Question(s) from Search: IIT

Fifteen coupons are numbered 1, 2, 3, .  .  .   ., 15 respectively. Seven coupons are selected at random one at a time with replacement. The probability that the largest number appearing on a selected coupon is 9 is

a)

b)

c)

d) None of these

Match the statement of column 1 and the properties of column 2

A circle C of radius 1 is inscribed in an equilateral triangle PQR. The point of contacts of C with its sides PQ, QR and RP are D, E, F respectively. The line PQ is given by  and the point D is . Further, it is given that the origin and the centre of C are on the same side of the line PQ. The equation of circle C is

a)

b)

c)

d)

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.

Consider a branch of the hyperbola
 
with vertex at the point A. Let B be one of the end points of its latus rectum. If C is the focus of the hyperbola nearest to the point A, then the area of triangle ABC is

a)

b)

c)

d)

Solve the following equation for x
 

a) −1

b)

c) 0

d) −1 and

For non-zero vectors a, b, c,  holds if and only if

a) a . b = 0, b . c = 0

b) b . c = 0, c . a = 0

c) c . a = 0, a . b = 0

d) a . b = 0, b . c = 0, c . a = 0

(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)

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

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?

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

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

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)

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)

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

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.