LemonmathSelf-study sheet

Grade 12 · Lesson 7.9

Graduation practice paper 1

Name: Date: Time spent: min

Questions

90 minutes, no notes. Part I: circle one letter; Part II: write T or F for each statement; Part III: write the answer in the boxes (a minus sign and a decimal point each take a box).

Part I. Multiple choice

Question 1.

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

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

The local maximum value of the function is

2

.B.

1

.C.

-3

.D.

0

Question 2. The horizontal asymptote of the graph of

y=\dfrac{4x+1}{2x-6}

is the line

y=4

.B.

x=3

.C.

y=2

.D.

y=-\dfrac16

Question 3. On which interval is the function

y=x^3-3x^2+2

decreasing?

(0, 2)

.B.

(-\infty, 0)

.C.

(2, +\infty)

.D.

(-2, 0)

Question 4. The indefinite integral of

f(x)=2\cos x-3x^2

-2\sin x-x^3+C

.B.

2\sin x-6x+C

.C.

2\sin x-x^3+C

.D.

2\sin x-3x^3+C

Question 5. Given

\displaystyle\int_1^3 f(x)\,dx=4

and

\displaystyle\int_1^3 g(x)\,dx=-1

, the value of

\displaystyle\int_1^3\left[2f(x)-3g(x)\right]dx

5

.B.

11

.C.

7

.D.

9

Question 6. In the

Oxyz

coordinate system, let

A(2, -1, 3)

and

B(4, 3, -1)

. The midpoint of the line segment

AB

(1, 2, -2)

.B.

(6, 2, 2)

.C.

(2, 4, -4)

.D.

(3, 1, 1)

Question 7. In the

Oxyz

coordinate system, the plane through the three points

M(2, 0, 0)

N(0, -3, 0)

P(0, 0, 6)

has equation

3x-2y+z-6=0

.B.

3x+2y+z-6=0

.C.

2x-3y+6z-6=0

.D.

3x-2y+z+6=0

Question 8. In the

Oxyz

coordinate system, the sphere

(S)\colon(x-1)^2+(y+2)^2+(z-3)^2=16

has centre

I

and radius

R

given by

I(-1, 2, -3)

R=4

.B.

I(1, -2, 3)

R=16

.C.

I(1, -2, 3)

R=4

.D.

I(-1, 2, -3)

R=16

Question 9. The pyramid

S.ABCD

has a square base of side

a

SA\perp(ABCD)

and

SA=a\sqrt6

. The angle between the line

SC

and the plane

(ABCD)

30^\circ

.B.

45^\circ

.C.

60^\circ

.D.

90^\circ

Question 10.

The daily home study time (minutes) of $40$ students is shown in the table:

Time (minutes)	$[0, 20)$	$[20, 40)$	$[40, 60)$	$[60, 80)$	$[80, 100)$
Number of students	$5$	$12$	$15$	$6$	$2$

The mean of this grouped data set is

34

minutes.B.

50

minutes.C.

44

minutes.D.

8

minutes.

Question 11. Events

A

and

B

satisfy

P(A)=0.6

P(B)=0.5

and

P(A\cap B)=0.3

. Then

P(A\mid B)

equals

0.5

.B.

0.15

.C.

0.3

.D.

0.6

Question 12. The solution set of the inequality

3^{2x-1}\le27

(-\infty, 2)

.B.

(-\infty, 2]

.C.

[2, +\infty)

.D.

(-\infty, 14]

Part II. True or false

Question 1. Let

f(x)=x^3-6x^2+9x+1

a) $f'(x)=3x^2-12x+9$ .	T / F
b) The function is increasing on the interval $(1, 3)$ .	T / F
c) The local maximum value of the function is $5$ .	T / F
d) On the interval $[0, 4]$ , the maximum value of the function is $5$ and it is reached at exactly one point.	T / F

Question 2. A model rocket is launched vertically from the ground. During the first

4

seconds its velocity is

v(t)=12t-3t^2

(m/s), where

t

is the number of seconds since launch. Let

h(t)

be the height (m) of the rocket at time

t

0\le t\le4

a) The velocity of the rocket at $t=1$ second is $9$ m/s.	T / F
b) The velocity of the rocket keeps increasing throughout the first $4$ seconds.	T / F
c) $h(t)=6t^2-t^3$ .	T / F
d) The greatest height the rocket reaches during the first $4$ seconds is $32$ m.	T / F

Question 3. In the

Oxyz

coordinate system, let

A(1, 0, 2)

B(3, 2, 0)

and the plane

(P)\colon 2x+y-z-4=0

a) $\overrightarrow{AB}=(2, 2, -2)$ .	T / F
b) The midpoint $I$ of the line segment $AB$ lies in the plane $(P)$ .	T / F
c) The perpendicular bisector plane of $AB$ has equation $x+y-z=0$ .	T / F
d) The angle between the line $AB$ and the plane $(P)$ is greater than $60^\circ$ .	T / F

Question 4. A factory has two production lines. Line I makes

60\%

of the products and line II makes

40\%

. The defect rate is

3\%

on line I and

5\%

on line II. One product of the factory is chosen at random.

a) The probability that the product is defective, given that it was made on line I, is $0.03$ .	T / F
b) The probability that the product is defective is $0.04$ .	T / F
c) The probability that the product was made on line I and is defective is $0.03$ .	T / F
d) Given that the chosen product is defective, the probability that it was made on line II is greater than $0.5$ .	T / F

Part III. Short answer

Question 1. A rectangular poster has a printed area that is a rectangle of area

384

cm². The top and bottom margins are

3

cm each and the left and right margins are

2

cm each. What is the smallest possible area of the poster, in cm²?

Question 2. The cuboid

ABCD.A'B'C'D'

has

AB=3

AD=4

AA'=12

. Find the distance from

A

to the plane

(A'BD)

(give your answer to 2 decimal places).

Question 3. Choose axes

Oxyz

(unit: metre) with the plane

(Oxy)

as the ground. A zip line is stretched straight from the point

A(0, 0, 20)

on a high platform to the point

B(30, 40, 5)

. Find the angle between the zip line and the ground (in degrees, to the nearest degree).

Question 4. A fire alarm system has three sensors that work independently. When there is a fire, the sensors detect it with probabilities

0.9

0.8

and

0.5

respectively. When there is a fire, find the probability that exactly two sensors detect it.

Question 5. Find the area of the region enclosed by the parabola

y=x^2-2x

and the line

y=x

Question 6. A freshly made cup of tea is at

95^\circ

C in a room at

25^\circ

C. Its temperature after

t

minutes is

T(t)=25+70\cdot e^{-kt}

(

^\circ

C), where

k

is a positive constant. After

5

minutes the tea is at

60^\circ

C. How many minutes after it was made is the tea at

30^\circ

C (to the nearest minute)?

Score: ........ / 10

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