Question(s) from Search: IIT

Suppose that the normals drawn at three different points on the

parabola  pass through the point (h, 0). Show that h > 2.

Through the vertex O of the parabola  chords OP and OQ are drawn at right angles. Show that for all positions of P, PQ cuts the axis of the parabola at a fixed point. Also find the locus of the midpoint of PQ.

For all x ε ( 0, 1 )

a)

b) ln (1 + x) < x

c) sinx > x

d) lnx > x

Given x = cy + bz, y = az + cx, z = bx + ay where x, y, z are not all zero, prove that  a² + b² + c² + 2abc = 1

Let and  are two complex numbers such that  then prove that .

The number of values of k for which the system of equations

(k + 1) x + 8y = 4k

kx + ( k + 3 ) y = 3k – 1

has infinitely many solutions is

a) 0

b) 1

c) 2

d) Infinity

Without expanding a determinant at any stage show that
 = Ax + B

where A, B are non-zero constants

True/False
If the complex numbers  represent the vertices of an equilateral triangle with  then .

a) True

b) False

The order of the differential equation whose general solution is given by  is

a) 5

b) 4

c) 3

d) 2

If f (x) =

a) f (x) is a strictly increasing function

b) f (x) has a local maxima

c) f (x) is a strictly decreasing function

d) f (x) is bounded

Let Δa =
Then show that  = c, a constant.

For any two complex numbers  and any real numbers  is equal to .  .  .  .

a)  

b)  

c)  

d)  

The locus of a variable point whose distance from  is  times its distance from the line  is

a) Ellipse

b) Parabola

c) Hyperbola

d) None of these

If  and  then  equals

a)

b)

c)

d) 1

The second degree polynomial satisfying f (0) = 0, f (1) = 1,  for all x ε [0, 1] is

a)

b) No such polynomial

c)

d)

For a > 0, d > 0, find the value of the determinant
 

a) 0

b) 1

c)

d)

Multiple choices

For real x, the function  will assume all real values provided

a)

b)

c)

d)

If the matrix A is equal to where a, b, c are real positive numbers, abc = 1 and A^TA = I then find the value of a³ + b³ + c³.

a) 1

b) 2

c) 3

d) 4

Prove if α, β are roots of the equation  and γ, δ are roots of  then show that
 

If the function f: [0, 4] → ℝ is differentiable, then for a, b ε [0, 4]

a) 8 f (a) f (b)

b) 8 f (a) f '(b)

c) 8 f '(a) f (b)

d) 8 f '(a) f '(b)

Find the equation of the common tangent in the first quadrant to the circle  and the ellipse   . Also find the length of the intercept of the tangent between the coordinate axis.

The value of the determinant  is …………

a) 0

b) 1

c) a² + b² + c² – abc

d) a² + b² + c² – 3abc

The equation  represents

a) An ellipse

b) A hyperbola

c) A circle

d) None of these

Find all the real values of x which satisfy  and  .

For the hyperbola  which of the following remains constant with change in α

a) Abscissae of vertices

b) Abscissae of focii

c) Eccentricity

d) Directrix