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
2020 konkur-riazi_1399_specialized_old-curriculum

37 maths questions

Q110 Addition & Double Angle Formulae Addition/Subtraction Formula Evaluation View
110. If the terminal side of arc $\alpha$ is in the second quadrant and $\sin\alpha = \dfrac{\sqrt{2}}{10}$, what is the value of $\cos\!\left(\dfrac{11\pi}{4} + \alpha\right)$?
$$-\frac{4}{5} \quad (1) \qquad -\frac{3}{5} \quad (2) \qquad \frac{3}{5} \quad (3) \qquad \frac{4}{5} \quad (4)$$
Q111 Addition & Double Angle Formulae Multi-Step Composite Problem Using Identities View
111. Assume $\sin\alpha = \dfrac{-3}{5}$ and the terminal side of arc $\alpha$ is in the third quadrant. What is the value of $\cos(\tan^{-1}(\sin 2\alpha))$?
$$\frac{25}{\sqrt{1201}} \quad (1) \qquad \frac{-25}{\sqrt{1201}} \quad (2) \qquad \frac{5}{\sqrt{51}} \quad (3) \qquad \frac{-5}{\sqrt{51}} \quad (4)$$
Q112 Standard trigonometric equations Solve trigonometric equation for solutions in an interval View
112. The sum of the solutions of the equation $\tan(3x)\tan(x) = 1$ in the interval $[\pi, 2\pi]$ is:
$$5\pi \quad (1) \qquad 6\pi \quad (2) \qquad \frac{9\pi}{2} \quad (3) \qquad \frac{11\pi}{2} \quad (4)$$
Q113 Sequences and Series Evaluation of a Finite or Infinite Sum View
113. We classify the natural numbers in groups such that each group contains consecutive numbers, i.e., $\ldots, \{4,5,6\}, \{2,3\}, \{1\}, \ldots$ What is the sum of the numbers in the group containing 350?
$$4125 \quad (1) \qquad 4050 \quad (2) \qquad 4015 \quad (3) \qquad 3980 \quad (4)$$
Q114 Exponential Equations & Modelling Threshold or Tipping-Point Calculation in Applied Exponential Models View
114. There are 24 grams of a radioactive element. If $\dfrac{1}{10}$ of the element decays every 30 days, after how many days will 8 grams remain? $(\log 3 = 0.48)$
$$360 \quad (1) \qquad 300 \quad (2) \qquad 270 \quad (3) \qquad 240 \quad (4)$$
Q115 4 marks Sequences and series, recurrence and convergence Convergence proof and limit determination View
115. The sequence $\{x_n\}$ is defined as follows. What is the limit of $\{x_n\}$?
$$x_0 = 3 \;,\quad x_{n+1} = \frac{3x_n^2 + 64}{4x_n^2} \;,\quad (n = 1, 2, \ldots)$$
$$2\sqrt{2} \quad (1) \qquad -2\sqrt{2} \quad (2) \qquad 2\sqrt[4]{2} \quad (3) \qquad -2\sqrt[4]{2} \quad (4)$$
Q116 Sign Change & Interval Methods View
116. What is $\displaystyle\lim_{x \to 1} \dfrac{2x - 7\sqrt{x} + 5}{2x - \sqrt{3x+1}}$?
$$-1.5 \quad (1) \qquad -1.2 \quad (2) \qquad -0.8 \quad (3) \qquad -0.6 \quad (4)$$
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Q117 Curve Sketching Finding Parameters for Continuity View
117. Suppose $f(x) = \begin{cases} (x-1)|x| & ; \ |x-1| < 1 \\ x^2 + ax + b & ; \ |x-1| \geq 1 \end{cases}$ is always a continuous function. What is the value of $a$?
(1) $\dfrac{3}{2}$ (2) $-1$ (3) $1$ (4) $\dfrac{5}{2}$
Q118 Curve Sketching Asymptote Determination View
118. The graph of $f(x) = \dfrac{-2x^2 + 3x}{ax^2 + bx + c}$ has asymptotes $y = -1$, $y = -2$, $x = -2$, and $x = 1$. What is $f(-1)$?
(1) $1.25$ (2) $1.5$ (3) $1.75$ (4) $-1.5$
Q119 Chain Rule Finding Composition Parameters from Derivative Conditions View
119. If $f$ is a differentiable function, $g(x) = f\!\left(\sqrt{1 + \tan^2 x}\right)$, and $g'\!\left(\dfrac{\pi}{3}\right) = \dfrac{\sqrt{3}}{2}$, what is the value of $f'(2)$?
(1) $-\dfrac{1}{2}$ (2) $\dfrac{1}{4}$ (3) $\dfrac{1}{2}$ (4) $1$
Q120 Stationary points and optimisation Find critical points and classify extrema of a given function View
120. The mean value theorem applies to the function $y = \sqrt{21 - x^2 + 4x}$ on the interval $[6,\ 8]$. For the instantaneous rate of change to equal the average rate of change of this function, what value of $x$ is required?
(1) $4 + \sqrt{7}$ (2) $3 + 2\sqrt{7}$ (3) $2 + \dfrac{3}{2}\sqrt{7}$ (4) $2 + \dfrac{5}{2}\sqrt{7}$
Q121 Tangents, normals and gradients Geometric properties of tangent lines (intersections, lengths, areas) View
121. The tangent line to the curve $f(x) = \dfrac{5x - 4}{\sqrt{x}}$ at the point $x = 4$. At which values does it intersect the $y$-axis?
(1) $-4$ (2) $-1$ (3) $2$ (4) $3$

