kyotsu-test

QI-Q1 Stationary points and optimisation Find absolute extrema on a closed interval or domain View

Suppose that $x$ and $y$ satisfy
$$3 x + y = 18 , \quad x \geqq 1 , \quad y \geqq 6 .$$
We are to find the maximum value and the minimum value of $x y$.
When we express $x y$ in terms of $x$, we have
$$x y = \mathbf { A B } ( x - \mathbf { C } ) ^ { 2 } + \mathbf { D E } .$$
Also, the range of values which $x$ can take is
$$\mathbf { F } \leqq x \leqq \mathbf { G } .$$
Hence, the value of $x y$ is maximized at $x = \mathbf { H }$ and its value there is $\mathbf { I J }$, and the value of $x y$ is minimized at $x = \mathbf { K }$ and its value there is $\mathbf { L M }$.

QI-Q2 Indices and Surds Conjugate Surds and Sum Evaluation via Identities View

Suppose that positive real numbers $a$ and $b$ satisfy
$$a ^ { 2 } = 3 + \sqrt { 5 } , \quad b ^ { 2 } = 3 - \sqrt { 5 } .$$
Let $c$ be the fractional portion of $a + b$. We are to find the value of $\frac { 1 } { c } - c$.
(1) We see that $( a b ) ^ { 2 } = \mathbf { N }$ and $( a + b ) ^ { 2 } = \mathbf { O P }$.
(2) Since $\mathbf { Q }$ $< a + b < \mathbf { Q } + 1$, the value of $c$ is $\sqrt { \mathbf { R S } } - \mathbf { T }$.
Thus we obtain $\frac { 1 } { c } - c = \mathbf { U }$.

QII-Q1 Probability Definitions Probability Distribution and Sampling View

There are nine cards on which the integers from 1 to 9 are written in a box. Two cards are taken simultaneously from this box. Let $S$ denote the sum of the numbers written on the two cards.
(1) The probability that $S$ is 5 or less is $\frac { \mathbf { A } } { \mathbf { B } }$. Let us assign a score to the result $S$.
When $S$ is 5 or less the score is $10 - S$, and when it is greater than 5 the score is 2. Then the expected value of the score is $\frac { \mathbf { C D } } { \mathbf { E F } }$.
(2) Let us perform the above trial twice, returning the two cards to the box before the second trial.
(i) The probability that $S$ is 5 or less in both trials is $\frac { \mathbf { G } } { \mathbf { H } }$.
(ii) The probability that $S$ is 5 or less in at least one trial is $\frac { \mathbf { J K } } { \mathbf { L M } }$.

QII-Q2 Solving quadratics and applications Counting solutions or configurations satisfying a quadratic system View

Let $a$ be a constant. For the two functions in $x$
$$\begin{aligned} & f ( x ) = 2 x ^ { 2 } + x + a - 2 \\ & g ( x ) = - 4 x - 5 \end{aligned}$$
we are to find the real values of $x$ for which $f ( x ) = g ( x )$ and also find the values of the two functions there.
(1) For each of $\mathbf { N } , \mathbf { O }$ and $\mathbf { P }$ in the following statements, choose the appropriate condition from (0) $\sim$ (8) below.
When $\mathbf { N }$, there are two real values of $x$ for which $f ( x ) = g ( x )$. When $\mathbf { O }$, there is only one real value of $x$ for which $f ( x ) = g ( x )$. When $\mathbf{P}$, there is no real value of $x$ for which $f ( x ) = g ( x )$.
(0) $a > \frac { 1 } { 8 }$
(1) $a = \frac { 17 } { 8 }$
(2) $a = \frac { 1 } { 6 }$
(3) $a < \frac { 1 } { 6 }$
(4) $a < \frac { 17 } { 8 }$
(5) $a < \frac { 1 } { 8 }$ (6) $a > \frac { 1 } { 6 }$ (7) $a = \frac { 1 } { 8 }$ (8) $a > \frac { 17 } { 8 }$
(2) When N, the values of $x$ for which $f ( x ) = g ( x )$ are $\frac { - \mathrm { Q } \pm \sqrt { \mathrm { R } - \mathbf { S } a } } { \mathbf{T} }$, and the values of the functions there are $\mp \sqrt { \mathbf { U } - \mathbf { V } a }$.
When O, the value of $x$ for which $f ( x ) = g ( x )$ is $- \frac { \mathrm { W } } { \mathrm { X} }$, and the value of the functions there is $\mathbf{Y}$.
(3) Consider the case where $f ( x ) = g ( x )$ and the absolute value of these functions there is greater than or equal to 3. The condition for this case is that $a \leqq - \mathbf { Z }$.

