1. Let A = \{1, 3, 4, 6, 7\}, B = \{1, 3, 5\} be two sets. Which of the following statements is
correct? ( )
(a) A\in B (b) A\subseteq B (c) B\in A (d) B\subseteq A
2. Let A = \{x \mid 3 < x < 5\} , B = \{x \mid x > 4\} be two sets. Then A \cap B = ( ) .
(a) \{x \mid 3 < x \leq 4\} (b) \{x \mid 4 \leq x < 5\} (c) \{x \mid 4 < x < 5\} (d) \{x \mid 3 < x < 4\}
3. Which of the following points is in the third quadrant? ( )
(a) (1, - 2) (b) (1, 2) (c) (- 1, - 2) (d) (- 1, 2)
4. The solution set of the inequality x^{2} - 2x - 3 \geq 0 is ( ).
(a) (- 1, 3) (b) (- \infty , - 1] \cup [3, +\infty) (c) (- \infty , - 1) \cup (3, +\infty) (d) [- 1, 3]
5. The domain of the function f(x) = \sqrt[3]{x} + 1 is ( ).
(a) (- \infty , 0] (b) (- \infty , +\infty) (c) [1, +\infty) (d) [0, +\infty)
6. In the Cartesian coordinate system, the horizontal coordinate of a point M is -4 , and its
distance to the x-axis is 5. The coordinates of M are ( ).
(a) (-4,5) or (-4,3) (b) (-4,5) or (-4,-5) (c) (5,-4) (d) (-5,-4)
7. \sin 120^{\circ} = ( )
(a) \frac{\sqrt{3}}{2} (b) \frac{-\sqrt{3}}{2} (c) \frac{1}{2} (d) \frac{1}{2}
8. Which of the following functions is an odd function? ( )
(a) y = \sqrt[3]{x} +2 (b) y = x^2 +3 (c) y = x^3 +1 (d) y = -x
9. The slope of the straight line passing through points A(-1,1) , B(2,3) is ( ).
(a) \frac{3}{2} (b) \frac{2}{3} (c) \frac{3}{2} (d) \frac{2}{3}
10. The inverse function of the function y = 10x + 3 is ( ). []‫تصویر‬
.11 Which of the following functions is a decreasing function on (- \infty , + \infty) ? ( )
[]‫تصویر‬
.12 The solution set of the inequality \frac{2x + 1}{x - 1} < 0 is ( ). []‫تصویر‬
.13 If -1, a, 15 is an arithmetic sequence, and a = ( ) .
(a) 9 (b) 5 (c) 7 (d) 11
14. If the angle of inclination of a straight line l is 60^{\circ} , then the slope of l is ( ). []‫تصویر‬
.15 If \{a_{n}\} is an arithmetic sequence, and a_{1} = 1 , a_{2} = 3 , then a_{5} = ( ).
(a) 9 (b) 5 (c) 7 (d) 11
16. If a point P(x,y) is on the terminal side of an angle \alpha , and the distance between P
and the origin is r ( r \neq 0 ), then \cos \alpha = ( ).
(a) \frac{y}{r} (b) \frac{y}{x} (c) \frac{x}{y} (d) \frac{x}{r}
17. If the general term of a sequence \{a_{n}\} is a_{n} = 2n - 1 , which of the following
statements is correct? ( )
(a) \{a_{n}\} is both an arithmetic and a geometric sequence
(b) \{a_{n}\} is neither an arithmetic nor a geometric sequence
(c) \{a_{n}\} is an arithmetic sequence
(d) \{a_{n}\} is a geometric sequence
18. Suppose that the distance from point A(0,3) to point B(x, - 1) is 5. Then x = ( ).
(a) \sqrt{21} (b) \pm 3 (c) 3 (d) -3
19. If a line l has a slope of -3 , and it passes through the point (1,2) , then the equation of l
is ( ).
(a) 3x + y - 5 = 0 (b) x + 3y - 7 = 0 (c) 3x + y + 1 = 0 (d) x + 3y + 5 = 0
20. Suppose that a circle has radius 3 and coordinates of the center at (-1, 2) . Then the
equation of the circle is ( ).
