Question(s) from Search: IIT

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 the expression  is real then the set of all possible values of x is .  .  .  .

a) x = 2nπ or mπ + π/4

b) x = nπ or mπ + π/4

c) x = 2nπ or 2mπ + π/4

d) x = nπ or 2mπ + π/4

(Assertion and reason)

The question contains statement – 1 (assertion) and statement 2 (reason). Of these statements mark correct choice if

a) Statement 1 and 2 are true. Statement 2 is a correct explanation for statement 1.

b) Statement 1 and 2 are true. Statement 2 is not a correct explanation for statement 1.

c) Statement 1 is true. Statement 2 is false.

d) Statement 1 is false. Statement 2 is true

Statement 1 – The curve  is symmetric with respect to the line x = 1

Statement 2 – The parabola is symmetric about its axis.

If  then

a)

b)

c)

d)

Let P = (x, y) be any point on  with focii  and  equals

a) 8

b) 6

c) 10

d) 12

Let α, β be roots of the equation are the roots of the equation  then the value of r is equal to

a)

b)

c)

d)

Let y = f (x) be a curve passing through (1, 1) such that the triangle formed by the coordinate axes and the tangent at any point of the curve lies in the first quadrant and has area 2. Find the differential equation and determine all such possible curves.

If  
then the two triangles with vertices (x₁, y₁), (x₂, y₂), (x₃, y₃), and (a₁, b₁), (a₂, b₂), (a₃, b₃) must be congruent.

a) True

b) False

If  then

a)

b)

c)

d) f and g cannot be determined

A curve passes through  and slope at the point  is

. Find the equation of the curve and the area between the

curve and the X-axis in the fourth quadrant.

Find the integral solutions of the following system of inequality
 

a) Ø

b) x = 1

c) x = 2

d) x = 3

Cosine of angle of intersection of curve y = 3^{x – 1}lnx and y = x^x – 1 is

Let A =

 
AU₁ =  , AU₂ =  and AU₃ =

a) −1

b) 0

c) 1

d) 3

If f : [1, ∞) → [2, ∞) is given by  then  equals

a)

b)

c)

d)