101. In a class of 39 students, 16 are in the sports group, 12 are in the newspaper group, and 9 are only in the sports group. How many of these students are in neither of these two groups? (1) $15$ (2) $16$ (3) $17$ (4) $18$
102. If $A = \sqrt[6]{4\sqrt[3]{16}}\left(\dfrac{1}{2}\right)^{-\frac{1}{3}}$, then $(2A)^{-\frac{4}{3}}$ equals which of the following? (1) $0.25$ (2) $0.5$ (3) $0.75$ (4) $1$
103. For which values of $m$, the equation $0 = (2m-1)x^2 + 6x + m - 2 = 0$ has two real roots? (1) $-2 < m < 2.5$ (2) $-2 < m < 3.5$ (3) $-1 < m < 3.5$ (4) $-1 < m < 2.5$
104. The graph of $y = -x^2 + 2x + 5$ is shifted 3 units to the right along the positive $x$-axis, then 2 units to the negative $y$-axis. What is the new vertex of the parabola? (1) $(3,4)$ (2) $(2,5)$ (3) $(3,5)$ (4) $(2,6)$
107. If $f = \{(1,2),(2,5),(3,4),(4,6)\}$ and $g = \{(2,3),(4,2),(5,6),(3,1)\}$, then $\dfrac{g}{\text{g} \circ f^{-1}}$ equals which of the following? (1) $\{(4,2),(5,2)\}$ (2) $\{(4,3),(3,5)\}$ (3) $\{(5,2),(2,4)\}$ (4) $\{(3,5),(2,4)\}$
108. The graph of $f(x) = -2 + \left(\dfrac{1}{2}\right)^{Ax+B}$ intersects the graph of $y = x^2 - x$ at two points with $x$-coordinates 1 and 2. What is $f(3)$? (1) $3$ (2) $4$ (3) $5$ (4) $6$ \rule{\textwidth}{0.4pt} Calculation Space %% Page 4
109. What is the value of $\tan\dfrac{11\pi}{4} + \sin\dfrac{15\pi}{4}\cos\dfrac{13\pi}{4}$? (1) $-\dfrac{3}{2}$ (2) $-\dfrac{1}{2}$ (3) $\dfrac{1}{2}$ (4) $\dfrac{3}{2}$
112. The graph shown is the graph of $y = 1 + a\sin bx \cos bx$. What is $a + b$? [Figure: Graph of $y = 1 + a\sin bx\cos bx$ showing a sinusoidal curve with amplitude markings at $\frac{3}{2}$ and $1$, and $x$-axis markings at $-\dfrac{\pi}{4}$ and $\dfrac{3\pi}{4}$] (1) $1$ (2) $\dfrac{3}{2}$ (3) $2$ (4) $3$
113. What is the solution set of the trigonometric equation $\sin^2 x + \cos^2 x = 1 - \dfrac{1}{2}\sin 2x$ on the interval $[0, 2\pi]$? (1) $\dfrac{5\pi}{2}$ (2) $\dfrac{7\pi}{2}$ (3) $2\pi$ (4) $3\pi$
116. The function with the formula $f(x) = \begin{cases} |x^2 - 2x| & ; \quad x < 2 \\ \dfrac{1}{2}x^2 + ax + b & ; \quad x \geq 2 \end{cases}$ is differentiable at $x = 2$. What is $a + b$? (1) $2$ (2) $3$ (3) $4$ (4) $5$
117. For the function with formula $f(x) = (x+2)\sqrt{4x+1}$, the average rate of change of the function on the interval $[0,2]$ is how much greater than the instantaneous rate of change at $x = \dfrac{3}{4}$? (1) $0.1$ (2) $0.15$ (3) $0.20$ (4) $0.25$
118. The graph shown is the graph of the function $f(x) = 3x^4 + ax^3 + bx^2 + cx$. What is $a$? [Figure: Graph of a function with a local minimum near $x=1$ on the positive x-axis, with the curve going to positive infinity on both sides] (1) $-8$ (2) $-7$ (3) $-5$ (4) $-4$
119. The absolute minimum point of the function $f(x) = \dfrac{x^2 + 2x}{(x-1)^2}$ is at what distance from its vertical asymptote? (1) $1$ (2) $\dfrac{4}{3}$ (3) $\dfrac{3}{2}$ (4) $2$
125. In the figure below, chord $AB$ equals the radius of the circle and $AB \parallel CD$, angle $\beta = 2\alpha$, and $CX$ is tangent to the circle. How many degrees is $\widehat{BD}$? \begin{minipage}{0.35\textwidth} [Figure: Circle with chord AB equal to radius, AB$\parallel$CD, tangent CX, angles $\alpha$ and $\beta$ marked] \end{minipage} \begin{minipage}{0.55\textwidth} (1) $50$ (2) $60$ (3) $70$ (4) $75$ \end{minipage}
