kyotsu-test

QCourse1-I-Q1 Completing the square and sketching Vertex and parameter conditions for a quadratic graph View

Consider the quadratic function
$$y = - x ^ { 2 } - a x + 3 .$$
(1) If $a > 0$ and the maximum value of function (1) is 7 , then $a = \square$. In this case, the equation of the axis of symmetry of the graph is $x = \mathbf { B C }$, and the $x$-coordinates of the points of intersection of this graph and the $x$-axis are $\mathbf { D E } \pm \sqrt { \mathbf { F } }.$
(2) If the curve obtained by translating the graph of function (1) by 2 in the $x$-direction and by $-3$ in the $y$-direction passes through $( - 3 , - 5 )$, then $a =$ $\square$ G.

QCourse1-I-Q2 Discriminant and conditions for roots Nature of roots given coefficient constraints View

For $\mathbf { H } , \mathbf { I }$ in question (1), and for $\mathbf { J }$, $\mathbf { K }$ in question (2), choose the appropriate answer from among (0) $\sim$ (3) at the bottom of this page.
For $\mathbf { L } \sim \mathbf { R }$ in question (3), enter the appropriate number.
Consider the following three possible conditions on two real numbers $x$ and $y$:
$p : x$ and $y$ satisfy the equation $( x + y ) ^ { 2 } = a \left( x ^ { 2 } + y ^ { 2 } \right) + b x y$, where $a$ and $b$ are real constants.
$$\begin{aligned} & q : x = 0 \text { and } y = 0 . \\ & r : x = 0 \text { or } y = 0 . \end{aligned}$$
(1) Suppose that in condition $p$, $a = b = 1$. Then $p$ is $\mathbf { H }$ for $q$, and $p$ is $\mathbf { I }$ for $r$.
(2) Suppose that in condition $p$, $a = b = 2$. Then $p$ is $\mathbf { J }$ for $q$, and $p$ is $\mathbf { K }$ for $r$.
(3) If in condition $p$ we set $a = 2$, we can transform the equation in $p$ into
$$\left( x + \frac { b - \mathbf { L } } { \mathbf { L } } y \right) ^ { 2 } + \left( \mathbf { L } - \frac { ( b - \mathbf { Q} ) ^ { 2 } } { \mathbf { L } } \right) y ^ { 2 } = 0 .$$
Hence $p$ is a necessary and sufficient condition for $q$ if and only if $b$ satisfies
$$\mathbf { Q } < b < \mathbf { R } .$$
(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

QCourse1-II-Q1 Discrete Probability Distributions Expectation and Variance from Context-Based Random Variables View

In a bag there are a total of nine balls: one white, three red and five black. The white ball is worth five points, a red ball is worth three points, and a black ball is worth one point. Two balls are taken from the bag together, and the trial is scored by the sum of the values of the two balls.
(1) The highest possible score is $\mathbf{A}$, and the probability that it happens is $\dfrac{\mathbf{B}}{\mathbf{CD}}$.
(2) The probability that the score is 6 is $\dfrac{\mathbf{E}}{\mathbf{F}}$.
(3) The expected value of the score is $\dfrac{\mathbf{GH}}{\mathbf{I}}$.

QCourse1-II-Q2 Proof Proof Involving Combinatorial or Number-Theoretic Structure View

A natural number $n$ is said to be a perfect square when there exists a natural number $x$ satisfying $n = x ^ { 2 }$. Similarly, $n$ is said to be a perfect cube when there exists a natural number $x$ satisfying $n = x ^ { 3 }$.
In the following two cases, find the natural number $n$ that satisfies the conditions.
(i) $n$ is a perfect square. Furthermore, the number obtained by adding 13 to $n$ is also a perfect square.
(ii) $n$ is a perfect cube. Furthermore, the number obtained by adding 61 to $n$ is also a perfect cube.
First, consider (i). From the definition of a perfect square number, $n$ can be expressed as $n = x ^ { 2 }$, where $x$ is a natural number. In addition, there exists a natural number $y$ such that
$$x ^ { 2 } + 13 = y ^ { 2 } .$$
Since $x < y$, $y - x = \square \mathbf { J }$ and $y + x = \mathbf { K L }$. It follows that
$$x = \mathbf { M } , \quad y = \mathbf { N } ,$$
and finally that $n = \mathbf { O P }$.
Next, consider (ii). Similar to (i), in (ii), there exists a natural number $x$ such that $n = x ^ { 3 }$, and there also exists a natural number $y$ such that
$$x ^ { 3 } + 61 = y ^ { 3 } .$$
When we solve this equation, we obtain
$$x = \mathbf { Q } , \quad y = \mathbf { R } ,$$
and hence the perfect cube $n = \mathbf{ST}$.