(a) x^{2} + y^{2} - 2x - 4y - 4 = 0 (b) x^{2} + y^{2} + 2x + 4y - 4 = 0 (c) x^{2} + y^{2} + 2x - 4y + 4
= 0 (d) x^{2} + y^{2} + 2x - 4y - 4 = 0
21. If \cos \alpha = \frac{2}{3} , and 0 < \alpha < \frac{\pi}{2} , then \tan \alpha = ( ) .
(a) \frac{\sqrt{3}}{3} (b) -\frac{\sqrt{3}}{2} (c) \frac{2\sqrt{5}}{5} (d) \frac{\sqrt{5}}{2}
22. If the sequence \{a_{n}\} is a geometric sequence, and a_{1} = -48 , a_{4} = 6 , then the
common ratio is q = ( ) .
(a) -\frac{1}{2} (b) -2 (c) \frac{1}{2} (d) 2
23. If a circle passes through the point A(2, 2) , and has the center at (2, 0) , then the
equation of the circle is ( ). []‫تصویر‬
.24 If a, b, c are rational numbers, and a > b, which of the following statements is correct?
( ) []‫تصویر‬
.25 If the exponential function y = a^x is monotonic decreasing, then the range of a is ( ).
[]‫تصویر‬
.26 If \sin \alpha = \frac{3}{4} , then \cos 2\alpha = ( ) . []‫تصویر‬
.27 The coordinates of the intersection of the lines l_1: 3x - y + 2 = 0 and l_2: 4x - y + 3 = 0
are ( ). []‫تصویر‬
.28 If the eccentricity of the hyperbola \frac{x^2}{a^2} - \frac{y^2}{3} = 1 ( a > 0 ) is e = 2 ,
then a = ( ).
(a) 2 (b) \frac{\sqrt{6}}{2} (c) \frac{\sqrt{5}}{2} (d) 1
29. If a sequence \{a_n\} is given by \frac{1}{2}, -\frac{1}{4}, \frac{1}{8}, \dots , then its
general term is a_n = ( ).
(a) \frac{1}{2^n} (b) (-1)^{n - 1} \frac{1}{2^n} (c) \frac{1}{2^{n - 1}} (d) (-1)^n \frac{1}{2^n}
30. (\log_{3} 27)(\log_{64} 16) = ( ).
(a) \frac{1}{6} (b) \frac{2}{3} (c) 6 (d) 5
31. The minimum value of the function y = \cos 5x + 4 is ( ).
(a) 5 (b) 3 (c) 4 (d) 2
32. If the parabola y = ax^2 passes through the point (1, 2) , then a = ( ).
(a) -\frac{1}{2} (b) 2 (c) \frac{1}{2} (d) 1
33. Which of the following straight lines is perpendicular to the line -2x + y - 2 = 0? []‫تصویر‬
.34 Given that \cos \alpha = - \frac{1}{2}, \alpha \in \left(\frac{\pi}{2}, \pi\right) , then the
value of \sin \frac{\alpha}{2} is ( ). []‫تصویر‬
.35 \cos 75^{\circ} = ( ) []‫تصویر‬
.36 If the function y = f(x) has a domain (-1,0) , then the domain of the function y = f(x^{2} - 1)
is ( ). []‫تصویر‬
.37 If \cos \alpha = - \frac{7}{25} , and \frac{\pi}{2} < \alpha < \pi , then \cos \frac{\alpha}{2} =
( ).
(a) - \frac{4}{5} (b) - \frac{4}{5} (c) - \frac{3}{5} (d) - \frac{3}{5}
38. Which of the following equations is correct? ( ) []‫تصویر‬
.39 The minimum positive period of the function y = 2 \tan \left(3x + \frac{\pi}{4}\right) is ( ).
