Question(s) from Search: IIT

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

2sinx + tanx > 3x where 0 ≤ x ≤

a) True

b) False

Let f(x) = (x + 1)² – 1, x ≥ −1 then the set {x : f(x) = f^-1(x)} is

a)

b) { 0, 1, −1}

c) {0, −1}

d) Ф

Suppose f (x) = (x + 1)² for x ≥ . If g (x) is the function whose graph is the reflection of the graph of f (x) with respect to the line y = x then g (x) equals

a) ,  0

b)

c)

d)

Let a, b, c be three positive real numbers and
 
Then tan θ = ………..

a) 0

b) 1

c) 2

d) 3

If X and Y are two sets and f : X  Y
If { f (c) = y, c ⊂ x, y ⊂ Y } then the true statement is

a)

b)

c) , a ⊂ X

d)

Let O (0, 0), P (3, 4), Q (6, 0) be the vertices of the triangle OPQ. The point inside the triangle OPQ is such that OPR, PQR, OQR are of equal area. The coordinates of R are

a)

b)

c)

d)

 If f be a one–one function with domain { x, y, z}and range { 1, 2, 3}. It is given that exactly one of the following statements is true and the remaining statements are false. Determine (1)

1. f(x) = 1

2. f(y) ≠ 1

3. f(z) ≠ 2

a) {0}

b) {1}

c) {y}

d) none of the above

One or more correct answers
In triangle ABC the internal angle bisector of ∠A meets the side BC in D. DE is a perpendicular to AD which meets AC in E and AB in F. Then

a) AE is harmonic mean of b and c

b) AD

c)

d) Δ AEF is isosceles