kyotsu-test

QCourse1-I-Q1 Completing the square and sketching Max/min of a quadratic function on a closed interval with parameter View

Let us consider the maximum value $M$ and the minimum value $m$ of the quadratic function
$$f ( x ) = x ^ { 2 } - 2 ( a + 1 ) x + 2 a ^ { 2 }$$
over $0 \leqq x \leqq 2$, where $a$ is a constant and $0 \leqq a \leqq 3$.
(1) The coordinates of the vertex of the graph of $y = f ( x )$ are
$$\left( a + \mathbf { A } , a ^ { 2 } - \mathbf { B } a - \mathbf { C } \right) .$$
(2) For $\mathbf { D } \sim$ H in the following sentences, choose the correct answers from among choices (0) $\sim$ (9) below.
Let us find the maximum value $M$ and the minimum value $m$ according to the position of the axis of symmetry. We have that if $0 \leqq a < \mathbf { D }$, then
$$M = \mathbf { E } , \quad m = \mathbf { F } ;$$
if $\mathrm { D } \leqq a \leqq 3$, then
$$M = \mathbf { G } , \quad m = \mathbf { H } .$$
(0) 0
(1) 1
(2) 2
(3) 3
(4) $a ^ { 2 } - 2 a$
(5) $a ^ { 2 } - 2 a - 1$ (6) $2 a ^ { 2 }$ (7) $2 a ^ { 2 } - 2 a - 1$ (8) $2 a ^ { 2 } - 4 a$ (9) $2 a ^ { 2 } - 6 a + 3$
(3) Thus, $m$ is maximized at $a = \square$ and the value of $m$ then is $\square \mathbf { J }$. Also, $m$ is minimized at $a = \mathbf { K }$ and the value of $m$ then is $\mathbf { L M }$.

QCourse1-I-Q2 Discrete Probability Distributions Probability Computation for Compound or Multi-Stage Random Experiments View

Let us throw one dice three times, and let the number that comes up on the first throw be $a$, on the second throw be $b$, and on the third throw be $c$. Using these $a , b$ and $c$, we consider the quadratic function $f ( x ) = a x ^ { 2 } + b x + c$.
(1) The probability that $b = 4$ and that the quadratic equation $f ( x ) = 0$ has two different real solutions is $\frac { \mathbf { N } } { \mathbf { O P Q } }$.
(2) Let us find the probability that $f ( 10 ) > 453$.
The number of the cases of $( a , b , c )$ such that $f ( 10 ) > 453$ is as follows: when $a = 4$ and $b = 5$, it is $\mathbf { R }$; when $a = 4$ and $b = 6$, it is $\mathbf{S}$; when $a = 5$, it is $\mathbf { T U }$; when $a = 6$, it is $\mathbf{VW}$. Hence, the probability that $f ( 10 ) > 453$ is $\frac { \mathbf { X } } { \mathbf { Y } }$.

QCourse1-II-Q1 Indices and Surds Conjugate Surds and Sum Evaluation via Identities View

Let $x = \frac { \sqrt { 3 } + 1 } { \sqrt { 3 } - 1 }$ and $y = \frac { \sqrt { 6 } - \sqrt { 2 } } { \sqrt { 6 } + \sqrt { 2 } }$.
(1) We have $x = \mathbf { A } + \sqrt { \mathbf { B } }$ and $y = \mathbf { C } - \sqrt { \mathbf { C } }$. Hence we have
$$x + y = \mathbf { E } , \quad x y = \mathbf { F } , \quad \frac { 1 } { x ^ { 2 } } + \frac { 1 } { y ^ { 2 } } = \mathbf { G H } .$$
Also we have
$$5 \left( x ^ { 2 } - 4 x \right) + 3 \left( y ^ { 2 } - 4 y + 1 \right) = \square \mathbf { I J } .$$
(2) The values of the integers $m$ and $n$ such that $\frac { m } { x } + \frac { n } { y } = 4 + 4 \sqrt { 3 }$ are
$$m = \mathbf { K L } , \quad n = \mathbf { M } .$$

