Consider the squares of an $8 \times 8$ chessboard filled with the numbers 1 to 64 as in the figure below. If we choose 8 squares with the property that there is exactly one from each row and exactly one from each column, and add up the numbers in the chosen squares, show that the sum obtained is always 260.
An isosceles triangle with base 6 cms. and base angles $30 ^ { \circ }$ each is inscribed in a circle. A second circle, which is situated outside the triangle, touches the first circle and also touches the base of the triangle at its midpoint. Find its radius.
If a circle intersects the hyperbola $y = 1 / x$ at four distinct points $\left( x _ { i } , y _ { i } \right) , i = 1,2,3,4$, then prove that $x _ { 1 } x _ { 2 } = y _ { 3 } y _ { 4 }$.
Show that the function $f ( x )$ defined below attains a unique minimum for $x > 0$. What is the minimum value of the function? What is the value of $x$ at which the minimum is attained? $$f ( x ) = x ^ { 2 } + x + \frac { 1 } { x } + \frac { 1 } { x ^ { 2 } } \quad \text { for } \quad x \neq 0 .$$ Sketch on plain paper the graph of this function.
Let $S = \{ 1,2 , \ldots , n \}$. Find the number of unordered pairs $\{ A , B \}$ of subsets of $S$ such that $A$ and $B$ are disjoint, where $A$ or $B$ or both may be empty.
Find the maximum value of $x ^ { 2 } + y ^ { 2 }$ in the bounded region, including the boundary, enclosed by $y = \frac { x } { 2 } , y = - \frac { x } { 2 }$ and $x = y ^ { 2 } + 1$.
Define $a _ { n } = \left( 1 ^ { 2 } + 2 ^ { 2 } + \ldots + n ^ { 2 } \right) ^ { n }$ and $b _ { n } = n ^ { n } ( n ! ) ^ { 2 }$. Recall $n !$ is the product of the first $n$ natural numbers. Then, (a) $a _ { n } < b _ { n }$ for all $n > 1$ (b) $a _ { n } > b _ { n }$ for all $n > 1$ (c) $a _ { n } = b _ { n }$ for infinitely many $n$ (d) none of the above.
Let $\mathbb { C }$ denote the set of all complex numbers. Define $$\begin{aligned}
& A = \{ ( z ,w ) \mid z , w \in \mathbb { C } \text { and } | z | = | w | \} \\
& B = \left\{ ( z , w ) \mid z , w \in \mathbb { C } , \text { and } z ^ { 2 } = w ^ { 2 } \right\}
\end{aligned}$$ Then, (a) $A = B$ (b) $A \subset B$ and $A \neq B$ (c) $B \subset A$ and $B \neq A$ (d) none of the above.
The set of all real numbers $x$ such that $x ^ { 3 } ( x + 1 ) ( x - 2 ) \geq 0$ is: (a) the interval $2 \leq x < \infty$ (b) the interval $0 \leq x < \infty$ (c) the interval $- 1 \leq x < \infty$ (d) none of the above.
Let $z$ be a non-zero complex number such that $\frac { z } { 1 + z }$ is purely imaginary. Then (a) $z$ is neither real nor purely imaginary (b) $z$ is real (c) $z$ is purely imaginary (d) none of the above.
In the interval $( 0,2 \pi )$, the function $\sin \left( \frac { 1 } { x ^ { 3 } } \right)$ (a) never changes sign (b) changes sign only once (c) changes sign more than once, but finitely many times (d) changes sign infinitely many times.
Let $f _ { 1 } ( x ) = e ^ { x } , f _ { 2 } ( x ) = e ^ { f _ { 1 } ( x ) }$ and generally $f _ { n + 1 } ( x ) = e ^ { f _ { n } ( x ) }$ for all $n \geq 1$. For any fixed $n$, the value of $\frac { d } { d x } f _ { n } ( x )$ is: (a) $f _ { n } ( x )$ (b) $f _ { n } ( x ) f _ { n - 1 } ( x )$ (c) $f _ { n } ( x ) f _ { n - 1 } ( x ) \ldots f _ { 1 } ( x )$ (d) $f _ { n + 1 } ( x ) f _ { n } ( x ) \ldots f _ { 1 } ( x ) e ^ { x }$.
