Question(s) from Search: IIT

The equation of the directrix of the parabola y² + 4y + 4x +2 = 0 is

a) x = − 1

b) x = 1

c)

d)

Let α, β be roots of the equation (x – a) (x – b) = c, c ≠ 0. Then the roots of the equation (x – α) (x – β) + c = 0 are

a) a, c

b) b, c

c) a, b

d) a + c, b + c

If = x + iy then

a) x = 3, y = 1

b) x = 1, y = 3

c) x = 0, y = 3

d) x = 0, y = 0

It is given that n is an odd integer greater than 3 and not a multiple of 3. Prove that  is a factor of  

The focal chord of  is tangent to  then the possible value of the slope of this chord are

a)   

b)  

c)   

d)   

If p, q ε {1, 2, 3, 4}. The number of equations of the form  having real roots is

a) 15

b) 9

c) 7

d) 8

If A =  and B = then the value of α for which A² = B is

a) 1

b) −1

c) 4

d) No real values

If   then show that |z| = 1.

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

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

The domain of the definition of the function y given by the equation  is

a) 0 < x < 1

b) 0 ≤ x ≤ 1

c) ∞ < x ≤ 0

d) ∞ < x ≤ 1

Solution of the differential equation is

Let A =

If U₁, U_2, U₃ are column matrices satisfying
AU₁ =  , AU₂ =  and AU₃ =

and U is a 3 x 3 matrix whose columns are U₁, U_2, U₃  then the value of [ 3  2  0 ] U  is

a)

b)

c)

d)