A circunferência de equação $x^2 + y^2 - 4x + 6y - 3 = 0$ tem centro e raio iguais a
(A) centro $(2, -3)$ e raio $4$ (B) centro $(-2, 3)$ e raio $4$ (C) centro $(2, -3)$ e raio $16$ (D) centro $(-2, 3)$ e raio $16$ (E) centro $(4, -6)$ e raio $3$

A circle has equation $x^2 + y^2 - 4x + 6y - 3 = 0$. What is its radius?
(A) 2
(B) 3
(C) 4
(D) 5
(E) 6

Solve the following two independent problems.
(i) Let $f$ be a function from domain $S$ to codomain $T$. Let $g$ be another function from domain $T$ to codomain $U$. For each of the blanks below choose a single letter corresponding to one of the four options listed underneath. (It is not necessary that each choice is used exactly once.) Write your answers as a sequence of four letters in correct order. Do NOT explain your answers.
If $g \circ f$ is one-to-one then $f$ $\_\_\_\_$ and $g$ $\_\_\_\_$ . If $g \circ f$ is onto then $f$ $\_\_\_\_$ and $g$ $\_\_\_\_$ .
Option A: must be one-to-one and must be onto. Option B: must be one-to-one but need not be onto. Option C: need not be one-to-one but must be onto. Option D: need not be one-to-one and need not be onto. Recall: $g \circ f$ is the function defined by $g \circ f ( a ) = g ( f ( a ) )$. The function $f$ is said to be one-to-one if, for any $a _ { 1 }$ and any $a _ { 2 }$ in $S , f \left( a _ { 1 } \right) = f \left( a _ { 2 } \right)$ implies $a _ { 1 } = a _ { 2 }$. The function $f$ is said to be onto if, for any $b$ in $T$, there is an $a$ in $S$ such that $f ( a ) = b$.
(ii) In the given figure $ABCD$ is a square. Points $X$ and $Y$, respectively on sides $BC$ and $CD$, are such that $X$ lies on the circle with diameter $AY$. What is the area of the square $ABCD$ if $AX = 4$ and $AY = 5$? (Figure is schematic and not to scale.)

8. The eccentricity of hyperbola $C_1$ is $e_1$. Both the semi-major axis $a$ and semi-minor axis $b$ (where $a \neq b$) are increased by $m$ units (where $m > 0$) to obtain hyperbola $C_2$ with eccentricity $e_2$. Then
A. For any $a, b$, we have $e_1 > e_2$
B. When $a > b$, $e_1 > e_2$; when $a < b$, $e_1 < e_2$
C. For any $a, b$, we have $e_1 < e_2$
D. When $a > b$, $e_1 < e_2$; when $a < b$, $e_1 > e_2$

Given that the eccentricity of the ellipse $\frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1 ( a > b > 0 )$ is $\frac { 1 } { 2 }$, then (A) $a ^ { 2 } = 2 b ^ { 2 }$ (B) $3 a ^ { 2 } = 4 b ^ { 2 }$ (C) $a = 2 b$ (D) $3 a = 4 b$

9. Let $S _ { n }$ denote the sum of the first $n$ terms of an arithmetic sequence $\left\{ a _ { n } \right\}$. Given $S _ { 4 } = 0 , a _ { 5 } = 5$, then
A. $a _ { n } = 2 n - 5$
B. $a _ { n } = 3 n - 10$
C. $S _ { n } = 2 n ^ { 2 } - 8 n$
D. $S _ { n } = \frac { 1 } { 2 } n ^ { 2 } - 2 n$

Mathematics (Science) Test Paper Page 2 (Total 5 Pages)

Let $a$ and $b$ be non-zero real numbers. Then, the equation $$\left( a x ^ { 2 } + b y ^ { 2 } + c \right) \left( x ^ { 2 } - 5 x y + 6 y ^ { 2 } \right) = 0$$ represents
(A) four straight lines, when $c = 0$ and $a , b$ are of the same sign
(B) two straight lines and a circle, when $a = b$, and $c$ is of sign opposite to that of $a$
(C) two straight lines and a hyperbola, when $a$ and $b$ are of the same sign and $c$ is of sign opposite to that of $a$
(D) a circle and an ellipse, when $a$ and $b$ are of the same sign and $c$ is of sign opposite to that of $a$

Let $\mathbb { R }$ denote the set of all real numbers. Let $z _ { 1 } = 1 + 2 i$ and $z _ { 2 } = 3 i$ be two complex numbers, where $i = \sqrt { - 1 }$. Let
$$S = \left\{ ( x , y ) \in \mathbb { R } \times \mathbb { R } : \left| x + i y - z _ { 1 } \right| = 2 \left| x + i y - z _ { 2 } \right| \right\}$$
Then which of the following statements is (are) TRUE?