QCourse1-II-Q2 Discriminant and conditions for roots Intersection/tangency conditions between two curves View

Let us consider the three quadratic functions
$$f ( x ) = - x ^ { 2 } - 2 x + 1 , \quad g ( x ) = - x ^ { 2 } + 4 x , \quad h ( x ) = 2 x ^ { 2 } + a x + b$$
(1) When we denote the discriminant of the quadratic equation $h ( x ) - f ( x ) = 0$ by $D _ { 1 }$ and the discriminant of the quadratic equation $h ( x ) - g ( x ) = 0$ by $D _ { 2 }$, we have
$$D _ { 1 } = \mathbf { N } , \quad D _ { 2 } = \mathbf { O }$$
(for N and O, choose the correct answers from among choices (0) $\sim$ (5) below). (0) $a ^ { 2 } + 4 a - 3 b + 7$
(1) $a ^ { 2 } - 8 a - 12 b + 16$
(2) $a ^ { 2 } + 4 a - 12 b + 16$
(3) $a ^ { 2 } + 8 a + 12 b + 16$
(4) $a ^ { 2 } - 4 a + 12 b + 16$
(5) $a ^ { 2 } - 8 a - 3 b + 7$
(2) The values of $a$ and $b$ such that both of the two equations $f ( x ) = h ( x )$ and $g ( x ) = h ( x )$ have only one real solution are
$$a = \mathbf { P } , \quad b = \frac { \mathbf { Q } } { \mathbf{4} } .$$
In this case, the solution of $f ( x ) = h ( x )$ is $x = - \frac { \mathbf { S } } { \mathbf{T} }$ and the solution of $g ( x ) = h ( x )$ is $x = \frac { \mathbf { U } } { \mathbf{4} }$.
(3) Let $b = 3$. Then the range of the values of $a$ such that both $f ( x ) < h ( x )$ and $g ( x ) < h ( x )$ hold for any $x$ is $\square$ W (for $\square$ W, choose the correct answer from among choices (0) $\sim$ (5) below). (0) $- 2 - 2 \sqrt { 6 } < a < 10$
(1) $a < - 2 - 2 \sqrt { 6 } , 10 < a$
(2) $a < - 1 - \sqrt { 6 } , 10 < a$
(3) $- 2 < a < - 1 + \sqrt { 6 }$
(4) $- 2 < a < - 2 + 2 \sqrt { 6 }$
(5) $- 1 - \sqrt { 6 } < a < 10$

QCourse1-III Number Theory Divisibility and Divisor Analysis View

Answer the following questions.
(1) The prime factorization of 1400 is $\mathbf{A}^{\mathbf{B}} \cdot \mathbf{C}^{\mathbf{D}} \cdot \mathbf{E}$ (give the answers in the order A/C).
(2) The number of the divisors of 1400 is $\mathbf{FG}$.
(3) Let $a$ and $b$ be any two divisors of 1400 satisfying $1 < a < b$. There are $\mathbf { H }$ pairs $( a , b )$ such that $a$ and $b$ are relatively prime and $a b = 1400$. Among them, $a$ and $b$ such that $b - a$ is maximized are
$$a = \mathbf { I } , \quad b = \mathbf { J K L } .$$
(4) For $a = \square$ and $b = \mathbf{JKL}$, consider the equation
We can transform (1) into the following equation:
$$y = \mathbf { M N } x + \frac { \mathbf { O } } { \mathbf { Q } } x - \frac { \mathbf { P } } { \mathbf { Q } } .$$
Therefore, among the pairs of positive integers $x$ and $y$ that satisfy equation (1), the pair such that $x$ is minimized is
$$x = \mathbf { R } , \quad y = \mathbf { S T } .$$