(a) \frac{\pi}{3} (b) 2\pi (c) \frac{\pi}{4} (d) \pi
40. If an ellipse C is centered at the origin, its foci are on the y-axis, the focal length is 4,
and the eccentricity is \frac{2}{3} , then the equation of C is ( ). []‫تصویر‬
.41 If a sequence \{a_{n}\} satisfies a_{n} = (-1)^{n}2a_{n - 1} (n\geq 2) and a_{1} = \frac{1}{3} ,
then a_{4} = ( ). []‫تصویر‬
.42 The domain of the function y = \frac{1}{\lg|x - 5|} is ( ). []‫تصویر‬
.43 Given that an ellipse C has the same foci as the ellipse \frac{x^{2}}{4} +\frac{y^{2}}{9} =
1 , and its minor axis has a length of 4\sqrt{5} , the equation of the ellipse C is ( ). []‫تصویر‬
.44 Let ABCD be a square, the point E is the midpoint of CD . Given that \overrightarrow{AB}
= \vec{a} and \overrightarrow{AD} = \vec{b} , then the vector \overrightarrow{BE} = ( ). []‫تصویر‬
.45 Suppose that the sum of the first n terms of a geometric sequence \{a_{n}\} is S_{n} ,
and 2S_{n} = 3a_{n + 1} - 3 (n = 1,2,\ldots) . Then S_{n} = ( ). []‫تصویر‬
.46 If a complex number z satisfies the equation (z - 2i)(2 - i) = 5 , then z = ( ). []‫تصویر‬
.47 If the sum of the first n terms of a sequence \{a_{n}\} is S_{n} = \frac{2}{3} a_{n} +
\frac{1}{3} , then the general term of \{a_{n}\} is a_{n} = ( ). []‫تصویر‬
.48 There are 12 balls of identical shape in an opaque bag: red, black, yellow, and green
balls. Xiaohong randomly draws one ball from the bag. The probability of drawing a red ball
is 1 / 3 . The probability of drawing a yellow or green ball is 5 / 12 . The probability of drawing
a black or yellow ball is also 5 / 12 . The probabilities of drawing a yellow ball, a black ball,
and a green ball, respectively, are ( ). [‫]تصویر‬
1. 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? (a) 9\in A (b) {6} \in A (c) 6\subseteq A (d) \Phi \in A
2. Let A = {x\mid -2\leq x\leq 2} and B = {x\mid x > 0} be two sets. Then A\cup B = (a)
{x\mid x\leq -2} (b) {x\mid 0< x\leq 2} (c) {x\mid x\geq -2} (d) {x\mid -2\leq x< 0}
3. The solution set of the inequality x^{2} - x - 2 > 0 is ( ). (a) {x\mid x< -1 or x > 2} (b)
{x\mid -1< x< 2} (c) {x\mid x< -2 or x > -1} (d) {x\mid -2< x< 1}
4. Let {a_{n}} be an arithmetic sequence, with the first term a_{1} = 2 and common
difference d = 3 . Then a_{100} = ( ). (a) 299 (b) 302 (c) 305 (d) 298
5. The domain of the function f(x) = \frac{1}{x} +\sqrt{1 - x} is ( ). (a) (- \infty ,1] (b) (\infty ,0)\cup (0,1) (c) (- \infty ,0)\cup (0,1] (d) [1, + \infty)
6. 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 ( ). (a) (2,1) (b) (1,2) (c) (-1,2) (d) (-1,-2)
7. [‫]سوال هفت مشخص نیست‬
8. About the function f(x) = x^{4} + 3 , which of the following statements is correct? ( ).
(a) The function is odd (b) None of the above (c) The function is neither odd nor even
(d) The function is even
9. The inverse function of the function y = x^{3} + 3 , x \in \mathbb{R} is ( ). y = \sqrt[3]{x
- 3}, x \in \mathbb{R}
y = \sqrt[3]{x + 3}, x \geq -3 y = \sqrt[3]{x + 3}, x \in \mathbb{R} y = \sqrt[3]{x - 3}, x \geq 3 10.
Which of the following points is in the third quadrant? ( ). (-1,2)
(-1,-2) (1,2) (1,-2) 11. Given that y = |x| , which of the following conclusions is correct? ( ). (a)
When x < 0 , the function is an increasing function (b) When x > 0 , the function is an
increasing function (c) When x \in \mathbb{R} , the function is a decreasing function (d)
When x \in \mathbb{R} , the function is an increasing function 12. The solution set of the
rational inequality \frac{2x + 1}{x - 2} \leq 0 is ( ). [.13 ]‫ تصویر‬Given that a = 2 - \sqrt{3} and b =
2 + \sqrt{3} , the arithmetic mean of a and b is ( ). (a) 1 (b) 2 (c) \pm 1 (d) -1 14. If the straight
line l passes through the points A(-2,3) and B(3,1) , then the slope of l is ( ). [.15 ]‫ تصویر‬If
\alpha is the angle between the x-axis and a line containing the point P(1,3) , then \sin
\alpha = ( ). [.16 ]‫ تصویر‬Among 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 17. The distance
from a point A(3, -2) to B(-5, -1) on the plane is |AB| = () . (a) 65 (b) \sqrt{65} (c) 13 (d)
\sqrt{13} 18. 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 ( ). (a) y = 2x - 2 (b) y = x + 2 (c) y = 2x + 2 (d) y
= x - 2 19. If the equation of a circle is x^{2} + y^{2} - 4x - 3 = 0 , then the center and radius
are given by ( ). (a) (0,2) and \sqrt{7} (b) (0,2) and 7 (c) (2,0) and \sqrt{7} (d) (2,0) and 7 20.
