Graduation practice paper 4

Question 1. Consider the function $y=\ln(x^2+1)-x$. Which statement is true?

Question 2. What is the total number of vertical and horizontal asymptotes of the graph of $y=\dfrac{x+1}{\sqrt{x^2-4}}$?

Question 3.

The function $y=f(x)$ is continuous on $\mathbb{R}$ and its derivative has the following sign table.

$x$	$-\infty$		$-2$		$1$		$3$		$+\infty$
$f'(x)$		$+$	$0$	$-$	$0$	$-$	$0$	$+$

Which statement is true?

Question 4. $F(x)$ is an antiderivative of $f(x)=2x+\sin x$ with $F(0)=3$. Then $F(x)$ is

Question 5. The value of $\displaystyle\int_0^{\ln2}\big(e^x+1\big)^2\,dx$ is

Question 6. In the $Oxyz$ coordinate system, the orthogonal projection of the point $M(3, 3, -1)$ onto the plane $(P)\colon x+2y-2z-2=0$ is the point

Question 7. In the $Oxyz$ coordinate system, the angle between the line $d\colon\begin{cases}x=1+t\\ y=2\\ z=-t\end{cases}$ and the plane $(P)\colon x+y-3=0$ is

Question 8. The pyramid $S.ABCD$ has a square base of side $a$, $SA$ is perpendicular to the base and $SA=\dfrac{a\sqrt6}{2}$. The dihedral angle $[S, BD, A]$ measures

Question 9. The right prism $ABC.A'B'C'$ has a base that is a right-angled isosceles triangle with the right angle at $A$, $BC=2a$, and the angle between the line $A'B$ and the base is $60^\circ$. The volume of the prism is

Question 10.

The ages of $40$ members of a tai chi club are shown in the table.

Age	$[40, 50)$	$[50, 60)$	$[60, 70)$	$[70, 80)$	$[80, 90)$
Number of members	$6$	$10$	$14$	$7$	$3$

The median of this grouped data set (to two decimal places) is

Question 11. Box I contains $3$ white and $2$ black balls; box II contains $2$ white and $4$ black balls. A fair coin is tossed: on heads a ball is drawn at random from box I, on tails from box II. The probability of drawing a white ball is

Question 12. The water level (metres) in a harbour $t$ hours after midnight is modelled by $h(t)=3+2\sin\dfrac{\pi t}{6}$, $0\le t\le24$. In one day, the total time for which the water level is at least $4$ m is

Question 1. Consider the function $f(x)=(x^2-3)e^x$.

Question 2. On a straight road, car A travels at a constant $16$ m/s. At time $t=0$ car A passes car B, which is at rest, and car B starts in the same direction with velocity $v_B(t)=2t$ (m/s) for the first $12$ seconds, then keeps a constant $24$ m/s.

Question 3. In the $Oxyz$ coordinate system (units in metres), the floor of a warehouse is the plane $(Oxy)$. A wireless transmitter at $I(1, -2, 3)$ covers the solid ball with centre $I$ and radius $5$ m.

Question 4. In a company's mailbox, $30\%$ of emails are spam. An automatic filter flags $92\%$ of spam emails and wrongly flags $4\%$ of ordinary emails. An email is chosen at random.

Question 1. A window frame is a rectangle with a semicircle on top, the diameter of the semicircle being the width of the rectangle. The perimeter of the whole frame (the bottom, the two sides and the arc) is $6$ m. What is the greatest possible area of the window, in square metres (give your answer to two decimal places)?

Question 2. The regular square pyramid $S.ABCD$ has base edge $4$ and height $3$. Find the distance from $A$ to the plane $(SCD)$ (give your answer to two decimal places).

Question 3. In the $Oxyz$ coordinate system (units in metres), two drones fly in straight lines at constant speeds at the same time. After $t$ seconds the first drone is at $P(2t, t, 10)$ and the second at $Q(20-t, 2t, 4+t)$. What is the least distance between the two drones, in metres (give your answer to two decimal places)?

Question 4. Two fair dice are rolled. Given that the total score is at least $9$, find the probability that at least one die shows a $6$.

Question 5. A machine part is the solid of revolution formed by rotating the region bounded by $y=1+\sqrt x$, $y=0$, $x=0$, $x=4$ about the $x$-axis (units on each axis in decimetres). Find the volume of the part (in $\text{dm}^3$, to one decimal place).

Question 6. At the start of every month, Lan pays $5$ million đồng into a savings account. The interest rate is $0.5\%$ per month, and the interest at the end of each month is added to the balance. After at least how many months (including the interest for the last month) does the balance reach $100$ million đồng or more?