Question(s) from Search: IIT

If in the expansion of (1 + x)^m (1 – x)ⁿ, the coefficients of x and x² are 3 and –6 respectively. then m is

a) 6

b) 9

c) 12

d) 24

Suppose  is an identity in x where  are constants and . Then the value of n = ……….

a) 4

b) 5

c) 6

d) 7

Prove that  is divisible by 25 for any natural number n.

The equation  has

a) No solution

b) One solution

c) More than one real solution

d) Cannot be said

The number of solutions of the equation

a) 0

b) 1

c) 2

d) Infinitely many

The number of values of x in the interval (0, 5π) satisfying the equation  is

a) 0

b) 5

c) 6

d) 10

Find the natural number a for which
 
where the function f satisfies the relation f (x + y) = f (x).f(y)for all natural numbers x and y and further f (1) = 2

If α + β =  and β + γ = α, then tanα equals

a) 2(tanβ + tanγ)

b) tanβ + tanγ

c) tanβ + 2tanγ

d) 2tanβ + tanγ

Let n be a positive integer and
(1 + x + x²)ⁿ = a₀ + a₁x + a₂x + a₂x² +  .  .  . + a_2nx²ⁿ then prove that
 

The larger of 99⁵⁰ + 100⁵⁰ and 101⁵⁰  is

Find all solutions of
 in

a)

b)

c)

d)

Let f (x) = sin x and g (x) = ln|x|. If the range of the composition

functions fog and gof are R₁ and R₂ respectively, then

a) R₁ = [ u : −1 ≤ u < 1], R₂ = [ v : − < v < 0 ]

b) R₁ = [ u : − < u < 0 ], R₂ = [ v : −1 ≤ v ≤ 0]

c) R₁ = [ u : −1 < u < 1], R₂ = [ v : − < v < 0 ]

d) R₁ = [ u : −1 ≤ u ≤ 1], R₂ = [ v : − < v ≤ 0 ]

a) True

b) False

Multiple choices
y = f ( x ) =  then

a) x = f (y)

b) f (1) = 3

c) y is increasing with x for x < 1

d) f is a rational function of x

Let f (x + y) = f (x) f (y) for all x, y. Suppose that f (5) = 2 and  (0) = 3. Find f (5).

a) 1

b) 2

c) 3

d) 6

One or more correct answers
In a triangle PQR, sin P, sin Q, sin R are in arithmetic progression then

a) Altitudes are in arithmetic progression

b) Altitudes are in harmonic progression

c) Medians are in geometric progression

d) Medians are in arithmetic progression

The external radii  of ΔABC are in harmonic progression then prove that a, b, c are in arithmetic progression

a) True

b) False

True / False

If f (x) = ( a – xⁿ )^1/n  where a > 0 and n is a positive integer then f ( f ( x ) ) = x.

a) True

b) False

Fill in the blank

The domain of the function f (x) =  is

a) [− 2, − 1]

b) [1, 2]

c) [− 2, − 1] ⋃ [1, 2]

d) None of the above

Both roots of the equation

( x – b) ( x – c) + (x – c) ( x – a) + (x – a) (x – b) = 0 are always

a) positive

b) negative

c) real

d) none of these

Two towns A and B are 60 meters apart. A school is to be built to serve 150 students in town A and 50 students in town B. If the total distance to be travelled by all the 200 students is to be as small as possible then the school should be built at

a) Town B

b) 45 km from town A

c) Town A

d) 45 km from town B

If then ab + bc + ca lies in the interval

a)  

b)  

c)  

d)  

Let α, β be roots of the equation (x – a) (x – b) = c, c ≠ 0. Then the roots of the equation (x – α) (x – β) + c = 0 are

a) a, c

b) b, c

c) a, b

d) a + c, b + c

If p, q ε {1, 2, 3, 4}. The number of equations of the form  having real roots is

a) 15

b) 9

c) 7

d) 8

For all x ε ( 0, 1 )

a)

b) ln (1 + x) < x

c) sinx > x

d) lnx > x