Question(s) from Search: IIT

One or more correct answers
In a triangle the length of the two larger sides are 10 and 9 respectively. If the angles are in arithmetic progression then the length of the third side can be

a)

b)

c) 5

d)

e) None of these

Let f (x) = Ax² + Bx + C where A, B , C are real numbers. Prove that if f (x) 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 are all integers then f ( x ) is an integer whenever x is an integer.

A ladder rests against a wall at an angle α to the horizontal. If its foot is pulled away from the wall through a distance a, so that it slides a distance b down the wall making an angle β with the horizontal, then .

a) True

b) False

Let be the vertices of an n sided regular polygon such that   . Then find n.

a) 5

b) 6

c) 7

d) 8

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 f(x) = 3x – 5 then  

a) is given by

b) is given by

c) does not exist because f is not one-one

d) does not exist because f is not onto

Find the integral solutions of the following system of inequality
 

a) x = 1

b) x = 2

c) x = 3

d) x = 4

Area bounded by  and

mn squares of equal size are arranged to form a rectangle of dimension m by n, where m and n are natural numbers. Two squares will be called neighbours if they have exactly one common side. A natural number is written in each square such that the number written in any square is the arithmetic mean of the numbers written in the neighbouring squares. Show that this is possible only if all the numbers used are equal.

Let A =
 
AU₁ =  , AU₂ =  and AU₃ =
 

a) 3

b) −3

c)  

d) 2

The domain of definition of  is

a)  

b)  

c)  

d)  

Let f : ℝ → ℝ be defined by f(x) = 2x + sinx for all x  ℝ. Then f is

a) One to one and onto

b) One to one but not onto

c) Onto but not one to one

d) Neither one to one nor onto

Range of    ;   x  ℝ is

a) (1, ∞)

b)

c)

d)

Let a, b, c, ε R and α, β be roots of  such that  and  then show that .