Question(s) from Search: IIT

Match the following  is

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

If  are three non–coplanar vectors, then

  equals

a) 0

b)

c)

d)

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)

Multiple choice

Let  be three vectors. A vector in the plane of b and c whose projection on a is of magnitude  is

a)

b)

c)

d)

Let A be vector parallel to the line of intersection of planes P₁ and P₂. Plane P₁ is parallel to the vectors   and  and that P₂ is parallel to  and , then the angle between vector A and a given vector  is

a)

b)

c)

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

A vector A has components A₁, A₂, A₃ in a right handed rectangular cartesian coordinate system OXYZ. The coordinate system is rotated about the X–axis through an angle . Find the components of A in the new co-ordinate system in terms of A₁, A₂, A₃.

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

In a triangle OAB, E is the midpoint of BO and D is a point on AB such that AD : DB = 2 : 1. If OD and AE intercept at P determine the ratio OP : PD using vector methods.

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

The position vectors of the vertices A, B, C of a tetrahedron are  respectively. The altitude from the vertex D to the opposite face ABC meets the median line through A of the triangle ABC at E. If the length of the side AD is 4 and the volume of the tetrahedron is . Find the position vector of E or all possible positions.

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

For any two vectors u and v prove that

i)

ii)

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

True/False

If  for some non zero vector X then  

a) True

b) False

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