Question(s) from Search: IIT

Fill in the blank
If f (x) = sin ln  then the domain of f (x) is ………….

a) (−2, −1)

b) (−2, 1)

c) (0, 1)

d) (1, ∞)

If x, y, z are real and distinct then
8u =
is always

a) Non–negative

b) Non–positive

c) Zero

d) None of these

If  are any real numbers and n is any positive integer then

a)

b)

c)

d) none of these

Let a + b + c = 0, then the quadratic equation  has

a) at least one root in (0, 1)

b) one root in (2, 3) and the other in

c) imaginary roots

d) none of these

If α and β are roots of  and  are roots of  then the equation  has always

a) Two real roots

b) Two positive roots

c) Two negative roots

d) One positive and one negative root

The number of points of intersection of the two curves y = 2sinx and y =  is

a) 0

b) 1

c) 2

d)

The roots of the equation  are real and less than 3, then

a) a < 2

b) 2 < a < 3

c) 3 ≤ a ≤ 4

d) a > 4

Let f(x) =  and m(b) be the minimum value of f(x). As b varies, range of m(b) is

a)

b) [ 0,

c) [

d)

The set of all real numbers x for which  is

a)

b)

c)

d)

If one root is square of the other root of the equation  then the relation between p and q is

a)

b)

c)

d)

If a ≠ p, b ≠ q, c ≠ r and
 = 0

Then find the value of
  +  +

a) 0

b) 1

c) 2

d) 3

The number of solutions of the pair of equations

in the interval [ 0, 2π ] is

a) 0

b) 1

c) 2

d) 4

The equation  has

a) At least one real solution

b) Exactly three real solutions

c) Has exactly one irrational solution

d) Complex roots

Show that for for any triangle with sides a, b, c
3 (ab + bc + ac) ≤ (a + b + c)² < 4 (ab + bc + ca)

The solution set of equation  = 0 is ……….

a) {0}

b) {1, 2}

c) {−1, 2}

d) {−1, −2}

The equation  has

a) no real solutions

b) one real solution

c) two real solutions

d) infinite real solutions

For positive numbers x, y and z the numerical value of the determinant
 is ………..

a) 1

b) –1

c) ±1

d) 0

If a > 0, b > 0, c > 0, prove that  

The third term of Geometric Progression is 4. The product of the five terms is

a)

b)

c)

d)

Find the set of all x for which

Sum of the first n terms of the series  is

a) 2ⁿ – n – 1

b) 1 – 2^{− n}

c) n + 2^{− n} – 1

d) 2ⁿ + 1

Let  be in Arithmetic Progression and
 be in Harmonic Progression. If  and
 then  is

a) 2

b) 3

c) 5

d) 6

If α, β are roots of  and  are roots of  for some constant δ, then prove that
 

Let the positive numbers a, b, c, d be in Arithmetic Progression. Then
abc, abd, acd, bcd are

a) Not in Arithmetic Progression/Geometric Progression/Harmonic Progression

b) In Arithmetic Progression

c) In Geometric Progression

d) In Harmonic Progression

If  is the area of a triangle with sides a, b, c then show that
 .
Also show that equality occurs if a = b = c