Question(s) from Search: IIT

For the function

The derivative from right  .  .  .  . and the derivative from the left  .  .  .  .

a) 0, 0

b) 0, 1

c) 1, 0

d) 1, 1

Let z₁ and z₂ be n^th roots of unity which subtend a right angle at the origin then n must be of the form

a) 4k + 1

b) 4k + 2

c) 4k + 3

d) 4k

If the triangle  another circle C₂ of radius 5 in such a manner that the common chord is of maximum length and a slope equal to  , then the coordinates of the centre of C₂ are . . . . .

Let ABC be a triangle such that
 

If A, B, C are in arithmetic progression, determine the values of A, B, C.

a) 30°, 60°, 90°

b) 30°, 75°, 75°

c) 45°, 60°, 75°

d) 60°, 60°, 60°

If f (x) =  and  

then (gof)(x) = …………

a) 0

b) 1

c) 2

d) 3

a) 0

b) 1

c) e³

d) e⁵

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.