Question(s) from Search: IIT

Given  and f(x) = cosx – x(x + 1). Find the range of f (A).

Multiple choices

If the first and  term of an Arithmetic Progression, a Geometric Progression and a Harmonic Progression are equal and their n^th term are a, b, c respectively then

a)

b)

c)

d)

Show that the value of  wherever defined, never lies between  and 3.

Let  where A, B, C are real numbers. Prove that if f(n) is an integer whenever n is an integer, then the numbers 2A, A + B and C are all integers. Conversely prove that if the numbers 2A, A + B and C all integers then f(n) is an integer whenever n is an integer.

Let  and  be three non-zero vectors such that c is a unit vector perpendicular to both the vectors a and b and the angle between the vectors a and b is  then
 is equal to

a) 1

b)

c)

d) None of these

Does there exist a Geometric Progression containing 27, 8 and 12 as three of its terms? If it exists, how many such progressions are possible?

The values of  lies in the interval .  .  .

If  and  then (gof)(x) is equal to

If 0 < x < 1, then  is equal to

The sum of integers from 1 to 100 that are divisible by 2 or 5 is

The minimum value of the expression  where  are real numbers satisfying  is

a) Positive

b) Zero

c) Negative

d) –3

Using the relation , or otherwise prove that

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

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

a) True

b) False