Let $0 < x < \frac { 1 } { 6 }$ be a real number. When a certain biased dice is rolled, a particular face $F$ occurs with probability $\frac { 1 } { 6 } - x$ and and its opposite face occurs with probability $\frac { 1 } { 6 } + x$; the other four faces occur with probability $\frac { 1 } { 6 }$. Recall that opposite faces sum to 7 in any dice. Assume that the probability of obtaining the sum 7 when two such dice are rolled is $\frac { 13 } { 96 }$. Then, the value of $x$ is:
(A) $\frac { 1 } { 8 }$
(B) $\frac { 1 } { 12 }$
(C) $\frac { 1 } { 24 }$
(D) $\frac { 1 } { 27 }$.

Let $a _ { 1 } < a _ { 2 } < a _ { 3 } < a _ { 4 }$ be positive integers such that
$$\sum _ { i = 1 } ^ { 4 } \frac { 1 } { a _ { i } } = \frac { 11 } { 6 }$$
Then, $a _ { 4 } - a _ { 2 }$ equals
(A) 11
(B) 10
(C) 9
(D) 8 .

An examination has 20 questions. For each question the marks that can be obtained are either $-1$ or $0$ or $4$. Let $S$ be the set of possible total marks that a student can score in the examination. Then, the number of elements in $S$ is
(A) 93
(B) 94
(C) 95
(D) 96 .

An urn contains 30 balls out of which one is special. If 6 of these balls are taken out at random, what is the probability that the special ball is chosen?
(A) $\frac { 1 } { 30 }$
(B) $\frac { 1 } { 6 }$
(C) $\frac { 1 } { 5 }$
(D) $\frac { 1 } { 15 }$

Let $j$ be a number selected at random from $\{1, 2, \ldots, 2024\}$. What is the probability that $j$ is divisible by 9 and 15?
(A) $\frac{1}{23}$
(B) $\frac{1}{46}$
(C) $\frac{1}{44}$
(D) $\frac{1}{253}$

A biased die, with faces numbered from 1 to 6, has the property that each even face appears with probability twice that of each odd face. Calculate the probabilities of obtaining, by rolling the die once, respectively:
－a prime number
－a number at least 3
－a number at most 3

3. In a college of 300 students, every student reads 5 newspapers and every newspaper is read by 60 students. The number of newspaper is:
(A) at least 30
(B) at most 20
(C) exactly 25
(D) none of these

10. If from each of the three boxes containing 3 white and 1 black, 2 white and 2 black, 1 white and 3 black balls, one ball is drawn at random, then the probability that 2 while and 1 black ball will be drawn is:
(A) $13 / 32$
(B) $1 / 4$
(C) $1 / 32$
(D) $3 / 16$

26. There are four machines and it is known that exactly two of them are faulty. They are tested, one by one, in a random order till both the faulty machines are identified. Then the probability that only two tests are needed is:
(A) $1 / 3$
(B) $1 / 6$
(C) $1 / 2$
(D) $1 / 4$

31. A fair coin is tossed repeatedly. If tail appears on first four tosses, then the probability of head appearing on fifth toss equals:
(A) $1 / 2$
(B) $1 / 32$
(C) $31 / 32$
(D) $1 / 5$

33. Seven white balls and three black balls are randomly placed in a row. The probability that no two black balls are placed adjacently equals :
(A) $1 / 2$
(B) $7 / 15$
(C) $2 / 15$
(D) $1 / 3$

