CSCA Examination
Questions with Detailed Solutions
Page 1: Sets and Functions
- Let $A = \{6, 7, 8, 9, 10\}$ be a set, and $\Phi$ to denote the empty set. Then which of the following statements is correct?
Explanation (a) $9 \in A$: True, because 9 is an element strictly listed inside the set definition.
(b) $\{6\} \in A$: False. $\{6\}$ is a set containing 6. It is a subset of A ($\{6\} \subseteq A$), not an element.
(c) $6 \subseteq A$: False. 6 is an element. The symbol $\subseteq$ is used for sets.
(d) $\Phi \in A$: False. The empty set is a subset of every set ($\Phi \subseteq A$), but it is not listed as an element inside A. Correct Answer: (a) - Let $A = \{x \mid -2 \le x \le 2\}$ and $B = \{x \mid x > 0\}$ be two sets. Then $A \cup B =$
Explanation Set A is the interval $[-2, 2]$. Set B is the interval $(0, +\infty)$.
The Union ($A \cup B$) includes everything in A OR B. Since A starts at -2 and B goes to infinity, the combination covers everything starting from -2.
Range: $[-2, +\infty)$, which is written as $\{x \mid x \ge -2\}$. Correct Answer: (c) - The solution set of the inequality $x^2 - x - 2 > 0$ is ( ).
Explanation Factor the quadratic: $(x - 2)(x + 1) > 0$.
Roots are $2$ and $-1$.
Since the inequality is "greater than 0" (positive), the solution lies on the outside of the roots.
Solution: $x < -1$ or $x > 2$. Correct Answer: (a) - Let $\{a_n\}$ be an arithmetic sequence, with the first term $a_1 = 2$ and common difference $d = 3$. Then $a_{100} =$ ( ).
Explanation General formula: $a_n = a_1 + (n-1)d$.
Substitute values: $a_{100} = 2 + (100-1)(3) = 2 + 99(3)$.
$a_{100} = 2 + 297 = 299$. Correct Answer: (a) - The domain of the function $f(x) = \frac{1}{x} + \sqrt{1-x}$ is ( ).
Explanation 1. Denominator restriction: $x \ne 0$.
2. Square root restriction: $1-x \ge 0 \Rightarrow 1 \ge x \Rightarrow x \le 1$.
Combine: $x \le 1$ AND $x \ne 0$.
Interval notation: $(-\infty, 0) \cup (0, 1]$. Correct Answer: (c)
Page 2: Coordinates and Functions
- Let $P(1, 2)$ be a point in the rectangular coordinate system. If a point $Q$ together with $P$ are symmetrical about the $x$ axis, then the coordinates of $Q$ are ( ).
Explanation Symmetry about the x-axis keeps the x-coordinate the same but negates the y-coordinate.
$P(x, y) \rightarrow Q(x, -y)$.
$P(1, 2) \rightarrow Q(1, -2)$. Correct Answer: (b) - [Question content missing in original image]
Note The content for this question was not visible in the source documents.
- About the function $f(x) = x^4 + 3$, which of the following statements is correct? ( ).
Explanation Test $f(-x)$: $f(-x) = (-x)^4 + 3 = x^4 + 3$.
Since $f(-x) = f(x)$, the function is Even.
(Odd functions satisfy $f(-x) = -f(x)$). Correct Answer: (d) - The inverse function of the function $y = x^3 + 3, \, x \in \mathbb{R}$ is ( ).
Explanation 1. Swap x and y: $x = y^3 + 3$.
2. Solve for y: $y^3 = x - 3$.
3. Take cube root: $y = \sqrt[3]{x-3}$.
The domain of an odd root (cube root) is all real numbers ($\mathbb{R}$). Correct Answer: (a) - Which of the following points is in the third quadrant? ( ).
Explanation Quadrant I: $(+, +)$, Quadrant II: $(-, +)$, Quadrant III: $(-, -)$, Quadrant IV: $(+, -)$.
Point $(-1, -2)$ has both coordinates negative, so it is in Q3. Correct Answer: (b) - Given that $y = |x|$, which of the following conclusions is correct? ( ).
Explanation The graph of $y=|x|$ is V-shaped.
For $x < 0$ (left side), the graph goes down as you move right (Decreasing).
For $x > 0$ (right side), the graph goes up as you move right (Increasing). Correct Answer: (b)
Page 3: Algebra and Geometry
- The solution set of the rational inequality $\frac{2x+1}{x-2} \le 0$ is ( ).
Explanation Critical points: Numerator $2x+1=0 \Rightarrow x=-1/2$. Denominator $x-2=0 \Rightarrow x=2$.
Test interval $-1/2 < x < 2$: Pos/Neg = Neg (Satisfies $\le 0$).
