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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- 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) $\quad \mathrm { b } _ { 0 } = 0 , \mathrm {~b} _ { 1 } = \mathrm { n } ^ { 2 } - 3 \mathrm { n } + 3$ - 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) $\quad \sin 15 ^ { \circ } \cos 75 ^ { \circ }$ - 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) $\quad y _ { 1 } y _ { 2 } y _ { 3 } y _ { 4 } = c ^ { 4 }$ - 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 - 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) $\vec { u } \times ( \vec { v } , \vec { w } )$ - 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) $\quad - \frac { 1 } { 2 }$ - 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.
- 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 $f ( x )$. - 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 $( 1 + x ) ( m + 1 ) p = [ ( 1 + x ) p ( 1 + x ) m p ]$
- 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.)
- Prove that a triangle $A B C$ is equilateral if and only if $\tan A + \tan B + \tan C = 3 \sqrt { } 3$.
- Using co-ordinate geometry, prove that the three altitudes of any triangle are concurrent.
- 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 .
- 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.
- 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 $\frac { y ^ { \prime } } { y } = \frac { 1 } { x } \left( \frac { a } { a - x } + \frac { b } { b - x } + \frac { c } { c - x } \right)$.
- Prove that $\int 01 \tan - 1 ( 1 - x + x 2 ) d x$.
- 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.
- 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]