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2021
eju-math__session1 12
2020
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2019
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2017
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2016
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2015
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2014
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2013
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2012
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2011
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2010
eju-math__session1 12
2014 eju-math__session2

12 maths questions

QCourse1-I-Q1 Solving quadratics and applications Determining quadratic function from given conditions View
Let $a$ and $b$ be real numbers, where $a > 0$. Consider the two quadratic functions
$$f ( x ) = 2 x ^ { 2 } - 4 x + 5 , \quad g ( x ) = x ^ { 2 } + a x + b .$$
We are to find the values of $a$ and $b$ when the function $g ( x )$ satisfies the following two conditions.
(i) The minimum value of $g ( x )$ is 8 less than the minimum value of $f ( x )$.
(ii) There exists only one $x$ which satisfies $f ( x ) = g ( x )$.
Since the minimum value of $f ( x )$ is $\mathbf { A }$, from condition (i), we derive the equality
$$b = \frac { a ^ { 2 } } { \mathbf { B } } - \mathbf { C } \text {. }$$
Hence the equation from which we can find the $x$ satisfying $f ( x ) = g ( x )$ is
$$x ^ { 2 } - ( a + \mathbf { D } ) x - \frac { a ^ { 2 } } { \mathbf { E } } + \mathbf { F G } = 0 .$$
Thus, since $a > 0$, from condition (ii) we obtain
$$a = \mathbf { H } , \quad b = \mathbf { I J } .$$
In this case, the $x$ satisfying $f ( x ) = g ( x )$ is $\square \mathbf{ K }$.
QCourse1-I-Q2 Probability Definitions Set Operations View
Consider the sets $A = \{ 4 m \mid m$ is a natural number $\}$ and $B = \{ 6 m \mid m$ is a natural number $\}$.
(1) For each of the following $\mathbf { L } \sim \mathbf{O}$, choose the correct answer from among (0) $\sim$ (3) below.
Let $n$ be a natural number.
(i) $n \in A$ is $\mathbf { L }$ for $n$ to be divisible by 2 .
(ii) $n \in B$ is $\mathbf { M }$ for $n$ to be divisible by 24 .
(iii) $n \in A \cup B$ is $\mathbf { N }$ for $n$ to be divisible by 3 .
(iv) $n \in A \cap B$ is $\square\mathbf{O}$ for $n$ to be divisible by 12 . (0) a necessary and sufficient condition
(1) a necessary condition but not a sufficient condition
(2) a sufficient condition but not a necessary condition
(3) neither a necessary condition nor a sufficient condition
(2) Let $C = \{ m \mid m$ is a natural number satisfying $1 \leqq m \leqq 100 \}$.
The number of elements which belong to $( \bar { A } \cup \bar { B } ) \cap C$ is $\mathbf { P Q }$, and the number of elements which belong to $\bar { A } \cap \bar { B } \cap C$ is $\mathbf { R S }$. Note that $\bar { A }$ and $\bar { B }$ denote the complements of $A$ and $B$, where the universal set is the set of all natural numbers.
QCourse1-II-Q1 Permutations & Arrangements Word Permutations with Repeated Letters View
Consider the permutations of the eight letters of the word ``POSITION''.
(1) The number of permutations in which the two I's are adjacent and the two O's also are adjacent is $\mathbf { A B C }$.
(2) The number of permutations such that the permutations both begin and end with the letter I and furthermore the two O's are adjacent is $\mathbf{DEF}$.
(3) The number of permutations that both begin and end with the letter I is $\mathbf{GHI}$.
(4) The number of permutations of the 4 letters I, I, O, O is $\mathbf { J }$. Also, the number of permutations of the 4 letters $\mathrm { N } , \mathrm { P } , \mathrm { S } , \mathrm { T }$ is $\mathbf { K L }$.
Hence the number of permutations of POSITION which begin or end with either I or O, and furthermore in which none of letters $\mathrm { N } , \mathrm { P } , \mathrm { S } , \mathrm { T }$ are adjacent to each other is $\mathbf{MNO}$.
QCourse1-II-Q2 Inequalities Simultaneous/Compound Quadratic Inequalities View
Suppose that an integer $x$ and a real number $y$ satisfy both the equation
$$2 ( y + 1 ) = x ( 8 - x ) \tag{1}$$
and the inequality
$$5 x - 4 y + 1 \leqq 0 . \tag{2}$$
We are to find $M$, the maximum value of $y$, and $m$, the minimum value of $y$.