Include numerator zero (-1/2), exclude denominator zero (2).
Result: $[-1/2, 2)$. Correct Answer: (a) - Given that $a = 2 - \sqrt{3}$ and $b = 2 + \sqrt{3}$, the arithmetic mean of $a$ and $b$ is ( ).
Explanation Arithmetic Mean $= \frac{a+b}{2}$.
$a+b = (2-\sqrt{3}) + (2+\sqrt{3}) = 4$.
Mean $= 4/2 = 2$. Correct Answer: (b) - If the straight line $l$ passes through the points $A(-2, 3)$ and $B(3, 1)$, then the slope of $l$ is ( ).
Explanation Slope formula: $m = \frac{y_2 - y_1}{x_2 - x_1}$.
$m = \frac{1 - 3}{3 - (-2)} = \frac{-2}{5} = -\frac{2}{5}$. Correct Answer: (c) - If $\alpha$ is the angle between the $x$-axis and a line containing the point $P(1, 3)$,
then $\sin \alpha =$ ( ).Explanation Coordinates $(x, y) = (1, 3)$.
Radius $r = \sqrt{x^2 + y^2} = \sqrt{1^2 + 3^2} = \sqrt{10}$.
$\sin \alpha = \frac{y}{r} = \frac{3}{\sqrt{10}} = \frac{3\sqrt{10}}{10}$. Correct Answer: (b) - Among the four sequences below, the number of arithmetic sequences is ( ).
(1) $7, 13, 19, 25$
(2) $2, 4, 7, 11$
(3) $-1, -3, -5, -7$
(4) $2, 4, 8, 16$Explanation (1) Diff +6 (Arithmetic).
(2) Diff +2, +3, +4 (Not Arithmetic).
(3) Diff -2 (Arithmetic).
(4) Ratio x2 (Geometric).
Total: 2. Correct Answer: (a)
Page 4: Analytic Geometry
- The distance from a point $A(3, -2)$ to $B(-5, -1)$ on the plane is $|AB| =$ ( ).
Explanation $d = \sqrt{(-5-3)^2 + (-1 - (-2))^2}$
$= \sqrt{(-8)^2 + (1)^2}$.
$d = \sqrt{64 + 1} = \sqrt{65}$. Correct Answer: (b) - If a straight line $l$ has an angle of inclination of $45^\circ$, and it passes through the point $(0, 2)$, then the equation of $l$ is ( ).
Explanation Slope $m = \tan(45^\circ) = 1$. Y-intercept $b = 2$.
Equation: $y = mx + b \Rightarrow y = x + 2$. Correct Answer: (b) - If the equation of a circle is $x^2 + y^2 - 4x - 3 = 0$, then the center and radius are given by ( ).
Explanation Rearrange: $(x^2 - 4x) + y^2 = 3$.
Complete square: $(x-2)^2 - 4 + y^2 = 3$
$ \Rightarrow (x-2)^2 + y^2 = 7$.
Center $(2, 0)$, Radius $\sqrt{7}$. Correct Answer: (c) - Given that $\sin \alpha = \frac{4}{5}$, and the angle $\alpha$ belongs to the second quadrant, then which of the following statements is correct? ( )
Explanation 3-4-5 triangle. In Q2, cosine is negative.
$\cos \alpha = -3/5$. Correct Answer: (b)
Page 5: Sequences and Functions
- In an arithmetic sequence $\{a_n\}$, if $a_2 = 1$ and $a_4 = 5$, then the first term $a_1$ and common difference $d$ are ( ) respectively.
Explanation $a_4 - a_2 = 2d \Rightarrow 5 - 1 = 4 $
$\Rightarrow 2d = 4 \Rightarrow d = 2$.
$a_2 = a_1 + d \Rightarrow 1 = a_1 + 2 \Rightarrow a_1 = -1$. Correct Answer: (a) - Let $a, b, c$ be three real numbers. Which of the following statements is correct? ( )
Explanation (a) False if $c < 0$.
(b) True, addition property of inequality.
(c) False if $c < 0$.
(d) False if numbers are negative. Correct Answer: (b) - About the exponential function $y = a^x$
($a > 0$, and $a \neq 1$), which of the following statements is incorrect? ( )Explanation Exponential functions are increasing ONLY if $a > 1$. If $0 < a < 1$, they are decreasing. Thus, statement (c) is not always true. Correct Answer: (c) - If $\sin \alpha = \frac{3}{5}$, and $\alpha$ belongs to the first quadrant, then $\sin 2\alpha =$ ( ).
Explanation In Q1, $\sin=3/5 \Rightarrow \cos=4/5$.
