Question(s) from Search: IIT

A variable plane at a distance of one unit from the origin cuts the coordinate axes at A, B and C. If the centroid D(x, y, z) of triangle ABC satisfies the relation  then the value of k is

a) 9

b)

c) 1

d) 3

Find the equation of the plane passing through the points (2, 1, 0), (4, 1, 1), (5, 0, 1). Find the point Q such that its distance from the plane is equal to the point P(2, 1, 6) from the plane and the line joining P and Q is perpendicular to the plane.

The unit vector perpendicular to the plane determined by
 is.

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.

Show that  =

For all A, B, C, P, Q, R show that
 = 0

Let f(x) = |x – 1|, then

a)

b)

c)

d) None of these

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

Let a, b, c be real numbers with a² + b² + c² = 1. Show that the equation represents a straight line
 = 0

Let , then the set  is

a)  

b)  

c)  

d)  ϕ

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  

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)

For the primitive differential equation
 

then  is

a) 3

b) 5

c) 1

d) 2

Consider the system of linear equations
 
 
 
Find the value of θ for which the systems of equations have non-trivial solutions.

The set of all solutions of the equation

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).

Multiple choices

If the first and  term of an Arithmetic Progression, a Geometric Progression and a Harmonic Progression are equal and their n^th term are a, b, c respectively then

a)

b)

c)

d)