102- The figure shows the graph of $y = f(x)$. The domain of $y = \sqrt{xf(x)}$ is which of the following? [Figure: Graph of $f(x)$ showing a curve passing through points $-3$, $1$, $2$ on the x-axis, with minimum near $x=-4$]
103- The figure shows part of the graph of $y = a\sin\pi\!\left(\dfrac{1}{2} + bx\right)$. What is $a \cdot b$? [Figure: Graph of a sinusoidal function with amplitude $2$, showing values at $x = 3.5$ and minimum $-2$, with dashed line at $y=2$]
104- From each of 6 regions of the country, 15 students are invited to a cultural center. In how many ways can 3 students be selected from among them such that no two of them are from the same region?
105- If $\alpha, \beta$ are the roots of the equation $2x^2 - 3x - 4 = 0$, the equation whose roots are $\left\{\dfrac{1}{\alpha}+1,\ \dfrac{1}{\beta}+1\right\}$ is:
108- The function $f(x) = x^2 + 2x + 1$ with domain $(-\infty, +\infty)$ is assumed to be invertible. The graphs of $f$ and $f^{-1}$ intersect at how many points?
110- What is the value of $\tan^{-1}\sqrt{x^2 + x} + \sin^{-1}\sqrt{x^2 + x + 1}$? (1) $\dfrac{\pi}{4}$ (2) $\dfrac{\pi}{2}$ (3) $\dfrac{3\pi}{4}$ (4) $\pi$
115- If $f(x) = [x] + [-x]$ and $g(x) = \begin{cases} f(x) & ; \ x \notin \mathbb{Z} \\ f(x) - 1 & ; \ x \in \mathbb{Z} \end{cases}$, then the number of points of discontinuity of $g$ on the interval $[4, -4]$ is which of the following? (1) $1$ (2) $2$ (3) $3$ (4) zero
117- The function $f(x) = \begin{cases} ax^3 + bx & ; \ x < 1 \\ 2\sqrt{4x - 3} & ; \ x \geq 1 \end{cases}$ is differentiable on the set of real numbers. What is $b$? (1) $\dfrac{1}{2}$ (2) $1$ (3) $\dfrac{3}{2}$ (4) $2$ %% Page 21
118- If $f(x) = \dfrac{x^3 - 2}{1 + x^3}$, $g(x) = \sqrt[3]{x-1}$, then $f'(g(x)) \cdot g'(x)$ is equal to which of the following? (1) $\dfrac{3}{x}$ (2) $\dfrac{3}{x^2}$ (3) $\dfrac{1}{3x}$ (4) $\dfrac{x-3}{x^2}$
119- If $f(x) = xe^x$; $x > 0$, then the tangent line to the graph of $f^{-1}$ at points located along $e$, intersects the $y$-axis at which value? (1) $\dfrac{1}{4}$ (2) $\dfrac{1}{3}$ (3) $\dfrac{1}{2}$ (4) $\dfrac{1}{e}$
120- For which set of values of $a$, the curve $y = x^4 + ax^3 + \dfrac{3}{2}x^2$ is always concave up? (1) $-1 < a < 1$ (2) $-1 < a < 2$ (3) $-2 < a < 1$ (4) $-2 < a < 2$
121- The set of lengths of inflection points of the curve $y = x|x^2 - 4x|$ is which of the following? (1) $\left\{\dfrac{4}{3}\right\}$ (2) $\left\{0, \dfrac{4}{3}, 4\right\}$ (3) $\left\{\dfrac{4}{3}, 4\right\}$ (4) $\left\{0, \dfrac{4}{3}\right\}$
122- The figure opposite shows the graph of $f(x) = \dfrac{x^3 + ax^2}{x^2 + bx + c}$. The value of $(bc - a)$ is which of the following? [Figure: Graph of a rational function with asymptotes, showing a curve with a local feature near the origin] (1) $-2$ (2) $-1$ (3) $1$ (4) $2$
123- In the figure below, the two shaded areas are equal, $C$ is which of the following? [Figure: Graph showing $y = \sqrt{x}$ with shaded region between $x = C$ and $x = 4$] (1) $\dfrac{4}{3}$ (2) $\dfrac{16}{9}$ (3) $2$ (4) $\dfrac{9}{4}$ \rule{\textwidth}{1pt} Scratch Work Area %% Page 22
130- Two circles with radii 4 and 5 are externally tangent. From the center of the smaller circle, a common external tangent to the larger circle is drawn. What is the length of this tangent segment? (1) $8$ (2) $4\sqrt{5}$ (3) $4\sqrt{6}$ (4) $15$
