iran-konkur

Papers (13)
2024
konkur-riazi_1403_ordibehesht_specialized 5
2023
konkur-riazi_1402_general 30 konkur-riazi_1402_tir_specialized 8
2022
konkur-riazi_1401_specialized 30
2021
konkur-riazi_1400_specialized 39
2020
konkur-riazi_1399_specialized_old-curriculum 37
2019
konkur-riazi_1398_specialized_new-curriculum 44
2018
konkur-riazi_1397_specialized 40
2017
konkur-riazi_1396_specialized 38
2016
konkur-riazi_1395_specialized 33
2015
konkur-riazi_1394 32
2014
konkur-riazi_1393 43
2013
konkur-riazi_1392 51
2015 konkur-riazi_1394

32 maths questions

Q109 Standard trigonometric equations Solve trigonometric equation for solutions in an interval View
109- What is the general solution of the trigonometric equation $\cot x = \dfrac{\sin x + \sin 2x}{\cos x + \cos 2x}$?

p{6cm}} (2) $\dfrac{2k\pi}{5}$(1) $\dfrac{k\pi}{5}$
[18pt] (4) $\dfrac{1}{5}(2k+1)\pi$(3) $\dfrac{3k\pi}{5}$

Q110 Composite & Inverse Functions Recover a Function from a Composition or Functional Equation View
110- The figure below shows the graph of the function $y = \sin^{-1}(U(x))$. What is the rule $U(x)$?
[Figure: graph of $y = \sin^{-1}(U(x))$ with a point marked at $x = -1$ and $x = 3$]
p{6cm}} (2) $\dfrac{2}{1-x}$(1) $\dfrac{2}{x-1}$
[18pt] (4) $\dfrac{1}{2-x}$(3) $\dfrac{1}{x-2}$


111- What is the value of the expression $169\sin\!\left(2\cos^{-1}\!\left(-\dfrac{5}{13}\right)\right)$?
p{6cm}} (2) $60$(1) $-120$
[6pt] (4) $120$(3) $-60$

Q112 5 marks Composite & Inverse Functions Evaluate Composition from Algebraic Definitions View
112- For which value of $a$ is the function $$f(x) = \begin{cases} \dfrac{a(1+\sqrt[5]{1-x})}{x^2 - 2x} & ; \ x > 2 \\[8pt] x - a & ; \ x \leq 2 \end{cases}$$ always continuous?
p{6cm}} (2) $1.6$(1) $1.2$
[6pt] (4) $2.2$(3) $2.4$

Q113 Sequences and series, recurrence and convergence Multiple-choice on sequence properties View
113- What is the limit of the sequence $a_n = \left(\dfrac{n+2}{n+1}\right)^{2n+2}$ as $n \to \infty$?
p{3cm} p{3cm} p{3cm}} (4) $3e^2$(3) $3e$(2) $e^2$(1) $3e$

Q114 Sign Change & Interval Methods View
114- What is the value of $\displaystyle\lim_{x \to 0}\left([2x]+[-2x]\right)\dfrac{1-\cos^2 x}{1-\sqrt{1+x^2}}$? (The symbol $[\,]$ denotes the floor function.)
p{4cm} p{4cm} p{3cm}} (1) $-2$(2) $2$(3) zero(4) does not exist.

Q115 Sign Change & Interval Methods View
115- One of the real roots of the equation $x^3 + 2x^2 - 4x - 3 = 0$ lies in which open interval?
p{6cm}} (2) $\left(-1, -\dfrac{2}{4}\right)$(1) $\left(-\dfrac{2}{4}, -\dfrac{1}{2}\right)$
[14pt] (4) $\left(0, \dfrac{1}{2}\right)$(3) $\left(-\dfrac{1}{2}, 0\right)$

