101- What is the value of $2\sqrt[4]{2} \cdot \sqrt[4]{2\sqrt{2}} \cdot (\sqrt{2-\sqrt{3}}+\sqrt{2+\sqrt{3}})$? (1) $\sqrt{3}$ (2) $2$ (3) $1+\sqrt{3}$ (4) $2\sqrt{3}$
102- Two functions with the sets $g=\{(2,5),(3,4),(1,6),(4,7),(8,1)\}$ and $f(x)=2x-5$ are given. If $(f^{-1}\circ g)(a)=6$, what is $a$? (1) $1$ (2) $2$ (3) $3$ (4) $4$
103- If $f(x)=1-\left(\dfrac{1}{2}\right)^x$, and the domain of $y=\sqrt{x\,f(x)}$ is given, what is the range? (1) $[-1,1]$ (2) $(-\infty,0)$ (3) $(-\infty,+\infty)$ (4) $(0,+\infty)$
107- For which values of $x$ in the domain of $y=\sqrt{5+4x-x^2}$, does the graph of $y=\sqrt{5+4x-x^2}$ lie above the graph of $y=|x-3|+2$? (1) $\left(\dfrac{3-\sqrt{17}}{2},\ 5\right)$ (2) $\left(2,\ \dfrac{3+\sqrt{17}}{2}\right)$ (3) $\left(2,\ \dfrac{4+\sqrt{15}}{2}\right)$ (4) $\left(2,\ 2+\sqrt{15}\right)$
109- What is the general solution of the trigonometric equation $\dfrac{\sin 3x}{\sin x}=2\cos^2 x$? (1) $\dfrac{k\pi}{2}$ (2) $\dfrac{k\pi}{2}+\dfrac{\pi}{4}$ (3) $k\pi-\dfrac{\pi}{4}$ (4) $k\pi+\dfrac{\pi}{4}$
112. The derivative of the function $y = \cos^2(\tan^{-1}x)$, at $x = 1$, is which of the following? (1) $-\dfrac{1}{2}$ (2) $-\dfrac{1}{4}$ (3) $\dfrac{1}{4}$ (4) $1$
113. For values $n \geq n_0$, if the distance of the terms of the sequence $\left\{\dfrac{fn+1}{rn-2}\right\}$ from its limit is less than $0.02$, what is the smallest value of $n_0$? (1) $61$ (2) $62$ (3) $63$ (4) $64$
117. The figure on the left shows the graph of the function $y = f(x)$. The graph of $f'(x)$ is in which form? [Figure: The main graph shows an S-shaped (sigmoid-like) increasing curve with two horizontal asymptotes.] [Option (1): Graph with a sharp peak (cusp) at the origin, symmetric, going to zero on both sides.] [Option (2): Graph with a smooth bell-shaped curve (positive hump).] [Option (3): Graph with a curve that dips below the x-axis on the left and rises above on the right, with horizontal asymptotes.] [Option (4): Graph with a smooth curve having a negative dip, symmetric about y-axis, with horizontal asymptotes.]
118. From the point $A(2,-1)$, two tangent lines to the curve $y = \dfrac{1}{2}x^2 - x$ are drawn. What is the angle between these two tangent lines? (1) $\dfrac{\pi}{4}$ (2) $\dfrac{\pi}{3}$ (3) $\dfrac{\pi}{2}$ (4) $\tan^{-1}2$
119. The right-hand derivative of the function with the rule $f(x) = ([x] - |x|)^5\!\sqrt{9x}$, at the point $x = -3$, is equal to: (1) $-\dfrac{16}{3}$ (2) $-5$ (3) $-4$ (4) $\dfrac{7}{3}$ %% Page 21 Mathematics 120-C Page 4
120. The tangent line to the curve of function $f$ at a point of length 3 on it, has equation $x + 2y = 7$. If $g(x) = \dfrac{1}{x} f^{-1}(x)$, then $g'(2)$ is which of the following? (1) $-\dfrac{7}{4}$ (2) $-\dfrac{5}{4}$ (3) $-\dfrac{3}{4}$ (4) $\dfrac{1}{4}$
121. For which domain, the function $f(x) = x^3 e^{-x}$ is increasing and its graph is concave upward? (1) $(0, 3-\sqrt{3})$ (2) $(3-\sqrt{3}, 3)$ (3) $(3, 3+\sqrt{3})$ (4) $(3+\sqrt{3}, +\infty)$
122. The figure shows the graph of the function with the formula $f(x) = \dfrac{a\sin 2x + b}{\sin x + \cos x}$, which has period one. What is $a$? [Figure: Graph of a periodic function with amplitude 2] (1) $-1$ (3) $\sqrt{2}$ (4) $2$ (1) $1$
