A square board is divided into 9 smaller identical squares by drawing lines. Three bullets are shot at the board randomly. The probability that at least 2 bullets hit the same small square is :

A and B complete a work in 30 days. B and C complete the same work in 24 days wherea C and A complete the same work in 28 days. Based on these statements which of the following conclusions is correct ?

Clock A loses 4 minutes every hour, clock B always shows the correct time and clock C gains 3 minutes every hour. On a Monday, all the three clock showed the same time, 8pm. On the following Wednesday, when the clock C shows 2 pm, what time will clock A show ?

In a class, there is one pencil for every two students, one eraser for every three students, and one ruler for every four students. If the total number of these stationery items required is 65 how many students are present in the class ?

The given table shows the numbers of active and recovered case of a certain disease. Assuming that the lienear trend for both continues, on which days will recovered cases be twice that of the active cases ?

Day	0	1	4	7	10
Active cases	990	1000	1030	1060	1090
Recovered cases	760	800	920	1040	1160

A boat weights 60kg, and oarsmen A and B weight 80 and 90kg, respectively. Rowing at a constant power, the time required to complete a course is proportional to the total weight. Rowing alone, A and B complete the course in 1 and 1½ hours, respectively. Assuming that their powers add up, how long will they ake to complete the course if the row together ?

The wavelength dependant absorbance of two compounds, A and B, is shown Absorbance of mixture is a linear function of the concentration of the two compounds. R is defined as a ratio of absorbance at 650nm to the abosrbance at 950 nm .

If the mixture contains 95% of compound A then R must be :

An epidemic is spreading in a population of size P. The rate of spread R of the disease at a given time is proportional to both, the number of people affected by the disease (N) and the number of people not yet affected by the disease. Which of the following graphs of R vs N is correct ?

The figure shows temperature and salinity of four samples of water. Which one of the samples has the highest density ?

Consider a parallelogram ABCD with centre O and E as the midpoint of side CD. The area of the triangle OAE, is :

The sum of the first n even numbers is :

A, B, C, D and E are the vertices of a regular pentagon as shown in the figure .

The angle ∠ABC is :

On a 200 m long straight road, maximum number of poles are fixed at 200 m interval. How many of the poles should be removed in order to have maximum number of poles at an interval of 40 m on the road ?

Let (En) be a squence of subsets of  .
Define

Which of the following statements is true ?

 be a bounded function. Which of the following statements is NOT true ?

Which of the following statements is true ?

The series  converges

Given (a_n)_{n ≥ 1} a sequence of real numbers, which of the following statements is true :

Given f, g are continuous function on [0,1] such that f(0) = f(1) = 0 ; = g (1) = 1 and . Which of the following statements is true ?

Let A be an n x n matrix such that the set of all its non-zero eigenvalues has exactly r elements. Which of the following statements is true ?

Let A and B be 2 x 2 metries. Then which on the following is true ?

If A = , then A²⁰ euqals

Let A be a 2 x 2 real matrix with det A = 1 and trace A = 3. What is the value of trace A² ?

For a, b ∈ , let p (x, y) = a²x₁y₁ + abx₂y₁ + abx₁y₂+b²x₂y₂, x = (x₁, x₂) y = (y₁, y₂) ∈ 
For what values of a and b does p :  x  →  define an inner product ?

Which of the following real quadratic froms on is positive definite ?

Let  be the positively oriented circle in the complex plane given by . Then  equals :

For a positive integer p, consider the holomorphic function 
For which values of p does there exist a holomorphic function  such that f(z) = g' (z) for  ?

Let  be the positively oriented circle in the complex plane given by . The line integral  equals :

Let p be a positie integer. Consider the closed curve r(t) = e^it, 0 ≤ t < 2π/. Let f be a function holomorphic in {z : |z| < 2 R} where R > 1. If f has a zero only at z₀, 0 < |z₀| < R, and it is of multiplicity q, then  equals

Which of the following statements is true ?

Let X be a non-empty set and P(X) be the set of all subsets of X. On P(X) defines two operations * and Δ as follows :
for A, B  P(X), A * B = A B ;
A Δ B = (A  B) (A  B) .
Which of the following statements is true ?

Let φ(n) be the cardinality of the set {a|1 ≤ a ≤ n, (a, n ) = 1} where (a, n) denotes the gcd of a and n. Which of the following is NOT true ?

For any two metric spaces (X, d_X), (Y, d_Y) a map  is said to be a closed map if whenever F is closed in X, then (F) is closed in Y. For any subset B of a metric space, B is given the induced metric. The metric on X x Y is given by d((x, y), (x', y')) = max {d_X(x, x'), d_Y(y, y')}. Which of the following are true ?

Let k be a positive integer. Consider the differential equation

Which of the following statements is true ?

Let y₀ > 0, Z₀ > 0 and α > 1 .
Consider the following tow differential equations :

We say that the solution to a differential equation exists globally if it exists for all t > 0 .
which of the following statements is true ?

The genral solution of the surface which are perpendicular to the family of surface
z² = kxy, k ϵ
is :

The general solution of the equation

Let be an infinitely differentiable real valued function on a bounded inerval I. Take n ≥ 1 interpolation points (x₀, x_i­, ..., x_n-1}. Take n additional interpolation points

where  is such that {x₀, x₁, ............... x_2n-1} are distinct .
Let P_2n-1 be the Lagrange interpolation polynomial of degree 2n - 1 with the interpolation points (x₀, x₁, ....., x_n-1} for the function .
Let q_2n-1 be the Hermite interpolation polynomial of degree 2n - 1 with the interpolation points {x₀, x₁, ...., x_n-1} for the function . In the  limit, the quantity

The extremal of the function

is :

The solution of the Fredholm integral equation
y(s) = s + 2  is :

Consider the solid S made of a material of constant density in the shape of a hemisphere of unit radius :
S = {x, y, z)|x²+y² +z² ≤ 1 z ≥ 0}
Which of the following statements is true ?

In an examination involving multiple choice questions, a student works out the solution in questions the student guesses the answer. However, when the answer is guessed the probbility that is is correct is 0.30. When the student works out the solutions it may be wrong with probability 0.10 .
If the answer to a particular question is correct, what is the probability that the student guessed the answer ?

Let X₁, X₂, .......... be i.i.d. randomly variables having a x² distribution with 5 degrees of freedom. Let a  be constant.Then the limiting distribution of a  is :

Consider a Markov Chain X₀, X₁, X₂, .......... with state space S. suppose i, j  S are two states which communicate with each other. Whcih of the following statements is NOT correct ?

Suppose that X has uniform distribution on the interval [0, 100]. Let Y denote the greatest integer smaller than or equal to X. Which of the following is true ?

Let X₁, X₂, ......... X_n be i.i.d random variables with common pdf , for 0 1 is an unknown parameter. Then the statistics is :

Consider the pdf given by

Based on one observation X with the above pdf, a UMP test of a size α for testing H₁ : θ ≤ θ₀ Versus H₁ : θ > θ₀ is :

Consider 35 i.i.d. observations X₁, X₂, ......... X₁₅ and Y₁, Y₂, .......... Y₂₀, Let R be the Wilicoxon's rank sum statistic based on the ranks of the X's in the combined sample. Then the expected value of R is :

Let I, J > 5. Consider two-way ANOVA, where the observations satisfy the linear model

In this set up

Let X₁ and X₂ be two i.i.d. p x 1 multivariate normal random vectors with mean µ and positive definite dispersion matrix Σ. Then which of the following random variables always has a central chi-square distribution .