101-- Geometric sequences with a common ratio greater than one that include 5 terms and are members of the set $\{1, 2, \ldots, 100\}$. How many of these sequences can be found whose terms are all members of the set $\{1, 2, \ldots, 100\}$? (1) $2$ (2) $4$ (3) $6$ (4) $7$
102-- The minimum value of the function $y = mx^2 - 12x + 5m - 1$ for $m = 2$ is the axis of symmetry of the parabola. What is $x$? (1) $x=2$ (2) $x=2.5$ (3) $x=3$ (4) $x=3.5$
106-- $\alpha$ and $\beta$ are roots of the equation $x^2 + 6x + a = 0$. If $0 < \alpha < \beta < 0$ and $12\sqrt{2} + 85 = 12\sqrt{2} + 85$, and $3\alpha^2 + 2\beta^2 = 12\sqrt{2} + 85$, what is the value of $a$? (1) $1$ (2) $\dfrac{13}{4}$ (3) $\dfrac{21}{5}$ (4) $2$
108. The function $f(x) = x^2\sqrt{x}$ is one-to-one on a domain. Which of the following is the inverse function on this domain? (1) $-\sqrt{x^2}\ ,\ x \leq 0$ (2) $-\sqrt[3]{x}\ ,\ x \leq 0$ (3) $-\sqrt{x^2}\ ,\ x \geq 0$ (4) $-\sqrt[3]{x}\ ,\ x \geq 0$
109. The distance of point $A$ from the line $x + y = a$ is $a$. The two points $B(-3, 2)$ and $C(-1, 4)$ are on this line, and $5$ is the distance. What is the value of $a$? (1) $2$ (2) $\dfrac{1}{2}$ (3) $-\dfrac{1}{2}$ (4) $-2$
111. Suppose $5^x = 10$ and $5^x = 20$. If $2^{f(x)} = 2$ holds, what is the function $f$? (1) $\dfrac{2x+1}{x+1}$ (2) $\dfrac{x-1}{2x-1}$ (3) $\dfrac{2x-1}{x-1}$ (4) $\dfrac{x+1}{2x+1}$
112. In triangle $ABC$, angle $A$ is $25$ degrees more than angle $B$. What is $2\cos A \sin B - \sin C$? (1) $\dfrac{\sqrt{2}}{2}$ (2) $-\dfrac{\sqrt{2}}{2}$ (3) $\dfrac{\sqrt{2}}{2}$ (4) $-\dfrac{\sqrt{2}}{2}$
113. The figure below shows a portion of the graph of $f(x) = a\cos(bx + c)$. If $b > 0$, $0 < c < \pi$, and $\dfrac{ac}{b} = 0$, what is the value? [Figure: Graph of a cosine function with amplitude $\frac{1}{3}$, showing one full cycle. The graph reaches a minimum of $-\frac{1}{3}$ and passes through key points at $x = \frac{3}{4}$ and $x = \frac{5}{4}$] (1) $\dfrac{1}{16}$ (2) $1$ (3) $\dfrac{1}{4\pi}$ (4) $\pi$ %% Page 5 Mathematics $\leftarrow$
114- The sum of the solutions of the trigonometric equation $\sqrt{7}\sin x + \sqrt{3}\cos x = \sqrt{7}$ on the interval $[-\pi, 2\pi]$ is which of the following? (1) $\dfrac{\pi}{2}$ (2) $\dfrac{7\pi}{3}$ (3) $\dfrac{9\pi}{4}$ (4) $\dfrac{11\pi}{6}$
117- The polynomial $p(x) = x^{2n+1} + 2x^{2n} + x^6 + 3x^5 + 16x + 16a$ is divisible by $x + 2$ for every natural number $n$. For $n=1$, what is the remainder when $p(x)$ is divided by $x^2 + 2x - 3$? (1) $-15x + 24$ (2) $-15x + 14$ (3) $-5x + 34$ (4) $-5x + 44$
120- At the intersection points of the curves $f(x) = \sin x + \dfrac{1}{2}\cos x$ and $g(x) = \dfrac{3}{2}\sin x$ on the interval $[0, \pi]$, a tangent line to the curve $f(x)$ is drawn. This tangent line intersects the $x$-axis at which interval? (1) $\dfrac{\pi}{4} - 1$ (2) $\dfrac{\pi}{4} - 2$ (3) $\dfrac{\pi}{4} + \dfrac{1}{\lambda}$ (4) $\dfrac{\pi}{4} + \dfrac{3}{\lambda}$ %% Page 6 121-- Function $f$ is differentiable and periodic with period 5. If $f'(-1)=\dfrac{3}{2}$ and $g(x)=f(x+1)+f(3x+10)$, then $g'(-2)$ is which of the following? (1) $3$ (2) $\dfrac{7}{2}$ (3) $6$ (4) $\dfrac{13}{2}$
Q122
4 marksDifferentiation from First PrinciplesView
122-- If $f(x)=(x-4)\sqrt[4]{x+3}$, then $\displaystyle\lim_{h\to 0}\dfrac{f^{2}(\Delta-h)-3f(\Delta-h)+2}{h(\Delta-h)}$ is which of the following? (1) $\dfrac{13}{30}$ (2) $-\dfrac{5}{12}$ (3) $\dfrac{5}{6}$ (4) $-\dfrac{13}{15}$
123-- Point $A(-1,1)$ is a relative extremum of the function $y=x^{2}|x|+3ax^{2}+b$. The value of $\dfrac{b}{a}$ is which of the following? (1) $-3$ (2) $-\dfrac{1}{3}$ (3) $3$ (4) $\dfrac{1}{3}$
124-- The locus of the intersection of the asymptotes of the hyperbola $y=\dfrac{ax+3}{(a+1)x+(a-1)}$ is $y=\dfrac{3}{2}x^{2}+x+\dfrac{5}{6}$. The graph of this hyperbola intersects the $x$-axis at which length? (1) $3$ (2) $-3$ (3) $\dfrac{3}{2}$ (4) $-\dfrac{3}{2}$
125-- How many five-digit natural numbers can be written with non-repeating digits such that among those digits, one is between two even and two odd digits? (1) $1850$ (2) $1950$ (3) $2150$ (4) $2500$
