

a) 1

b) −1

c) 4

d) No real values