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
2016 konkur-riazi_1395_specialized

33 maths questions

Q108 Standard trigonometric equations Solve trigonometric equation for solutions in an interval View
108. The sum of all solutions of the equation $\sin 4x = \sin^2 x - \cos^2 x$, in the interval $[0, \pi]$, equals which of the following?
(4) $\dfrac{11\pi}{2}$ (3) $\dfrac{5\pi}{2}$ (2) $\dfrac{9\pi}{4}$ (1) $\dfrac{7\pi}{4}$
Q109 Curve Sketching Number of Solutions / Roots via Curve Analysis View
109. The graph of $y = \cos(\tan^{-1} x)$ and the line $y = mx$, for which set of values of $m$, share exactly one point in common?
(4) $(0, +\infty)$ (3) $(-\infty, 0)$ (2) $(-\infty, +\infty)$ (1) $(-\infty, +\infty) - \{0\}$
Q113 Sequences and series, recurrence and convergence Multiple-choice on sequence properties View
113. The sequence $\left\{\dfrac{n^2 + (-1)^n}{2n^2 + 2}\right\}$ is of what type?
(1) Divergent -- divergent (2) Non-increasing -- convergent (3) Decreasing -- convergent (4) Increasing -- divergent
Q115 Sequences and series, recurrence and convergence Multiple-choice on sequence properties View
115. The largest lower bound of the sequence $\left\{\dfrac{2n+1}{3n+1}\right\}$ is which of the following?
(4) $1$ (3) $\dfrac{2}{4}$ (2) $\dfrac{5}{7}$ (1) $\dfrac{2}{3}$
Q116 2 marks Curve Sketching Asymptote Determination View
116. The oblique asymptote of the curve $y = \sqrt[2]{8x^2 + 2x^2}$ intersects the $y$-axis at which point?
(4) $\dfrac{5}{6}$ (3) $\dfrac{2}{3}$ (2) $\dfrac{1}{2}$ (1) $\dfrac{1}{6}$
Q117 Composite & Inverse Functions Find or Apply an Inverse Function Formula View
117. If $f(x) = \dfrac{1}{2}(x + \sqrt{x^2 + 4})$, then $f^{-1}(x) + f^{-1}\!\left(\dfrac{1}{x}\right)$ equals which of the following?
(4) zero (3) $x^2 - 1$ (2) $\dfrac{2}{x}$ (1) $2x$
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Q118 Tangents, normals and gradients Geometric properties of tangent lines (intersections, lengths, areas) View
118- The tangent line to the graph of $f(x) = (x+2)e^{1-x}$ at the point $x = 1$ meets the line connecting this point to the origin. What is $\tan\alpha$?
(1) $0.5$ (2) $1$ (3) $1.5$ (4) $2$
Q119 Tangents, normals and gradients Find tangent line equation at a given point View
119- The line $y = 3x - 2$ at the point $x = 2$ is tangent to the curve $y = f(x)$. What is $\displaystyle\lim_{x \to 2} \dfrac{f^2(x) - 4f(x)}{x - 2}$?
(1) $2$ (2) $6$ (3) $12$ (4) $15$
Q120 3 marks Stationary points and optimisation Find concavity, inflection points, or second derivative properties View
120- What is the length of the inflection point of the graph of $y = (5-x)\sqrt[3]{x^2}$?
(1) $-1$ (2) zero (3) $1$ (4) $2$
Q121 4 marks Applied differentiation Applied modeling with differentiation View
121- In manufacturing a cone in the shape of a right circular cone with volume $\dfrac{\pi}{3}$, at what height is the least material consumed?
(1) $\dfrac{\sqrt{2}}{2}$ (2) $1$ (3) $\sqrt[4]{2}$ (4) $\sqrt{2}$
Q122 Curve Sketching Asymptote Determination View
122- The figure opposite shows part of the graph of $f(x) = \dfrac{x^2 + ax + b}{x + c}$. What is $b$?
[Figure: Graph of a rational function with vertical asymptote and oblique asymptote; marked points at $x = -2$ and $x = 7$ on the x-axis]
(1) $1$
(2) $4$
(3) $6$
(4) $9$
Q123 Indefinite & Definite Integrals Average Value of a Function View
123- What is the mean value of $f(x) = \dfrac{x^2 - 2}{x^2}$ on the interval $[2, 4]$?
