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1. Let f be a real valued function defined on the interval (0, ∞) by f (x) = λ n x +

1) The document contains 6 problems related to probability, geometry, and linear algebra. 2) Problem 1 involves finding the area of a circle based on angles and distances between points related to chords and diameters of the circle. 3) Problem 2 asks to show that the absolute value of the derivative of a function f is less than 1 on the entire interval [0,1] given conditions on f and its second derivative.

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Jitender Gupta
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63 views1 page

1. Let f be a real valued function defined on the interval (0, ∞) by f (x) = λ n x +

1) The document contains 6 problems related to probability, geometry, and linear algebra. 2) Problem 1 involves finding the area of a circle based on angles and distances between points related to chords and diameters of the circle. 3) Problem 2 asks to show that the absolute value of the derivative of a function f is less than 1 on the entire interval [0,1] given conditions on f and its second derivative.

Uploaded by

Jitender Gupta
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
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1. A circle passes through three points A, B and C with the line segment AC as its diameter.

A
line passing through A intersects the chord BC at a point D inside the circle. If angles DAB
and CAB are and respectively and the distance between the point A and the mid point
of the line segment DC is d, prove that the area of the circles is
 

      (


)

2. For all x in [0, 1], let the second derivative f(x) of a function f(x) exist and satisfy |f(x)| <
1. If f(0)= f(1), then show that |f(x)| < 1 for all x in [0, 1].
2010
1. Let f be a real valued function defined on the interval (0, ) by f(x) = n x +

h  1 + sin t dt. Then which of the following statement(s) is (are) true?
(a)f(x) exists for all x (0 )
(b)f(x) exists for all x (0, ) and f is continuous on (0, )
(c)there exists > 1 such that |f(x)|<|f(x)| for all x (, )
(d)there exists > 1 such that |f(x)| + |f(x)| for all x (0 )
3. A coin has probability p of showing head when tossed. It is tossed n times. Let p denote the
probability that no two (or more) consecutive heads occur. Prove that p1=1, p2 =1-p2 and pn
= (1- p). pn-1 + p(l - p) pn - 2 for all n 3.
4. An urn contains m white and n black balls. A ball is drawn at random and is put back into
the urn along with k additional balls of the same colour as that, of the ball drawn. A ball is
again drawn at random. What is the probability that the ball drawn now is white?
5. An unbiased die, with faces numbered 1, 2, 3, 4, 5, 6, is thrown n times and the list of n
numbers showing up is noted. What is the probability that, among the numbers 1, 2, 3, 4, 5,
6, only three numbers appear in this list?
6. Eight players P1, P2 P8 play a knock-out tournament. It is known that whenever the
players Pi and Pj. play, the player P1 will win if i < j. Assuming that the players are paired
at random in each round, what is the probability that the player P4 reaches the final?
Let p be an odd prime number and T be the following set of 2 x 2 matrices: T =
a b
! : a, b, c {0, 1, 2, , p 1}+.
A = 
c d
1. The number of A in T such that A is either symmetric or skew-symmetric or both, and
det(A) divisible by p is
(a) (p 1)
(b) 2(p 1)
(c) (p 1) + 1
(d) 2p 1
2. The number of A in T such that the trace of A is not divisible by p but det(A) is divisible
by p is
(a) (p 1) (p p + 1) (b) p, (p 1)
(c) (p 1)
(d) (p 1)( p
2)
3. The number of A in T such that det(A) is not divisible by p is
(a) 2p
(b) p, 5p
(c) p, 3p
(d) p, p
1. Let S be a square of unit area. Consider any quadrilateral which has one vertex on each side
of S. If a, b, c and d denotes the length of the sides of the quadrilateral, prove that 2 a2 +
b2 + c2 + d2 4.
2. Let 0 < < /2 be a fixed angle. If P = (cos , sin ) & Q ={cos (-), sin (-)}, then Q is
obtained from P by
(a) clockwise rotation around origin through an angle
(b) anti clockwise rotation around origin through an angle
(c) reflection in the line through origin with slope tan
(d) reflection in the line through origin with slope tan /2

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