Mathematics Checkpoint — IB1 AA HL — Higher Difficulty

Maximum score: 10 marks

Instructions
Calculators are not permitted.
Answer all questions. Give exact answers unless otherwise stated.
In Section I, select exactly one answer for each question.
In Section II, show sufficient working to justify your answers.
All angles are measured in radians unless otherwise stated.

SECTION I. MULTIPLE CHOICE (3,0 marks)

Each question is worth 0,2 marks. Suggested time: 15 minutes.

Question 1 (0,2 marks). Let $a=\log_2 3$ and $b=\log_2 5$. Which expression is equal to $\log_6 45$?

Question 2 (0,2 marks). The equation $9^x-4\cdot3^x+3=0$ has two real solutions. What is their sum?

Question 3 (0,2 marks). What is the domain of the function

$$f(x)=\log_2\left(\dfrac{x-1}{3-x}\right)?$$

Question 4 (0,2 marks). For which values of $m$ is

$$x^2-2mx+m+2>0$$

true for every real number $x$?

Question 5 (0,2 marks). For which values of $k$ does the equation $kx^2-4x+4=0$ have exactly one distinct real solution?

Question 6 (0,2 marks). An arithmetic sequence has $S_{10}=150$ and $S_{20}=500$, where $S_n$ is the sum of its first $n$ terms. Find $u_{15}$.

Question 7 (0,2 marks). Find the sum of all positive terms of the arithmetic sequence

$$50,\ 46,\ 42,\ \ldots$$

Question 8 (0,2 marks). Find the coefficient of $x^3$ in the expansion of $(1-2x)^4(1+x)^2$.

Question 9 (0,2 marks). Find the constant term in the expansion of

$$\left(x^2-\dfrac{2}{x}\right)^6,\qquad x\ne0.$$

Question 10 (0,2 marks). Given that $\sin\theta=p$ and $0<\theta<\dfrac{\pi}{2}$, express $\tan\left(\dfrac{\pi}{2}+\theta\right)$ in terms of $p$.

Question 11 (0,2 marks). How many distinct solutions does $\sin 2x=\dfrac12$ have in the interval $0\leq x<2\pi$?

Question 12 (0,2 marks). Which option gives the range and the least positive period of $f(x)=-2\cos 3x+1$?

Question 13 (0,2 marks). The point $A(1;4)$ is reflected in the line $y=x+1$. Find the coordinates of its image.

Question 14 (0,2 marks). An account pays a nominal annual interest rate of $6\%$, compounded quarterly. Which expression gives its effective annual interest rate?

Question 15 (0,2 marks). The diagram shows the graph of a quadratic function $f$. Its only zeros are $-1$ and $3$.

[Đồ thị]

What is the domain of $g(x)=\ln(f(x))$?

SECTION II. SHORT WRITTEN RESPONSE (7,0 marks)

Suggested time: 30 minutes. Questions 1–6 are worth 0,5 marks each. Questions 7–10 are worth 1,0 mark each.

Question 1 (0,5 marks). Solve

$$\log_2(x-1)+\log_2(x+1)=3.$$

Question 2 (0,5 marks). Find all positive real solutions of

$$x^{\ln x}=e^9.$$

Question 3 (0,5 marks). The line $y=2x+k$ is tangent to the parabola $y=x^2-4x+7$. Find $k$ and the coordinates of the point of contact.

Question 4 (0,5 marks). An arithmetic sequence satisfies

$$u_4+u_{10}=50,\qquad u_{10}-u_4=18.$$

Find the sum of its first $12$ terms.

Question 5 (0,5 marks). Find the coefficient of $x^2$ in the expansion of

$$(1+x)^5(1-2x)^3.$$

Question 6 (0,5 marks). Solve

$$2\cos^2x+\sin x-1=0,\qquad 0\leq x<2\pi.$$

Question 7 (1,0 mark). Find all real values of $m$ for which

$$x^2-2(m+1)x+m^2-1=0$$

has two distinct positive real roots. Justify your conditions.

Question 8 (1,0 mark). Solve

$$(\log_2x)(\log_4x)-3\log_8x=2.$$

Question 9 (1,0 mark). A student saves $40$ USD at the end of the first month and increases the amount saved by $5$ USD each subsequent month. The savings earn no interest.

Find the smallest number of months needed for the total savings to reach at least $1000$ USD. Justify why your answer is the smallest possible.

Question 10 (1,0 mark). Let $n\geq2$ be an integer and let $a$ be a non-zero real number. The expansion of $(1+ax)^n$ begins

$$(1+ax)^n=1+12x+60x^2+\cdots.$$

a) Find $n$ and $a$.
b) Hence find the coefficient of $x^3$.