(1) $\dfrac{5}{8}$ (2) $\dfrac{11}{16}$ (3) $\dfrac{3}{4}$ (4) $\dfrac{7}{8}$
Q124 Standard Integrals and Reverse Chain Rule Definite Integral Evaluation via Substitution or Standard Forms View
124- What is $\displaystyle\int_{\frac{\pi}{4}}^{\frac{\pi}{2}} \dfrac{1 + \cos 2x}{2\sin^2 x}\, dx$?
(1) $1 - \sqrt{2}$ (2) $1 - \dfrac{\pi}{4}$ (3) $\dfrac{\pi}{2} - 1$ (4) $\dfrac{2}{4}$
%% Page 6 Mathematics 120-C Page 5
Q125 Circles Circle-Line Intersection and Point Conditions View
125. In a square with side 2 units, a circle with center at one vertex and radius $2.5$ units cuts two sides of the square. What is the distance from the nearest vertex of the square to the intersection points?
(1) $\dfrac{1}{\mathfrak{f}}$ (2) $\dfrac{1}{\mathfrak{r}}$ (3) $\dfrac{\sqrt{\mathfrak{r}}}{\mathfrak{r}}$ (4) $\dfrac{\sqrt{\mathfrak{r}}}{\mathfrak{r}}$
Q131 Linear transformations View
131. Consider line $\Delta$ with equation $3x + 2y = 6$. Under rotation about the origin by $\dfrac{\pi}{2}$, in the direction of line $\Delta'$, the equation of line $\Delta'$ under the translation $T(x,y) = (x-3, y+1)$ is:
(1) $3y - 2x = 12$ (2) $3y - 2x = 15$ (3) $2y - 3x = 8$ (4) $2y + 2x = 9$
Q133 Vectors Introduction & 2D Angle or Cosine Between Vectors View
133. Given $\mathbf{a} = (3, m, 5)$ and $\mathbf{b} = (3-m, 7, \circ)$. For a value of $m$, the two vectors $\mathbf{a}+\mathbf{b}$ and $\mathbf{a}-\mathbf{b}$ are perpendicular to each other. What is the angle between vectors $\mathbf{a}$ and $\mathbf{b}$ in this case?
(1) $30$ (2) $45$ (3) $60$ (4) $90$
%% Page 7 Mathematics 120-C Page 6
Q134 Vectors 3D & Lines Distance from a Point to a Line (Show/Compute) View
134- What is the initial distance from the line passing through point $(1, 2, -3)$ parallel to the vector with components $(4, -3, -5)$?
(1) $\dfrac{\sqrt{5}}{2}$ (2) $\sqrt{3}$ (3) $\sqrt{5}$ (4) $2\sqrt{3}$
Q135 Vectors 3D & Lines Shortest Distance Between Two Lines View
135- What is the length of the common perpendicular of the line $\dfrac{x-1}{4} = \dfrac{y+2}{3} = \dfrac{z-3}{1}$ and the $z$-axis?
(1) $2/2$ (2) $2/4$ (3) $2/5$ (4) $2/6$
Q136 Circles Circles Tangent to Each Other or to Axes View
136- A circle passing through point $(-9, -2)$ is tangent to both coordinate axes. What is the radius of the larger circle?
(1) $14$ (2) $15$ (3) $17$ (4) $19$
Q137 Conic sections Eccentricity or Asymptote Computation View
137- In the hyperbola $4x^2 - y^2 + 4y = 12$, $8x^2 - y^2 + 4y = 12$. What is the distance from a focus to an asymptote?
(1) $\sqrt{3}$ (2) $2$ (3) $2\sqrt{3}$ (4) $3$
Q138 Linear transformations View
138- Matrix $A = \begin{bmatrix} 1 & -\sqrt{3} \\ \sqrt{3} & 1 \end{bmatrix}$ is given. If matrix $A^2$ acts on point $(1, 2, -2)$, what are the coordinates of the resulting point?
(1) $(-16, 8)$ (2) $(-8, 16)$ (3) $(8, -16)$ (4) $(16, -8)$
Q139 Matrices Matrix Power Computation and Application View
139- If two matrices $A$ and $(I - A)$ are inverses of each other, then matrix $A^4$ equals which of the following?
(1) $A$ (2) $-A$ (3) $I$ (4) $-I$
Q140 Simultaneous equations View
140- In solving the system of equations $\begin{cases} x + y - z = 7 \\ 4x - y + 5z = 3 \\ 5x + y + z = 17 \end{cases}$ using Gaussian elimination, the matrix $\begin{bmatrix} 1 & 0 & 0 & a \\ 0 & 1 & 0 & b \\ 0 & 0 & 1 & c \end{bmatrix}$ is obtained. What is $b$?