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Q116 Curve Sketching Asymptote Determination View
116. The extensions of the asymptotes of the graph of the function $f(x) = \sqrt{x^2 + 2x} - \sqrt{x^2 - 2x}$, the first and third bisectors intersect at two points A and B. What is the length of AB?
(1) $2\sqrt{7}$ (2) $4$ [6pt] (3) $2\sqrt{5}$ (4) $4\sqrt{7}$
Q117 Tangents, normals and gradients Find tangent line equation at a given point View
117. If $\theta$ is the angle between the left and right tangents to the graph of the function $f(x) = \left[x + \frac{1}{2}\right]x + x^2$, at the point $x = \frac{1}{2}$, what is $\tan\theta$?
(1) $\dfrac{1}{4}$ (2) $\dfrac{1}{2}$ (3) $\dfrac{2}{3}$ (4) $\dfrac{3}{4}$
Q118 Stationary points and optimisation Find concavity, inflection points, or second derivative properties View
118. From the relation $x^2y - y^2 - 2\sqrt{x} + 4 = 0$, the value of $\dfrac{d^2y}{dx^2}$ at the point $(1, 2)$ is which of the following?
(1) $\dfrac{7}{6}$ (2) $\dfrac{8}{6}$ [6pt] (3) $\dfrac{11}{6}$ (4) $\dfrac{13}{6}$
Q119 Stationary points and optimisation Determine intervals of increase/decrease or monotonicity conditions View
119. If $f(x) = x^3 - x^2 + 2x$, the equation of the line perpendicular to the curve of $f^{-1}$ at the point $x = 2$ is which of the following?
(1) $y + 3x = 7$ (2) $y - 3x = -5$ [6pt] (3) $3y + x = 5$ (4) $3y - x = 1$
120. For the graph of $y = |x| \cdot e^{-x}$, on which interval is it decreasing and concave down?
(1) $(-\infty, 2)$ (2) $(0, 1)$ [6pt] (3) $(1, 2)$ (4) $(2, +\infty)$
Q121 Applied differentiation Kinematics via differentiation View
121. In triangle ABC, $BC = 20$, $AH = 12$, and $AB = 12$ units. Line $\Delta$ parallel to BC moves at a constant speed of $0.2$ units per second. The rate of increase of the area of the trapezoid at the moment when the distance between the two parallel lines is 9 units is which of the following?
[Figure: Triangle ABC with vertex A at top, base BC at bottom, height AH drawn, points D and E on sides AB and AC respectively forming a trapezoid BCED, with H the foot of the altitude on BC]
(1) $0.8$ [4pt] (2) $0.9$ [4pt] (3) $1$ [4pt] (4) $1.2$
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Q122 Stationary points and optimisation Determine parameters from given extremum conditions View
122- The figure below shows the graph of the function with equation $f(x) = -x^4 + 4x^3 + ax^2 + b$. What is $a$?
[Figure: Graph of a polynomial function with a local maximum and minimum]
  • [(1)] $-18$
  • [(2)] $-15$
  • [(3)] $-12$
  • [(4)] $-9$
Q123 Connected Rates of Change Table-Based Estimation with Rate of Change Interpretation View
123- If $G(x) = x^2 \int_{2}^{\sqrt{x}} \dfrac{\ln(t+2)}{t^2}\, dt$, and $G'(4)$ equals how many times $\ln 2$?
  • [(1)] $1$
  • [(2)] $2$
  • [(3)] $1.5$
  • [(4)] $3$
Q124 Indefinite & Definite Integrals Piecewise/Periodic Function Integration View
124- What is the value of $\displaystyle\int_{0}^{2} \left[\frac{x}{2}\right] \frac{\sqrt{x}-1}{x}\, dx$?
  • [(1)] $4 - 2\sqrt{2} - \ln 2$
  • [(2)] $4 - 2\sqrt{2} + \ln 2$
  • [(3)] $2 + \sqrt{2} - \ln 2$
  • [(4)] $2 - \sqrt{2} + \ln 2$
Q129 Circles Circles Tangent to Each Other or to Axes View
129. Two circles with centers $O$ and $O'$ are externally tangent. A circle with diameter $\overline{OO'}$ is drawn with external common tangency to these two circles. What is the position of the two circles?
(1) Intersecting(2) Tangent(3) External(4) Indeterminate
Q133 Vectors 3D & Lines Section Division and Coordinate Computation View
133. Points $O(0,0,0)$, $B(-1,2,4)$, $A(5,-4,1)$ are given, and $\overrightarrow{AM} = \dfrac{2}{3}\overrightarrow{AB}$ and $\overrightarrow{AB}$ are known. The value of $|\overrightarrow{OM}|$ is:
(2) $\sqrt{11}$(1) $\sqrt{10}$
(4) $\sqrt{14}$(3) $\sqrt{13}$
Q134 Vectors 3D & Lines Shortest Distance Between Two Lines View
134. The distance between the two lines $\dfrac{x-1}{2} = \dfrac{y+2}{1} = \dfrac{z}{-1}$ and $\left(x = 2y+1,\ z = -y+2\right)$ is:
(2) $2\sqrt{2}$(1) $\sqrt{6}$
(4) $3\sqrt{2}$(3) $2\sqrt{3}$

