Question(s) from Search: IIT

Consider the lines

 ;

 
The shortest distance between L₁ and L₂ is

a) 0

b)

c)

d)

Let ABCD is the base of parallelopiped T and Aʹ.BʹCʹDʹ be the upper face. The parallelopiped is compressed so that the vertex Aʹ shifts to Aʹʹ on a parallelepiped S. If the volume of the new parallelopiped is 90% of the parallelopiped T, prove that the locus of Aʹʹ is a plane.

The differential equation representing the family of curves  where c is a positive parameter, is of

a) Order 1

b) Order 2

c) Degree 3

d) Degree 4

A normal is drawn at a point  of a curve meeting X-axis at Q. If PQ is of constant length k, then show that the differential equation of the curve is  

Area bounded by  and

Let a, b, c be real numbers. Then the following system of equations in x, y, z

  + −  = 1

  − +  = 1

−  + +  = 1  has

a) No solution

b) Unique solution

c) Infinitely many solutions

d) Finitely many solutions

The domain of definition of the function  is

a)  excluding  

b) [0, 1] excluding 0.5

c)  excluding x = 0

d) None of these

If  then

a)

b)

c)

d) f and g cannot be determined

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

a)

b)

c)

d)

Multiple choices with one or more than one correct answers
  then

a) x = f(y)

b) f(1) = 3

c) y increases with x for x < 1

d) f is a rational function of x

Given  and f(x) = cosx – x(x + 1). Find the range of f (A).

Show that the value of  wherever defined, never lies between  and 3.

Let  where A, B, C are real numbers. Prove that if f(n) is an integer whenever n is an integer, then the numbers 2A, A + B and C are all integers. Conversely prove that if the numbers 2A, A + B and C all integers then f(n) is an integer whenever n is an integer.

The values of  lies in the interval .  .  .

If  and  then (gof)(x) is equal to

If 0 < x < 1, then  is equal to

Let  where 0 ≤ x ≤ 1. Determine the area bounded by y = f (x), X–axis, x = 0 and x = 1.

a)

b)

c)

d)

A curve C has the property that the tangent drawn at any point P on C meets the co-ordinate axes at A and B, and P is the mid-point of AB. The curve passes through the point (1, 1). Determine the equation of the curve.

a) x²y = 1

b) x = y

c) xy = 1

d) x² = y

For a positive integer n, define
 then

a) a(100) ≤ 100

b) a(100) > 100

c) a(200) ≤ 100

d) a(200) > 100

Let –1 ≤ p ≤ 1. Show that the equation 4x³ – 3x – p = 0 has a unique root in the interval  and identify it.

a) p

b) p/3

c)

d)

Find the coordinates of all points P on the ellipse , for which the area of △PON is maximum where O denotes the origin and N the feet of perpendicular from O to the tangent at P.

If p is a natural number then prove that p^{n + 1} + (p + 1)^{2n – 1} is divisible by p² + p + 1 for every positive integer n.

Prove by mathematical induction that
 for every positive integer n.

Prove that  is an integer for every positive integer.

Let a hyperbola pass through the foci of the ellipse  . The transverse and conjugate axes of the hyperbola coincide with the major and minor axes of the given ellipse. Also the product of the eccentricity of the given ellipse and hyperbola is 1 then

a) Equation of the hyperbola is

b) Equation of the hyperbola is

c) Focus of the hyperbola is (5, 0)

d) Vertex of the hyperbola is