Question(s) from Search: IIT

For the primitive differential equation
 

then  is

a) 3

b) 5

c) 1

d) 2

If  and  then f is

a) One-one and onto

b) One-one but not onto

c) Onto but not one-one

d) Neither one-one nor onto

If
and
Then f – g is

a) Neither one to one nor onto

b) One to one and onto

c) One to one and into

d) Many one and onto

Subjective Problems
Let f (x + y) = f (x) . f (y) for all x, y. Suppose f (5) = 2 and  = 3. Find f (5).

Find the natural number a for which  where the function f satisfies the relation f(x + y) = f(x) f(y) for all natural numbers x and y and further f(1) = 2.

If where a > 0 and n is a positive integer then f(f(x)) = x.

a) True

b) False

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  , 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

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

If  P =  , A =  and Q = PAP^T

then P^T (Q²⁰⁰⁵) P is equal to

a)

b)

c)

d)

Consider the lines

 ;

 
The distance of the point (1, 1, 1) from the plane through the point (−1, −2, −1) and whose normal is perpendicular to both lines L₁ and L₂ is

a)

b)

c)

d)

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)

A curve  passes through  and the tangent at  cuts the X-axis and Y-axis at A and B respectively such that then

a) Equation of the curve is

b) Normal at  is

c) Curve passes through

d) Equation of the curve is

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

Let y = f (x) be a curve passing through (1, 1) such that the triangle formed by the coordinate axes and the tangent at any point of the curve lies in the first quadrant and has area 2. Find the differential equation and determine all such possible curves.

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

a) 0

b) 1

c) 2

d) infinite

A curve passes through  and slope at the point  is

. Find the equation of the curve and the area between the

curve and the X-axis in the fourth quadrant.

If  are three non–coplanar vectors, then

  equals

a) 0

b)

c)

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)

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.

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.

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.

For any two vectors u and v prove that

i)

ii)