Graduation practice paper 2

Question 1. The function $y=f(x)$ has derivative $f'(x)=(x+1)(x-2)^2(x-4)$ for all $x\in\mathbb{R}$. The function has a local minimum at

Question 2. The oblique asymptote of the graph of $y=\dfrac{2x^2+3x-1}{x+2}$ is the line

Question 3. Which of the following functions is increasing on $\mathbb{R}$?

Question 4. On the interval $(0, +\infty)$, the indefinite integral of $f(x)=3^x+\dfrac1x$ is

Question 5. The value of $\displaystyle\int_0^3|x-2|\,dx$ is

Question 6. A geometric sequence $(u_n)$ has $u_1=2$ and $u_4=-54$. Its common ratio is

Question 7. In the $Oxyz$ coordinate system, the plane through $A(1, -1, 2)$ perpendicular to the line $d\colon\dfrac{x-1}{2}=\dfrac{y}{-1}=\dfrac{z+3}{3}$ has equation

Question 8. In the $Oxyz$ coordinate system, the sphere with centre $I(2, -1, 1)$ passing through $A(0, 1, 2)$ has equation

Question 9. $ABCD.A'B'C'D'$ is a cube. The angle between the lines $AB'$ and $BC'$ is

Question 10. The pyramid $S.ABCD$ has a rectangular base with $AB=3$, $AD=4$, $SA\perp(ABCD)$, and the angle between the line $SC$ and the base is $45^\circ$. The volume of the pyramid $S.ABCD$ is

Question 11.

The masses (kg) of $40$ boys in year 12 are shown in the table:

Mass (kg)	$[40, 50)$	$[50, 60)$	$[60, 70)$	$[70, 80)$	$[80, 90)$
Number of students	$4$	$9$	$15$	$10$	$2$

The mode of this grouped data set (to 2 decimal places) is

Question 12. An and Binh each throw one ball at a basket, independently of each other. An scores with probability $0.7$ and Binh with probability $0.6$. The probability that exactly one of them scores is

Question 1. Let $(C)$ be the graph of $y=\dfrac{x+3}{x-1}$.

Question 2. A train starts from rest at a station. During the first $20$ seconds its acceleration is $a(t)=2-0.1t$ (m/s²), where $t$ is the number of seconds since it started. Let $v(t)$ be the velocity (m/s) of the train.

Question 3. Choose axes $Oxyz$ (unit: metre) with the plane $(Oxy)$ as the ground. A vertical aerial mast has its foot at the origin $O$ and its top at $A(0, 0, 12)$. A cable is stretched straight from the top $A$ to an anchor point $B(4, 6, 0)$ on the ground.

Question 4. Box I contains $4$ red and $2$ blue marbles; box II contains $3$ red and $5$ blue marbles. One marble is taken at random from box I and put into box II, and then one marble is taken at random from box II.

Question 1. A hotel has $50$ rooms. At a price of $400$ thousand dong per room per day every room is taken; each increase of $20$ thousand dong leaves one more room empty. Each occupied room costs $40$ thousand dong per day to service. To make the daily profit as large as possible, what price per room (in thousand dong) should the hotel charge?

Question 2. The regular square pyramid $S.ABCD$ has base edge $6$ and height $4$. Find the distance from $A$ to the plane $(SCD)$.

Question 3. Choose axes $Oxyz$ (unit: metre). After $t$ seconds, the first drone is at $M(3t, 4t, 20)$ and the second drone is at $N(40-2t, 10+t, 30-t)$. Find the smallest distance between the two drones (to 1 decimal place).

Question 4. In a multiple-choice test, each question has $4$ options and exactly one is correct. For each question, the probability that Lan knows the answer is $0.6$; if she does not know it, she picks an option at random. Given that Lan answers a question correctly, find the probability that she really knew the answer (to 2 decimal places).

Question 5. A glass paperweight is a solid of revolution, formed by rotating the region bounded by the curve $y=4-x^2$, the $x$-axis and the $y$-axis (with $0\le x\le2$) about the $x$-axis (unit on the axes: cm). Find the volume of the paperweight in cm³ (to 1 decimal place).

Question 6. The number of bacteria in a culture after $t$ hours is $N(t)=N_0\cdot a^t$, where $N_0$ and $a>0$ are constants. After $2$ hours there are $300$ bacteria, and after $5$ hours there are $2400$. How many hours after the start does the culture contain $10\,000$ bacteria (to 1 decimal place)?