Test

Let f(x) = (1 – x)/(1 + x), x ≠ 1. Then f(f(1/x)), x ≠ 0 and x ≠ -1, is

What is the value of the following definite integral?

2∫[(e^x).Cosx dx,limits of integral 0 to π/2 

Let f : R → R be a function defined as follows:

f(x) = |x − 1| + (x − 1).

Which of the following is not true for f?

Population of a city is 40 % male and 60 % female. Suppose also that 50 % of males and 30 % of females in the city smoke. The probability that a smoker in the city is male is closest to

A blue and a red die are thrown simultaneously. We define three events
as follows:
• Event E: the sum of the numbers on the two dice is 7.
• Event F: the number on the blue die equals 4.
• Event G: the number on the red die equals 3.
Which of the following statements is true?

The range of the function f(x) = 4^x + 2 ^x + 4 ^-x + 2 ^-x + 3 , where x ∈ (−∞, ∞), is

A “good” word is any seven letter word consisting of letters from{A,B,C} (some letters may be absent and some letter can be present
more than once), with the restriction that A cannot be followed by B,B cannot be followed by C, and C cannot be followed by A. How manygood words are there?

Let n be a positive integer and 0 < a < b < ∞ The total number of
real roots of the equation (x − a)^{2n + 1} + (x − b)^{2n + 1} = 0 is

Consider the optimization problem below:
max x + y
x,y
subject to 2x + y ≤ 14
−x + 2y ≤ 8
2x − y ≤ 10
x, y ≥ 0.
The value of the objective function at optimal solution of this optimization problem:

A random variable X is distributed in [0, 1]. Mr. Fox believes that X follows a distribution with cumulative density function (cdf) F : [0, 1] → [0, 1] and Mr. Goat believes that X follows a distribution with cdf G : [0, 1] → [0, 1]. Assume F and G are differentiable, F≠ G and F(x) ≤ G(x) for all x ∈ [0, 1]. Let E_F [X] and E_G[X] be the expected values of X for Mr. Fox and Mr. Goat respectively. Which of the following is true?

Let f : [0, 2] →[0, 1] be a function defined as follows:
f(x)    = x if x ≤ α
=  1/2 if x ∈ (α, 2].
where α ∈ (0, 2). Suppose X is a random variable distributed in [0, 2] with probability density function f. What is the probability that the realized value of X is greater than 1?

Let f and g be differentiable functions for 0 < x < 1 and f (0) = g(0) = 0, f (1) = 6. Suppose that for all x ∈ (1, 0) , the equality f ‘(x) = 2g ‘(x) holds. Then g(1) equals

13. Consider the following system of inequalities.

x₁ − x₂ ≤ 3

x₂ − x₃ ≤ −2

x₃  − x₄ ≤ 10

x₄  − x₂ ≤ α

x₄  − x₃ ≤ −4,
where α is a real number. A value of α for which this system has a
solution is

A fair coin is tossed infinite number of times. The probability that a head turns up for the first time after even number of tosses is

An entrance examination has 10 “true-false” questions. A student answers all the questions randomly and his probability of choosing the correct answer is 0.5. Each correct answer fetches a score of 1 to the student, while each incorrect answer fetches a score of zero. What is the probability that the student gets the mean score?

Consider the real valued function f(x) = ax² + bx + c defined on [1, 2]. Then it is always possible to get a k ∈ (1, 2) such that

In a sequence the first term is 1/3 .The second term equals the first term divided by 1 more than the first term. The third term equals the second term divided by 1 more than the second term, and so on. Then the 500^th term is

Let α ∈  (0, 1) and f(x) = x^α + (1-x)^α for all x ∈ [0, 1]. Then the
maximum value of f is

In how many ways can three persons, each throwing a single die once, make a score of 10?

The first term of an arithmetic progression is a and common difference is d ∈ (0, 1). Suppose the k-th term of this arithmetic progression equals the sum of the infinite geometric progression whose first term is a and common ratio is d. If a > 2 is a prime number, then which of the following is a possible value of d?

In period 1, a chicken gives birth to 2 chickens (so, there are three chickens after period 1). In period 2, each chicken born in period 1 either gives birth to 2 chickens or does not give birth to any chicken. If a chicken does not give birth to any chicken in a period, it does not give birth in any other subsequent periods. Continuing in this manner, in period (k + 1), a chicken born in period k either gives birth to 2 chickens or does not give birth to any chicken. This process is repeated for T periods – assume no chicken dies. After T periods, there are in total 31 chickens. The maximum and the minimum possible values of T are respectively

Consider the polynomial P(x) = ax³ + bx² + cx + d, where a, b, c, d ∈ {1, 2,…, 9}. If P(10) = 5861, then the value of c is

For what value of α does the equation (x − 1)( x² − 7x + α) = 0 have exactly two unique roots?

You are given five observations x₁, x₂, x₃, x₄, x₅ on a variable x, ordered from lowest to highest. Suppose x₅ is increased. Then,

There are 20 persons at a party. Each person shakes hands with some of the persons at the party. Let K be the number of persons who shook hands with odd number of persons. What is a possible value of K?

Suppose the sum of coefficients in the expansion (x + y)ⁿ is 4096. The largest coefficient in the expansion is:

Let f : R → R be a function that satisfies for all x, y ∈ R f(x + y)f(x − y) = .( f(x) + f(y))² − 4x²f(y) . Which of the following is not possible for f?

Consider the following function f : R → Z, where R is the set of all real numbers and Z is the set of all integers. f(x) = ⌈x⌉,where ⌈x⌉ is the smallest integer that is larger than x. Now, define anew function g as follows. For any x ∈ R, g(x) = |f(x)|−f(|x|), where |x| gives the absolute value of x. What is the range of g?

The value of

lim  (x + 1)/|x + 1|  is.

x→ -1

Let f : R → R be a function such that f(x) = 2 if x ≤ 2 and f(x) = a² − 3a if x > 2, where a is a positive integer. Which of the following is true?