QIII Completing the square and sketching Vertex and parameter conditions for a quadratic graph View

Let $a$ be a constant. Consider the quadratic function in $x$
$$y = 2 x ^ { 2 } + a x + 3 .$$
Suppose that the vertex of the graph of (1) is in the first (upper right-hand) quadrant.
(1) The range of values which $a$ can take is
$$\mathbf { A B } \sqrt { \mathbf { C } } < a < \mathbf { D } ,$$
and the least integer $a$ satisfying this inequality is $\mathbf{EF}$.
(2) Let $a = \mathrm { EF }$ in (1). Let
$$y = 2 x ^ { 2 } + p x + q$$
be the equation of the graph which is obtained by translating the graph of (1) by $- \frac { 1 } { n }$ in the $x$-direction and by $\frac { 6 } { n ^ { 2 } }$ in the $y$-direction. Then
$$p = \frac { \mathbf { G } } { n } - \mathbf { H } , \quad q = \frac { \mathbf { I } } { n ^ { 2 } } - \frac { \mathbf { J } } { n } + \mathbf { K } .$$
(3) The total number of natural numbers $n$ for which $p$ in (2) is an integer is $\mathbf { L }$. Among these $n$, consider the ones such that the value of $q$ is also an integer. Then $\mathbf { M }$ is the value of the $n$ for which $q$ takes the minimum value $\mathbf { N }$.

QIV Sine and Cosine Rules Cyclic quadrilateral or inscribed polygon problem View

A quadrangle $ABCD$ which is inscribed in a circle $O$ satisfies
$$\mathrm { AB } = \mathrm { BC } = \sqrt { 2 } , \quad \mathrm { BD } = \frac { 3 \sqrt { 3 } } { 2 } , \quad \angle \mathrm { ABC } = 120 ^ { \circ } ,$$
where
$$AD > CD .$$
(1) Then $\mathrm { AC } = \sqrt { \mathbf { A } }$, and the radius of circle O is $\sqrt { \mathbf { B } }$.
(2) Set $x = \mathrm { AD }$. Since $\angle \mathrm { ADB } = \mathbf { CD }^\circ$, $x$ satisfies
$$4 x ^ { 2 } - \mathbf { E } \mathbf { F } x + \mathbf { G H } = 0 .$$
Also, set $y = \mathrm { CD }$. In the same way, it follows that $y$ satisfies
$$4 y ^ { 2 } - \mathbf { I J } y + \mathbf { K } \mathbf { L } = 0 .$$
Thus, noting (1), we obtain $AD$ and $CD$.

QC2-I-Q1 Stationary points and optimisation Find absolute extrema on a closed interval or domain View

Suppose that $x$ and $y$ satisfy
$$3 x + y = 18 , \quad x \geqq 1 , \quad y \geqq 6$$
We are to find the maximum value and the minimum value of $x y$.
When we express $x y$ in terms of $x$, we have
$$x y = \mathbf { A B } ( x - \mathbf { C } ) ^ { 2 } + \mathbf { D E } .$$
Also, the range of values which $x$ can take is
$$\mathbf { F } \leq x \leqq \mathbf { G }$$
Hence, the value of $x y$ is maximized at $x = \mathbf { H }$ and its value there is $\mathbf { I J }$, and the value of $x y$ is minimized at $x = \mathbf { K }$ and its value there is $\mathbf { L M }$.

QC2-I-Q2 Indices and Surds Conjugate Surds and Sum Evaluation via Identities View

Suppose that positive real numbers $a$ and $b$ satisfy
$$a ^ { 2 } = 3 + \sqrt { 5 } , \quad b ^ { 2 } = 3 - \sqrt { 5 } .$$
Let $c$ be the fractional portion of $a + b$. We are to find the value of $\frac { 1 } { c } - c$.
(1) We see that $( a b ) ^ { 2 } = \mathbf { N }$ and $( a + b ) ^ { 2 } = \mathbf { O P }$.
(2) Since $\mathbf { Q }$ $< a + b < \mathbf { Q } + 1$, the value of $c$ is $\sqrt { \mathbf { R S } } - \mathbf { T }$.
Thus we obtain $\frac { 1 } { c } - c = \mathbf { U }$.

