Question(s) from Search: IIT

Prove that  = 2[cosx + cos3x + cos5x + … + cos(2k−1)x] for any positive integer k. Hence prove that  =

The function
f(x) =|px – q| + r |x|, x ε (−, )
where p > 0, q > 0, r > 0 assumes minimum value on one point if

a) p ≠ q

b) r = q

c) r ≠ p

d) r = p = q

Let f : R → R be any function defined g : R → R by g (x) = |f (x)| for all x. Then g is

a) onto if f is onto

b) one to one if f is one to one

c) continuous if f is continuous

d) differentiable if f is differentiable

If f : [ 1,  → [ 2, ] is given by f (x) = x +  then ( x ) is given by

a)

b)

c)

d) 1 +

The function of f : R → R be defined by f (x) = 2x + sinx for x ε R . Then f is

a) one-one and onto

b) one-one but not onto

c) onto but not one-one

d) neither one-one nor onto

Multiple choice

There exists a triangle ABC satisfying the conditions

a) bsinA = a, A <

b) bsinA > a, A >

c) bsinA > a, A <

d) bsinA < a, A <, b > a

e) bsinA < a, A >, b = a

With usual notation if in a triangle ABC,  then

 .

a) True

b) False

If in a triangle ABC, cosA cosB + sinA sinB sin C = 1 then show that  a : b : c = 1 : 1 :

a) True

b) False

If the lines  and  intersect then the value of k is

a)

b)

c)

d)

The area of a triangle whose vertices are
 is

The parameter on which the value of the determinant
Δ =
does not depend upon is

a) a

b) p

c) d

d) x

Consider the lines

 ;

 
The unit vector perpendicular to both L₁ and L₂ is

a)

b)

c)

d)

If b > a then the equation ( x – a ) ( x – b )1 = 0 has

a) Both roots in [ a, b ]

b) Both roots in ( , a )

c) Both roots in (  )

d) One root in ( , a ) and other in ( )

For what value of m does the system of equations 3x + my = m, 2x − 5y = 20 have a solution satisfying the condition x > 0, y > 0.

a) m  (−∞, ∞)

b) m  (−∞, −15) ∪ (30, ∞)

c)  

d)  

If α is a repeated root of a quadratic equation f(x) = 0 and A(x), B(x), C(x) be polynomials of degree 3, 4, 5 respectively, Then show that
 

is divisible by f(x) where prime denotes the derivatives.

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