126-- In a random experiment, $S=\{x,y,z\}$ is a sample space. If $P(x)$, $P(y)$, and $P(z)$ form a geometric sequence and together they are less than one unit and their geometric mean is $\dfrac{1}{5}$, then the smallest simple event in $S$ is how much? (1) $\dfrac{2-\sqrt{2}}{5}$ (2) $\dfrac{2-\sqrt{2}}{5}$ (3) $\dfrac{2-\sqrt{3}}{10}$ (4) $\dfrac{2-\sqrt{3}}{10}$
127-- In a bag there are 16 balls numbered 1 to 16. Two balls are drawn randomly and without replacement. If we know that the number of the second ball is less than the number of the first ball, what is the probability that the number of the first ball is 16? (1) $\dfrac{1}{16}$ (2) $\dfrac{1}{12}$ (3) $\dfrac{1}{8}$ (4) $\dfrac{1}{4}$ %% Page 7
128. To estimate the mean income of individuals in a community, we randomly select two samples. We use the standard deviation of the second sample as an estimate for the mean of the first sample, which equals $\frac{2}{\overline{x}}$ times the calculated value for the first sample. The size of the second sample is how many times the size of the first sample? (1) $1/5$ (2) $2/25$ (3) $2/75$ (4) $3/5$
129. The mean of six statistical data is a natural number, and the variance of these data is $1$, $9$, $b^2$, $5$, $\pi^2$, $9$. If the variance of these data equals $4$, what is the value of $ab$? $(a, b \in \mathbb{Z})$ (1) $-4$ (2) $4$ (3) $2$ (4) $-2$ *130. In isosceles triangle $ABC$, point $M$ is the midpoint of $AB$, and the perpendicular bisector of $AB$ cuts side $AC$ at point $N$. If $\widehat{NBC} = 54°$, what is the measure of angle $\widehat{MNB}$? (1) $48$ (2) $56$ (3) $66$ (4) $78$
136- In rectangle $ABCD$, point $(5,3)$ is vertex $B$ and the lengths of sides $C$ and $D$ are $5/8$ and $3$ respectively. If vertex $D$ is reflected over the $x$-axis, the distance from the image of vertex $C$ to the line $BD$ from the origin of coordinates is how much? \[
\text{(1)}\ 2/5 \qquad \text{(2)}\ \sqrt{6/5} \qquad \text{(3)}\ \sqrt{6} \qquad \text{(4)}\ 2
\]
137- The internal bisector of angle $A$ in triangle $ABC$ divides the opposite side into segments of $3/5$ and $2/5$ units. If the measure of angle $C$ is $60$ degrees, the smaller side of the triangle is how many units? \[
\text{(1)}\ 3/75 \qquad \text{(2)}\ 4/25 \qquad \text{(3)}\ 4/75 \qquad \text{(4)}\ 5/25
\]
138- If $A = \begin{bmatrix} x & -1 & -x \\ 0 & 0 & 4 \\ y & z & z \end{bmatrix}$, $B = \begin{bmatrix} yz & \frac{1}{2} & 2 \\ yz & 0 & -4y \\ 0 & \frac{1}{2} & 0 \end{bmatrix}$ and matrix $AB$ is scalar for every $y \in \mathbb{Z}$, the value of $xy$ is which? \[
\text{(1)}\ -1 \qquad \text{(2)}\ -2 \qquad \text{(3)}\ 1 \qquad \text{(4)}\ 2
\]
140- For every $m$, the equation $y = 6$, $(m+1)x + (m-2)y = 6$ is the equation of a chord of circle $C$. If point $A(-1,1)$ lies on circle $C$, the circumference of circle $C$ is which? \[
\text{(1)}\ 2\sqrt{3}\pi \qquad \text{(2)}\ 2\pi \qquad \text{(3)}\ 3\pi \qquad \text{(4)}\ 2\sqrt{7}\pi
\] %% Page 9
142- Three vectors $\vec{a} = (1,1,0)$, $\vec{b} = (-1,2,0)$, and $\vec{c}$ are non-coplanar, and $\vec{h} = (x,y,4)$ is the altitude vector of the parallelepiped formed by these three vectors. If $\vec{a} \cdot \vec{c} = 1$ and $\vec{b} \cdot \vec{c} = 5$, what is the magnitude of vector $\vec{c}$? (4) $\sqrt{21}$ (3) $\sqrt{19}$ (2) $4$ (1) $5$
144- The point $(a, b)$ lies on the curve $y = \dfrac{3x-1}{x+2}$. If $a, b \in \mathbb{Z}$, how many points with this property lie on this curve? (4) $4$ (3) $3$ (2) $2$ (1) $1$
150- At minimum, how many subsets must be chosen from the set $\{7, \ldots, 3, 2, 1\}$ so that we are certain that two subsets share a common element? (4) $46$ (3) $45$ (2) $64$ (1) $65$ Place for Calculations %% Page 10 Control Code: 122 A Download of questions and descriptive answer keys of the national entrance exam from the Riazi Sara website www.riazisara.ir National University Entrance Exam for Universities and Higher Education Institutions of the Country --- Year 1401 Mathematical and Technical Sciences GroupSpecialized Exam