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1. Let n be an odd integer. If $\sin \mathrm { n } \theta = \sum _ { r = 0 } ^ { n } b _ { r } \sin ^ { r } \theta$, for every value of $\theta$, then :
   (A) $\mathrm { b } _ { 0 } = 1 , \mathrm {~b} _ { 1 } = 3$
   (B) $\quad \mathrm { b } _ { 0 } = 0 , \mathrm {~b} _ { 1 } = \mathrm { n }$
   (C) $\quad \mathrm { b } _ { 0 } = - 1 , \mathrm {~b} _ { 1 } = \mathrm { n }$
   (D) $\quad \mathrm { b } _ { 0 } = 0 , \mathrm {~b} _ { 1 } = \mathrm { n } ^ { 2 } - 3 \mathrm { n } + 3$
2. Which of the following number(s) is/are rational?
   (A) $\quad \sin 15 ^ { \circ }$
   (B) $\quad \cos 15 ^ { \circ }$
   (C) $\quad \sin 15 ^ { \circ } \cos 15 ^ { \circ }$
   (D) $\quad \sin 15 ^ { \circ } \cos 75 ^ { \circ }$
3. If the circle $x ^ { 2 } + y ^ { 2 } = a ^ { 2 }$ intersects the hyperbola $x y = c ^ { 2 }$ in four points $P \left( x _ { 1 } , y _ { 1 } \right) , Q \left( x _ { 2 } , y _ { 2 } \right) , R \left( x _ { 3 } , y _ { 3 } \right) , S \left( x _ { 4 } , y _ { 4 } \right)$, then:
   (A) $x _ { 1 } + x _ { 2 } + x _ { 3 } + x _ { 4 } = 0$
   (B) $y _ { 1 } + y _ { 2 } + y _ { 3 } + y _ { 4 } = 0$
   (C) $\quad x _ { 1 } x _ { 2 } x _ { 3 } x _ { 4 } = c ^ { 4 }$
   (D) $\quad y _ { 1 } y _ { 2 } y _ { 3 } y _ { 4 } = c ^ { 4 }$
4. If $E$ and $F$ are events with $P ( E ) \leq P ( F )$ and $P ( E \cap F ) > 0$, then:
   (A) occurrence of $\mathrm { E } \Rightarrow$ occurrence of F
   (B) occurrence of $\mathrm { F } \Rightarrow$ occurrence of E
   (C) non-occurrence of $\mathrm { E } \Rightarrow$ non-occurrence of F
   (D) none of the above implications holds
5. Which of the following expressions are meaningful question
   (A) $\vec { u } \cdot ( \vec { v } \times \vec { w } )$
   (B) $( \vec { u } , \vec { v } ) , \vec { w }$
   (C) $( \vec { u } , \vec { v } ) \vec { w }$
   (D) $\vec { u } \times ( \vec { v } , \vec { w } )$
6. If $\int _ { 0 } ^ { x } f ( \mathrm { t } ) \mathrm { dt } = \mathrm { x } + \int _ { x } ^ { 1 } t \mathrm { f } ( \mathrm { t } ) \mathrm { dt }$, then the value of $\mathrm { f } ( 1 )$ is:
   (A) $\frac { 1 } { 2 }$
   (B) 0
   (C) 1
   (D) $\quad - \frac { 1 } { 2 }$
7. Let $h ( x ) = f ( x ) - ( f ( x ) ) 2 + ( f ( x ) ) 3$ for every real number $x$. Then:
   (A) $h$ is increasing whenever $f$ is increasing
   (B) $h$ is increasing whenever $f$ is decreasing
   (C) $h$ is decreasing whenever $f$ is decreasing
   (D) nothing can be said in general

SECTION II

Instructions

There are 15 questions in this section. Each questions carries 8 marks. At the end of the answer to a question, leave 3 cm blank space, draw a horizontal line and start the answer to the next question. The corresponding question number must be written in the left margin. Answer all parts of a question at one place only.

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The use of only Arabic numerals $( 0,1,2 , \ldots \ldots , 9 )$ is allowed in answering the questions irrespective of the language in which you answer.

