Question(s) from Search: IIT

If |z| = 1 and  then Re (w) is

a) 0

b)

c)

d)

The equation of the locus of the midpoints of the chord of the circle  that subtends an angle of  at the centre is  . . . . .

Find the area bounded by the X–axis, part of the curve  and the ordinates at x = 2 and x = 4. If the ordinate x = a divides the area in two equal parts, find a.

a)

b)

c)

d)

The chord of contact of the pair of tangents drawn from each point on the line  to the circle  passes through the point . . . . .

The largest interval for which  is

a)

b)

c)

d)

Find the tangents to the curve
y = cos(x + y), − 2π ≤ x ≤ 2π
that are parallel to the line x + 2y = 0

The equation  has

a) No root

b) One root

c) Two equal roots

d) Infinitely many roots

If w ( ≠1 ) is cube root of unity, then
 

a) 0

b) 1

c) - 1

d) w

Let a, b, c be real numbers, a ≠ 0. If α is a root of β is a root of  and 0 < α < β then the equation  has a root γ that always satisfies

a) γ =

b) γ =

c) γ = α

d) α < γ < β

The determinant
 = 0 if

a) x, y, z are in arithmetic progression

b) x, y, z are in geometric progression

c) x, y, z are in harmonic progression

d) xy, yz, zx are in arithmetic progression

If  are the n roots of unity then show that .

The equation of the common tangent to the curves  and  is

a)

b)

c)

d)

If p, q, r are positive and are in arithmetic progression the roots of the quadratic  are all real for

a)

b)

c)

d)

The number of distinct roots of
 = 0
in the interval   ≤ x ≤   is

a) 0

b) 2

c) 1

d) 3

(Multiple choice)

The equation of common tangent to the parabolas  and  is/are

a)

b)

c)

d)

If α and β (α < β) are roots of the equation  where c < 0 < b then

a) 0 < α < β

b) α < 0 < β < | α |

c) α < β < 0

d) α < 0 < | α | < β

If A =  and | A³| = 125 then the value of α is

a) ± 1

b) ±2

c) ± 3

d) ± 5

Let and  be the roots of the equation  where the coefficients p and q may be complex numbers. Let A and B represent  in the complex plane. If  and OB = OA where O is the origin, prove that .

Three normals are drawn from the point (c, 0) to the curve . Show that c must be greater than. One normal is always the X-axis. Find c for which the other two normals are perpendicular.

For the equation  if one of the roots is square of the other then p is equal to

a)

b)

c) 3

d)

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