Question(s) from Search: IIT

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

Let A and B be square matrices of equal degree, then which one is correct amongst the following

a) A + B = B + A

b) A + B = A – B

c) A – B = B – A

d) AB = BA

The edges of a parallelepiped are of unit length and are parallel to non-coplanar unit vectors  such that . Then the volume of the parallelepiped is

a)

b)

c)

d)

If  P =  , A =  and Q = PAP^T

then P^T (Q²⁰⁰⁵) P is equal to

a)

b)

c)

d)

Consider three planes
P₁ : x – y + z = 1

P₂ : x + y – z = −1

P₃  : x – 3y + 3z = 2

Let L₁, L₂, L₃ be lines of intersection of planes P₂ and P₃, P₃ and P_1, and P₁ and P₂ respectively.

Statement 1 – At least two of the lines L₁, L₂, L₃ are non parallel

Statement 2 – The three planes do not have a common point.

a) Statement 1 is true. Statement 2 is true. Statement 2 is a correct explanation of statement 1.

b) Statement 1 is true. Statement 2 is true. Statement 2 is not a correct explanation of statement 1.

c) Statement 1 is true. Statement 2 is false.

d) Statement 1 is false. Statement 2 is true.

Show that the system of equations
3x – y + 4z = 3
x + 2y − 3z = −2
6x + 5y + λz = −3
has at least one solution for any real number λ ≠ −5. Find the set of solutions if λ = −5

a)

b)

c)

d)

The solution of primitive equation
 is . If  and  then is

a)

b)

c)

d)

If  then prove that

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

Let f(x) be defined for all x > 0 and be continuous. If f(x) satisfies  for all x, y and f(e)=1 then

a) f(x) is bounded

b)

c) x f(x) → 1 as x → 0

d) f(x) = lnx

The number of values of x where the function  attains its maximum is

a) 0

b) 1

c) 2

d) infinite

  is

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

a)

b)

c)

d)

Domain of definition of the function   for real values of x is

a)

b)

c)

d)

Let λ and α be real. Find the set of all values of λ for which the system of linear equations
 
 
 
has a non-trivial solution. For λ = 1 find the value of α.