$\sin 2\alpha = 2 \sin \alpha \cos \alpha $
$= 2 \times \dfrac{3}{5} \times \dfrac{4}{5} = \dfrac{24}{25}$. Correct Answer: (b) - The intersection of the lines $l_1 : 3x - y + 8 = 0$ and $l_2 : x + 2y - 9 = 0$ has coordinates ( ).
Explanation $l_1 \Rightarrow y = 3x + 8$. Substitute into $l_2$:
$x + 2(3x + 8) - 9 = 0
\Rightarrow 7x + 7 = 0 \Rightarrow x = -1$.
$y = 3(-1) + 8 = 5$. Intersection: $(-1, 5)$. Correct Answer: (d)
Page 6: Conics and Logarithms
- Given that the equation of a hyperbola is $\frac{x^2}{64} - \frac{y^2}{16} = 1$, which of the following statements is correct? ( )
Explanation $a^2=64, b^2=16 \Rightarrow c^2 = 80 \Rightarrow c = 4\sqrt{5}$.
Focal distance $2c = 8\sqrt{5}$. Correct Answer: (a) - Which of the following inequalities is correct? ( )
Explanation Power function $x^{2/3}$ increases for $x>0$. Since $2.1 > 1.2$, $2.1^{2/3} > 1.2^{2/3}$. Correct Answer: (d)
- A circle has center at $(-3, 2)$ and radius 4. Then the equation of the circle is ( ).
Explanation Formula $(x-h)^2 + (y-k)^2 = r^2$.
$(x - (-3))^2 + (y - 2)^2 = 4^2$
$\Rightarrow (x+3)^2 + (y-2)^2 = 16$. Correct Answer: (d) - Let $a > 0$, and $b > 0$ be real numbers. Then which of the following statements is incorrect? ( )
Explanation The change of base formula is $\log_a b = \dfrac{\ln b}{\ln a}$. Statement (b) is inverted. Correct Answer: (b)
- Given the sine function $y = \sin x$, which of the following statements is incorrect? ( )
Explanation $\sin(-x) = -\sin(x)$, so sine is an Odd function. Correct Answer: (c)
Page 7: Geometry and Functions
- Let the focus of the parabola $y^2 = 4x$ be $F$. Suppose that a point $P$ is on the parabola, and the horizontal coordinate of $P$ is 4. Then $|PF| =$ ( ).
Explanation $4p=4 \Rightarrow p=1$. Directrix $x=-1$.
Distance to focus = Distance to directrix
= $x_p - (-1)
= 4 + 1 = 5$. Correct Answer: (d) - Given two straight lines $l_1 : ax + (a - 1)y + 3 = 0$ and $l_2 : 2x + ay - 1 = 0$. If $l_1 \perp l_2$, then the value of $a$ is ( ).
Explanation Perpendicular condition: $A_1A_2 + B_1B_2 = 0$.
$a(2) + (a-1)a = 0 $
$\Rightarrow 2a + a^2 - a = 0$
$\Rightarrow a(a+1) = 0$.
$a=0$ or $a=-1$. Correct Answer: (d) - The distance between the points $P(-1, 2)$ and $Q(3, 1)$ is ( ).
Explanation $d = \sqrt{(3 - (-1))^2 + (1 - 2)^2}$
=$ \sqrt{16 + 1} = \sqrt{17}$. Correct Answer: (b) - Let $\cos \alpha = -\dfrac{\sqrt{5}}{5}$, and $\alpha \in (\dfrac{\pi}{2}, \pi)$. Then $\sin(\frac{\pi}{6} + \alpha) =$ ( ).
Explanation $\sin \alpha = \sqrt{1 - 5/25} = 2\sqrt{5}/5$.
$\sin(\dfrac{\pi}{6} + \alpha) = (\dfrac{1}{2})(-\dfrac{\sqrt{5}}{5}) + (\dfrac{\sqrt{3}}{2})(2\times \dfrac{\sqrt{5}}{5}) $
$= \dfrac{2\sqrt{15}-\sqrt{5}}{10}$. Correct Answer: (c) - Which of the following statements is correct? ( )
Explanation (b) $\frac{x^4-1}{x^2+1} = \frac{(x^2-1)(x^2+1)}{x^2+1} = x^2-1$. Since denominator $x^2+1 \ne 0$, domains are equal ($\mathbb{R}$). Same function. Correct Answer: (b)
Page 8: Trigonometry
- Let $\sin \alpha = -\frac{12}{13}$, and $\alpha \in (\pi, \frac{3}{2}\pi)$. Then $\cos \frac{\alpha}{2} =$ ( ).
Explanation $\alpha \in Q3 \Rightarrow \cos \alpha = -5/13$.
$\alpha/2 \in Q2 \Rightarrow \cos(\alpha/2) < 0$.