QCourse1-IV Sine and Cosine Rules Find a side length using the cosine rule View

Let the quadrangle ABCD be a rhombus where the length of the sides is $\sqrt { 2 }$ and $\angle \mathrm { ABC } = 30 ^ { \circ }$.
(1) We have
$$\mathrm { AC } ^ { 2 } = \mathbf { A } - \mathbf { B } \sqrt { \mathbf { C } } , \quad \mathrm { BD } ^ { 2 } = \mathbf { E } + \mathbf{F} \sqrt{\mathbf{E}} .$$
Now, for any positive numbers $a$ and $b$, we have
$$( \sqrt { a } \pm \sqrt { b } ) ^ { 2 } = a + b \pm 2 \sqrt { a b } \quad \text { (double-sign correspondence). }$$
Using this formula, we obtain
$$\mathrm { AC } = \sqrt { \mathbf { G } } - \mathbf { H } , \quad \mathrm { BD } = \sqrt { \mathbf { I } } + \mathbf{I} . \mathbf { J } .$$
(2) Let us draw four circles, each centered on one vertex of rhombus ABCD, with the following conditions:
The radii of the circles centered on vertices A and C are of length $r$, and those centered on vertices B and D are of length $\sqrt { 2 } - r$.
Circles centered on opposite vertices (A and C, B and D) may touch each other but may not intersect.
Let us denote the area of the region common to rhombus ABCD and these four circles by $S$. We have
$$S = \pi \left( r ^ { 2 } - \frac { \sqrt { \mathbf { K } } } { \mathbf { L } } r \right)$$
where the range of $r$ is
$$\sqrt { \mathbf { O } } - \frac { \sqrt { \mathbf { P } } + \mathbf { Q } } { \mathbf { R } } \leqq r \leqq \frac { \sqrt { \mathbf { S } } - \square \mathbf { T } } { \square \mathbf { U } }$$
Hence $S$ is minimized when $r = \frac { \sqrt { \mathbf { V } } } { \mathbf { W } }$, and the value of $S$ then is $\frac { \mathbf{X} } { \mathbf { Y } } \pi$.

QCourse2-I-Q1 Completing the square and sketching Max/min of a quadratic function on a closed interval with parameter View

Let us consider the maximum value $M$ and the minimum value $m$ of the quadratic function
$$f ( x ) = x ^ { 2 } - 2 ( a + 1 ) x + 2 a ^ { 2 }$$
over $0 \leqq x \leqq 2$, where $a$ is a constant and $0 \leqq a \leqq 3$.
(1) The coordinates of the vertex of the graph of $y = f ( x )$ are
$$\left( a + \mathbf { A } , a ^ { 2 } - \mathbf { B } a - \mathbf { C } \right) .$$
(2) For $\mathbf { D } \sim$ H in the following sentences, choose the correct answers from among choices (0) $\sim$ (9) below.
Let us find the maximum value $M$ and the minimum value $m$ according to the position of the axis of symmetry. We have that if $0 \leqq a < \mathbf { D }$, then
$$M = \mathbf { E } , \quad m = \mathbf { F } ;$$
if $\mathrm { D } \leqq a \leqq 3$, then
$$M = \mathbf { G } , \quad m = \mathbf { H } .$$
(0) 0
(1) 1
(2) 2
(3) 3
(4) $a ^ { 2 } - 2 a$
(5) $a ^ { 2 } - 2 a - 1$ (6) $2 a ^ { 2 }$ (7) $2 a ^ { 2 } - 2 a - 1$ (8) $2 a ^ { 2 } - 4 a$ (9) $2 a ^ { 2 } - 6 a + 3$
(3) Thus, $m$ is maximized at $a = \square$ and the value of $m$ then is $\square$ J. Also, $m$ is minimized at $a = \mathbf { K }$ and the value of $m$ then is $\mathbf { L M }$.

QCourse2-I-Q2 Discrete Probability Distributions Probability Computation for Compound or Multi-Stage Random Experiments View

Let us throw one dice three times, and let the number that comes up on the first throw be $a$, on the second throw be $b$, and on the third throw be $c$. Using these $a , b$ and $c$, we consider the quadratic function $f ( x ) = a x ^ { 2 } + b x + c$.
(1) The probability that $b = 4$ and that the quadratic equation $f ( x ) = 0$ has two different real solutions is $\frac { \mathbf { N } } { \mathbf { O } \mathbf { P Q } }$.
(2) Let us find the probability that $f ( 10 ) > 453$.
The number of the cases of $( a , b , c )$ such that $f ( 10 ) > 453$ is as follows: when $a = 4$ and $b = 5$, it is $\mathbf { R }$; when $a = 4$ and $b = 6$, it is $\mathbf{S}$; when $a = 5$, it is $\mathbf{TU}$; when $a = 6$, it is $\mathbf{VW}$. Hence, the probability that $f ( 10 ) > 453$ is $\frac { \mathbf { X } } { \mathbf { Y } }$.