First of all, let us transform (1) into
$$y = - \frac { 1 } { \mathbf { P } } ( x - \mathbf { Q } ) ^ { 2 } + \mathbf{R} .$$
Also, from (1) and (2) we obtain the inequality in $x$
$$2 x ^ { 2 } - \mathbf { S T } x + \mathbf { U V } \leqq 0 . \tag{3}$$
Thus when $x$ is an integer satisfying (3) if we consider the range of values which $y$ can take, we see that $y$ is maximized at $x = \square \mathbf{ V }$ and is minimized at $x = \square \mathbf{ W }$, and hence that
$$M = \mathbf { X } , \quad m = \frac { \mathbf { Y } } { \mathbf{Z} } .$$
QCourse1-III Inequalities Simultaneous/Compound Quadratic Inequalities View
For each of $\mathbf{A} \sim \mathbf{D}$ in the following questions, choose the correct answer from among (0) $\sim$ (5) below each question.
Consider the three quadratic inequalities
$$\begin{aligned} x ^ { 2 } + 3 x - 18 & < 0 \tag{1}\\ x ^ { 2 } - 2 x - 8 & > 0 \tag{2}\\ x ^ { 2 } + a x + b & < 0 . \tag{3} \end{aligned}$$
(1) The range of $x$ which satisfies both of the inequalities (1) and (2) is $\mathbf { A }$. Also, the range of $x$ which satisfies neither inequality (1) nor (2) is $\mathbf{B}$. (0) $3 \leqq x \leqq 4$
(1) $- 6 \leqq x \leqq - 2$
(2) $3 < x < 4$
(3) $2 < x < 6$
(4) $- 6 < x < - 2$
(5) $- 4 \leqq x \leqq - 3$
(2) The range of $x$ that satisfies at least one of the inequalities (1) and (3) will be $- 6 < x < 7$, if and only if $a$ and $b$ satisfy the equation $\square \mathbf{C}$, and $a$ satisfies the inequality $\square \mathbf{D}$. (0) $b = 6 a - 36$
(1) $b = 7 a - 49$
(2) $b = - 7 a - 49$
(3) $- 10 < a \leqq - 3$
(4) $- 10 < a \leqq - 1$
(5) $- 1 \leqq a < 3$
QCourse1-IV Sine and Cosine Rules Cyclic quadrilateral or inscribed polygon problem View
Suppose that a quadrangle ABCD which is inscribed in a circle has the side lengths
$$\mathrm { AB } = \sqrt { 2 } , \quad \mathrm { BC } = \mathrm { CD } = 2 , \quad \mathrm { DA } = \sqrt { 6 } .$$
(1) Let us set $\theta = \angle \mathrm { BAD }$. We have the two equalities
$$\begin{aligned} & \mathrm { BD } ^ { 2 } = \mathbf { A } - \mathbf { B } \sqrt { \mathbf { C } } \cos \theta , \\ & \mathrm { BD } ^ { 2 } = \mathbf { D } + \mathbf { E } \cos \theta . \end{aligned}$$
Hence,
$$\theta = \mathbf { F G } { } ^ { \circ } , \quad \mathrm { BD } = \mathbf { H } \text {. } \mathbf { I } \text {. }$$
(2) Furthermore, we have
$$\angle \mathrm { BAC } = \mathbf { J K } ^ { \circ } , \quad \angle \mathrm { BCA } = \mathbf { L M } ^ { \circ } \text { and } \mathrm { AC } = \mathbf { N } + \sqrt { \mathbf { O } } \text {. }$$
We also have
$$\sin \angle \mathrm { ADC } = \frac { \sqrt { \mathbf { P } } } { \mathbf{Q} } ( \sqrt { \mathbf { R } } + \mathbf { S } )$$
(3) Let us denote the point of intersection of the straight line AD and the straight line BC by E. We have $\mathrm { EB } = \mathbf { T } + \mathbf { U } \sqrt { \mathbf { V } }$.
QCourse2-I-Q1 Solving quadratics and applications Determining quadratic function from given conditions View
Let $a$ and $b$ be real numbers, where $a > 0$. Consider the two quadratic functions
$$f ( x ) = 2 x ^ { 2 } - 4 x + 5 , \quad g ( x ) = x ^ { 2 } + a x + b .$$
We are to find the values of $a$ and $b$ when the function $g ( x )$ satisfies the following two conditions.
(i) The minimum value of $g ( x )$ is 8 less than the minimum value of $f ( x )$.