Let $f ( x ) = a _ { 0 } + a _ { 1 } | x | + a _ { 2 } | x | ^ { 2 } + a _ { 3 } | x | ^ { 3 }$, where $a _ { 0 } , a _ { 1 } , a _ { 2 } , a _ { 3 }$ are constants. Then (a) $f ( x )$ is differentiable at $x = 0$ whatever be $a _ { 0 } , a _ { 1 } , a _ { 2 } , a _ { 3 }$ (b) $f ( x )$ is not differentiable at $x = 0$ whatever be $a _ { 0 } , a _ { 1 } , a _ { 2 } , a _ { 3 }$ (c) $f ( x )$ is differentiable at $x = 0$ only if $a _ { 1 } = 0$ (d) $f ( x )$ is differentiable at $x = 0$ only if $a _ { 1 } = 0 , a _ { 3 } = 0$.
If $f ( x ) = \cos ( x ) - 1 + \frac { x ^ { 2 } } { 2 }$, then (a) $f ( x )$ is an increasing function on the real line (b) $f ( x )$ is a decreasing function on the real line (c) $f ( x )$ is increasing on the interval $- \infty < x \leq 0$ and decreasing on the interval $0 \leq x < \infty$ (d) $f ( x )$ is decreasing on the interval $- \infty < x \leq 0$ and increasing on the interval $0 \leq x < \infty$.
The number of roots of the equation $x ^ { 2 } + \sin ^ { 2 } x = 1$ in the closed interval $\left[ 0 , \frac { \pi } { 2 } \right]$ is (a) 0 (b) 1 (c) 2 (d) 3
The number of maps $f$ from the set $\{ 1,2,3 \}$ into the set $\{ 1,2,3,4,5 \}$ such that $f ( i ) \leq f ( j )$ whenever $i < j$ is (a) 60 (b) 50 (c) 35 (d) 30
Let $a$ be a real number. The number of distinct solutions $( x , y )$ of the system of equations $( x - a ) ^ { 2 } + y ^ { 2 } = 1$ and $x ^ { 2 } = y ^ { 2 }$, can only be (a) $0,1,2,3,4$ or 5 (b) 0, 1 or 3 (c) 0, 1, 2 or 4 (d) $0,2,3$, or 4
The set of values of $m$ for which $m x ^ { 2 } - 6 m x + 5 m + 1 > 0$ for all real $x$ is (a) $m < \frac { 1 } { 4 }$ (b) $m \geq 0$ (c) $0 \leq m \leq \frac { 1 } { 4 }$ (d) $0 \leq m < \frac { 1 } { 4 }$.
A lantern is placed on the ground 100 feet away from a wall. A man six feet tall is walking at a speed of 10 feet/second from the lantern to the nearest point on the wall. When he is midway between the lantern and the wall, the rate of change in the length of his shadow is (a) $3.6 \mathrm { ft } . / \mathrm { sec }$. (b) $2.4 \mathrm { ft } . / \mathrm { sec }$. (c) $3 \mathrm { ft } . / \mathrm { sec }$. (d) $12 \mathrm { ft } . / \mathrm { sec }$.
Let $n \geq 3$ be an integer. Assume that inside a big circle, exactly $n$ small circles of radius $r$ can be drawn so that each small circle touches the big circle and also touches both its adjacent small circles. Then, the radius of the big circle is: (a) $r \operatorname { cosec } \frac { \pi } { n }$ (b) $r \left( 1 + \operatorname { cosec } \frac { 2 \pi } { n } \right)$ (c) $r \left( 1 + \operatorname { cosec } \frac { \pi } { 2 n } \right)$ (d) $r \left( 1 + \operatorname { cosec } \frac { \pi } { n } \right)$.
The function $x ( \alpha - x )$ is strictly increasing on the interval $0 < x < 1$ if and only if (a) $\alpha \geq 2$ (b) $\alpha < 2$ (c) $\alpha < - 1$ (d) $\alpha > 2$.
For any integer $n \geq 1$, define $a _ { n } = \frac { 1000 ^ { n } } { n ! }$. Then the sequence $\left\{ a _ { n } \right\}$ (a) does not have a maximum (b) attains maximum at exactly one value of $n$ (c) attains maximum at exactly two values of $n$ (d) attains maximum for infinitely many values of $n$.
The equation $x ^ { 3 } y + x y ^ { 3 } + x y = 0$ represents (a) a circle (b) a circle and a pair of straight lines (c) a rectangular hyperbola (d) a pair of straight lines.
Let $P$ be a variable point on a circle $C$ and $Q$ be a fixed point outside $C$. If $R$ is the mid-point of the line segment $P Q$, then the locus of $R$ is (a) a circle (b) an ellipse (c) a line segment (d) segment of a parabola.