QCourse2-II-Q1 Sequences and Series Evaluation of a Finite or Infinite Sum View

The sequence $\left\{ a _ { n } \right\}$ is defined by
$$a _ { 1 } = \frac { 2 } { 9 } , \quad a _ { n } = \frac { ( n + 1 ) ( 2 n - 3 ) } { 3 n ( 2 n + 1 ) } a _ { n - 1 } \quad ( n = 2,3,4 , \cdots ) .$$
We are to find the general term $a _ { n }$ and the infinite sum $\sum _ { n = 1 } ^ { \infty } a _ { n }$.
(1) For A $\sim$ E in the following sentences, choose the correct answers from among (0) $\sim$ (9) below.
First, when we set $b _ { n } = \frac { n + 1 } { 3 ^ { n } a _ { n } }$ and express $\frac { b _ { n } } { b _ { n - 1 } }$ in terms of $n$, we have
$$\frac { b _ { n } } { b _ { n - 1 } } = \frac { \mathbf { A } } { \mathbf { B } } \cdot \frac { a _ { n - 1 } } { a _ { n } } = \frac { \mathbf { C } } { \mathbf { D } }$$
From this equation, we have
$$a _ { n } = \frac { n + 1 } { 3 ^ { n } ( \mathbf { E } ) ( 2 n + 1 ) } .$$
(0) $n - 1$
(1) $n$
(2) $n + 1$
(3) $2 n - 1$
(4) $2 n + 1$
(5) $2 n - 3$ (6) $2 n + 3$ (7) $3 n - 1$ (8) $3 n$ (9) $3 n + 1$
(2) Next, let $c _ { n } = \frac { 1 } { 3 ^ { n } ( 2 n + 1 ) } ( n = 0,1,2 , \cdots )$. When we set $a _ { n } = A c _ { n - 1 } + B c _ { n }$, we see that $A = \frac { \mathbf { F } } { \mathbf { G } }$ and $B = \frac { \mathbf { H I } } { \mathbf { G } }$. Using this result to find $S _ { n } = \sum _ { k = 1 } ^ { n } a _ { k }$, we have
$$S _ { n } = \frac { \mathbf { K } } { \mathbf { L } } \left( \mathbf { M } \right)$$
Hence we obtain
$$\sum _ { n = 1 } ^ { \infty } a _ { n } = \lim _ { n \rightarrow \infty } S _ { n } = \frac { \mathbf { N } } { \mathbf { O } }$$

QCourse2-II-Q2 Circles Tangent Lines and Tangent Lengths View

Let $C$ be a circle with a radius of 4, centered at the point $( 5,0 )$ on the $x$-axis.
(1) If $\mathrm { P } ( p , q )$ is a point on circle $C$, then
$$p ^ { 2 } - \mathbf { PQ } p + q ^ { 2 } + \mathbf { R } = 0 .$$
Also, the equation of the tangent to circle $C$ at point $\mathrm { P } ( p , q )$ is
$$( p - \mathbf { S } ) x + q y = \mathbf { T } p - \mathbf { U } .$$
(2) Let us draw a line tangent to circle $C$ from point $\mathrm { A } ( 0 , a )$ on the $y$-axis, where $a \geqq 0$, and let $\mathrm { P } ( p , q )$ be the tangent point.
The length of the segment AP is minimized at $a = \mathbf { V }$, and the length in this case is $\mathbf { W }$.
Furthermore, the two tangents to circle $C$ from point A are orthogonal when the length of AP is $\mathbf { X }$. In this case, the value of $a$ is $a = \sqrt { \mathbf { Y } }$.

