If α, β are roots of  and γ, δ are roots of  , show that .

A waterman rows a place 48 km distant and back in 14 hours. He finds that he can row 4 km with the stream in the same time as 3 km against the stream. Find the rate of the stream.

Compute  where

The value of x for which  is

a)

b) 1

c) 0

d)

In ΔABC, AD is an altitude from A. Given b > c, ∠C = 23 and
AD = then ∠B is equal to

a) 90°

b) 102°

c) 107°

d) 113°

Compute
 

Ascertain the existence of the limits
 
Can L’Hospital’s rule be applied in computing them? Does the formal application lead to the correct answer?

The base of the triangle is fixed; find the locus of the vertex when one of the base angles is double of the other.

Prove that the distances of two points P and Q, each from the polar of the other with respect to a circle, are to each other as the distance of the points from the centre of the circle.

A man takes a step forward with probability 0.4 and backward with probability 0.6. Find the probability that at the end of eleven steps he is one step away from the starting point.

(1 + ax)ⁿ = 1 + 8x + 24x² + .  .  . then a = .  . ., n = .  .  .

Let n be a positive integer. If the coefficient of the 2^nd, 3^rd and 4^th terms in the expansion of (1 + x)ⁿ are in arithmetic progression then n = …..

The sum of the rational terms in the expansion of  is

If  and = and f(0) = 0. Find the value of . Given that 0 < <

a)

b)

c)

d) 1

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

is equal to

a) 0

b) 4

c) 6

d) −4

If the function f: [0, 4] → ℝ is differentiable, then for a, b ε [0, 4]

a) 8 f (a) f (b)

b) 8 f (a) f '(b)

c) 8 f '(a) f (b)

d) 8 f '(a) f '(b)

An ellipse has eccentricity  and one of the focus at the point  It’s one directrix is the common tangent near to the point P to the circle  and the hyperbola . Then the equation of the ellipse in the statement form is . . . . .

Total number of ways in which six ‘+’ and four ‘’ signs can be arranged in a line so that no two ‘’signs occur together is …..

One Indian and four American men and their wives are to be seated randomly around a circular table. Then the conditional probability that Indian man is seated adjacent to his wife given that each American man is seated adjacent to his wife is

a)

b)

c)

d)

The larger of 99⁵⁰ + 100⁵⁰ and 101⁵⁰  is

Having given  , expand x in form
   and show that
 

To complete a certain work A alone would take m times as many days as B and C together; B would take n times as many days as A and C together; C alone would take p times as many days as A and B together. Show that the number of days in which each do it alone are as . Prove also .

Solve the equation
 

Solve
 
 

If the sum of m terms of an arithmetic progression is equal to sum of either the next n terms or the next p terms, prove that
 

A geometric and harmonic progression have the same p^th, q^th, r^th terms a, b, c respectively. Show that

Find all complex numbers such that

Find all positive integer n such that  

Find the roots of  and show that the points which represent them are collinear.

 is equal to

a)

b)

c)

d)

The expression
 
 is equal to

a) 0

b) 1

c) 3

d) sin4α + cosα

 and
where α, β ε [π, π]. Values of α, β which satisfy both the equations is/are

a) 0

b) 1

c) 2

d) 4

The equation  has

a) No solution

b) One solution

c) More than one real solution

d) Cannot be said

The general solution of
 is

a)

b)

c)

d)

The maximum value of cos₁ cos₂ cos_{3 ……} cos_nunder the restriction 0  _{1 , 2} , ₃ …. , _n   and cot₁ cot₂ cot_{3 ……} cot_n= 1 is

a)

b)

c)

d)

If   are altitudes of a triangle from the vertices A, B, C and Δ the area of the triangle then  

a) True

b) False

Find the number of solutions of  

a) 0

b) 1

c) 2

d) Infinitely many

Find all solutions of
 in

a)

b)

c)

d)

The number of solutions of  is

a) 0

b) One

c) Two

d) Infinite

Find limit
  (a ≠ kπ where k is an integer)

a)

b) cota

c) e^sina

d) e^cota

A right circular cylinder is inscribed in a right circular cone. Show that the curved surface of the cylinder will be maximum when the diameter of the cylinder is equal to the radius of the base of the cone.