QCourse1-III Inequalities Quadratic Inequality Holding for All x (or a Restricted Domain) View

Let $a$ be a constant. Consider the quadratic inequality
$$x ^ { 2 } - 2 ( a + 2 ) x + 25 > 0 . \tag{1}$$
The left-hand side of inequality (1) can be transformed into
$$( x - a - \mathbf { A } ) ^ { 2 } - a ^ { 2 } - \mathbf { B } a + \mathbf { C D } .$$
Hence, we have the following results.
(1) The condition under which inequality (1) holds for all real numbers $x$ is
$$\mathbf { E F } < a < \mathbf { G } .$$
(2) The condition under which inequality (1) holds for all real numbers $x$ satisfying $x \geqq - 1$ is
$$\mathbf { H I J } < a < \mathbf { K } .$$

QCourse1-IV Sine and Cosine Rules Multi-step composite figure problem View

Suppose that in the figure to the right
$$\mathrm { AB } = 4 , \quad \mathrm { AC } = 5 , \quad \cos \angle \mathrm { BAC } = \frac { 1 } { 8 }$$
and
$$\angle \mathrm { BAD } = \angle \mathrm { ACB } , \quad \angle \mathrm { CAE } = \angle \mathrm { ABC } .$$
(1) When we denote the area of $\triangle \mathrm { ABC }$ by $S$, we have
$$S = \frac { \square \mathbf { A B } \sqrt { \square \mathbf { C } } } { \square } .$$
Also $\mathrm { BC } = \mathbf { E }$.
(2) Furthermore, when we denote the areas of $\triangle \mathrm { ABD }$ and $\triangle \mathrm { ACE }$ by $S _ { 1 }$ and $S _ { 2 }$, respectively, we have
$$S : S _ { 1 } : S _ { 2 } = 1 : \frac { \mathbf { F } } { \mathbf{G} } : \frac { \mathbf { H I } } { \mathbf { J } } .$$
(3) When we denote the area of $\triangle \mathrm { ADE }$ by $T$, we have
$$T = \frac { \mathbf { L M } \sqrt { \mathbf { N } } } { \mathbf { O P } } .$$
Also $\mathrm { DE } = \dfrac { \mathbf { Q } } { \mathbf{R} }$.

QCourse2-I-Q1 Completing the square and sketching Vertex and parameter conditions for a quadratic graph View

Consider the quadratic function
$$y = - x ^ { 2 } - a x + 3 .$$
(1) If $a > 0$ and the maximum value of function (1) is 7 , then $a = \square$. In this case, the equation of the axis of symmetry of the graph is $x = \mathbf { B C }$, and the $x$-coordinates of the points of intersection of this graph and the $x$-axis are $\mathbf { D E } \pm \sqrt { \mathbf { F } }.$
(2) If the curve obtained by translating the graph of function (1) by 2 in the $x$-direction and by $-3$ in the $y$-direction passes through $( - 3 , - 5 )$, then $a =$ $\square$ G.

QCourse2-I-Q2 Discriminant and conditions for roots Nature of roots given coefficient constraints View

For $\mathbf { H } , \mathbf { I }$ in question (1), and for $\mathbf { J }$, $\mathbf { K }$ in question (2), choose the appropriate answer from among (0) $\sim$ (3) at the bottom of this page.
For $\mathbf { L } \sim \mathbf { R }$ in question (3), enter the appropriate number.
Consider the following three possible conditions on two real numbers $x$ and $y$:
$p : x$ and $y$ satisfy the equation $( x + y ) ^ { 2 } = a \left( x ^ { 2 } + y ^ { 2 } \right) + b x y$, where $a$ and $b$ are real constants.
$$\begin{aligned} & q : x = 0 \text { and } y = 0 . \\ & r : x = 0 \text { or } y = 0 . \end{aligned}$$
(1) Suppose that in condition $p$, $a = b = 1$. Then $p$ is $\mathbf { H }$ for $q$, and $p$ is $\mathbf { I }$ for $r$.
(2) Suppose that in condition $p$, $a = b = 2$. Then $p$ is $\mathbf { J }$ for $q$, and $p$ is $\mathbf { K }$ for $r$.
(3) If in condition $p$ we set $a = 2$, we can transform the equation in $p$ into
$$\left( x + \frac { b - \mathbf { L } } { \mathbf { L } } y \right) ^ { 2 } + \left( \mathbf { N } - \frac { ( b - \mathbf { Q} ) ^ { 2 } } { \mathbf { P } } \right) y ^ { 2 } = 0 .$$
Hence $p$ is a necessary and sufficient condition for $q$ if and only if $b$ satisfies $\mathbf { Q } < b < \mathbf { R }$.
(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

QCourse2-II Geometric Sequences and Series Geometric Series with Trigonometric or Functional Terms View