1. Suppose $f ( x )$ is a function satisfying the following conditions :
   (A) $\quad \mathrm { f } ( 0 ) = 2 , \mathrm { f } ( 1 ) = 1$.
   (B) f has a minimum value at $\mathrm { x } = \frac { 5 } { 2 }$, and
   (C) for all $\mathrm { X } , \mathrm { f } ^ { \prime } ( \mathrm { x } ) = \left| \begin{array} { c c c } 2 a x & 2 a x - 1 & 2 a x + b + 1 \\ b & b + 1 & - 1 \\ 2 ( a x + b ) & 2 a x + 2 b + 1 & 2 a x + b \end{array} \right|$ where $\mathrm { a } , \mathrm { b }$ are some constants. Determine the constants $\mathrm { a } , \mathrm { b }$ and the function $f ( x )$.
2. Let $p$ be a prime and $m$ a positive integer. By mathematical induction on $m$, or otherwise, prove that whenever $r$ is an integer such that p does not divide $\mathrm { r } , \mathrm { p }$ divides mpCr . [0pt] [Hint : You may use the fact that $( 1 + x ) ( m + 1 ) p = [ ( 1 + x ) p ( 1 + x ) m p ]$
3. A bird flies in a circle on a horizontal plane. An observer stands at a point on the ground. Suppose 600 and 300 are the maximum and the minimum angles of elevationof the bird and that they occur when the bird is at the points P and $Q$ respectively on its path. Let $q$ be the angle of elevation of the bird when it is at a point on the arc of the circle exactly midway between P and Q . Find the numerical value of $\tan 2 \mathrm { q }$. (Assume that the observer is not inside the vertical projection of the path of the bird.)
4. Prove that a triangle $A B C$ is equilateral if and only if $\tan A + \tan B + \tan C = 3 \sqrt { } 3$.
5. Using co-ordinate geometry, prove that the three altitudes of any triangle are concurrent.
6. C 1 and C 2 are two concentric circles, the radius of C 2 being twice that of C 1 . From a point P on C 2 . Tangents PA and PB are drawn to C 1 . Prove that the centroid of the triangle PAB lies on C 1 .
7. The angle between a pair of tangents drawn from a point $P$ to the parabola $y 2 = 4 a x$ is 450 . Show that the locus of the point P is a hyperbola.
8. if $\mathrm { y } = \frac { a x ^ { 2 } } { ( x - a ) ( x - b ( x - c ) } + \frac { b x } { ( x - b ) ( x - c ) } + \frac { c } { ( x - c ) } + 1$, prove that $\frac { y ^ { \prime } } { y } = \frac { 1 } { x } \left( \frac { a } { a - x } + \frac { b } { b - x } + \frac { c } { c - x } \right)$.
9. Prove that $\int 01 \tan - 1 ( 1 - x + x 2 ) d x$.
10. A curve $C$ has the property that if the tangent drawn at any point $P$ on Cmeet the coordinate axes at $A$ and $B$, then $P$ is the mid-point of $A B$. The curve passes through the point $( 1,1 )$. Determine the equation of the curve.
11. Three players $A$, $B$ and $C$, toss a coin cyclically in that order (that is $A , B , C , A , B , C , A$, $\mathrm { B } , \ldots \ldots \ldots$. ) till a head shows. Let p be the probability that the coin shows a head. Let $\mathrm { a } , \mathrm { b }$ and

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$y$ be, respectively, the probability that $A , B$ and $C$ gets the first head. Prov that $b = ( 1 -$ p) a. Determine a, b and y (in terms of p). 12. Prove, by vector methods or otherwise, that the point of intersection of the diagonals of a trapezium lies on the line passing through the mid-points of the parallel sides. (You may assume that the trapezium is not a parallelogram.) 13. For any two vectors $\vec { u }$ and $\vec { v }$ prove that
(A) $\quad ( \vec { u } . \vec { v } ) ^ { 2 } + | \vec { u } \times \vec { v } | ^ { 2 } = | \vec { u } | ^ { 2 } | \vec { v } | ^ { 2 }$ and
(B) $\quad \left( 1 + | \vec { u } | ^ { 2 } \right) \left( 1 + | \vec { v } | ^ { 2 } \right) = ( 1 - \vec { u } \cdot \vec { v } ) ^ { 2 } | \vec { u } + \vec { v } + ( \vec { u } \times \vec { v } ) | ^ { 2 }$ 14. Let $f ( x ) = A x 2 + B x + C$ where $A , B , C$ are real numbers. Prove that if $f ( x )$ is an integer whenever $x$ is an integer, then the numbers $2 A , A + B$ and $C$ are all integers. Conversely, prove that if the numbers $2 A , A + B$ and $C$ are all integers then $f ( x )$ is an integer whenever $x$ is an integer. 15. Let $C 1$ and $C 2$ be the graphs of the function $y = x 2$ and $y = 2 x , 0 \pounds x \pounds 1$ respectively. Let C3 be the graph of a function $y = f ( x ) , 0 \pounds x \pounds 1 , f ( 0 ) = 0$. For a point Pon C1, let the lines through P , parallel to the axes, meet C 2 and C 3 at Q and R respectively (see figure). If the for every position of P (on C1), the areas of the shaded regions OPQ and ORP are equal, determine the function $f ( x )$. [Figure]

25. If the integers $m$ and $n$ are chosen at random between 1 and 100 , then the probability that a number of the form $7 m + 7 n$ is divisible by 5 equals :
(A) $1 / 4$
(B) $1 / 7$
(C) $1 / 8$
(D) $1 / 49$

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DIRECTIONS : Question numbers $26 - 35$ carry 3 marks each and may have more than one correct answers. All correct answers must be marked to get any credit in these questions.