Find the area of the greatest rectangle that can be inscribed in the ellipse   +  = 1

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  then the values of a, b are

a) a  R, b  R

b) a = 1, b  R

c) a  R, b = 2

d) a = 1, b = 2

 is

a)

b) 0

c)

d) ∞

Let f(a) = g(a) = k and their n^th derivatives f⁽ⁿ⁾(a), g⁽ⁿ⁾(a) exists and are not equal for some n. Further if  then k =

a) 4

b) 2

c) 1

d) 0

 is equal to

a) e⁴

b) e²

c) e³

d) e

Let y =  Find

a)

b)

c)

d) 0

Let ℝ be the set of real numbers and f : ℝ → ℝ such that for all x and y in ℝ, . Then f (x) is a constant.

a) True

b) False

Let P(asecθ, btanθ) and Q(asecɸ, btanɸ) where θ + ɸ =  be two points on the hyperbola . If (h, k) be the point of intersection of the normals at P and Q then k is equal to

a)

b)

c)

d)

Multiple choice

The function  

has local minimum at x =

a) 0

b) 1

c) 2

d) 3

Multiple choice

For which value of m, is the area of the region bounded by the curve y = x –x² and the line y = mx equal to

a) – 4

b) – 2

c) 2

d) 4

Sketch the region bounded by the curves y = x² and  . Find the area.

a)

b)

c)

d)

(Fill in the blanks) The function y = 2x² – ln|x| is monotonically increasing for values of x (≠0) satisfying the inequalities .  .  .  . and monotonically decreasing for values of x satisfying the inequalities .  .  .  .

a)

b)

c)

d)

Suppose f(x) is a function satisfying the following conditions

i) f(0) = 2, f(1) = 1

ii) f has a minimum value at x = 5/2 and

iii) for all x

 
where a, b are constants. Determine the constants a and b, and the function f(x).

a)

b)

c)

d)

The value of the integral
 is

a) sin⁻¹ x – 6tan⁻¹(sin⁻¹ x) + c

b) sin⁻¹x – 2(sinx)⁻¹ + c

c) sin⁻¹x – 2(sinx)⁻¹ − 6tan⁻¹(sin⁻¹x) + c

d) sin⁻¹x – 2(sinx)⁻¹ + 5tan⁻¹(sin⁻¹x) + c

 =

a) True

b) False

Show that the integral
 =

For any natural number m, show that
 

Solve  

Solve (2x + y + 1) dx + (4x + 2y – 1) dy = 0

Solve  

Solve (x³e^x – my²) dx + mxy dy = 0

Find the orthogonal trajectory of the family of curves  

Radium disintegrates at a rate proportional to the amount present. If the half life of radium is 1600 years, find the percentage remaining after 100 years.

Uranium disintegrates at a rate proportional to the amount present at any time. If m₁ and m₂ grams of uranium are present at time t₁ and t₂ respectively then show that the half life of uranium is  

A tank is initially filled with 100 litres of salt solution containing 0.1 kg of salt per litre. Fresh brine containing 0.2 kg of salt per litre runs into the tank at the rate of 5 litres / minute and the mixture, assumed to be kept uniform by stirring, runs out at the same rate. Find the amount of salt at any time and determine how long will it take for this amount to reach 15 kg.

Solve  

Solve cos (x + y) dy = dx

Show that the diagonals of the parallelogram formed by the four straight lines  are perpendicular to each other.

Find the equation of the line passing through the point (a, b) and the intersection of the lines  and  

A straight line is such that the algebraic sum of the perpendiculars let fall upon it from any number of fixed points is zero ; show that it always passes through a fixed point.

Prove that the straight lines joining the origin to the point of intersection of the straight line x – y = 2 and the curve  make equal angles with the axes.

Find the locus of the vertex of a triangle, given its base and the sum of m times the square of one side and n times the square of the other side.

Find the equation to the circle which cuts orthogonally each of the three circles
x² + y² + 2x + 17y + 4 = 0
x² + y² + 7x + 6y + 11 = 0
x² + y² – x +22y + 3 = 0

A line of fixed length (a + b) moves so that its ends are always on two fixed perpendicular lines; prove that the locus of a point which divides the line into portions of length and b, is an ellipse.

ABC is an isosceles triangle in a circle of radius r. If AB = AC and h is the altitude from A to BC then the triangle ABC has perimeter , area A = . . . . .

Also  . . . . .