Let the sequence $\left\{ a _ { n } \right\} ( n = 1,2,3 , \cdots )$ be an arithmetic progression satisfying
$$a _ { 2 } = 2 , \quad a _ { 6 } = 3 a _ { 3 } .$$
Then, consider the series $\displaystyle\sum _ { n = 1 } ^ { \infty } \frac { 3 ^ { n } } { r ^ { a _ { n } } }$, where $r$ is a positive real number.
(1) When we denote the first term of $\left\{ a _ { n } \right\}$ by $a$, and the common difference by $d$, we have
$$a = \mathbf { A B } , \quad d = \mathbf { C } .$$
(2) The series $\displaystyle\sum _ { n = 1 } ^ { \infty } \frac { 3 ^ { n } } { r ^ { a _ { n } } }$ is an infinite geometric series where the first term is $\square r^{\mathbf{E}}$, and the common ratio is $\dfrac { \mathbf { F } } { r^{\mathbf{G}} }$. Hence, this series converges when
$$r > 3 ^ { \frac { \mathbf { H } } { \mathbf{I} } } ,$$
and its sum $S$ is
$$S = \frac { \square \mathbf { J } \, r ^ { \mathbf { K } } } { r ^ { \mathbf { L } } - \mathbf { M } } .$$
(3) This sum $S$ is minimized at
$$r = \mathbf { N } ^ { \frac { \mathbf { O } } { 2 } } .$$

QCourse2-III Quadratic trigonometric equations View

Consider the function
$$f ( x ) = \sin 2 x - 3 ( \sin x + \cos x )$$
on the interval $- \dfrac { \pi } { 3 } \leqq x \leqq \dfrac { \pi } { 3 }$.
(1) Let $t = \sin x + \cos x$. Find the range of the values which $t$ can take.
(2) The function $f ( x )$ takes its minimum value $\mathbf { E } - \mathbf { F } \sqrt{\mathbf{G}}$ at $x = \dfrac { \mathbf { H } } { \mathbf { I } }$.

QCourse2-IV-Q1 Differentiating Transcendental Functions Monotonicity or convexity of transcendental functions View

For each of $\mathbf{A} \sim \mathbf{I}$ in the following sentences, choose the appropriate answer from among (0) $\sim$ (9) at the bottom of this page.
We are to compare the magnitudes of $a ^ { a + 1 }$ and $( a + 1 ) ^ { a }$ by using the properties of the function $f ( x ) = \dfrac { \log x } { x }$, where $a > 0$.
(1) Since the derivative of $f ( x )$ is
$$f ^ { \prime } ( x ) = \frac { \mathbf { A } - \log x } { x^{\mathbf{B}} } ,$$
the interval on $x$ in which $f ( x )$ monotonically increases is
$$\mathbf { C } < x \leqq \mathbf { D }$$
and the interval on $x$ in which $f ( x )$ monotonically decreases is
$$\mathbf { E } \leq x .$$
(2) When we set $p = a ^ { a + 1 }$, $q = ( a + 1 ) ^ { a }$, we have
$$\log p - \log q = \left( a ^ { \mathbf { F } } + a \right) \{ f ( a ) - f ( a + \mathbf { G } ) \} .$$
Hence we see that
$$\text { if } \quad 0 < a < \tfrac{3}{2} \quad \text { then } \quad p \quad \mathbf { H } \quad q ,$$
and
$$\text { if } \quad 3 < a \quad \text { then } \quad p \quad \mathbf { I } \quad q .$$
(0) 0\quad (1) 1\quad (2) 2\quad (3) 3\quad (4) $e$\quad (5) $e + 1$\quad (6) $\dfrac{1}{e}$\quad (7) $>$\quad (8) $=$\quad (9) $<$

QCourse2-IV-Q2 Areas by integration View

Let $0 < a < 1$. Let $S ( a )$ denote the sum of the areas of two regions, one region bounded by the curve $y = x e ^ { 2 x }$, the $x$-axis, and the straight line $x = a - 1$, and the other region bounded by the curve $y = x e ^ { 2 x }$, the $x$-axis, and the straight line $x = a$. We are to find the value of $a$ at which $S ( a )$ is minimized.
The indefinite integral of $x e ^ { 2 x }$ is to be determined, where $C$ is the constant of integration.
The value of $x e ^ { 2 x }$ is $x e ^ { 2 x } < 0$ for $x < 0$ and $x e ^ { 2 x } \geqq 0$ for $x \geqq 0$. Hence we have
$$S ( a ) = \frac { \mathbf { L M } } { \mathbf { N } } \left\{ \mathbf { O } + \left( \mathbf { P } a - \mathbf { Q } \right) e ^ { 2 ( a - 1 ) } + ( \mathbf { R } a - 1 ) e ^ { 2 a } \right\} .$$
Further, since
$$S ^ { \prime } ( a ) = ( a - \mathbf { S } ) e ^ { 2 ( a - 1 ) } + a e ^ { 2 a } ,$$
the value of $a$ at which $S ( a )$ is minimized is $a = \dfrac { \square \mathbf { T } } { e ^ { 2 } + \mathbf { U } }$, which satisfies $0 < a < 1$.

kyotsu-test

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2013 eju-math__session2