QC2-II Laws of Logarithms Characteristic and Mantissa of Common Logarithms View

Given a sequence $\left\{ a _ { n } \right\}$ that satisfies the following conditions
$$\begin{aligned} & a _ { 1 } = 1 \\ & a _ { n + 1 } = 2 a _ { n } ^ { 2 } \quad ( n = 1,2,3 , \cdots ) , \end{aligned}$$
we are to find the number of natural numbers $n$ satisfying $a _ { n } < 10 ^ { 60 }$. (For the value of $\log _ { 10 } 2$, use the approximation 0.301.)
In this sequence we note that $a _ { n } > 0$ for all natural numbers $n$. Thus when we consider common logarithms of both sides of (1), we have
$$\log _ { 10 } a _ { n + 1 } = \log _ { 10 } \mathbf { A } + \mathbf { B } \log _ { 10 } a _ { n } .$$
When we set $b _ { n } = \log _ { 10 } a _ { n } + \log _ { 10 } \mathbf{A}$, the sequence $\left\{ b _ { n } \right\}$ is a geometric progression such that the common ratio is $\mathbf { C }$. Then
$$\log _ { 10 } a _ { n } = \left( ( \mathbf { D } ) ^ { n - 1 } - \mathbf { E } \right) \log _ { 10 } \mathbf { F } .$$
Furthermore, since $a _ { n } < 10 ^ { 60 }$,
$$\mathbf{D}^{ n - 1 } < \frac { \mathbf { G H } } { \log _ { 10 } \mathbf { F } } + \mathbf { E }$$
Since $\mathbf{IJK}$ is the least natural number which is larger than the value of the right side of (2), the number of natural numbers $n$ satisfying $a _ { n } < 10 ^ { 60 }$ is $\mathbf{L}$.

QC2-III Conic sections Conic Identification and Conceptual Properties View

Consider the following two equations
$$\begin{gathered} \left( \log _ { 4 } 2 \sqrt { x } \right) ^ { 2 } + \left( \log _ { 4 } 2 \sqrt { y } \right) ^ { 2 } = \log _ { 2 } ( \sqrt [ 4 ] { 2 } \cdot x \sqrt { y } ) \\ \sqrt [ 3 ] { x } \cdot \sqrt [ 4 ] { y } = 2 ^ { k } \end{gathered}$$
We are to find the range of values which the constant $k$ can take so that there exist positive real numbers $x , y$ which satisfy (1) and (2) simultaneously.
Set $X = \log _ { 2 } x$ and $Y = \log _ { 2 } y$. Let us express (1) and (2) in terms of $X$ and $Y$. First we consider (1). Since
$$\log _ { 4 } 2 \sqrt { x } = \frac { \log _ { 2 } x + \mathbf { A } } { \mathbf { B } }$$
and
$$\log _ { 2 } ( \sqrt [ 4 ] { 2 } \cdot x \sqrt { y } ) = \frac { \mathbf { C } } { \mathbf { D } } + \log _ { 2 } x + \frac { \log _ { 2 } y } { \mathbf { E } } ,$$
(1) reduces to
$$( X - \mathbf { F } ) ^ { 2 } + ( Y - \mathbf { G } ) ^ { 2 } = \mathbf { H I } .$$
In the same way, (2) reduces to
$$4 X + \mathbf { J } Y = \mathbf { K } \mathbf { L } k .$$
Since the distance $d$ from the center of the circle (3) to the straight line (4) on the $XY$-plane is given by
$$d = \frac { | \mathbf { M N } - \mathbf { O P } k | } { \mathbf { Q } } ,$$
the range of values which $k$ can take is
$$\mathbf { R } \leq k \leqq \mathbf { S } .$$

QC2-IV-Q1 Chain Rule Chain Rule Combined with Fundamental Theorem of Calculus View