14. (a) 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?
(b) 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?

For a student to qualify, he must pass at least two out of three exams. The probability that he will pass the 1st exam is P . If he fails in one of the exams then the probability of his passing in the next exam is $\mathrm { P } / 2$ otherwise it remains the same. Find the probability that he will qualify.

10. Two numbers are selected randomly from the set $S = \{ 1,2,3,4,5,6 \}$ without replacement one by one. The probability that minimum of the two numbers is less than 4 is :
(a) $[ 1 / 15 ]$
(b) $[ 14 / 15 ]$
(c) $[ 1 / 5 ]$
(d) $[ 4 / 5 ]$

19. A bag contains 12 red balls and 6 white balls. Six balls are drawn one by one without replacement of which atleast 4 balls are white. Find the probability that in the next two draws exactly one white ball is drawn. (leave the answer in terms of ${ } ^ { \mathrm { n } } \mathrm { C } _ { \mathrm { r } }$ ).
Sol. Let $\mathrm { P } ( \mathrm { A } )$ be the probability that atleast 4 white balls have been drawn. $P \left( A _ { 1 } \right)$ be the probability that exactly 4 white balls have been drawn. $\mathrm { P } \left( \mathrm { A } _ { 2 } \right)$ be the probability that exactly 5 white balls have been drawn. $P \left( A _ { 3 } \right)$ be the probability that exactly 6 white balls have been drawn. $\mathrm { P } ( \mathrm { B } )$ be the probability that exactly 1 white ball is drawn from two draws. $\mathrm { P } ( \mathrm { B } / \mathrm { A } ) = \frac { \sum _ { \mathrm { i } = 1 } ^ { 3 } \mathrm { P } \left( \mathrm { A } _ { \mathrm { i } } \right) \mathrm { P } \left( \mathrm { B } / \mathrm { A } _ { \mathrm { i } } \right) } { \sum _ { \mathrm { i } = 1 } ^ { 3 } \mathrm { P } \left( \mathrm { A } _ { \mathrm { i } } \right) } = \frac { \frac { { } ^ { 12 } \mathrm { C } _ { 2 } { } ^ { 6 } \mathrm { C } _ { 4 } } { { } ^ { 18 } \mathrm { C } _ { 6 } } \cdot \frac { { } ^ { 10 } \mathrm { C } _ { 1 } { } ^ { 2 } \mathrm { C } _ { 1 } } { { } ^ { 12 } \mathrm { C } _ { 2 } } + \frac { { } ^ { 12 } \mathrm { C } _ { 1 } { } ^ { 6 } \mathrm { C } _ { 5 } } { { } ^ { 18 } \mathrm { C } _ { 6 } } \cdot \frac { { } ^ { 11 } \mathrm { C } _ { 1 } { } ^ { 1 } \mathrm { C } _ { 1 } } { { } ^ { 12 } \mathrm { C } _ { 2 } } } { \frac { { } ^ { 12 } \mathrm { C } _ { 2 } { } ^ { 6 } \mathrm { C } _ { 4 } } { { } ^ { 18 } \mathrm { C } _ { 6 } } + \frac { { } ^ { 12 } \mathrm { C } _ { 1 } { } ^ { 6 } \mathrm { C } _ { 5 } } { { } ^ { 18 } \mathrm { C } _ { 6 } } + \frac { { } ^ { 12 } \mathrm { C } _ { 0 } { } ^ { 6 } \mathrm { C } _ { 6 } } { { } ^ { 18 } \mathrm { C } _ { 6 } } }$ $= \frac { { } ^ { 12 } \mathrm { C } _ { 2 } { } ^ { 6 } \mathrm { C } _ { 4 } { } ^ { 10 } \mathrm { C } _ { 1 } { } ^ { 2 } \mathrm { C } _ { 1 } + { } ^ { 12 } \mathrm { C } _ { 1 } { } ^ { 6 } \mathrm { C } _ { 5 } { } ^ { 11 } \mathrm { C } _ { 1 } { } ^ { 1 } \mathrm { C } _ { 1 } } { { } ^ { 12 } \mathrm { C } _ { 2 } \left( { } ^ { 12 } \mathrm { C } _ { 2 } { } ^ { 6 } \mathrm { C } _ { 4 } + { } ^ { 12 } \mathrm { C } _ { 1 } { } ^ { 6 } \mathrm { C } _ { 5 } + { } ^ { 12 } \mathrm { C } _ { 0 } { } ^ { 6 } \mathrm { C } _ { 6 } \right) }$