The number of unit vectors perpendicular to the vectors  and  is

a) 1

b) 2

c) 3

d) Infinitely many

e) None of these

Let a, b, c be three non co–planar vectors and p, q, r are vectors defined by the relations
  
Then the value of the expression
 is equal to

a) 0

b) 1

c) 2

d) 3

The value of a so that the volume of parallelopiped formed by  becomes minimum is

a)  

b)  3

c)  

d)  

Let  . A vector in the plane of a and b whose projection on c is  is

a)

b) 3

c)

d)

The number of distinct real values of λ for which   are coplanar is

a) Zero

b) One

c) Two

d) three

If c be a given non-zero scalar and A and B be given non-zero vectors such that A is perpendicular to B, find the vector X which satisfies the equations

Let OABC be a parallelogram with O as the origin and OC a diagonal. Let D be the midpoint of OA. Using vector method, prove that BD and CO intersect in the same ratio.

Let A = . Determine a vector R satisfying  and .

Determine the value of c so that for all real x the vector cx and  make an obtuse angle with each other.

If the vectors
 

are coplanar then the value of  . . . . . .

Ten different letters of an alphabet are given. Words with five letters are formed from the given letters. Then the number of words which have at least one letter repeated is

a) 69760

b) 30240

c) 99748

d) None of these

The value of  is equal to

a)

b)

c)

d) None of these

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

How many different 9 digit numbers can be formed from the numbers 223355888 by rearranging its digits so that the odd digits occupy even positions

a) 16

b) 36

c) 60

d) 180

If

a)

b) [2, ∞)

c)

d)

A student is allowed to select at most n books from a collection of (2n + 1) books. If the total number of ways in which he can select at least one book is 63, find the value of n?

For a fixed value of n
D =
Then show that  is divisible by n

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

Let p be a prime and m be a positive integer. By mathematical induction on m, or otherwise, prove that whenever r is an integer such that p does not divide r, p divides

The sides AB, BC and CA of a triangle ABC have 3, 4 and 5 interior points respectively on them. The number of triangles that can be constructed using these interior points as vertices is .  .  .  .

India played two matches each with Australia and West indies. In any match the probability of India getting the points 0, 1, and 2 are 0.45, 0.05 and 0.50 respectively. Assuming that the outcomes are independent, the probability of India getting at least seven points is

a) 0.8730

b) 0.0875

c) 0.0625

d) 0.0250

An unbiased die with faces marked 1, 2, 3, 4, 5 and 6 is rolled 4 times. Out of four face values obtained, the probability that the minimum face value is not less than 2 and the maximum face value is not greater than 5 is

a) 16/81

b) 1/81

c) 80/81

d) 65/81

Three of the vertices of a regular hexagon are chosen at random. The probability that the triangle with three vertices is equilateral equals

a)

b)

c)

d)

For the three events A, B, C,

P(exactly one of A or B occurs) = P(exactly one of B or C occurs) = P(exactly one of C or A occurs) = p and P(all the three events occur simultaneously =  where . Then the probability of at least one of A, B, C occurring is

a)

b)

c)

d)

If from each of the three boxes containing 3 white and one black; 2 white and 2 black; 1 white and 3 black balls, one ball is drawn at random then the probability that 2 white and 1 black ball will be drawn is

a)

b)

c)

d)

The numbers are selected from the set S = {1, 2, 3, 4, 5, 6} without replacement one by one. Probability that the minimum of the two numbers is less than 4 is

a)

b)

c)

d)

(One or more correct answers)
Let E and F are two independent events. The probability that both E and F happen is  and the probability that neither E nor F happens is  , then

a)   

b)    

c)        

d)     

(One or more correct answers)
There are four machines and it is known that exactly two of them are faulty. They are tested one by one, in a random order till both the faulty machines are identified. Then the probability that only two tests are needed

a)

b)

c)

d)

The probability that a student passes in Mathematics, Physics and Chemistry are m, p and c respectively. Of these subjects, the student has 75% chances of passing in at least one, a 50% chance of passing in at least two and 40% chance of passing in exactly two. Which of the following relations is true?

a)

b)

c)

d)

An anti–aircraft gun can take a maximum of 4 shots at an enemy plane moving away from it. The probability of hitting the plane at the first shot, 2^nd, 3^rd and 4^th shots are 0.4, 0.3, 0.2 and 0.1 respectively. What is the probability that the gun hits the plane?