(ii) There exists only one $x$ which satisfies $f ( x ) = g ( x )$.
Since the minimum value of $f ( x )$ is $\mathbf{A}$, from condition (i), we derive the equality
$$b = \frac { a ^ { 2 } } { \mathbf { B } } - \mathbf { C } \text {. }$$
Hence the equation from which we can find the $x$ satisfying $f ( x ) = g ( x )$ is
$$x ^ { 2 } - ( a + \mathbf { D } ) x - \frac { a ^ { 2 } } { \mathbf { E } } + \mathbf { F G } = 0 .$$
Thus, since $a > 0$, from condition (ii) we obtain
$$a = \mathbf { H } , \quad b = \mathbf { I J } .$$
In this case, the $x$ satisfying $f ( x ) = g ( x )$ is $\mathbf { K }$.
QCourse2-I-Q2 Probability Definitions Set Operations View
Consider the sets $A = \{ 4 m \mid m$ is a natural number $\}$ and $B = \{ 6 m \mid m$ is a natural number $\}$.
(1) For each of the following $\mathbf { L } \sim \square \mathbf { O }$, choose the correct answer from among (0) $\sim$ (3) below.
Let $n$ be a natural number.
(i) $n \in A$ is $\mathbf { L }$ for $n$ to be divisible by 2 .
(ii) $n \in B$ is $\mathbf { M }$ for $n$ to be divisible by 24 .
(iii) $n \in A \cup B$ is $\mathbf { N }$ for $n$ to be divisible by 3 .
(iv) $n \in A \cap B$ is $\mathbf{O}$ for $n$ to be divisible by 12 . (0) a necessary and sufficient condition
(1) a necessary condition but not a sufficient condition
(2) a sufficient condition but not a necessary condition
(3) neither a necessary condition nor a sufficient condition
(2) Let $C = \{ m \mid m$ is a natural number satisfying $1 \leqq m \leqq 100 \}$.
The number of elements which belong to $( \bar { A } \cup \bar { B } ) \cap C$ is $\mathbf { P Q }$, and the number of elements which belong to $\bar { A } \cap \bar { B } \cap C$ is $\mathbf { R S }$. Note that $\bar { A }$ and $\bar { B }$ denote the complements of $A$ and $B$, where the universal set is the set of all natural numbers.
QCourse2-II Vectors 3D & Lines Vector Algebra and Triple Product Computation View
Let $S$ be a circle with its center at point O and a radius of 1. Let $\triangle \mathrm { ABC }$ be a triangle such that all its vertices are on $S$ and $\mathrm { AB } : \mathrm { AC } = 3 : 2$. As shown in the figure, let D be a point on the extension of side BC and $k$ be the number where
$$\mathrm { BC } : \mathrm { CD } = 2 : k .$$
Moreover, set
$$\overrightarrow { \mathrm { OA } } = \vec { a } , \quad \overrightarrow { \mathrm { OB } } = \vec { b } , \quad \overrightarrow { \mathrm { OC } } = \vec { c }$$
Answer the following questions.
(1) When we express $\overrightarrow { \mathrm { OD } }$ in terms of $\vec { b } , \vec { c }$ and $k$, we have
$$\overrightarrow { \mathrm { OD } } = \left( \frac { k } { \mathbf { A } } + \mathbf { B } \right) \vec { c } - \frac { k } { \mathbf { C } } \vec { b }$$
(2) Since the equality
$$| \vec { b } - \vec { a } | = \frac { \mathbf { D } } { \mathbf { E } } | \vec { c } - \vec { a } |$$
holds, by expressing the inner product $\vec { a } \cdot \vec { b }$ in terms of the inner product $\vec { a } \cdot \vec { c }$, we have
$$\vec { a } \cdot \vec { b } = \frac { \mathbf { F } } { \mathbf { G } } \vec { a } \cdot \vec { c } - \frac { \mathbf { H } } { \mathbf { I } }$$
(3) It follows that when the tangent to $S$ at the point A passes through the point D,
$$k = \frac { \mathbf { J } } { \mathbf { K } } .$$
QCourse2-III Exponential Equations & Modelling Solve Exponential Equation for Unknown Variable View
Let $p > 1$ and $q > 1$. Consider an equation in $x$
$$e ^ { 2 x } - a e ^ { x } + b = 0 \tag{1}$$
such that the equation in $t$ obtained by setting $t = e ^ { x }$ in (1)
$$t ^ { 2 } - a t + b = 0 \tag{2}$$
has the solutions $\log _ { q ^ { 2 } } p$ and $\log _ { p ^ { 3 } } q$. We are to find the minimum value of $a$ and the solution of equation (1) at this minimum.
