Question(s) from Search: IIT

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 .

If  where
. Given F(5) = 5, then f(10) is equal to

a) 5

b) 10

c) 0

d) 15

Subjective problems
Let .  Find all real values of x for which y takes real values.

a) [− 1, 2)

b)  [3, ∞)

c) [− 1, 2) ∪ [3, ∞)

d) None of the above

Let R be the set of real numbers and f : R → R be such that for all x and y in R, . Prove that f(x) is constant.

If f₁(x) and f₂(x) are defined by domains D₁ and D₂ respectively then f₁(x) + f₂(x) is defined as on D₁ ⋂ D₂

a) True

b) False

If  then the domain of f(x) is

The real numbers x₁, x₂, x₃ satisfying the equation x³ – x² + βx + γ = 0 are in Arithmetic Progression. Find the interval in which β and γ lie.

Let p, q, r be three mutually perpendicular vectors of the same magnitude. If x satisfies the equation p  ((x – q)  p) + q ((x – r)  q) + r  ((x – p)  r) = 0 then x is given by

a)

b)

c)

d)

Let f(x) be a non constant differentiable function defined on (−∞, ∞) such that f(x) = f(1 – x) and  then

a)  vanishes at twice an (0, 1)

b)

c)

d)