3. (a) In any triangle ABC , prove that $$\operatorname { Cot } \mathrm { A } / \mathrm { b } + \cot \mathrm { B } / 2 + \cot \mathrm { C } / 2 = \mathrm { A } / 2 \cot \mathrm {~B} / 2 \cot \mathrm { C } / 2$$ (b) Let ABC be a triangle with incentre I and inradius r . Let $\mathrm { D } , \mathrm { E } , \mathrm { F }$ be the feet of the perpendiculars from I to the sides $\mathrm { BC } , \mathrm { CA }$ and AB respectively. If $\mathrm { r } 1 , \mathrm { r } 2$ and B are the radii of circles inscribed in the quadrilaterals AFIE, BDIF and CEID respectively, prove that r1/ ( $\mathrm { r } - \mathrm { r } 1$ ) + உ / ( $\mathrm { r } - \mathrm { r } 2$ ) + r3 / ( $\mathrm { r } - \mathrm { r } 3$ ) = ( r 1 B B ) / ( $\mathrm { r } - \mathrm { r } 1$ ) ( $\mathrm { r } - \mathrm { r } 2$ ) ( $\mathrm { r } - \mathrm { r } 3$ )
4. For points $\mathrm { P } = ( \mathrm { x } 1 , \mathrm { y } 1 )$ and $\mathrm { Q } = ( \mathrm { x } 2 , \mathrm { y } 2 )$ of the coordinate plane, a new distance $\mathrm { d } ( \mathrm { P } , \mathrm { Q } )$ is defined by $\mathrm { d } ( \mathrm { P } , \mathrm { Q } ) = { } _ { \mid } ^ { \mid } \mathrm { x } 1 - \left. \mathrm { x } 2 \right| _ { \mid } ^ { \mid } + { } _ { \mid } \mathrm { y } 1 - \left. \mathrm { y } 2 \right| _ { \mid } ^ { \mid }$. Let $\mathrm { O } = ( 0,0 )$ and $\mathrm { A } = ( 3,2 )$. Prove that the set of points in the first quadrant which are equidistant (with respect to the new distance) from O and A consists of the union of line segment of finite length and an infinite ray. Sketch this set in a labelled diagram. 5. (a) Prove that for all values of $\theta$ $$\left| \begin{array} { c c c }
\sin \theta & \cos \theta & \sin 2 \theta \\
\sin \left( \theta + \frac { 2 \pi } { 3 } \right) & \cos \left( \theta + \frac { 2 \pi } { 3 } \right) & \sin \left( 2 \theta + \frac { 4 \pi } { 3 } \right) \\
\sin \left( \theta - \frac { 2 \pi } { 3 } \right) & \cos \left( \theta - \frac { 2 \pi } { 3 } \right) & \sin \left( 2 \theta - \frac { 4 \pi } { 3 } \right)
\end{array} \right| = 0 .$$ (b) Let ABC be an equilateral triangle inscribed in the circle $\mathrm { x } 2 + \underset { 2 } { 2 } = a 2$. Suppose perpendiculars from $\mathrm { A } , \mathrm { B } , \mathrm { C }$ to the major axis of the ellipse $\mathrm { x } 2 / \mathrm { a } 2 + \mathrm { J } / \mathrm { b } 2 = 1 , ( \mathrm { a } > \mathrm { b } )$ meets the ellipse respectively at $\mathrm { P } , \mathrm { Q } , \mathrm { R }$ so that $\mathrm { P } , \mathrm { Q } , \mathrm { R }$ lie on the same side of the major axis as $\mathrm { A } , \mathrm { B } , \mathrm { C }$ respectively. Prove that the normals to the ellipse drawn at the points $\mathrm { P } , \mathrm { Q }$ and R are concurrent.
6. Let Cl and C 2 be, respectively, the parabolas $\mathrm { x } 2 = \mathrm { y } - 1$ and $\mathrm { y } 2 = \mathrm { a }$. Let P be any point on C 1 and Q be any point on C 2 . Let P 1 and Q 1 be the reflections of P and Q , respectively, with respect to the line y $= \mathrm { x }$. Prove that P 1 lies on $\mathrm { C } 2 , \mathrm { Q } 1$ lies on Cland PQ $\geq \min \{ \mathrm { PP } 1 , \mathrm { QQ } 1 \}$. Hence or otherwise, determine points P0 and Q on the parabolas Cl and C 2 respectively such that $\mathrm { P } 0 \mathrm { Q } \leq \mathrm { P } 0$ for all pairs of points $( \mathrm { P } , \mathrm { Q } )$ with P on C 1 and Q on C . 7. (a) $$\begin{aligned}
& \text { Suppose } P ( x ) = a _ { 0 } + a _ { 1 } x + a _ { 2 } x ^ { 2 } + \ldots \ldots . . + a _ { n } x ^ { 7 } . \text { If } | p ( x ) | \leq \left| e ^ { x - 1 } - 1 \right| \text { for all } \\
& x \geq 0 \text {, prove that } \left| a _ { 1 } + 2 a _ { 2 } + \ldots \ldots . + n a _ { n } \right| \leq 1
\end{aligned}$$ (b) For $x > 0$, let $f ( x ) = \int _ { 1 } ^ { x } \frac { \ln t } { 1 + \tau } d t$. Find the function $f ( x ) + f ( 1 / x )$ and show that $f ( e ) + f ( 1 / e ) = 1 / 2$. Here $\ln t = \log _ { e } t$.
8. A country has food deficit of $10 \%$. Its population grows continuously at a rate of $3 \%$ per year. Its annual food production every year is $4 \%$ more than that of the last year. Assuming that the average food requirement per person remains constant, prove that the country will become self-sufficient in food after n years, where n is the smallest integer bigger than or equal to (In $\mathbf { 1 0 } - \mathbf { I n } \mathbf { 9 } ) / ( \mathbf { I n } ( \mathbf { 1 . 0 4 } ) - \mathbf { 0 . 0 3 } )$.
9. (a) A coin has probability p of showing head when tossed. It is tossed n times. Let pn , denote the probability that no two (or more) consecutive heads occur. Prove that $\mathrm { p } 1 = 1 , \mathrm { p } ^ { 2 } = 1 - \mathrm { p } _ { 2 }$ and $\mathrm { p } = ( 1 -$ p ). $\mathrm { pn } - 1 + \mathrm { p } ( 1 - \mathrm { p } ) \mathrm { pn } - 2$ for all $\mathrm { n } \geq 3$. (b) In (a), prove by induction on n , that $\mathrm { pn } = \mathrm { A } \alpha \mathrm { n } + \mathrm { B } \beta \mathrm { n }$ for all $\mathrm { n } \geq 1$, where $\alpha$ and $\beta$ are the roots of the quadratic x2-(1-p) $x - p ( 1 - p ) = 0$ and $$A = \frac { p ^ { 2 } + \beta - 1 } { \alpha \beta - \alpha ^ { 2 } } , B = \frac { p ^ { 2 } + \alpha - 1 } { \alpha \beta - \beta ^ { 2 } } .$$
Let ABC and PQR be any two triangles in the same plane. Assume that the perpendiculars from the points $\mathrm { A } , \mathrm { B } , \mathrm { C }$ to the sides $\mathrm { QR } , \mathrm { RP } , \mathrm { PQ }$ respectively are concurrent. Using vector methods or otherwise, prove that the perpendiculars from $\mathrm { P } , \mathrm { Q } , \mathrm { R }$ to $\mathrm { BC } , \mathrm { CA } , \mathrm { AB }$ respectively are also concurrent.