123. The area of the region bounded by the graph of the function $y = x^2|x|$ and the line $y = 8$ is which of the following? (1) $16$ (2) $18$ (3) $22$ (4) $24$
127. In an equilateral triangle with side $2\sqrt{3}$ units, the volume of the solid obtained by rotating both shaded regions about the altitude $AH$ is which of the following? [Figure: Equilateral triangle with altitude AH and shaded regions (inscribed circle area)] (1) $\dfrac{4\pi}{3}$ (2) $\dfrac{3\pi}{2}$ [6pt] (3) $2\pi$ (4) $\dfrac{5\pi}{3}$ %% Page 22 Download Exam Questions from Riazisara Website \begin{flushright} Mathematics 120-C Page 5 \end{flushright}
131- The image of the line $2x + 3y = 6$ under the transformation $T(x,y) = (2y-1, x+3)$ passes through which point? (1) $(-3, 2)$ (2) $(1,-1)$ (3) $(5, \circ)$ (4) $(7, \circ)$
133- Three points $A(2,1,\circ)$, $B(3,-1,2)$, $C(-1,1,3)$ are vertices of a triangle. What is $\cos A$? (1) $\dfrac{\sqrt{2}}{6}$ (2) $\dfrac{\sqrt{2}}{4}$ (3) $\dfrac{\sqrt{2}}{6}$ (4) $\dfrac{\sqrt{3}}{4}$
134- Two vectors $\mathbf{a} = (1,-2,3)$ and $\mathbf{b} = (2,1,-1)$ are given. The volume of the parallelepiped built on vectors $\mathbf{a}$, $\mathbf{b}$, and $\mathbf{a} \times \mathbf{b}$ is: (1) $54$ (2) $72$ (3) $75$ (4) $80$
135- What is the length of the common perpendicular of the two lines $$\frac{x-1}{1} = \frac{y+2}{-1} = \frac{z}{3} \quad \text{and} \quad \begin{cases} x = 2y - 1 \\ z = 3y - 2 \end{cases}$$? (1) $\sqrt{3}$ (2) $\sqrt{6}$ (3) $2\sqrt{3}$ (4) $2\sqrt{6}$
137- The two lines $y = -2x$ and $y = 2x + 4$ are the asymptotes of a hyperbola, and $M\!\left(\dfrac{3}{2},\, 5\right)$ is one of its points. The distance between the two foci of this hyperbola is: (1) $2\sqrt{3}$ (2) $2\sqrt{5}$ (3) $4\sqrt{3}$ (4) $4\sqrt{5}$ \begin{flushright} \fbox{Workspace} \end{flushright} %% Page 23 Mathematics120-CPage 6
138- If determinant $D = \begin{vmatrix} 1 & 1 & 1 \\ bc & ac & ab \\ ac & ab & bc \end{vmatrix}$, then what is the value of $\begin{vmatrix} a+b & b & ab \\ b+c & c & bc \\ a+c & a & ac \end{vmatrix}$? (1) $-D$ (2) $D$ (3) $(a+b+c)D$ (4) $abcD$
139- If matrix $A$ has the transformation $T(x,y) = (2x - y,\ 3x - 4y)$ and $I$ is the identity matrix, and $\alpha$ and $\beta$ are two real numbers such that $\alpha A + \beta I = A^{-1}$, what is the value of $\beta$? (1) $-\dfrac{3}{5}$ (2) $-\dfrac{1}{5}$ (3) $\dfrac{2}{5}$ (4) $\dfrac{4}{5}$
140- Three planes are given by the following matrix equation: $$\begin{bmatrix} 2 & -1 & 1 \\ 1 & 3 & -1 \\ 1 & -11 & 5 \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 6 \\ 4 \\ 2 \end{bmatrix}$$ What is the relationship of the two planes with respect to each other? (1) Parallel (2) Coincident (3) Perpendicular (4) Lacking a common line of intersection
142- The scores of a technical skills test for two workers $A$ and $B$ are as follows: $A: 15, 14, 15, 16, 17, 19$ $B: 16, 14, 17, 14, 17, 18$ Which worker has greater precision? (1) $A$ (2) $B$ (3) Equal (4) Unpredictable
143- We place each of the integers from 1 to 30 on 30 balls and put them in a bag. We draw at least how many balls to be certain that at least two of the drawn numbers have a greatest common divisor greater than 1? (1) $15$ (2) $11$ (3) $12$ (4) $13$