19. A bag contains 12 red balls and 6 white balls. Six balls are drawn one by one without replacement of which atleast 4 balls are white. Find the probability that in the next two draws exactly one white ball is drawn. (leave the answer in terms of ${ } ^ { \mathrm { n } } \mathrm { C } _ { \mathrm { r } }$ ).
Sol. Let $\mathrm { P } ( \mathrm { A } )$ be the probability that atleast 4 white balls have been drawn. $P \left( A _ { 1 } \right)$ be the probability that exactly 4 white balls have been drawn. $\mathrm { P } \left( \mathrm { A } _ { 2 } \right)$ be the probability that exactly 5 white balls have been drawn. $P \left( A _ { 3 } \right)$ be the probability that exactly 6 white balls have been drawn. $\mathrm { P } ( \mathrm { B } )$ be the probability that exactly 1 white ball is drawn from two draws. $\mathrm { P } ( \mathrm { B } / \mathrm { A } ) = \frac { \sum _ { \mathrm { i } = 1 } ^ { 3 } \mathrm { P } \left( \mathrm { A } _ { \mathrm { i } } \right) \mathrm { P } \left( \mathrm { B } / \mathrm { A } _ { \mathrm { i } } \right) } { \sum _ { \mathrm { i } = 1 } ^ { 3 } \mathrm { P } \left( \mathrm { A } _ { \mathrm { i } } \right) } = \frac { \frac { { } ^ { 12 } \mathrm { C } _ { 2 } { } ^ { 6 } \mathrm { C } _ { 4 } } { { } ^ { 18 } \mathrm { C } _ { 6 } } \cdot \frac { { } ^ { 10 } \mathrm { C } _ { 1 } { } ^ { 2 } \mathrm { C } _ { 1 } } { { } ^ { 12 } \mathrm { C } _ { 2 } } + \frac { { } ^ { 12 } \mathrm { C } _ { 1 } { } ^ { 6 } \mathrm { C } _ { 5 } } { { } ^ { 18 } \mathrm { C } _ { 6 } } \cdot \frac { { } ^ { 11 } \mathrm { C } _ { 1 } { } ^ { 1 } \mathrm { C } _ { 1 } } { { } ^ { 12 } \mathrm { C } _ { 2 } } } { \frac { { } ^ { 12 } \mathrm { C } _ { 2 } { } ^ { 6 } \mathrm { C } _ { 4 } } { { } ^ { 18 } \mathrm { C } _ { 6 } } + \frac { { } ^ { 12 } \mathrm { C } _ { 1 } { } ^ { 6 } \mathrm { C } _ { 5 } } { { } ^ { 18 } \mathrm { C } _ { 6 } } + \frac { { } ^ { 12 } \mathrm { C } _ { 0 } { } ^ { 6 } \mathrm { C } _ { 6 } } { { } ^ { 18 } \mathrm { C } _ { 6 } } }$ $= \frac { { } ^ { 12 } \mathrm { C } _ { 2 } { } ^ { 6 } \mathrm { C } _ { 4 } { } ^ { 10 } \mathrm { C } _ { 1 } { } ^ { 2 } \mathrm { C } _ { 1 } + { } ^ { 12 } \mathrm { C } _ { 1 } { } ^ { 6 } \mathrm { C } _ { 5 } { } ^ { 11 } \mathrm { C } _ { 1 } { } ^ { 1 } \mathrm { C } _ { 1 } } { { } ^ { 12 } \mathrm { C } _ { 2 } \left( { } ^ { 12 } \mathrm { C } _ { 2 } { } ^ { 6 } \mathrm { C } _ { 4 } + { } ^ { 12 } \mathrm { C } _ { 1 } { } ^ { 6 } \mathrm { C } _ { 5 } + { } ^ { 12 } \mathrm { C } _ { 0 } { } ^ { 6 } \mathrm { C } _ { 6 } \right) }$