127. If the area of a regular hexagon inscribed in a circle is $6\sqrt{3}$, then the area of a regular hexagon circumscribed about this circle is how many times $\sqrt{3}$? (1) $7/2$ (2) $7/5$ (3) $8$ (4) $9$
132- If $A$ is a $3 \times 3$ matrix and $|A| = 4$, then the determinant of matrix $A \cdot A$ is which of the following? (1) $64$ (2) $96$ (3) $128$ (4) $256$
133- The common chord of circle $C$ with equation $x^2 + y^2 - 4x + y^2 = 6$ is tangent to the first region of circle $C$. If the point $(-1, 4)$ lies on it, the equation of the common chord is which of the following? (1) $x^2 + y^2 - y + 3x = 6$ (2) $x^2 + y^2 + 3y - x = 6$ (3) $x^2 + y^2 - 2y + x = 6$ (4) $x^2 + y^2 - 3y - x = 6$
135- In an ellipse with semi-major axis $2\sqrt{5}$ and 2 foci, the two foci and the two ends of the minor axis form a square. The sum of the squares of the focal radii of point $M$ on the ellipse is which of the following? (1) $12$ (2) $16$ (3) $18$ (4) $20$
136- For which value of $m$, the three vectors $\vec{a} = (-1,2,3)$, $\vec{b} = (2,\circ,1)$, $\vec{c} = (-4,m,5)$ are coplanar? (1) $-2$ (2) $2$ (3) $3$ (4) $4$
138- In two boxes there are respectively 20 and 12 bulbs. In the first box 4 bulbs are defective and in the second box 3 bulbs are defective. From the first box 5 bulbs are randomly taken and placed in the second box. With what probability is a randomly selected bulb from the new second box defective? (1) $\dfrac{5}{24}$ (2) $\dfrac{11}{48}$ (3) $\dfrac{13}{48}$ (4) $\dfrac{7}{24}$ %% Page 8 Mathematics121-APage 7
139. Let A and B be two independent events. If $P(A \cap B) = 0.6$ and $P(A \cap B) = 0.2$, then $P(A \cup B')$ is equal to: (1) $0.7$ (2) $0.75$ (3) $0.85$ (4) $0.9$
141. The unemployment rate of a country over 10 years is given below. What is the value of $\dfrac{Q_1 + Q_3 - 2Q_2}{Q_3 - Q_1}$? \fbox{$11.5,\ 12.8,\ 13.5,\ 11.2,\ 12.3,\ 12.6,\ 11.9,\ 10.6,\ 10.2,\ 30,\ 12.7$} (1) $-0.225$ (2) $-0.125$ (3) $0.175$ (4) $0.275$
142. If a number leaves remainders 6 and 11 when divided by 5 and 7 respectively, then when divided by 66, what is the remainder? (1) $29$ (2) $32$ (3) $40$ (4) $41$
143. For some values of $n \in \mathbb{N}$, if $3 \mid 13n + 3$ and $7 \mid n + 4$ and $\alpha \neq 1$, then what is the smallest sum of digits of $n$? (1) $7$ (2) $8$ (3) $9$ (4) $10$
147. In how many ways can 11 identical balls be distributed among 5 people such that each person has at least one ball? (1) $160$ (2) $180$ (3) $210$ (4) $220$ %% Page 9 Download of Descriptive Exam Questions and Answers from Riazisara Website ریاضیات 121-A صفحه ۸
148. The number of surjective (onto) functions from a set with 6 elements to a set with 3 elements is which of the following? (1) $360$ (2) $450$ (3) $480$ (4) $540$
156. A particle starts from rest at the origin on the $x$-axis with constant acceleration, and at moment $t = 5\text{s}$ reaches position $x = -122.5\,\text{m}$. How many meters per second does the speed of the particle reach at this moment? (1) $19.6$ (2) $33.4$ (3) $45.0$ (4) $49.0$