$\cos(\alpha/2) = -\sqrt{\frac{1-5/13}{2}} = -\sqrt{\frac{4}{13}} = -\frac{2}{\sqrt{13}}$. Correct Answer: (c) - Suppose that $\cos \alpha = -\frac{1}{2}, \alpha \in (\frac{\pi}{2}, \pi)$, then the value of $\sin \frac{\alpha}{2}$ is ( ).
Explanation $\alpha = 120^\circ \Rightarrow \alpha/2 = 60^\circ$.
$\sin(60^\circ) = \sqrt{3}/2$. Correct Answer: (b) - Which of the following statements is correct? ( )
Explanation Cosine is an even function, so $\cos(-\alpha) = \cos(\alpha)$. Correct Answer: (d)
- About the tangent function $y = \tan x$, which of the following statements is correct? ( )
Explanation The period of $\tan x$ is $\pi$. Range is $\mathbb{R}$. It is an odd function. Correct Answer: (d)
Page 9: Sequences and Conics
- The general formula for the sequence $-\frac{1}{5}, \frac{1}{7}, -\frac{1}{9}, \frac{1}{11}, \cdots$ is $a_n =$ ( ).
Explanation Alternating signs starting negative: $(-1)^n$.
Denominators 5, 7, 9... is arithmetic $2n+3$.
Term: $\frac{(-1)^n}{2n+3}$. Correct Answer: (b) - If the graph of the function $f(x) = 2 + \log_a(x - 3)$ ($a > 0$, and $a \neq 1$) passes through a fixed point $P$, then the coordinates of $P$ are ( ).
Explanation Set argument $x-3 = 1 \Rightarrow x=4$.
$y = 2 + \log_a(1) = 2$. Point $(4, 2)$. Correct Answer: (d) - The equation of the directrix of the parabola $y^2 = -x$ is ( ).
Explanation $4p = 1 \Rightarrow p=1/4$. Opens Left.
Directrix is vertical to the right: $x = 1/4$. Correct Answer: (a) - Suppose that the foci of the ellipse $\frac{x^2}{4} + \frac{y^2}{2} = 1$ are $F_1, F_2$, and the point $P$ is on the ellipse. Then $|PF_1| + |PF_2| =$ ( ).
Explanation Definition: Sum of focal radii = $2a$.
$a^2=4 \Rightarrow a=2$. Sum = 4. Correct Answer: (b)
Page 10: Vectors and Complex Numbers
- Given that $\vec{AB} = \vec{a} + 5\vec{b}$, $\vec{BC} = -2\vec{a} + 8\vec{b}$, $\vec{CD} = 3\vec{a} - 3\vec{b}$. Thus, ( ).
Explanation $\vec{BD} = \vec{BC} + \vec{CD} = \vec{a} + 5\vec{b}$.
$\vec{AB} = \vec{BD}$, so they are parallel and share point B. A, B, D are collinear. Correct Answer: (b) - Suppose that a straight line $l$ is perpendicular to another line $l_1 : x + 2y + 4 = 0$, and passes through the intersection of $l_2 : x + y + 1 = 0$ and $l_3 : 2x + y - 1 = 0$. The equation of $l$ is ( ).
Explanation Intersection of $l_2, l_3$: $x=2, y=-3$.
Perp to $l_1$ (slope -1/2) $\Rightarrow$ Slope 2.
$y - (-3) = 2(x - 2) $
$\Rightarrow y+3 = 2x-4 $
$\Rightarrow 2x-y-7=0$. Correct Answer: (a) - On the complex plane, the point representing a complex number $z$ is on the straight line $x - y = 0$. If $z$ is a root of a quadratic equation $x^2 + mx + 4 = 0$, then $m =$ ( ).
Explanation Roots are conjugates $z, \bar{z}$. Since $z$ is on $y=x$, $z=k(1+i)$.
$z\bar{z} = 4 \Rightarrow 2k^2=4 \Rightarrow k=\pm\sqrt{2}$.
$z+\bar{z} = -m \Rightarrow 2k = -m \Rightarrow m = \mp 2\sqrt{2}$. Correct Answer: (b) - Suppose that, in a sequence $\{a_n\}$, we have $a_1 = 1$, and $\frac{1}{a_n} + \frac{2}{a_{n+1}} = 0$ ($n = 1, 2, \cdots$). Suppose that another sequence $\{b_n\}$ satisfies $b_n = |a_n|$ ($n = 1, 2, \cdots$). Then, the sum of the first $n$ terms of the sequence $\{b_n\}$ is $S_n =$ ( ).
Explanation $a_{n+1} = -2a_n$. $b_n = |a_n|$ is geometric with ratio 2.
Sum $S_n = \frac{1(2^n - 1)}{2-1} = 2^n - 1$. Correct Answer: (c)
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