(A)	$S$ is a circle with centre $\left( - \frac { 1 } { 3 } , \frac { 10 } { 3 } \right)$
(B)	$S$ is a circle with centre $\left( \frac { 1 } { 3 } , \frac { 8 } { 3 } \right)$
(C)	$S$ is a circle with radius $\frac { \sqrt { 2 } } { 3 }$
(D)	$S$ is a circle with radius $\frac { 2 \sqrt { 2 } } { 3 }$

Two sets $A$ and $B$ are as under: $A = \{ ( a , b ) \in R \times R : | a - 5 | < 1$ and $| b - 5 | < 1 \}$; $B = \left\{ ( a , b ) \in R \times R : 4 ( a - 6 ) ^ { 2 } + 9 ( b - 5 ) ^ { 2 } \leq 36 \right\}$. Then :
(1) neither $A \subset B$ nor $B \subset A$
(2) $B \subset A$
(3) $A \subset B$
(4) $A \cap B = \phi$ (an empty set)

Choose the correct statement about two circles whose equations are given below: $x ^ { 2 } + y ^ { 2 } - 10 x - 10 y + 41 = 0$ $x ^ { 2 } + y ^ { 2 } - 22 x - 10 y + 137 = 0$
(1) circles have same centre
(2) circles have no meeting point

The set of values of $k$ for which the circle $C : 4 x ^ { 2 } + 4 y ^ { 2 } - 12 x + 8 y + k = 0$ lies inside the fourth quadrant and the point $\left( 1 , - \frac { 1 } { 3 } \right)$ lies on or inside the circle $C$ is
(1) An empty set
(2) $\left( 6 , \frac { 95 } { 9 } \right]$
(3) $\left[ \frac { 80 } { 9 } , 10 \right)$
(4) $\left( 9 , \frac { 92 } { 9 } \right]$

In a group of 100 persons 75 speak English and 40 speak Hindi. Each person speaks at least one of the two languages. If the number of persons who speak only English is $\alpha$ and the number of persons who speaks only Hindi is $\beta$, then the eccentricity of the ellipse $25\left(\beta^{2}x^{2} + \alpha^{2}y^{2}\right) = \alpha^{2}\beta^{2}$ is
(1) $\frac{\sqrt{119}}{12}$
(2) $\frac{\sqrt{117}}{12}$
(3) $\frac{3\sqrt{15}}{12}$
(4) $\frac{\sqrt{129}}{12}$

On the coordinate plane, there is a figure $\Gamma$ with equation $(x-1)^2 + (y-1)^2 = 101$. Select the correct options.
(1) $\Gamma$ intersects the negative $x$-axis and negative $y$-axis at $(-9, 0)$ and $(0, -9)$ respectively
(2) The point on $\Gamma$ with the maximum $x$-coordinate is $(11, 0)$
(3) The maximum distance from a point on $\Gamma$ to the origin is $\sqrt{2} + \sqrt{101}$
(4) Points on $\Gamma$ in the third quadrant can be expressed in polar coordinates as $[9, \theta]$, where $\pi < \theta < \frac{3}{2}\pi$
(5) After a rotational linear transformation, the figure can still be expressed by a quadratic equation in two variables without an $xy$ term

On the coordinate plane, which of the following equations represents a circle passing through the point $(1,1)$?
(1) $( x - 1 ) ^ { 2 } + y ^ { 2 } = 1$
(2) $( x - 1 ) ^ { 2 } + ( y - 1 ) ^ { 2 } = 1$
(3) $3 ( x - 1 ) ^ { 2 } + y ^ { 2 } = 1$
(4) $x ^ { 2 } + y ^ { 2 } = 1$
(5) $x ^ { 2 } + 3 y = 4$