Question(s) from Search: IIT

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

For a triangle ABC it is given that  , then Δ ABC is equilateral.

a) True

b) False

True / False

The function f (x) =  is not one to one.

a) True

b) False

Find the set of all values of a such that  are sides of a triangle.

a) (0, 3)

b) (3, ∞)

c) (0, 5)

d) (5, ∞)

Fill in the blank

Let A be the set of n distinct elements then the total number of distinct functions from A to A is ……… and out of these …… are onto

a) n!, 1

b) nⁿ, n!

c) nⁿ, 1

d) none of the above

In a triangle of base a the ratio of the other two sides is  r (< 1). Then the altitude of the triangle is less than or equal to  .

a) True

b) False

The value of k such that  lies in the plane
  is

a) 7

b) – 7

c) No real value

d) 4

If ABCD are four points in a space, prove that

If a, b, c are distinct positive numbers then the expression
( b + c – a ) ( c + a – b ) ( a + b – c ) –abc is

a) Positive

b) Negative

c) Non–positive

d) None of these

If M is a 3 x 3 matrix where det (M) = 1 and MM^T = I, then prove that det (M – I) = 0.

Let A =

If U₁, U_2, U₃ are column matrices satisfying
AU₁ =  , AU₂ =  and AU₃ =

and U is a 3 x 3 matrix whose columns are U₁, U_2, U₃  then the value of [ 3  2  0 ] U  is

a)

b)

c)

d)

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.

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.

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

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 = .  .  .

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

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