Given that \sin \alpha = \frac{4}{5} , and the angle \alpha belongs to the second quadrant,
then which of the following statements is correct? ( ) (a) \cos \alpha = \frac{3}{5} (b) \cos
\alpha = -\frac{3}{5} (c) \tan \alpha = -\frac{3}{4} (d) \tan \alpha = \frac{4}{3} 21. 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. (a) -1,2 (b) 1, -2 (c) -1, -2 (d) 1,2 22. Let a,b,c be
three real numbers. Which of the following statements is correct? ( ) (a) If a > b , then a c >
b c (b) If a > b , then a + c > b + c (c) If a c > b c , then a > b (d) If a > b , then a^{2} > b^{2} 23.
About the exponential function y = a^{x} ( a > 0 , and a \neq 1 ), which of the following
statements is incorrect? ( ) (a) The range is (0, + \infty) (b) The graph passes through the
point (0,1) (c) The function is increasing over the domain (d) The domain is \mathbb{R} 24. If
\sin \alpha = \frac{3}{5} , and \alpha belongs to the first quadrant, then \sin 2\alpha = ( ).
[.25 ]‫ تصویر‬The intersection of the lines l_{1}:3x - y + 8 = 0 and l_{2}:x + 2y - 9 = 0 has
coordinates ( ). [.26 ]‫ تصویر‬Given that the equation of a hyperbola is \frac{x^2}{64} \frac{y^2}{16} = 1, which of the following statements is correct? ( ) (a) The focal distance is
8\sqrt{5} (b) The semi-major axis has length 4 (c) The eccentricity of the hyperbola is
\sqrt{5} (d) The semi-minor axis has length 8 27. Which of the following inequalities is
correct? ( ) (a) 2.1^{-2} > 1.2^{-2} (b) 0.75^{-0.2} > 0.75^{-0.4} (c) 3^{2.25} > 3^{3} (d)
2.1^{\frac{2}{3}} > 1.2^{\frac{2}{3}} 28. A circle has center at (-3, 2) and radius 4. Then the
equation of the circle is ( ). (a) (x + 3)^2 + (y - 2)^2 = 4 (b) (x - 3)^2 + (y + 2)^2 = 4 (c) (x - 3)^2 +
(y + 2)^2 = 16 (d) (x + 3)^2 + (y - 2)^2 = 16 29. Let a > 0, and b > 0 be real numbers. Then
which of the following statements is incorrect? ( ) (a) \ln(ab) = b \ln a (b) \log_a b = \frac{\ln
a}{\ln b} (a \neq 1, b \neq 1) (c) \ln\left(\frac{a}{b}\right) = \ln a - \ln b (d) \ln(ab) = \ln a + \ln b
30. Given the sine function y = \sin x, which of the following statements is incorrect? ( ) (a)
The maximum and minimum values are 1 and -1 respectively (b) The function is periodic
function (c) The function is even (d) The domain is \mathbb{R} 31. 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| = ( ). (a) 2 (b) 3 (c) 4 (d) 5 32. 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 ( ). (a) -1 (b) -1
or 1 (c) 1 (d) 0 or -1 33. The distance between the points P(-1,2) and Q(3,1) is ( ). (a) \sqrt{5}
(b) \sqrt{17} (c) \sqrt{13} (d) 5 34. Let \cos \alpha = -\frac{\sqrt{5}}{5}, and \alpha \in
\left(\frac{\pi}{2}, \pi\right). Then \sin \left(\frac{\pi}{6} + \alpha\right) = ( ). (a) \frac{2\sqrt{5}
- \sqrt{15}}{10} (b) \frac{2\sqrt{15} + \sqrt{5}}{10} (c) \frac{2\sqrt{15} - \sqrt{5}}{10} (d)
\frac{2\sqrt{5} + \sqrt{15}}{10} 35. Which of the following statements is correct? ( ) (a) y =
(\sqrt{x})^{2} and y = \sqrt{x^{2}} are the same function (b) y = \frac{x^{4} - 1}{x^{2} + 1} and y
= x^{2} - 1 are the same function (c) y = 1 and y = x^{2} are the same function (d) y = x and y =
\sqrt{x^{2}} are the same function 36. Let \sin \alpha = - \frac{12}{13}, and \alpha \in
