Question(s) from Search: IIT

If the matrix A is equal to where a, b, c are real positive numbers, abc = 1 and A^TA = I then find the value of a³ + b³ + c³.

a) 1

b) 2

c) 3

d) 4

Prove if α, β are roots of the equation  and γ, δ are roots of  then show that
 

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)

Find the equation of the common tangent in the first quadrant to the circle  and the ellipse   . Also find the length of the intercept of the tangent between the coordinate axis.

A determinant is chosen at random from the set of all determinants of order 2 with elements 0 or 1 only. The probability that the value of the determinant chosen is positive is

a)

b)

c)

d)

If one root of  is equal to the power of the other then show that
 

For 0 < a < x the minimum value of the function log_ax + log_xa is 2.

a) True

b) False

The curve described parametrically by   ,  represents

a) A pair of straight lines

b) An ellipse

c) A parabola

d) A hyperbola

If a, b, c are in Geometric Progression then the equations  
have a common root if  are in

a) Arithmetic Progression

b) Geometric Progression

c) Harmonic Progression

d) None of these

The circle  intersects hyperbola  in four points then

a)

b)

c)

d)

Solve for x in the following equation

Let T_r be the r^th term of an Arithmetic Progression for  If for some positive integers m, n we have  and  then T_mn equals

a)

b)

c)

d)

The probability that at least one of the events A and B occurs is 0.6. If A and B occur simultaneously with probability 0.2 then  is

a) 0.4

b) 0.8

c) 1.2

d) 1.4

Consider an infinite geometric series with first term a and common ratio r. If its sum is four and the second term is  then

a)

b)

c)

d)

The probability of India winning a test match against West Indies is ½. Assuming independence of outcomes in each match, the probability that in a 5 test match series India’s second win will occur in the third test is

a)

b)

c)

d)

For every positive integer n, prove that
 
Hence or otherwise prove that
 
Where [ ] denotes greatest integer not exceeding x.

If  are positive real numbers whose product is a fixed number c then the minimum value of
  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)

Let a and b the roots of the equation  and those of  are c and d, then find the value of a + b + c + d when a ≠ b ≠ c ≠ d.

 =

a) True

b) False

The scalar  equals

a) 0

b)

c)

d) None of these

Fill in the blanks

If  is a root of the equation  where p and q are real then (p, q)  …………

If the m^th, n^th and p^th term of an Arithmetic Progression and a Geometric Progression are equal and are x, y, z then prove that
 

A fair die is rolled. The probability that 1 occurs at the even number of trail is

a)

b)

c)

d)

Fill in the blank

If the quadratic equation
  and  have a common root then the numerical value of a + b is …………