12. A fair die is rolled. The probability that the first time 1 occurs at the even throw is:
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(b) $5 / 11$
(c) $6 / 11$
(d) $5 / 36$

21. If $\mathrm { P } \left( \mathrm { u } _ { \mathrm { i } } \right) \propto \mathrm { i }$, where $\mathrm { i } = 1,2,3 , \ldots \mathrm { n }$, then $\lim _ { \mathrm { n } \rightarrow \infty } \mathrm { P } ( \mathrm { w } )$ is equal to
(A) 1
(B) $\frac { 2 } { 3 }$
(C) $\frac { 3 } { 4 }$
(D) $\frac { 1 } { 4 }$
Sol. (B)
$$\begin{aligned} & \mathrm { P } \left( \mathrm { u } _ { \mathrm { i } } \right) = \mathrm { ki } \\ & \Sigma \mathrm { P } \left( \mathrm { u } _ { \mathrm { i } } \right) = 1 \\ & \Rightarrow \mathrm { k } = \frac { 2 } { \mathrm { n } ( \mathrm { n } + 1 ) } \\ & \lim _ { \mathrm { n } \rightarrow \infty } \mathrm { P } ( \mathrm { w } ) = \lim _ { \mathrm { n } \rightarrow \infty } \sum _ { \mathrm { i } = 1 } ^ { \mathrm { n } } \frac { 2 \mathrm { i } ^ { 2 } } { \mathrm { n } ( \mathrm { n } + 1 ) ^ { 2 } } = \lim _ { \mathrm { n } \rightarrow \infty } \frac { 2 \mathrm { n } ( \mathrm { n } + 1 ) ( 2 \mathrm { n } + 1 ) } { \mathrm { n } ( \mathrm { n } + 1 ) ^ { 2 } 6 } = \frac { 2 } { 3 } \end{aligned}$$

1. If $\mathrm { P } \left( \mathrm { u } _ { \mathrm { i } } \right) = \mathrm { c }$, where c is a constant then $\mathrm { P } \left( \mathrm { u } _ { \mathrm { n } } / \mathrm { w } \right)$ is equal to
   (A) $\frac { 2 } { \mathrm { n } + 1 }$
   (B) $\frac { 1 } { n + 1 }$
   (C) $\frac { n } { n + 1 }$
   (D) $\frac { 1 } { 2 }$

Sol. (A)
$$\mathrm { P } \left( \frac { \mathrm { u } _ { \mathrm { n } } } { \mathrm { w } } \right) = \frac { \mathrm { c } \left( \frac { \mathrm { n } } { \mathrm { n } + 1 } \right) } { \mathrm { c } \left( \frac { \Sigma \mathrm { i } } { ( \mathrm { n } + 1 } \right) } = \frac { 2 } { \mathrm { n } + 1 }$$

1. If n is even and E denotes the event of choosing even numbered urn $\left( \mathrm { P } \left( \mathrm { u } _ { \mathrm { i } } \right) = \frac { 1 } { \mathrm { n } } \right)$, then the value of $\mathrm { P } ( \mathrm { w } / \mathrm { E } )$ is
   (A) $\frac { n + 2 } { 2 n + 1 }$
   (B) $\frac { n + 2 } { 2 ( n + 1 ) }$
   (C) $\frac { n } { n + 1 }$
   (D) $\frac { 1 } { n + 1 }$

Sol. (B)

$$\mathrm { P } \left( \frac { \mathrm { w } } { \mathrm { E } } \right) = \frac { 2 + 4 + 6 + \cdots \mathrm { n } } { \frac { \mathrm { n } ( \mathrm { n } + 1 ) } { 2 } } = \frac { \mathrm { n } + 2 } { 2 ( \mathrm { n } + 1 ) }$$