157. The velocity–time graph of a particle moving in a straight line is shown in the figure below. The distance traveled by this particle in the time interval $0\,\text{s}$ to $20\,\text{s}$ is how many meters? \begin{minipage}{0.45\textwidth} [Figure: V(m/s) vs t(s) graph. The graph starts at $V=0$ at $t=0$, decreases to $V=-8$ at some time, then increases linearly to $V=22$ at $t=15\,\text{s}$, then decreases back to $V=0$ at $t=20\,\text{s}$.] \end{minipage} \begin{minipage}{0.45\textwidth}
158. A bullet is fired from height $h$ with speed $V$. The bullet passes through the ground at height $9\,\text{m}$ and reaches the ground with speed $\dfrac{3}{2}V$. $h$ is how many meters? (Ignore air resistance and $g = 10\,\dfrac{\text{m}}{\text{s}^2}$) (1) $16.2$ (2) $18$ (3) $33.4$ (4) $36$
159. According to the position–time graph below, the motion of the particle is uniform. The speed of the particle at moment $t = 8\,\text{s}$ is how many meters per second? \begin{minipage}{0.45\textwidth} [Figure: $x$(m) vs $t$(s) graph. The curve starts at $x=0$, goes to a minimum near $t=4\,\text{s}$, then rises to $x=12$ and continues increasing in a parabolic (uniform acceleration) shape.] \end{minipage} \begin{minipage}{0.45\textwidth}
160. A driver with a car of mass 2 tons, moving at speed $36\,\dfrac{\text{km}}{\text{h}}$ on a straight horizontal road, applies the brakes upon seeing a red light. The car stops after traveling $4\,\text{m}$. How many Newtons is the braking friction force applied to the car? (1) $7500$ (2) $12500$ (3) $15000$ (4) $25000$ \begin{flushright} Calculation Space \end{flushright} www.riazisara.ir %% Page 11 Physics121-APage 10
161. A uniform rope of mass $40\,\text{kg}$, as shown in the figure below, rests against a frictionless wall. If the wall exerts a force of $300\,\text{N}$ on the rope, how many newtons does the surface exert horizontally on the rope? $$\left(g = 10\,\frac{\text{N}}{\text{kg}}\right)$$ [Figure: A rope leaning against a wall at an angle, with the base on a horizontal surface]
162. A satellite of mass $500\,\text{kg}$ orbits the Earth at an altitude of $1600\,\text{km}$ above the Earth's surface. How many newtons of gravitational force acts on the satellite? $$\left(g = 10\,\frac{\text{m}}{\text{s}^2} \text{ and } R_e = 6400\,\text{km}\right)$$
163. A scale is fixed to the floor of an elevator. During motion, the scale shows a person's weight greater than the rest value. How is the elevator moving?
[(1)] Necessarily accelerating upward
[(2)] Necessarily accelerating downward
[(3)] Accelerating upward or decelerating downward
[(4)] Decelerating upward or accelerating downward
164. According to the figure below, a body is at rest on a horizontal surface. A horizontal force $F = 40\,\text{N}$ is applied to the body. After $5$ seconds, the force $F$ decreases to $30\,\text{N}$. How does the body move after that? $$\left(g = 10\,\frac{\text{m}}{\text{s}^2}\right)$$ [Figure: A block of $2\,\text{kg}$ on a surface with $F = 40\,\text{N}$ applied horizontally; $\mu_s = 0.6$ and $\mu_k = 0.5$]
[(1)] The body stops at that instant.
[(2)] The body moves with acceleration $1\,\dfrac{\text{m}}{\text{s}^2}$.
[(3)] The body moves with acceleration $2\,\dfrac{\text{m}}{\text{s}^2}$.
[(4)] The body continues moving at constant speed.