144- If $A = \{x \in \mathbb{N},\ 5 < x^2 < 50\}$ and $B = \{3k-2 \mid k \in \mathbb{Z},\ 1 \leq k \leq 4\}$, then the number of elements of $(A \times B) \cap (B \times A)$ is: (1) $4$ (2) $8$ (3) $16$ (4) $32$
146- Is the relation $ad = bc \Leftrightarrow (a,b)\,R\,(c,d)$ an equivalence relation on $\mathbb{R}^2$? If it is an equivalence relation, which point does the graph $[(2,6)]$ pass through? (1) It is not an equivalence relation. $(1,2)$ (2) $(1,3)$ (3) $(2,3)$ (4)
147- We roll two dice together. With which probability is the sum of the two numbers rolled an odd number? $$\frac{5}{12} \ (1) \hspace{2cm} \frac{4}{9} \ (2) \hspace{2cm} \frac{5}{9} \ (3) \hspace{2cm} \frac{7}{12} \ (4)$$
148- In the equation $ax^2 + bx = 5$, coefficient $a$ is chosen randomly from the interval $[1,3]$ and coefficient $b$ is chosen randomly from the interval $[-3, 0]$. With which probability is the set of solutions of this equation more than $\dfrac{2}{3}$? $$\frac{4}{9} \ (1) \hspace{2cm} \frac{5}{9} \ (2) \hspace{2cm} \frac{7}{12} \ (3) \hspace{2cm} \frac{5}{6} \ (4)$$
150- Seven times a six-digit number $\overline{abcabc}$ is a perfect square. What is the largest value of the number $\overline{abc}$? $$14 \ (1) \hspace{2cm} 15 \ (2) \hspace{2cm} 16 \ (3) \hspace{2cm} 17 \ (4)$$
151- Two natural numbers equal to $N = \overline{abc}$, written in the form $\varphi(a \circ bc)$ with a change of base, give the value $N$. What is the largest value of $N$, at least how many units less than a perfect square? $$1 \ (1) \hspace{2cm} 2 \ (2) \hspace{2cm} 3 \ (3) \hspace{2cm} 4 \ (4)$$
152- For how many natural numbers $n$, are the two numbers in the forms $5n-2$ and $7n+3$ not coprime? $$3 \ (1) \hspace{2cm} 4 \ (2) \hspace{2cm} 5 \ (3) \hspace{2cm} 6 \ (4)$$
154- The number of ordered triples, with non-negative integer and non-positive integer coordinates, such that the sum of every three coordinates of each set equals $10$ and each coordinate is less than $6$, is which of the following? $$17 \ (1) \hspace{2cm} 18 \ (2) \hspace{2cm} 20 \ (3) \hspace{2cm} 21 \ (4)$$
155- In a container there are 5 white marbles and 3 black marbles; in another container there are 4 white marbles and 2 black marbles. We randomly draw 4 marbles from each container. With which probability are the 4 marbles drawn all the same color? $$0.12 \ (1) \hspace{2cm} 0.15 \ (2) \hspace{2cm} 0.18 \ (3) \hspace{2cm} 0.24 \ (4)$$ %% Page 25 Physics120-CPage 8
164- In the figure below, a pendulum ball is released from point $A$ and passes through the lowest point of the path with speed $V$. When the ball's speed reaches $\dfrac{\sqrt{2}}{2}\ V$, what angle does the string make with the vertical? $$\left(g = 10\ \frac{\text{m}}{\text{s}^2},\ \cos 53^\circ = 0.6\right)$$ [Figure: Pendulum of length $1\ \text{m}$ released from point $A$ at $53^\circ$ from vertical] (Air resistance is neglected.)
165- A person of mass $80\ \text{kg}$ is inside an elevator. At the moment the elevator begins moving downward with constant acceleration $2\ \dfrac{\text{m}}{\text{s}^2}$, how many Newtons of force does the person exert on the elevator? $$\left(g = 10\ \frac{\text{m}}{\text{s}^2}\right)$$
[(1)] $960$ (2) $800$ (3) $160$ (4) $640$
\rule{\textwidth}{0.4pt} Workspace for Calculations %% Page 27 Physics120-CPage 10