Question(s) from Search: IIT

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)

The number of solutions of  is

a) 3

b) 1

c) 2

d) 0

If a, b, c be positive and not all equal, show that the value of the determinant  is negative.

Match the following

Normals are drawn at the points P, Q and R lying on the parabola  which intersect at (3, 0) then

If a polynomial of degree 3, then  equals

a)

b)

c)

d) a constant

If ε then  is always greater than or equal to

a) 2 tan

b) 1

c) 2

d)

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