122. If $\tan\alpha$ and $\tan\beta$ are the roots of the equation $2x^2 + 3x - 1 = 0$, what is $\tan(\alpha + \beta)$?
(1) $1$ (2) $\dfrac{3}{2}$ (3) $-3$ (4) $-1$
Q129 Second order differential equations Verifying a particular solution satisfies a second-order ODE View
129. The function with the rule $f(x) = \displaystyle\lim_{n \to +\infty} \left(1 - \dfrac{3x}{n}\right)^n$ is defined for every real number $x$. Which statement is correct?
  • [(1)] $f''(x) + 6f'(x) + 9f(x) = 0$
  • [(2)] $f''(x) + 3f'(x) + 2f(x) = 0$
  • [(3)] $f''(x) - 6f'(x) + 9f(x) = 0$
  • [(4)] $f''(x) - 3f'(x) + 2f(x) = 0$
Q130 Volumes of Revolution Volume of Revolution about a Horizontal Axis (Evaluate) View
130. The solid of revolution obtained by rotating right triangle $ABC$ with legs $AB$ and $AC$ of lengths 5 and $2\sqrt{6}$ respectively, one unit about the axis passing through vertex $C$ and parallel to side $AB$, is:
  • [(1)] $60\pi$
  • [(2)] $70\pi$
  • [(3)] $75\pi$
  • [(4)] $80\pi$