131- The images of points $A(2,4)$ and $B(-6,2)$ under the transformation $D(x,y) = \left(-\dfrac{1}{2}y,\ \dfrac{1}{2}x+1\right)$ are called $A'$ and $B'$. What is the angle between lines $AB$ and $A'B'$? (1) $30°$ (2) $60°$ (3) $90°$ (4) $180°$
133- If $\mathbf{a} = \mathbf{i} - 2\mathbf{j}$, $\mathbf{b} = 3\mathbf{j} + 2\mathbf{k}$, and $\mathbf{c} = 4\mathbf{i} + \mathbf{j} - 2\mathbf{k}$, then the image of vector $(\mathbf{a} \times \mathbf{b}) \times \mathbf{c}$ on the $x$-axis is which of the following? (1) $1$ (2) $2$ (3) $3$ (4) $4$
134- From point $A(5,-2,1)$, a line perpendicular to the plane with equation $x = t+1$, $y = -2t+1$, $z = 2t-3$ is drawn. What are the coordinates of the intersection point of this line and the plane? (1) $(2,-1,-1)$ (2) $(1,1,-3)$ (3) $(4,5,3)$ (4) $(3,-3,1)$
135- The plane passing through the two intersecting lines $(D): \begin{cases} 2x+y=3 \\ 2y-z=0 \end{cases}$ and $(D'): \dfrac{x+1}{2}=\dfrac{y}{1}=\dfrac{z+1}{3}$. Which value does the $z$-axis intercept cut? (1) $-0.8$ (2) $-0.6$ (3) $0.8$ (4) $1.2$
136- The center of a circle is on the first-quadrant angle bisector. If this circle passes through point $A(6,3)$ and is tangent to the line $y = 2x$, what is its radius? (1) $\sqrt{5}$ (2) $\sqrt{6}$ (3) $2\sqrt{2}$ (4) $\sqrt{10}$
140- If $A = \begin{bmatrix} 0 & -\tan\alpha \\ \tan\alpha & 0 \end{bmatrix}$ and $I$ is the identity matrix of order 2, the first row of $(I+A)(I-A)^{-1}$ is which of the following? (1) $[\cos 2\alpha \;\; -\sin 2\alpha]$ (2) $[\cos 2\alpha \;\; \sin 2\alpha]$ (3) $[\sin 2\alpha \;\; \cos 2\alpha]$ (4) $[-\sin 2\alpha \;\; \cos 2\alpha]$
141- All the data in the stem-and-leaf plot below are multiplied by 3, then 40 units are subtracted from each of them. What is the new mean of the data?
142- In 12 statistical data, the total sum of all data is 72 and the sum of their squares is 480. What is the coefficient of variation of this data? (1) $\dfrac{1}{4}$ (2) $\dfrac{1}{9}$ (3) $\dfrac{1}{3}$ (4) $\dfrac{2}{5}$
145- If $A_i = \left[-i,\, \dfrac{9-i}{2}\right]$, $i \in \{1,2,3,\ldots,9\}$, then the set $(A_1 \cap A_2) - (A_2 \cap A_5)$ is which of the following? (1) $(-2,-1) \cup (1,2]$ (2) $[-2,-1] \cup [1,2]$ (3) $[-1,1]$ (4) $\phi$
146- If $A = \{2k-1 \mid k \in \mathbb{Z},\, 1 \leq k \leq 5\}$ and $B = \{k \in \mathbb{Z} : |k-3| \leq 2\}$, then the set $(A \times B) \cap (B \times A)$ has how many elements? (1) $6$ (2) $8$ (3) $9$ (4) $16$ \hrule Workspace %% Page 25
147. Inside a regular hexagon with side $2\sqrt{3}$ units, a point is chosen at random. What is the probability that the distance from this point to each side of the hexagon is greater than one unit? (1) $\dfrac{4}{9}$ (2) $\dfrac{2}{3}$ (3) $\dfrac{5}{9}$ (4) $\dfrac{3}{4}$
148. If $A$ and $B$ are two events from sample space $S$ such that $P(A) = 0.6$, $P(B) = 0.7$, and $P(A \cap B') = 0.2$, then $P(A' \cap B)$ is equal to? (1) $0.1$ (2) $0.3$ (3) $0.4$ (4) $0.5$
150. The four-digit number $\overline{aabb}$ is a perfect square. The remainder when the two-digit number $\overline{ab}$ is divided by 13 is which of the following? (1) $9$ (2) $10$ (3) $11$ (4) $12$
153. In how many ways can 9 identical books be placed in 5 shelves such that at least one book is placed on each shelf? (1) $35$ (2) $42$ (3) $56$ (4) $70$
154. Five white marbles numbered 1 to 5 and five black marbles numbered 1 to 5 are placed in two separate containers. Two marbles are drawn at random from each container. If the sum of the two marbles from each container is 6, what is the probability that both marbles are the same color? (1) $\dfrac{2}{5}$ (2) $\dfrac{4}{9}$ (3) $\dfrac{5}{9}$ (4) $\dfrac{3}{5}$