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Q135 Vectors: Lines & Planes Find Cartesian Equation of a Plane View
135- The plane passing through the line with equation $\dfrac{x+1}{2} = \dfrac{y}{3} = \dfrac{z-2}{-1}$ and the point $(0,3,0)$ intersects the $Z$-axis at what elevation?
(1) $-2$ (2) $-3$ (3) $2$ (4) $3$
Q136 Circles Circles Tangent to Each Other or to Axes View
136- Circles $C$ and $C'$ are tangent at point $(0,1)$ and their common internal tangent lines with respect to circle $C$ are equidistant from point $(-3, 2)$. If circle $C'$ with radius $\sqrt{5}$ passes through $(-3,2)$, what is the center of circle $C'$?
(1) $(-1, 3)$ (2) $(-1, 2)$
(3) $(1, -2)$ (4) $(1, -1)$
Q137 Conic sections Focal Distance and Point-on-Conic Metric Computation View
137- A parabola with focus $F(3,2)$ and a line with equation $x = -1$ intersect the $x$-axis at point $A$. What is the distance from point $A$ to the focus of the parabola?
(1) $2.75$ (2) $2.5$
(3) $2.75$ (4) $3$
Q138 Conic sections Conic Identification and Conceptual Properties View
138- The rotation matrix $A$, with the relation $\begin{bmatrix} x \\ y \end{bmatrix} = A \cdot \begin{bmatrix} x' \\ y' \end{bmatrix}$, transforms the conic equation $5x^2 + 24xy - 2y^2 = 12$ into standard form with respect to $x'$ and $y'$. What is the tangent of the rotation angle?
(1) $\dfrac{2}{3}$ (2) $\dfrac{3}{4}$ (3) $\dfrac{4}{3}$ (4) $\dfrac{3}{2}$
Q139 Matrices Matrix Algebra and Product Properties View
139- If $A = [a_{ij}]_{r \times 3}$ and $B = [b_{ij}]_{r \times 2}$, which of the following matrix products is defined?
(1) $AB$ (2) $A^t B$ (3) $B^t A^t$ (4) $AB^t$
Q140 Matrices Linear System and Inverse Existence View
140- If $A = \begin{bmatrix} 1 & -1 & 2 \\ 0 & 2 & -1 \\ 0 & 0 & 3 \end{bmatrix}$, what is the sum of the entries of the second column of $A^{-1}$?
(1) $-\dfrac{1}{3}$ (2) $\dfrac{2}{3}$
(3) $1$ (4) zero
Q141 Measures of Location and Spread View
141- To the statistical data shown in the frequency polygon, two data values $29$ and $32$ are added. What are the new median values?
[Figure: Frequency polygon with x-axis values 24, 27, 30, 33, 36 and y-axis (f) values approximately 8, 9, 15, 11, 12]
(1) $23$ (2) $24$
(3) $25$ (4) $26$
%% Page 26 Mathematics 120-C Page 8
Q142 Measures of Location and Spread View
142- If the mean of grouped data equals 16, to determine the fourth class frequency, which value is correct?
Class representative1214161820
Frequency5710$a$3

(1) $4/\Lambda\Delta$(2) $4/9\Upsilon$(3) $\Delta/\Delta\Delta$(4) $\Delta/V4$

Q143 Proof by induction Prove a summation inequality by induction View
143- In proving the inequality $3^{n+1} > n!$ by mathematical induction, after finding a suitable number $m$, the inductive step relation for $k \geq m$ is which of the following?
(1) $k+1 > 2,\ m = 5$(2) $k+1 > 2,\ m = 6$
(3) $(2k+1) > 4,\ m = 5$(4) $(2k+1) > 4,\ m = 6$

Q144 Number Theory Combinatorial Number Theory and Counting View
144- If $S$ is a subset of natural numbers with 115 elements, when dividing the elements of $S$ by 27, at least how many elements certainly have the same remainder?
(1) $4$(2) $5$(3) $6$(4) $7$

Q147 Probability Definitions Finite Equally-Likely Probability Computation View
147- Each of the numbers $1, 2, 3, 4, 5, 6$ is written on six equally likely balls. Consecutively, one ball is drawn from the box. What is the probability that an odd or even number appears among them?
(1) $0/1$(2) $0/12$(3) $0/15$(4) $0/2$

Q148 Geometric Probability View
148- A point is chosen randomly inside an equilateral triangle with side $\sqrt{2\pi\sqrt{3}}$. What is the probability that the distance from this point to each vertex of the triangle is more than 1 unit?
(1) $\dfrac{1}{3}$(2) $\dfrac{1}{2}$(3) $\dfrac{2}{3}$(4) $\dfrac{3}{4}$

Q150 Number Theory Modular Arithmetic Computation View
150- How many three-digit numbers exist that are multiples of 11 and whose remainders when divided by both 4 and 5 equal 1?
(1) $3$(2) $4$(3) $5$(4) $6$

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Q155 Probability Definitions Conditional Probability and Bayes' Theorem View
155. In two containers there are respectively 24 and 18 identical balls. In the first container there are 6 white balls and in the second container there are 3 white balls. From the first container 7 balls and from the second container 5 balls are randomly drawn and placed in another container. Then from the last container one ball is drawn. What is the probability that this ball is white?
(1) $\dfrac{13}{72}$ (2) $\dfrac{7}{36}$
(3) $\dfrac{15}{72}$ (4) $\dfrac{31}{144}$
%% Page 28 Physics 120-C Page 10
Q164 Work done and energy Energy conservation with friction or dissipative forces View
164. A body of mass $2\,\text{kg}$ slides on an inclined surface that makes an angle of $30°$ with the horizontal, moving downward at constant speed. If in this motion the body is displaced $2\,\text{m}$, how much work (in Joules) does the friction force do? $\left(g = 10\,\dfrac{\text{m}}{\text{s}^2}\right)$
$-25\sqrt{3}\ (1$ $-10\sqrt{3}\ (2$ $-10\ (3$ $-20\ (4$