\left(\pi , \frac{3}{2}\pi\right). Then \cos \frac{\alpha}{2} = ( ). (a) \frac{3\sqrt{13}}{13} (b)
\frac{-3\sqrt{13}}{13} (c) \frac{-2\sqrt{13}}{13} (d) \frac{2\sqrt{13}}{13} 37. Suppose that \cos
\alpha = - \frac{1}{2}, \alpha \in \left(\frac{\pi}{2}, \pi\right), then the value of \sin
\frac{\alpha}{2} is ( ). [.38 ]‫ تصویر‬Which of the following statements is correct? ( ) [.39 ]‫تصویر‬
About the tangent function y = \tan x, which of the following statements is correct? ( ) (a)
The range is [-1, 1] (b) The domain is \mathbb{R} (c) The function is even (d) The minimum
positive period of the function is \pi 40. The general formula for the sequence \frac{1}{5},
\frac{1}{7}, -\frac{1}{9}, \frac{1}{11}, \dots is a_n = (). [.41 ]‫ تصویر‬If 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 ( ). [.42 ]‫ تصویر‬The equation of the directrix of the parabola y^2 = -x is
( ). [.43 ]‫ تصویر‬Suppose 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| = (). (a) 2 (b) 4 (c) 8 (d) 6 44. Given
that \overrightarrow{AB} = \vec{a} + 5\vec{b}, \overrightarrow{BC} = -2\vec{a} + 8\vec{b},
\overrightarrow{CD} = 3\vec{a} - 3\vec{b}. Thus, ( ). (a) A,C,D are on the same line (b) A,B,D
are on the same line (c) B,C,D are on the same line (d) A,B,C are on the same line 45.
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 ( ). 2x
-y-7=0
2x + y - 5 = 0 x - 2y - 1 = 0 x + 2y - 3 = 0 46. 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 = ( ). {\sqrt{2}}\mathrm{or}2{\sqrt{2}}
{2{\sqrt{2}}\mathrm{or}-2{\sqrt{2}}} {-\sqrt{2}\mathrm{or}-2{\sqrt{2}}} {\sqrt{2}\mathrm{or}\sqrt{2}} 47. 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,\dots). Suppose that another sequence {b_{n}} satisfies b_{n}
= |a_{n}| (n = 1,2,\dots). Then, the sum of the first n terms of the sequence {b_{n}} is S_{n} =
( ). [.48 ]‫ تصویر‬In daily life, people enjoy using running apps to track and share their exercise
routes. To understand the usage of running apps among high school and college students
in a certain region, 200 high school students and 80 college students were randomly
chosen, and their favorite running apps were recorded as follows:
App A App B App C App D High school students 80 40 60 20 College students 30 20 20 10
Assuming that college students' and high school students' preferences for running apps
are independent of each other, if one person is randomly selected from each group, the
probability that their favorite running apps are different is ( ). [‫]تصویر‬
‫ در زیر متن کامل سواالت فایل‬.‫ حتما‬PDF ‫( ریاضی سوم‬Maths-December-CSCA) ‫بدون تغییر کپی شده است‬:
1. If the sets A = \{1,2\} and B = \{1,2,3\} , then [‫]تصویر‬
2. If the sets A = (-1,3) and B = [2,4) , then A \cap B equals [‫]تصویر‬
3. The solution set of the quadratic inequality x^{2} - 5x + 4 < 0 is [‫]تصویر‬
4. The domain of the function y = \frac{1}{\sqrt{x} + 1} is [‫]تصویر‬
5. If \{a_{n}\} is an arithmetic progression with first term a_{1} = 2 and common difference d
= 3 , then a_{100} equals [‫]تصویر‬
6. The point (-1,3 - \pi) lies in the
(a) third quadrant
(b) first quadrant
7. Compute \left(\sin {\frac{\pi}{4}} + \cos {\frac{\pi}{4}}\right)\cdot \tan {\frac{\pi}{4}} =
(a) \frac{2}{\sqrt{2}} (b) 2 (c) \sqrt{2} (d) 1
