Question(s) from Search: IIT

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}

An ellipse has eccentricity  and one of the focus at the point  It’s one directrix is the common tangent near to the point P to the circle  and the hyperbola . Then the equation of the ellipse in the statement form is . . . . .

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

The equation  represents

a) No locus if k > 0

b) An ellipse if k < 0

c) A point if k = 0

d) A hyperbola if k > 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)

If the line  touches the hyperbola  then the point of contact is

a)

b)

c)

d)

Let  then one of the possible value of k is

a) 1

b) 2

c) 4

d) 16

Two events A and B have probabilities 0.25 and 0.50 respectively. The possibility of both A and B occur simultaneously is 0.14 then the probability that neither A nor B occur is

a) 0.39

b) 0.25

c) 0.11

d) None of these

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

The value of the integral
 is

a) sin⁻¹ x – 6tan⁻¹(sin⁻¹ x) + c

b) sin⁻¹x – 2(sinx)⁻¹ + c

c) sin⁻¹x – 2(sinx)⁻¹ − 6tan⁻¹(sin⁻¹x) + c

d) sin⁻¹x – 2(sinx)⁻¹ + 5tan⁻¹(sin⁻¹x) + c

Three identical dice are rolled. The probability that the same number will appear on each of them is

a)

b)

c)

d)

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

a) 2

b) 3

c) 5

d) 6

Integrate

a)

b)

c)

d)

An unbiased die with faces marked 1, 2, 3, 4, 5 and 6 is rolled 4 times. Out of four face values obtained, the probability that the minimum face value is not less than 2 and the maximum face value is not greater than 5 is

a) 16/81

b) 1/81

c) 80/81

d) 65/81

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 f(x) be the interval of  find

a) ½

b) 1

c) 2

d) 4

For the three events A, B, C,

P(exactly one of A or B occurs) = P(exactly one of B or C occurs) = P(exactly one of C or A occurs) = p and P(all the three events occur simultaneously =  where . Then the probability of at least one of A, B, C occurring is

a)

b)

c)

d)

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

An infinite Geometric Progression has first term x and sum 5 then

a)

b)

c)

d)

 =

a)

b)

c)

d)