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Q131 Circles Chord Length and Chord Properties View
131- In the figure below, segment $AC$ equals chord $AB$. Which of the following is necessarily true?
[Figure: Circle with points A, B, C, D where AC is a chord equal to chord AB]
  • [(1)] $BC = BA$
  • [(2)] $BD = AC$
  • [(3)] $BC = BD$
  • [(4)] $DA = DC$
Q132 Circles Inscribed/Circumscribed Circle Computations View
132- Which of the following quadrilaterals can be inscribed in a circle with diameter $(x+2)$?
[Figure: Trapezoid with sides labeled x, y, 4, 9 and angle $60^\circ$]
  • [(1)] $\sqrt{51}$
  • [(2)] $\sqrt{55}$
  • [(3)] $\sqrt{57}$
  • [(4)] $\sqrt{59}$
Q133 Circles Circle Equation Derivation View
133- The smallest circle passing through the two points $A(2,5)$ and $B(-4,1)$ intersects the $x$-axis at what length?
  • [(1)] $1\ ,\ -3$
  • [(2)] $5\ ,\ -3$
  • [(3)] $2\ ,\ -1$
  • [(4)] $3\ ,\ -2$
Q134 Circles Circle Equation Derivation View
134- Among the circles passing through the point $A(-4\ ,\ 1)$ and tangent to the lines $4x + 3y = 0$ and the $y$-axis, the one with the largest radius is:
  • [(1)] $\dfrac{5}{3}$
  • [(2)] $\dfrac{17}{9}$
  • [(3)] $\dfrac{7}{3}$
  • [(4)] $\dfrac{22}{9}$
Q135 Conic sections Focal Distance and Point-on-Conic Metric Computation View
135- In an ellipse with semi-axes $8$ and $2\sqrt{7}$, and foci $F$ and $F'$, a circle with diameter $F'F$ intersects the ellipse at point $M$. The distance from point $M$ to the nearest focus is:
  • [(1)] $4 - 3\sqrt{2}$
  • [(2)] $7.5$
  • [(3)] $4 - \sqrt{2}$
  • [(4)] $3$
Q136 Conic sections Equation Determination from Geometric Conditions View
136- If the point $F(-0.25\ ,\ -2)$ is the focus of the parabola $y^2 + ay + bx + 1 = 0$, what is the smallest value of $b$?
  • [(1)] $-4$
  • [(2)] $-3$
  • [(3)] $-2$
  • [(4)] $2$
Q137 3x3 Matrices Matrix Algebraic Properties and Abstract Reasoning View
137- If $A = \begin{bmatrix} 2 & 1 & 5 \\ -3 & 0 & 4 \\ 1 & 0 & 2 \end{bmatrix}$, what are the entries of the first row of $A^3$?
  • [(1)] $\begin{bmatrix} 30 & 6 & 94 \end{bmatrix}$
  • [(2)] $\begin{bmatrix} 30 & 6 & 78 \end{bmatrix}$
  • [(3)] $\begin{bmatrix} 24 & 8 & 46 \end{bmatrix}$
  • [(4)] $\begin{bmatrix} 30 & 6 & 46 \end{bmatrix}$