8. The following function is odd:
y = x\sin x
y = x^4 y = \cos x y = x + x^3
9. The inverse function of y = x^{3} is
y = \frac{x}{3}
y = 3x y = x^3 y = \sqrt[3]{x}
10. The following function is monotonically increasing: [‫]تصویر‬
11. The following point lies in the third quadrant:
(-1, -2)
(-1,2) (1,2) (1, -2)
12. The solution set of the fractional inequality \frac{x - 1}{x - 2} \leq 1 is [‫]تصویر‬
13. It is known that a, b, c form an arithmetic progression and satisfy a + c = 20 . Then b
equals
(a) 10 (b) 5 (c) 20 (d) 0
14. The angle of inclination of the line y = \sqrt{3} x + 10 equals
(a) 60^{\circ} (b) 30^{\circ} (c) 45^{\circ} (d) 120^{\circ}
15. The following inequality is true:
(a) 0.7^{-0.2}< 0.7^{-0.4} (b) 2.1^{-2} > 1.2^{-2} (c) 3^{2.25}< 3^{3} (d) 2.1^{2 / 3}< 1.2^{2 /
3}
16. If the point P(1, y) lies on the terminal side of an angle \alpha and \tan \alpha = 2 , then y
equals
(a) 1 (b) \frac{1}{2} (c) \sqrt{3} (d) 2
17. It is known that \{a_{n}\} is an arithmetic progression and a_{1} = 1 , a_{2} = 2 . Then
a_{2025} equals
(a) 2 (b) 2025 (c) 2024 (d) 20
18. If the center of a circle has coordinates (-3,2) and its radius equals 4, then the equation
of the circle is
(x + 3)^2 +(y - 2)^2 = 4
(x - 3)^2 +(y + 2)^2 = 4 (x - 3)^2 +(y +2)^2 = 16 (x + 3)^2 +(y - 2)^2 = 16
19. Find the distance between the two points P(-1, -1) and Q(1,1) .
(a) 2 (b) 2\sqrt{2} (c) 4 (d) \sqrt{2}
20. Find the equation of the line passing through the two points P(1,2) and Q(2,4) .
(a) y = x + 1 (b) y = 2x + 2 (c) y = x + 2 (d) y = 2x
21. Let the equation of a circle be x^{2} + y^{2} + 2x - 2y = 0 . Then the radius of this circle
equals
(a) 2 (b) \frac{1}{2} (c) \sqrt{2} (d) 1
22. If \alpha is an acute angle and \sin \alpha = \frac{\sqrt{3}}{2} , then \tan \alpha equals
(a) \frac{1}{2} (b) \frac{\sqrt{3}}{2} (c) \frac{1}{4} (d) \frac{\sqrt{3}}{3}
23. It is known that the geometric progression \{a_{n}\} has the first three terms 1, 2, and 4.
Then the general term of this progression is [‫]تصویر‬
24. It is known that a > b . Then
\frac{1}{a} < \frac{1}{b}
a^2 > b^2 5^a > 5^b \sin a > \sin b
25. The distance from the point P(-1,2) to the point Q(3,1) equals
\frac{\sqrt{17}}{\sqrt{13}}
\sqrt{5}
26. The following statement about the function y = \left(\frac{1}{2}\right)^x is true:
(a) This function is symmetric with respect to the origin.
(b) This function is symmetric with respect to the x-axis.
(c) This function is monotonically increasing.
(d) This function is monotonically decreasing.
27. If \sin \alpha = \frac{1}{4} , then \cos 2\alpha equals [‫]تصویر‬
28. The point of intersection of the line l_{1}:y = 2x + 1 and the line l_{2}:x + y + 1 = 0 is
\left(\frac{1}{3},\frac{2}{3}\right)
29. The coordinates of the foci of the hyperbola x^{2} - y^{2} = 1 are [‫]تصویر‬
30. The value of \log_{2}32 + \log_{0.5}32 equals
(a) -5 (b) 10 (c) 0 (d) 5
31. The following statement about the function y = \cos x is true:
(a) \cos (x + \pi) = \cos x (b) This function is odd. (c) This function is monotonically
increasing on the interval [0, \pi ] . (d) This function is periodic with period 2\pi .
32. The following statement about the parabola y^{2} = 4x is true:
(a) It passes through the point (-4,4) .