QCourse2-III Stationary points and optimisation Construct or complete a full variation table View

Given the function
$$f ( x ) = x ^ { 3 } - 3 a x ^ { 2 } - 3 ( 2 a + 1 ) x + a + 2 ,$$
answer the following questions.
(1) For $\mathbf { G }$ $\sim$ $\mathbf { K }$, choose the correct answers from among (0) $\sim$ (5) below, and for the other $\square$, enter the correct numbers.
Since
$$f ^ { \prime } ( x ) = \mathbf { A } ( x - \mathbf { B } a - \mathbf { C } ) ( x + \mathbf { D } ) ,$$
we see that
(i) when $a > \mathbf { EF }$, $f ( x )$ is $\mathbf { G }$ at $x = - \square \mathbf { D }$ and is $\square$ H at $x =$ $\square$ B $a +$ $\square$ C;
(ii) when $a =$ $\square$ EF, $f ( x )$ is always $\square$ I;
(iii) when $a < \mathbf{EF}$, $f ( x )$ is $\square$ J at $x = -$ $\square$ D and is $\square$ K at $x =$ $\square$ B $a +$ $\square$ C. (0) locally maximized
(1) locally minimized
(2) increasing
(3) decreasing
(4) maximized
(5) minimized
(2) When we express the minimum value $m$ of $f ( x )$ over the range $- 1 \leqq x \leqq 1$ in terms of $a$, we have that
(i) when $a \geqq \mathbf { L }$, $m = \mathbf { MN } a$;
(ii) when $\mathbf { OP } \leqq a < \mathbf { L }$, $m = \mathbf { QR } \left( a ^ { 3 } + \mathbf { S } a ^ { 2 } + \mathbf { T } a \right)$;
(iii) when $a < \mathbf{OP}$, $m = \mathbf { U } a + \mathbf { V }$.
(3) The value of $m$ in (2) is maximized at $a = \frac { - \mathbf { W } + \sqrt { \mathbf { X } } } { \square \mathbf { Y } }$.

QCourse2-IV Integration by Parts Area or Volume Computation Requiring Integration by Parts View

Consider the two functions
$$y = x \log a x , \tag{1}$$ $$y = 2 x - 3 , \tag{2}$$
where $a > 0$, and where $\log$ is the natural logarithm.
(1) Let us find $a$ such that the graph of (1) is tangent to the graph of (2).
The equation of the tangent to the graph of (1) at the point $( t , t \log a t )$ is $\mathbf { A }$ (for A, choose the correct answer from among choices (0) $\sim$ (3) below). (0) $y = ( \log a t + 1 ) x - t$
(1) $y = ( \log a t + a ) x - t$
(2) $y = ( a \log t + 1 ) x + t$
(3) $y = ( a \log t + a ) x + t$
Hence, the graph of (1) is tangent to the graph of (2) when $a = \frac { e } { \square \mathbf{B} }$, and the coordinates of the tangent point are ( $\mathbf { C }$, $\mathbf { D }$ ).
(2) When $a = \frac { e } { \mathbf{B} }$, function (1) is minimized at $x = \square e ^ { - \mathbf { F } }$, and in this case the minimum value is $- \mathbf { G } \cdot e ^ { - \mathbf { H } }$.
(3) When $a = \frac { e } { \mathbf{B} }$, let us find the area $S$ of the region bounded by the graphs of (1) and (2) and the $x$-axis.
For the following indefinite integral, we have
$$\int x \log a x \, d x = \square + C , \quad \text { where } C \text { is an integral constant }$$
(for I, choose the correct answer from among (0) $\sim$ (3) below). (0) $\frac { 1 } { 2 } x ^ { 2 } \log a x - \frac { 1 } { 2 } x ^ { 2 }$
(1) $2 x ^ { 2 } \log a x - 2 x ^ { 2 }$
(2) $\frac { 1 } { 2 } x ^ { 2 } \log a x - \frac { 1 } { 4 } x ^ { 2 }$
(3) $2 x ^ { 2 } \log a x - 4 x ^ { 2 }$
Hence we obtain
$$S = \frac { \mathbf { J } } { \mathbf { K } }$$

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2018 eju-math__session2