Question(s) from Search: IIT

Let a, b, c, d be real numbers in geometric progression. If u, v, w satisfy the system of equations

 
 
 
Then show that the roots of the equation
 
 
and  are reciprocal of each other.

Subjective Problems
Let f (x + y) = f (x) . f (y) for all x, y. Suppose f (5) = 2 and  = 3. Find f (5).

Find the natural number a for which  where the function f satisfies the relation f(x + y) = f(x) f(y) for all natural numbers x and y and further f(1) = 2.

The interior angles of a polygon are in Arithmetic Progression. The smallest angle is 120° and the common difference is 5. Find the number of sides of the polygon.

If where a > 0 and n is a positive integer then f(f(x)) = x.

a) True

b) False

A vector a has components 2p and 1 with respect to a rectangular cartesian system. This system is rotated through a certain angle about the origin in the counter clockwise sense. If with respect to new system a has components p + 1 and 1 then

a) p ≠ 0

b) p = 1 or p =

c) p = −1 or

d) p = 1 or p = −1

e) None of these

The domain of the function  is

If f is an even function defined on (−5, 5) then the four real values of x satisfying the equation  are

Let a₁, a₂, … a_n be positive real numbers in Geometric Progression. For each n let A_n, G_n, H_n be respectively the arithmetic mean, geometric mean and harmonic mean of a₁, a₂, .  .  .  ., a_n. Find the expressions for the Geometric mean of G₁, G₂, .  .  .  .G_n in terms of A₁, A₂, .  .  .  .,A_n; H₁, H₂, .  .  .  .H_n

Let  , 0 < x < 2 are integers m ≠ 0, n > 0 and let p be the left hand derivative of |x − 1| at x = 1. If , then

a) n = −1, m = 1

b) n = 1, m = −1

c) n = 2, m = 2

d) n > 2, n = m

For three vectors  which of the following expressions is not equal to any of the remaining three

a)

b)

c)

d)

If total number of runs scored in n matches is
 where n > 1 and the runs scored in the k^th match are given by k.2^{n + 1 – k}  where 1 ≤ k ≤ n. Find n.

In a triangle ABC if cotA, cotB, cotC are in Arithmetic Progression then a, b, c are in .  .  .  .  . Progression.

For any odd integer n ≥ 1,
n³ – (n – 1)³ + .  .  . + (−)^{n – 1} 1³ = .  .  .

A unit vector which is orthogonal to the vectors  and

coplanar with the vectors  and  is

a)

b)

c)

d)

The area of the equilateral triangle which contains three coins of unit radius is

a)  square units

b)  square units

c)  square units

d)  square units

a) True

b) False

a) True

b) False

Match the following  is

If the vectors b, c, d, are not coplanar then prove that a is parallel to the vector  

Prove by vector method or otherwise, that the point of intersection of the diagonals of a trapezium lies on the line passing through the mid points of the parallel sides (you may assume that the trapezium is not a parallelogram)

True / False

Let  are unit vectors. Suppose that  and the angle between B and  then

a) True

b) False

If  then  

a) True

b) False

Let  and  where O, A and B are non-collinear points. Let p denote the area of the quadrilateral OABC and let q denote the area of the quadrilateral with OA and OC as adjacent sides. If p = kq then k = .  .  .  .  .

Prove that  = 2[cosx + cos3x + cos5x + … + cos(2k−1)x] for any positive integer k. Hence prove that  =