155. The probability function is defined as $P(X = x) = \dfrac{\dbinom{5}{x}}{A}$\,;\; $x = 0, 1, 2, 3, 4, 5$. By calculating the value of $A$, what is $P(X = 2 \text{ or } X = 3)$? (1) $\dfrac{3}{8}$ (2) $\dfrac{7}{16}$ (3) $\dfrac{9}{16}$ (4) $\dfrac{5}{8}$ %% Page 26 Physics120APage 9
156- Three forces $\vec{F}_1$, $\vec{F}_2$, $\vec{F}_3$ make pairwise angles of $120^\circ$ with each other. If the magnitudes of the forces are 5, 10, and 15 newtons respectively, find their resultant. How many newtons is it? (1) zero (2) $5$ (3) $5\sqrt{3}$ (4) $10$
157- Train A, with length 200 m, is moving at constant speed $4\,\dfrac{\text{m}}{\text{s}}$. Train B, with length 225 m, has stopped on the adjacent track. At the moment train A completely passes train B, train B starts moving in the same direction as train A with constant acceleration $2\,\dfrac{\text{m}}{\text{s}^2}$ and brings its speed up to $50\,\dfrac{\text{m}}{\text{s}}$ and continues at that speed. How many seconds after the start of motion does train B completely pass train A? (1) $57.5$ (2) $82.5$ (3) $80$ (4) $105$
158- The equation of motion of a particle in SI is $x = \dfrac{2}{3}t^3 - 6t^2 + 20t$. What is the minimum speed (in meters per second) that this particle reaches along its path? (1) zero (2) $1$ (3) $2$ (4) $4$
159- A ball is thrown vertically upward from a height of 20 m above the ground with initial speed $V_0$. It reaches a height of 65 m above the ground. The speed of the ball becomes zero at ground level. If $g = 10\,\dfrac{\text{m}}{\text{s}^2}$, how many meters per second is $V_0$? (Air resistance is negligible) (1) $35$ (2) $30$ (3) $13\sqrt{10}$ (4) $10\sqrt{13}$
160- The initial velocity vector of a projectile in SI is $\vec{V}_0 = 15\vec{i} + 20\vec{j}$. What is the displacement vector of this projectile in the first 3 seconds in SI? ($g = 10\,\dfrac{\text{m}}{\text{s}^2}$ and air resistance is negligible.) (1) $45\vec{i} + 15\vec{j}$ (2) $15\vec{i} - 10\vec{j}$ (3) $45\vec{i} - 10\vec{j}$ (4) $10\vec{i} + 45\vec{j}$
162. The velocity of a 2kg ball changes from $\vec{V}_1 = 10\hat{i} - 8\hat{j}$ to $\vec{V}_2 = 6\hat{i} - 5\hat{j}$ (in SI units) under a constant force. If the time of force application is $\frac{1}{10}$ seconds, the magnitude of the force is how many Newtons? (1) $10$ (2) $12$ (3) $15$ (4) $20$
163. In the figure, a weight is thrown upward from the bottom of an inclined surface with initial velocity $V_0$, tangent to the surface. The weight goes up and returns to the starting point. If the friction force is $\frac{2}{10}$ of the weight, the time to go up equals the time to come down by how much? ($g = 10\,\frac{m}{s^2}$) [Figure: inclined surface at $30^\circ$ with initial velocity $V_0$ along the surface] (1) $\sqrt{\dfrac{4}{3}}$ (2) $\sqrt{\dfrac{3}{7}}$ (3) $\dfrac{3}{5}$ (4) $\dfrac{5}{3}$
164. A pulley whose string length is 2 meters and ball mass is 2kg, starting from a position where the string makes a $53^\circ$ angle with the vertical, is released from rest. The tension in the string at the moment it makes a $37^\circ$ angle with the vertical is how many Newtons? ($\sin 37^\circ = 0.6$, air resistance is negligible, $g = 10\,\frac{m}{s^2}$) (1) $16$ (2) $20$ (3) $24$ (4) $36$