(b) Its axis of symmetry is x = 0 .
(c) The coordinates of its focus are (\pm 1,0) .
(d) The coordinates of its focus are (1,0) .
33. The following line is perpendicular to the line l: x + y + 3 = 0 :
(a) y = -2x + 1 (b) y = x + 3 (c) y = 1 - x (d) y = 2x + 1
34. It is known that \cos \alpha = -\frac{1}{2} , \alpha \in \left(\frac{\pi}{2}, \pi\right) . Then
the value of \sin \frac{\alpha}{2} equals [‫]تصویر‬
35. It is known that \sin \frac{\pi}{4} = \frac{\sqrt{2}}{2} , \sin \frac{\pi}{6} = \frac{1}{2} . Then
\cos \frac{\pi}{12} equals
\frac{\sqrt{6}}{4}
\frac{\sqrt{6} + \sqrt{2}}{4} \frac{\sqrt{6} - \sqrt{2}}{4} \frac{1 + \sqrt{3}}{4}
36. The following function coincides with the function y = |x| :
y = \sqrt[3]{x^3}
y = (\sqrt{x})^2 y = x y = \sqrt{x^2}
37. If \alpha is an acute angle and \cos \alpha = \frac{1}{2} , then \sin \frac{\alpha}{2} equals
[‫]تصویر‬
38. Regarding the trigonometric reduction formulas, the following formula is incorrect:
\cos (2\pi -\alpha) = \cos \alpha
\tan (\pi +\alpha) = -\tan \alpha \sin (\pi -\alpha) = \sin \alpha
39. The equation of the directrix of the parabola y^{2} = -x is [‫]تصویر‬
40. It is known that \frac{\cos\alpha - \sin\alpha}{\cos\alpha + \sin\alpha} = \frac{1}{3} .
Then \tan \alpha equals [‫]تصویر‬
41. If the sequence \{a_{n}\} satisfies a_{1} = 1 and
a_{n} = \frac{1}{1 + \frac{1}{a_{n - 1}}},\quad n\geq 2,
then a_{100} equals [‫]تصویر‬
42. The following statement about the logarithmic function y = \log_{a}x is true:
(a) The domain of this function is (- \infty , + \infty) .
(b) For a > 1 it is monotonically decreasing on its entire domain.
(c) For a > 1 and any x > 0 it holds that \log_{a}x > 1 .
(d) For 0< a< 1 it is monotonically decreasing on its entire domain.
43. The following statement about the ellipse \frac{x^{2}}{4} +\frac{y^{2}}{20} = 1 is true:
[‫]تصویر‬
44. If a line is perpendicular to the line l_{1}:x + 2y + 4 = 0 and passes through the
intersection point of the lines l_{2}:x + y + 1 = 0 and l_{3}:2x + y - 1 = 0 , then the equation of
this line is
x + 2y - 3 = 0
2x + y - 5 = 0 2x - y - 7 = 0 x - 2y - 1 = 0
45. Let O be the origin, A and B be two points in the plane, and P be the midpoint of the
segment AB . Then the vector \overrightarrow{OP} equals
\frac{1}{2} (\overrightarrow{OA} +\overrightarrow{OB})
\overrightarrow{OA} +\overrightarrow{OB} \overrightarrow{OB} -\overrightarrow{OA}
\frac{1}{2} (\overrightarrow{OB} -\overrightarrow{OA})
46. If the complex number z satisfies z^{3} = 1 and z \neq 1 , then
1 + z + z^{2} + \dots +z^{999} =
[‫]تصویر‬
47. It is known that the arithmetic progression \{a_{n}\} satisfies
3(a_{2} + a_{6}) + 2(a_{6} + a_{10} + a_{14}) = 24.
Then the sum S_{13} equals
(a) 104 (b) 52 (c) 39
48. The twelve zodiac signs, also known as the animal symbols of the years, are twelve
animals used in China to denote years, namely: Rat, Ox, Tiger, Rabbit, Dragon, Snake,
Horse, Goat, Monkey, Rooster, Dog, and Pig. For example, people born in 1984, 1996, and
2008 were born in the Year of the Rat, and people born in 1985, 1997, and 2009 were born
in the Year of the Ox, and so on. The probability that among three randomly chosen people
at least two have the same zodiac sign equals
\frac{17}{72}
\frac{1}{12} \frac{55}{72} \frac{143}{144}