Question(s) from Search: IIT

Let α, β, γ be distinct real numbers. The points with position vectors
 

a) Are collinear

b) Form an equilateral triangle

c) Form a scalene triangle

d) Form a right angled triangle

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)

Let  If c is a vector such that  and the angle between  and c is 30° then  is equal to

a)

b)

c) 2

d) 3

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?

A and B are independent events. The probability that both A and B occur is  and probability that neither of them occur is . Find the probability of the occurrence of A.

If a and b are two unit vectors such that  are perpendicular to each other then the angle between a and b is

a) 45°

b) 60°

c)

d)

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.

If  are non-coplanar vectors and
  then a.b₁ and a.are orthogonal.

Let A be a set containing n elements. A subset P of A is constructed at random. The set A is reconstructed by replacing the elements of P. A subset of Q of A is again chosen at random. Find the probability that P and Q have no elements in common.

If  then

a) Re(z) = 0

b) Im(z) = 0

c) Re(z) = 0, Im(z) > 0

d) Re(z) > 0, Im(z) < 0

Let z and ω be two non zero complex numbers such that |z| = |ω| and Arg(z) + Arg(ω) = π then z equals

a)  ω

b)  

c)  

d)   

The set of lines  where  is concurrent at the point . . .

If the algebraic sum of the perpendicular distance from the point
(2, 0), (0, 2) and (1, 1) to a variable straight line be zero then the line passes through a fixed point whose coordinates are

The equation of the circles through (1, 1) and the point of intersection of
 
is

a)

b)

c)

d) None of these

If a circle passes through the points (a, b) and cuts the circle  orthogonally, then the equation of the locus of its centre is

a)

b)

c)

d)

The locus of the centre of circles which touches externally  and which touches the Y-axis is given by the equation

a)

b)

c)

d)

A, B, C , D are four points in a plane with position vectors a, b, c, d respectively, such that . The point D then is the  . . . . . . .  of the triangle ABC.

If the vectors
 

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

A unit vector coplanar with  and  and perpendicular to  is . . . . .

The centre of the circle inscribed in the square formed by the lines  and

a) (4, 7)

b) (7, 4)

c) (9, 4)

d) (4, 9)

Let a, b, c be non-zero real numbers such that
 
 
Then the quadratic function  has

a) no root in (0, 2)

b) at least one root in (1, 2)

c) a double root in (0, 2)

d) two imaginary roots

Let  be a polynomial in a real variable x with 0< then the function p(x) has

a) neither maximum nor minimum

b) only one maximum

c) only one minimum

d) only one maximum and only one minimum

e) none of these

Let a given line L₁ intersect the X-axis and Y-axis at P and Q respectively. Let another line L₂ perpendicular to L₁ cut the X and Y axis at R and S respectively. Show that the locus of the point of intersection of the lines PS and QR is a circle passing through the origin.

The function defined by  is

a) Decreasing for all x

b) Decreasing in  and increasing in

c) Increasing for all x

d) Decreasing in  and increasing in  

Let a circle be given by . Find the condition on a and b if two chords each bisected by the X–axis can be drawn from .