Question(s) from Search: IIT

(One or more correct answers)
If E and F are independent events such that 0 < P (E) < 1 and 0 < P (F) < 1 then

a) E and F are mutually exclusive

b) E and  are independent

c)  are independent

d)

A lot contains 20 articles. The probability that the lot contains exactly 2 defective articles is 0.4 and the probability that the lot contains exactly three defective articles is 0.6. Articles are drawn from the lot at random one by one without replacement and tested till defective articles are found. What is the probability that the testing will end at the 12^th testing?

The points  in the complex plane are the vertices of a parallelogram if and only if

a)

b)

c)

d) None of these

If ω(≠1) is a cube root of unity and  then A and B are respectively

a) 0, 1

b) 1, 1

c) 1, 0

d) – 1, 1

Let V be the volume of the parallelepiped formed by the vectors  and . If a_r, b_r, c_r where r = 1, 2, 3 are non-negative real numbers and , show that V ≤ L³

The locus of the midpoint of a chord of the circle  which subtend a right angle at the origin is

a)

b)

c)

d)

If the normal to the curve y = f(x) at the point (3, 4) makes an angle  with the positive X–axis then

a) – 1

b)

c)

d) 1

A circle passes through points A, B and C with the line segment AC as its diameter. A line passing through A intersects the chord BC at D inside the circle. If ∠DAB and ∠CAB are α and β respectively and the distance between the point A and the midpoint of the line segment DC is d, prove that the area of the circle is
 

C₁ and C₂ are two concentric circles, the radius of C₂ being twice of C₁ . From a point on C₂ tangents PA and PB are drawn to C₁. Prove that the centroid of ΔPAB lies on C₁.

In [0, 1], Lagrange’s Mean Value theorem is not applicable to

a)

b)

c)

d)

For a positive integer n, define
 then

a) a(100) ≤ 100

b) a(100) > 100

c) a(200) ≤ 100

d) a(200) > 100

If p is a natural number then prove that p^{n + 1} + (p + 1)^{2n – 1} is divisible by p² + p + 1 for every positive integer n.

Prove by mathematical induction that
 for every positive integer n.

Prove that  is an integer for every positive integer.

Let a, b, c be positive real numbers such that b² – 4ac > 0 and let α₁ = c. Prove by induction that
 

Is well defined and  for n=1, 2, …

Here well defined means that the denominator in the expression of  is not zero.

Solve the following equation for x
 

a) −1

b)

c) 0

d) −1 and

Let E = {1, 2, 3, 4} and F = {1, 2} then the number of onto functions from E to F is

a) 14

b) 16

c) 12

d) 8

Which of the following pieces of data does not uniquely determine an acute angled triangle ABC (R being the radius of the circumcircle).

a) a, sinA, sinB

b) a, b , c

c) a, sinB, R

d) a, sinA, R

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

a) 1

b) 2

c) 3

d) 4

In Δ ABC the median to the side BC is of length  and divides ∠A into 30° and 45°. Then find the length of side BC.

a) 1

b) 2

c)

d)

If f is an even function defined on (−5, 5) then the real values of x satisfying the equation f (x) =  are ……………

a)

b)

c)

d)

The number of all possible triplets  such that
 for all x is

a) Zero

b) One

c) Three

d) Infinite

e) None

Two rays in the first quadrant x + y = |a| and ax – y = 1 intersect each other in the interval a ε (a₀, ∞). The value of a₀ is

then tan t =

The domain of the function y(x) given by the equation  is

a) 0 < x ≤ 1

b) 0 ≤ x ≤ 1

c)  < x ≤ 0

d)  < x < 1