(1) $2$ (2) $3$ (3) $4$ (4) $5$
Q141 Measures of Location and Spread View
141- Referring to the frequency histogram below, what is the total variance of the data?
\begin{minipage}{0.45\textwidth} [Figure: Frequency histogram with x-axis values 3, 5, 7, 9, 11 and y-axis (frequency) values up to 8, showing a roughly triangular distribution with peak around x=7] \end{minipage} \begin{minipage}{0.45\textwidth} (1) $4.5$
(2) $4.8$
(3) $4.92$
(4) $5.12$ \end{minipage}
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Q142 Measures of Location and Spread View
142. The data $x_i = 1, 2, 3, 4, 5$ are given. Suppose the data are transformed by $u_i = 12x_i + 6$. What is $\bar{u}_i$?
(1) $0/4$ (2) $0/48$ (3) $0/52$ (4) $0/6$
Q143 Proof by induction Prove a summation identity by induction View
143. We know that the sum of cubes of consecutive odd numbers starting from 1 equals the square of the sum of those numbers. What is the sum of cubes of consecutive odd numbers starting from 1 and ending at 19?
(1) $18800$ (2) $18900$ (3) $19800$ (4) $19900$
Q144 Binomial Distribution Find Minimum n for a Probability Threshold View
144. We toss a fair coin at least a few times so that we are more than 99\% certain that the result of three heads has occurred at least once. How many times at minimum must we toss?
(1) $12$ (2) $12$ (3) $18$ (4) $19$
Q146 Combinations & Selection Basic Combination Computation View
146. How many subsets does the set $A = \{1, 2, 3, 4, 5, 6\}$ have that contain exactly two elements?
(1) $8$ (2) $10$ (3) $12$ (4) $15$
Q147 Geometric Probability View
147. Inside an equilateral triangle with side length 8 units, we randomly select a point. What is the probability that this point is more than $\sqrt{3}$ units away from every side of the triangle?
(1) $\dfrac{1}{16}$ (2) $\dfrac{1}{9}$ (3) $\dfrac{1}{8}$ (4) $\dfrac{3}{16}$
Q148 Probability Definitions Probability Involving Algebraic or Number-Theoretic Conditions View
148. From the set of consecutive integers $\{300, 301, \ldots, 51\}$... wait: $\{300, 301, \ldots, 51\}$, i.e., $\{300, 301, \ldots, 451\}$, a number is chosen at random. What is the probability that this number is divisible by 6 but not by 7 and not by 42?
(1) $0/24$ (2) $0/26$ (3) $0/28$ (4) $0/31$
Q154 Discrete Probability Distributions Multiple Choice: Direct Probability or Distribution Calculation View
154- In a box there are $8$ light bulbs, two of which are defective. The bulbs are tested randomly one by one and the good bulb is set aside, until the first defective bulb is found. In the third test, what is the probability of finding the first defective bulb?
(1) $\dfrac{5}{28}$ (2) $\dfrac{4}{21}$ (3) $\dfrac{3}{14}$ (4) $\dfrac{5}{21}$
Q155 Discrete Probability Distributions Multiple Choice: Direct Probability or Distribution Calculation View
155- In a bag there are $5$ white beads, $4$ blue beads, and $3$ red beads. We draw three beads from the bag. What is the probability that at most $2$ of the drawn beads are the same color?
(1) $\dfrac{17}{22}$ (2) $\dfrac{19}{22}$ (3) $\dfrac{39}{44}$ (4) $\dfrac{41}{44}$
%% Page 10 Physics 120-C Page 9
Q164 Work done and energy Work done by gravity in specific scenarios View
164. According to the figure below, a object of mass $250\,\text{g}$ is placed on top of a spring whose spring constant is $2.5\,\dfrac{\text{N}}{\text{cm}}$, is released, and after hitting the spring, the spring compresses $12\,\text{cm}$. The work done by gravity on the object from the moment of release to the moment the spring reaches maximum compression is how many joules? (Air resistance is negligible and $g = 10\,\dfrac{\text{m}}{\text{s}^2}$.)
[Figure: A block resting on a spring attached to the ground]
  • [(1)] $0.3$
  • [(2)] $1.2$
  • [(3)] $1.8$
  • [(4)] $3.6$