(1) First of all, we see that
$$b = \frac { \mathbf { A } } { \mathbf { A B } }$$
and
$$a = \frac { \mathbf { C } } { \mathbf{D} } \log _ { q } p + \frac { \mathbf { E } } { \mathbf { F } } \log _ { p } q .$$
(2) As long as $p > 1$ and $q > 1$, it always follows that $\log _ { p } q > \mathbf { G }$. Hence, $a$ takes the minimum value $\frac { \sqrt { \mathbf { H } } } { \mathbf { I } }$ when $\log _ { p } q = \frac { \sqrt { \mathbf { J } } } { \mathbf { K } }$. In this case, the solution of (1) is
$$x = - \frac { \mathbf { L } } { \mathbf { M } } \log _ { e } \mathbf { N } .$$
QCourse2-IV-Q1 Areas by integration View
Let $a$ and $t$ be positive real numbers. Let $\ell$ be the tangent to the graph $C$ of $y = a x ^ { 3 }$ at a point $\mathrm { P } \left( t , a t ^ { 3 } \right)$, and let Q be the point at which $\ell$ intersects the curve $C$ again. Further, let $p$ be the line passing through the point P parallel to the $x$-axis; let $q$ be the line passing through the point Q parallel to the $y$-axis; and let R be the point of intersection of $p$ and $q$.
Also, let us denote by $S _ { 1 }$ the area of the region bounded by the curve $C$, the straight line $p$ and the straight line $q$, and denote by $S _ { 2 }$ the area of the region bounded by the curve $C$ and the tangent $\ell$. We are to find the value of $\frac { S _ { 1 } } { S _ { 2 } }$.
First, since the equation of the tangent $\ell$ is
$$y = \mathbf { A } a t ^ { \mathbf{B} } x - \mathbf { C } a t ^ { \mathbf{D} } \text {, }$$
the $x$-coordinate of Q is $- \mathbf { E } t$.
Hence, $S _ { 1 }$ is
$$S _ { 1 } = \frac { \mathbf { F G } } { \mathbf { H } } a t ^ { \mathbf { I } } .$$
Also, since $S _ { 2 }$ is obtained by subtracting $S _ { 1 }$ from the area of the triangle PQR, we have
$$S _ { 2 } = \frac { \mathbf { J K } } { \mathbf { L } } a t ^ { \mathbf { M } } .$$
Hence, the value of $\frac { S _ { 1 } } { S _ { 2 } }$ is always
$$\frac { S _ { 1 } } { S _ { 2 } } = \mathbf { N } ,$$
independent of the values of $a$ and $t$.
QCourse2-IV-Q2 Reduction Formulae Evaluate a Closed-Form Expression Using the Reduction Formula View
Given the function in $x$
$$f _ { n } ( x ) = \sin ^ { n } x \quad ( n = 1,2,3 , \cdots ) ,$$
answer the following questions.
(1) Consider the cases in which the equality
$$\lim _ { x \rightarrow 0 } \frac { a - x ^ { 2 } - \left( b - x ^ { 2 } \right) ^ { 2 } } { f _ { n } ( x ) } = c$$
holds for three real numbers $a , b$ and $c$.
(i) We have $a = b$.
(ii) When $n = 2$, if $c = 6$, then $b = \frac { \mathbf { P } } { \mathbf { Q } }$.
(iii) When $n = 4$, then $b = \frac { \mathbf { R } } { \mathbf { S } }$ and $c = - \mathbf { T }$.
(2) For this $f _ { n } ( x )$, consider the definite integral
$$I _ { n } = \int _ { 0 } ^ { \frac { \pi } { 2 } } f _ { n } ( x ) \sin 2 x \, d x \quad ( n = 1,2,3 , \cdots )$$
When the integral is calculated, we have
$$I _ { n } = \frac { \mathbf { U } } { n + \mathbf { V } } .$$
Hence we obtain
$$\lim _ { n \rightarrow \infty } \left( I _ { n - 1 } + I _ { n } + I _ { n + 1 } + \cdots + I _ { 2 n - 2 } \right) = \int _ { 0 } ^ { \mathbf { W } } \frac { \mathbf { X } } { \mathbf { Y } + x } \, dx = \log \mathbf { Z }$$