--------- HẾT ---------

Candidate name: ................................................ Candidate number: ........................

ANSWERS, WORKED SOLUTIONS AND MARKING GUIDE

A. GENERAL GUIDANCE

Award full credit for any mathematically valid alternative method.
In Section I, award 0,2 marks for each correct answer and no marks for an incorrect answer or more than one selected answer.
In Section II, award the indicated marks for correct reasoning and results.
Bold scores in opening rows are question totals, not additional marks.
Exact equivalent forms are acceptable.

B. ANSWERS AND MARKS

SECTION I. MULTIPLE CHOICE (3,0 marks)

Each correct answer earns 0,2 marks.

Explanations

Question 1. By change of base,

$$\log_6 45=\dfrac{\log_2(3^2\cdot5)}{\log_2(2\cdot3)}=\dfrac{2a+b}{1+a}.$$

Question 2. Put $t=3^x>0$. Then

$$t^2-4t+3=(t-1)(t-3)=0.$$

Thus $x=0$ or $x=1$, and the sum is $1$.

Question 3. The logarithm requires

$$\dfrac{x-1}{3-x}>0.$$

The numerator and denominator are both positive precisely when $1<x<3$. They cannot both be negative. Hence the domain is $(1;3)$.

Question 4. Completing the square gives

$$x^2-2mx+m+2=(x-m)^2+m+2-m^2.$$

The minimum value must be strictly positive:

$$m+2-m^2>0\iff(m-2)(m+1)<0\iff-1<m<2.$$

Question 5. If $k=0$, the equation is linear and has the unique solution $x=1$. If $k\ne0$, exactly one distinct real solution requires

$$\Delta=16-16k=0,$$

so $k=1$. Both values are valid.

Question 6. Let the first term be $u_1$ and the common difference be $d$. Then

$$2u_1+9d=30,\qquad 2u_1+19d=50.$$

Subtracting gives $d=2$, and hence $u_1=6$. Therefore

$$u_{15}=6+14\cdot2=34.$$

Question 7. The general term is $u_n=54-4n$. Therefore $u_n>0$ exactly for the positive integers $n\leq13$. The last positive term is $u_{13}=2$, so

$$S_{13}=\dfrac{13(50+2)}2=338.$$

Question 8. The relevant coefficients in $(1-2x)^4$ are $-8$, $24$ and $-32$ for $x$, $x^2$ and $x^3$, respectively. Since $(1+x)^2=1+2x+x^2$, the required coefficient is

$$-32+2\cdot24-8=8.$$

Question 9. A general term is

$$C_6^k(x^2)^{6-k}\left(-\dfrac2x\right)^k=C_6^k(-2)^kx^{12-3k}.$$

For a constant term, $12-3k=0$, so $k=4$. The term is

$$C_6^4(-2)^4=15\cdot16=240.$$

Question 10. Since $\theta$ is acute, $\cos\theta=\sqrt{1-p^2}$. Thus

$$\tan\left(\dfrac\pi2+\theta\right)=-\cot\theta=-\dfrac{\sqrt{1-p^2}}p.$$

Question 11. The interval for $2x$ is $[0;4\pi)$. The sine equation has two solutions in each complete period, giving four solutions. Explicitly,

$$x\in\left\{\dfrac\pi{12};\dfrac{5\pi}{12};\dfrac{13\pi}{12};\dfrac{17\pi}{12}\right\}.$$

Question 12. From $-1\leq\cos3x\leq1$, it follows that $-1\leq f(x)\leq3$. The least positive period is $\dfrac{2\pi}{3}$.

Question 13. Translate the plane down by one unit so that the mirror line becomes $y=x$. The point becomes $(1;3)$, whose reflection is $(3;1)$. Translating back gives $(3;2)$.

Equivalently, the midpoint of $(1;4)$ and $(3;2)$ is $(2;3)$, which lies on $y=x+1$, and the segment joining the two points has slope $-1$.

Question 14. The quarterly interest rate is $\dfrac{0{,}06}{4}$. After four quarters, the annual growth factor is

$$\left(1+\dfrac{0{,}06}{4}\right)^4.$$

Subtracting one and converting to a percentage gives option B.

Question 15. The logarithm is defined only when $f(x)>0$. From the graph, this occurs outside the two zeros, excluding the zeros themselves. Hence

$$D_g=(-\infty;-1)\cup(3;+\infty).$$

SECTION II. SHORT WRITTEN RESPONSE (7,0 marks)

Students using other correct methods receive full credit.