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Q138 3x3 Matrices Solving a 3×3 Linear System Explicitly View
138- From the matrix relation $\begin{bmatrix} 2 & 3 \\ 4 & 5 \end{bmatrix} X \begin{bmatrix} 2 & -1 \\ -4 & 3 \end{bmatrix} = \begin{bmatrix} 4 & 0 \\ 0 & 8 \end{bmatrix}$, what is matrix $X$?
\[ (1)\quad \begin{bmatrix} 7 & 9 \\ 4 & 4 \end{bmatrix} \qquad (2)\quad \begin{bmatrix} 7 & 9 \\ 2 & -2 \end{bmatrix} \] \[ (3)\quad \begin{bmatrix} -9 & 7 \\ -4 & -4 \end{bmatrix} \qquad (4)\quad \begin{bmatrix} -9 & -7 \\ 4 & 4 \end{bmatrix} \]
Q139 3x3 Matrices Determinant of Parametric or Structured Matrix View
139- What are the solutions of the equation $0 = \begin{vmatrix} -4 & 1 & 1 \\ 1 & 2-x & 1 \\ 3 & 2 & 3-x \end{vmatrix}$?
\[ (1)\quad 1,\ -4 \qquad (2)\quad 1,\ 4 \qquad (3)\quad 1,\ 5 \qquad (4)\quad 2,\ 5 \]
Q140 Conic sections Conic Identification and Conceptual Properties View
140- The area of the graph of the curve $0 = 3x^2 + \sqrt{3}xy + 2y^2 - 10 = 0$ is:
\[ (1)\quad 6\pi \qquad (2)\quad 7\pi \qquad (3)\quad \frac{10\pi}{3} \qquad (4)\quad \frac{20\pi}{\sqrt{21}} \]
Q141 Solving quadratics and applications Geometric or real-world application leading to a quadratic equation View
141- The side lengths of a right triangle are $x+1$, $2x+1$, and $2x+3$. The area of the triangle is:
\[ (1)\quad 60 \qquad (2)\quad 56 \qquad (3)\quad 45 \qquad (4)\quad 39 \]
Q142 Permutations & Arrangements Forming Numbers with Digit Constraints View
142- How many four-digit natural numbers are divisible by 5, with non-repeating digits?
\[ (1)\quad 948 \qquad (2)\quad 952 \qquad (3)\quad 968 \qquad (4)\quad 972 \]
Q143 Combinations & Selection Counting Integer Solutions to Equations View
143- The number of terms in the expansion of $(a+b+c)^{12}$ is:
\[ (1)\quad 72 \qquad (2)\quad 78 \qquad (3)\quad 84 \qquad (4)\quad 91 \]
Q144 Principle of Inclusion/Exclusion View
144- In a group of 7 literature books, 2 art books, and 10 mathematics books. At least how many books must we take from this group so that we are certain that, with at least 4 books, both subjects are represented?
\[ (1)\quad 10 \qquad (2)\quad 9 \qquad (3)\quad 8 \qquad (4)\quad 7 \]
Q145 Probability Definitions Probability Involving Algebraic or Number-Theoretic Conditions View
145- A two-digit natural number is chosen at random. What is the probability that the chosen number is a multiple of 3 or 5?
\[ (1)\quad \frac{2}{5} \qquad (2)\quad \frac{3}{5} \qquad (3)\quad \frac{7}{15} \qquad (4)\quad \frac{8}{15} \]
Q146 Probability Definitions Probability Using Set/Event Algebra View
146- We have three urns. The first urn contains 9 white and 4 black marbles, the second contains 9 black and 4 white marbles, and the third contains 5 white and 5 black marbles. One marble is randomly drawn from one urn. What is the probability that at least one of these two marbles is black?
\[ (1)\quad \frac{1}{3} \qquad (2)\quad \frac{11}{18} \qquad (3)\quad \frac{25}{36} \qquad (4)\quad \frac{13}{18} \]
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Q147 Measures of Location and Spread View
147 -- Referring to the relative frequency histogram of grouped quantitative data, what is the mean?
[Figure: A relative frequency histogram with data values 7, 12, 13, 17, 19 on the horizontal axis and relative frequencies approximately 12, 35, 18, 25, 10 on the vertical axis]
  • [(1)] $13$
  • [(2)] $13.8$
  • [(3)] $14$
  • [(4)] $14.2$
Q148 Permutations & Arrangements Linear Arrangement with Constraints View
148 -- In a competition, 3 drivers participate for three consecutive days with 3 cars on routes A, B, and C. Each driver selects only one route and one car per day, and the scheduling is done in the form of a Latin square. In how many ways can the scheduling be done such that on the first day, no one selects car A?
231
312
123

  • [(1)] $1$
  • [(2)] $2$
  • [(3)] $3$
  • [(4)] $4$
Q149 Number Theory Combinatorial Number Theory and Counting View
149 -- How many natural multiples of 9 exist such that when divided by 430, the remainder equals the integer part of the quotient?
  • [(1)] $4$
  • [(2)] $5$
  • [(3)] $6$
  • [(4)] $7$
Q150 Number Theory GCD, LCM, and Coprimality View
150 -- The least common multiple of two numbers is 60 and the greatest common divisor of them is 6. If the sum of these two numbers is 136, what is their difference?
  • [(1)] $42$
  • [(2)] $48$
  • [(3)] $52$
  • [(4)] $56$
Q151 Number Theory Modular Arithmetic Computation View
151 -- If the number $2^n - 1$ is divisible by 217, how many two-digit values does $n$ have?
  • [(1)] $4$
  • [(2)] $5$
  • [(3)] $6$
  • [(4)] $7$
Q152 Solving quadratics and applications Geometric or real-world application leading to a quadratic equation View
152 -- The four-digit number $\overline{aabb}$, whose square root is the two-digit number $\overline{cc}$, and $\overline{cc} = a - b$. What is $a - b$?
  • [(1)] $2$
  • [(2)] $3$
  • [(3)] $4$
  • [(4)] $5$
Q155 Vectors 3D & Lines Line-Plane Intersection View
155 -- Suppose point $M$ is the intersection of the line passing through points $A(2,2,1)$ and $B(2,-1,5)$ with the plane $x + y + z = 1$. What is the distance from point $M$ to point $B$?
  • [(1)] $25$
  • [(2)] $27$
  • [(3)] $10\sqrt{5}$
  • [(4)] $5\sqrt{13}$

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