Comprehension II

Suppose we define the definite integral using the following formula $\int _ { a } ^ { b } f ( x ) d x = \frac { b - a } { 2 } ( f ( a ) + f ( b ) )$, for more accurate result for $\mathrm { c } \in ( \mathrm { a } , \mathrm { b } ) \mathrm { F } ( \mathrm { c } ) = \frac { \mathrm { c } - \mathrm { a } } { 2 } ( \mathrm { f } ( \mathrm { a } ) + \mathrm { f } ( \mathrm { c } ) ) + \frac { \mathrm { b } - \mathrm { c } } { 2 } ( \mathrm { f } ( \mathrm { b } ) + \mathrm { f } ( \mathrm { c } ) ) \quad$. When $\mathrm { c } = \frac { \mathrm { a } + \mathrm { b } } { 2 } , \int _ { \mathrm { a } } ^ { \mathrm { b } } \mathrm { f } ( \mathrm { x } ) \mathrm { dx } = \frac { \mathrm { b } - \mathrm { a } } { 4 } ( \mathrm { f } ( \mathrm { a } ) + \mathrm { f } ( \mathrm { b } ) + 2 \mathrm { f } ( \mathrm { c } ) )$.

Consider the system of equations $a x + b y = 0 , c x + d y = 0$, where $a , b , c , d \in \{ 0,1 \}$. STATEMENT-1 : The probability that the system of equations has a unique solution is $\frac { 3 } { 8 }$. and STATEMENT-2 : The probability that the system of equations has a solution is 1.
(A) STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is a correct explanation for STATEMENT-1
(B) STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is NOT a correct explanation for STATEMENT-1
(C) STATEMENT-1 is True, STATEMENT-2 is False
(D) STATEMENT-1 is False, STATEMENT-2 is True

Let $\omega$ be a complex cube root of unity with $\omega \neq 1$. A fair die is thrown three times. If $r _ { 1 } , r _ { 2 }$ and $r _ { 3 }$ are the numbers obtained on the die, then the probability that $\omega ^ { r _ { 1 } } + \omega ^ { r _ { 2 } } + \omega ^ { r _ { 3 } } = 0$ is
A) $\frac { 1 } { 18 }$
B) $\frac { 1 } { 9 }$
C) $\frac { 2 } { 9 }$
D) $\frac { 1 } { 36 }$

Four persons independently solve a certain problem correctly with probabilities $\frac { 1 } { 2 } , \frac { 3 } { 4 } , \frac { 1 } { 4 } , \frac { 1 } { 8 }$. Then the probability that the problem is solved correctly by at least one of them is
(A) $\frac { 235 } { 256 }$
(B) $\frac { 21 } { 256 }$
(C) $\frac { 3 } { 256 }$
(D) $\frac { 253 } { 256 }$

A box $B _ { 1 }$ contains 1 white ball, 3 red balls and 2 black balls. Another box $B _ { 2 }$ contains 2 white balls, 3 red balls and 4 black balls. A third box $B _ { 3 }$ contains 3 white balls, 4 red balls and 5 black balls.
If 1 ball is drawn from each of the boxes $B _ { 1 } , B _ { 2 }$ and $B _ { 3 }$, the probability that all 3 drawn balls are of the same colour is
(A) $\frac { 82 } { 648 }$
(B) $\frac { 90 } { 648 }$
(C) $\frac { 558 } { 648 }$
(D) $\frac { 566 } { 648 }$

Of the three independent events $E _ { 1 } , E _ { 2 }$ and $E _ { 3 }$, the probability that only $E _ { 1 }$ occurs is $\alpha$, only $E _ { 2 }$ occurs is $\beta$ and only $E _ { 3 }$ occurs is $\gamma$. Let the probability $p$ that none of events $E _ { 1 } , E _ { 2 }$ or $E _ { 3 }$ occurs satisfy the equations $( \alpha - 2 \beta ) p = \alpha \beta$ and $( \beta - 3 \gamma ) p = 2 \beta \gamma$. All the given probabilities are assumed to lie in the interval $( 0,1 )$.
$$\text { Then } \frac { \text { Probability of occurrence of } E _ { 1 } } { \text { Probability of occurrence of } E _ { 3 } } =$$

Three boys and two girls stand in a queue. The probability, that the number of boys ahead of every girl is at least one more than the number of girls ahead of her, is
(A) $\frac{1}{2}$
(B) $\frac{1}{3}$
(C) $\frac{2}{3}$
(D) $\frac{3}{4}$