Question(s) from Search: IIT

The value of . Given that a, x, y, z, b are in Arithmetic Progression while the value of . If a, x, y, z, b are in Harmonic Progression then find a and b.

If S₁, S₂, .  .  .  .,S_n are the sums of infinite geometric series whose first terms are 1, 2, 3,   .  .  ., n and whose common ratios are  respectively, then find the value of

Let a, b are real positive numbers. If a, A₁, A₂, b are in Arithmetic Progression, a, G₁, G₂, b are in Geometric Progression and a, H₁, H₂, b are in Harmonic Progression show that
 

Consider the lines

 ;

 
The unit vector perpendicular to both L₁ and L₂ is

a)

b)

c)

d)

The differential equation  determines a family of circles with

a) Variable radii and a fixed centre ( 0, 1)

b) Variable radii and a fixed centre ( 0, -1)

c) Fixed radius and a variable centre along the X-axis

d) Fixed radius and a variable centre along the Y-axis

If   and  , then show that
 

Let u (x) and v (x) satisfy the differential equations and  where p (x), f (x) and g (x) are continuous functions. If u (x₁) > v (x₁) for some x₁ and f (x) > g (x) for all x > x₁, prove that at any point (x, y) where x > x₁ does not satisfy the equations y = u (x) and y = v (x)

  is

Let f(x) = |x – 1|, then

a)

b)

c)

d) None of these

Let , then the set  is

a)  

b)  

c)  

d)  ϕ

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

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)

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

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)

Let n be an odd integer. If sin nθ =  for every value of θ, then

a) = 1, = 3

b) = 0, = n

c) = −1, = n

d) = 1, =

Multiple choices
Let and  (x is measured in radians) then x lies in the interval

a)

b)

c)

d)

 lies between –4 and 10.

a) True

b) False

Let  and  be three non-zero vectors such that c is a unit vector perpendicular to both the vectors a and b and the angle between the vectors a and b is  then
 is equal to

a) 1

b)

c)

d) None of these

Determine the smallest positive value of x (in degrees) for which  

a) 30°

b) 50°

c) 55°

d) 60°