Graduation practice paper 3

Question 1. The function $y=f(x)$ has derivative $f'(x)=x^2(x-1)(x+2)^3$ for all $x\in\mathbb{R}$. How many turning points does the function have?

Question 2. What is the total number of vertical and horizontal asymptotes of the graph of $y=\dfrac{2x^2-3x+1}{x^2-1}$?

Question 3.

The function $y=f(x)$ has the following table of variations.

$x$	$-\infty$		$-1$		$3$		$+\infty$
$f'(x)$		$+$	$0$	$-$	$0$	$+$
$f(x)$	$-\infty$	$\nearrow$	$2$	$\searrow$	$-4$	$\nearrow$	$+\infty$

How many real solutions does the equation $2f(x)+5=0$ have?

Question 4. The family of antiderivatives of $f(x)=e^{2x}-\dfrac1x$ on the interval $(0, +\infty)$ is

Question 5. Given $\displaystyle\int_0^2 f(x)\,dx=3$ and $\displaystyle\int_0^5 f(x)\,dx=7$, the value of $\displaystyle\int_2^5\big[2f(x)-1\big]\,dx$ is

Question 6. In the $Oxyz$ coordinate system, let $A(1, 2, -1)$ and $B(3, 0, 3)$. The perpendicular bisector plane of the segment $AB$ has equation

Question 7. In the $Oxyz$ coordinate system, consider the sphere $(S)\colon x^2+y^2+z^2-4x+2y-6z-2=0$ and the plane $(P)\colon 2x-y+2z+1=0$. Which statement is true?

Question 8. The pyramid $S.ABCD$ has a square base of side $a$, $SA$ is perpendicular to the base and $SA=a\sqrt2$. The angle between the line $SC$ and the plane $(SAB)$ is

Question 9.

The daily self-study time of $40$ students is shown in the table.

Time (minutes)	$[0, 10)$	$[10, 20)$	$[20, 30)$	$[30, 40)$	$[40, 50)$
Number of students	$5$	$12$	$15$	$6$	$2$

The mode of this grouped data set is

Question 10.

The lengths (cm) of $10$ leaves are shown in the table.

Length (cm)	$[4, 6)$	$[6, 8)$	$[8, 10)$
Number of leaves	$3$	$5$	$2$

The standard deviation of this grouped data set is

Question 11. Events $A$ and $B$ satisfy $P(A)=0.5$, $P(B)=0.4$ and $P(A\cap B)=0.12$. The value of $P(A\mid B)$ is

Question 12. The solution set of the inequality $\log_{\frac12}(x-1)\ge-2$ is

Question 1. Consider the function $f(x)=\dfrac{x^2-3x+6}{x-1}$.

Question 2. A tank holds $200$ litres of water. From time $t=0$ a tap fills it at the rate $v(t)=16+6t-t^2$ (litres per minute), $0\le t\le8$, with $t$ in minutes.

Question 3. In the $Oxyz$ coordinate system (units in metres), a drone flies in a straight line at constant speed $3$ m/s from the point $M(1, 2, 3)$ in the direction of $\vec u=(2, -1, 2)$. The plane $(Oxy)$ is the ground. The control station is at $K(7, 2, 6)$.

Question 4. A factory has two production lines. Line I makes $60\%$ of the products with a defect rate of $3\%$; line II makes $40\%$ of the products with a defect rate of $5\%$. A product of the factory is chosen at random.

Question 1. A workshop makes closed cylindrical cans with volume $1000\ \text{cm}^3$. The material for the top and bottom costs twice as much per $\text{cm}^2$ as the material for the curved surface. What base radius, in centimetres, makes the cost of material as small as possible (give your answer to one decimal place)?

Question 2. The pyramid $S.ABCD$ has a square base of side $2$, $SA$ is perpendicular to the base and $SA=2$. Find the distance between the skew lines $BD$ and $SC$ (give your answer to two decimal places).

Question 3. In the $Oxyz$ coordinate system (units in metres), the ground is the plane $(Oxy)$. A zip line is a taut straight cable from the point $A(0, 0, 20)$ on a tower to the point $B(40, 30, 5)$ on a lower post. Find the angle between the cable and the ground (in degrees, to one decimal place).

Question 4. A box contains $5$ red balls and $4$ blue balls. Two balls are taken out at random at the same time. Given that at least one of them is red, find the probability that both are red (give your answer to two decimal places).

Question 5. In a coordinate plane (units in metres), a flower bed is the region bounded by the parabola $y=\dfrac{x^2}{2}$ and the line $y=x+4$. Planting costs $250\,000$ đồng per square metre. Find the cost of planting the whole bed (in millions of đồng).

Question 6. A cup of hot water is left in a room at $25^\circ\text{C}$. The temperature of the water after $t$ minutes is $T(t)=25+70e^{-kt}$ ($^\circ$C), where $k>0$ is a constant. After $10$ minutes the water is at $60^\circ\text{C}$. How many minutes after the start does the water reach $35^\circ\text{C}$ (give your answer to one decimal place)?