We are to differentiate
$$f ( x ) = \int _ { 0 } ^ { 2 x } \left( t ^ { 2 } - x ^ { 2 } \right) \sin 3 t \, d t$$
with respect to $x$.
(1) We know that if $g ( t )$ is a continuous function and $G ( t )$ is one of its primitive functions, then
$$\int _ { 0 } ^ { 2 x } g ( t ) d t = G ( 2 x ) - G ( 0 )$$
By differentiating both sides of this equality with respect to $x$, we have
$$\frac { d } { d x } \int _ { 0 } ^ { 2 x } g ( t ) d t = \mathbf { A }$$
where $\mathbf{A}$ is the appropriate expression from among the following (0) $\sim$ (7).
(0) $g ( x )$
(1) $\frac { 1 } { 2 } g ( x )$
(2) $2 g ( x )$
(3) $g ( 2 x )$
(4) $\frac { 1 } { 2 } g ( 2 x )$
(5) $2 g ( 2 x )$ (6) $g ( x ) - g ( 0 )$ (7) $g ( 2 x ) - g ( 0 )$
(2) We know that $f ( x ) = \int _ { 0 } ^ { 2 x } t ^ { 2 } \sin 3 t \, d t - \int _ { 0 } ^ { 2 x } x ^ { 2 } \sin 3 t \, d t$.
Since
$$\frac { d } { d x } \int _ { 0 } ^ { 2 x } t ^ { 2 } \sin 3 t \, d t = \mathbf { B } x ^ { 2 } \sin \mathbf { C } x$$
and
$$\frac { d } { d x } \int _ { 0 } ^ { 2 x } x ^ { 2 } \sin 3 t \, d t = \frac { \mathbf { D } } { \mathbf { E } } x ( - \cos \mathbf { F } x + \mathbf { G } + \mathbf { H } x \sin \mathbf { I } x )$$
we obtain
$$f ^ { \prime } ( x ) = \frac { \mathbf { D } } { \mathbf { E } } x ( \cos \mathbf { J } x - \mathbf { K } + \mathbf { L } x \sin \mathbf { M } x )$$

QC2-IV-Q2 Integration by Parts Area or Volume Computation Requiring Integration by Parts View

Let $a$ be a positive real number. Let P denote the point of intersection of the following two curves
$$\begin{aligned} & C _ { 1 } : y = \frac { 3 } { x } \\ & C _ { 2 } : y = \frac { a } { x ^ { 2 } } , \end{aligned}$$
and let $\ell$ denote the tangent to $C _ { 2 }$ at P. Then we are to find the area $S$ of the region bounded by $C _ { 1 }$ and $\ell$.
Since the coordinates of P are $\left( \frac { a } { \mathbf { N } } , \frac { \mathbf { O } } { a } \right)$, the equation of $\ell$ is
$$y = - \frac { \mathbf { P Q } } { a ^ { 2 } } x + \frac { \mathbf { R S } } { a }$$
When we set
$$p = \frac { a } { \mathbf { T } } , \quad q = \frac { a } { \mathbf { U } } \quad ( p < q )$$
$S$ is obtained by calculating
$$S = [ \mathbf { V } ] _ { p } ^ { q }$$
where $\mathbf{V}$ is the appropriate expression from among (0) $\sim$ (5) below.
(0) $\frac { 18 } { a ^ { 2 } } x ^ { 2 } - \frac { 27 } { a } x + 3 \log | x |$
(1) $\frac { 9 } { a ^ { 2 } } x ^ { 2 } - \frac { 9 } { a } x + 3 \log | x |$
(2) $- \frac { 27 } { a ^ { 2 } } x ^ { 2 } + \frac { 18 } { a } x - 3 \log | x |$
(3) $- \frac { 27 } { a ^ { 2 } } x ^ { 2 } + \frac { 27 } { a } x - 3 \log | x |$
(4) $\frac { 27 } { a ^ { 2 } } x ^ { 2 } - \frac { 27 } { a } x + 3 \log | x |$
(5) $- \frac { 18 } { a ^ { 2 } } x ^ { 2 } + \frac { 27 } { a } x - 3 \log | x |$
Hence we obtain
$$S = \frac { \mathbf { W } } { \mathbf { X } } - 3 \log \mathbf { Y }$$

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2011 eju-math__session1