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Secondary 4

INSTRUCTIONS

Belilios Public School

Half-Yearly Examination, 2015-2016

MATHEMATICS Compulsory Part

2. This paper consists of THREE sections, A(1), A(2) and B.

Time Allowed: 1/2 hours

1. After the announcement of the start of the examination, you should first write your Name, Class Number

and Class in the spaces provided.

Maximum Marks: 105

Math Class: C / M

3. Attempt ALL questions in this paper. Write your answers in the spaces provided in this

Question-Answer Book.

5. Unless otherwise specified, all working must be clearly shown.

4. Graph paper and supplementary answer sheets will be supplied on request.

7. The diagrams in this paper are not necessarily drawn to scale.

6. Unless otherwise specified, numerical answers should be either exact or correct to 3 significant figures.

Answers written in the margins will not be marked.

(a) Solve the equation 92x-1= 27x+4.

(b) Solve the equation 5*-2 +5* - 130 = 0.

Answers written in the margins will not be marked.

Answers written in the margins will not be marked.

13. If a and are the roots of the equation 2x² − 3x + 1 = 0, find

(a) the value of a² + B

(b) a quadratic equation in x whose roots are o + 2 and ß² + 2.

Answers written in the margins will not be marked.

Answers written in the margins will not be marked.

SECTION B (35 marks)

Answer ALL questions in this section and write your answers in the spaces provided.

Express z in the form of a + bi, where a, b are real numbers.

If z is a real number, find

(i) the value of k,

where k is a real number.

Answers written in the margins will not be marked.

Answers written in the margins will not be marked.

In Figure 4, L₁ is the straight line passing through A(-3, 2)

and B(2, 7). L₂ is perpendicular to L₁ and passes through B

and C. L3 is a horizontal line which intersects L₁ and L2 at

A and C respectively.

(a) Find the equation of L₂ in the general form, and the

coordinates of C.

(b) D is a point on the line segment AC such that

area of AABD = 4 × area of ACBD.

(i) Find the coordinates of D.

(ii) Find the equation of BD in the general form.

Answers written in the margins will not be marked.

Answers written in the margins will not be marked.

Answers written in the margins will not be marked.

Answers written in the margins will not be marked.

As shown in Figure 5, in AABC, ZABC = 90°, AB = 8 cm and BC = 6 cm. The rectangle BDEF is

inscribed in AABC. Let DB = EF = x cm. Let the area of rectangle BDEF be S cm².

(a) Express the length of DE in terms of x.

(b) Express Sin terms of x.

(c) A student thinks that the area of BDEF is maximum when D is the mid-point of AB.

Do you agree? Explain your answer.

Answers written in the margins will not be marked.

Answers written in the margins will not be marked.

Answers written in the margins will not be marked.

Answers written in the margins will not be marked.

In a city, the number N of babies born in the t-th year after 2008 can be represented by the

following formula:

N = a - b(0.9)', where a and b are constants.

The number of babies born in 2010 was 43 480. The number of babies born in 2011 was 44 452.

(a) Find the values of a and b.

Find the number of babies born in 2013.

(c) In 2016, the government of the city will give a subsidy of $5 000 to each baby

born. Will the

total amount of subsidy in 2016 exceed $200 000 000? Explain your answer.

(Give the answers correct to the nearest integer if necessary.)

Answers written in the margins will not be marked.

END OF PAPER

Answers written in the margins will not be marked.

Answers written in the margins will not be marked.

Belilios Public School

Half-Yearly Examination, 2015-2016

S4 Mathematics Compulsory Part Marking Scheme

SECTION A(1) (35 marks)

1. Make y subject of the formula z =

2. Simplify

z(x - y) = xy

xz = zy + xy

xz = y(x + z)

and express your answer with positive indices.

4-1 (- 9)-3

1M for collecting like

IM for √√a = an

1M for a xa" = a

can be omitted

Answers written in the margins will not be marked.

SECTION A(1)

1. Make y the subject of the formula _z =

and express your answer with positive indices.

Answers written in the margins will not be marked.

(a) x²-x²y + 18x²,

(b) x²-x²y + 18x2² - x +y -18.

x² - xy + 18x²

= x²(x -y +18)

(b) x²-x²y + 18x2²-

18x² − x +y −18

= x²(x-y+18) - (x-y+18)

= (x² − 1)(x-y+18)

= (x+1)(x-1)(x − y +18)

(a) Selling price of the vase = $280(1-40%)

(b) Cost = $280:(1+75%)

As cost of the vase < selling price of the vase,

There will be a gain after selling the vase.

4. The marked price of a vase is $280. The vase is sold at a discount of 40% of the marked price.

(a) Find the selling price of the vase.

(b) If the marked price of the vase is 75% above its cost, determine whether there will be a gain or

a loss after selling the vase. Explain your answer.

1M (use (a))

5. Consider the graph of the quadratic function y = −3(x+5)² +7.

(a) State the direction of opening.

(b) Find coordinates of the vertex.

(c) Find the y-intercept of the graph.

(a) It opens downwards.

(b) Coordinates of the vertex are (-5,7)

(c) When x = 0, y = -68

The y-intercept = -68

(b) 2(x + 1)(x − 1) = x

Solve the following quadratic equations. (Leave your answers in surd form if necessary.)

(a) x²-12x - 45 = 0

(a) x² -12x-45= 0

(x −15)(x + 3) = 0

x = 15 or −3

(b) 2(x+1)(x - 1) = x

2(x²-1) = x

- (-1) ± √(-1)² - 4(2)(−2)

1± √√17

1M can be omitted

It is given that f(x) = ax² + 24x + 7 and ƒ(-1) = −13.

(b) If g(x) = f(-/-)₁

Find the value of a.

(i) find the algebraic representation of g(x).

(ii) Determine the nature of roots of the equation g(x) = 0.

a (-1)² + 24(-1) + 7 = -13

= x² + 12x +7

(ii) A = 12²-4(1)(7)

There are two distinct real roots.

It is given that the quadratic equation x(x-7)= k-7 has real roots.

(a) Find the range of values of k.

Solve the equation when k is the smallest positive integer in (a).

x² - 7x+7-k=0

" The quadratic equation x(x-7)=k-7 has real roots.

(-7)² - 4(1)(7-k) ≥0

49-28+4k ≥0

.". The range of values of k is k≥--

(b) When k is the smallest positive integer in (a), k = 1.

When k = 1, the corresponding quadratic equation is

x² -7x+7-1=0

x² - 7x+6=0

(x-1)(x-6)=0

x=1 or x = 6

1M for A≥0 +1A

SECTION A(2) (35 marks)

In Figure 1, the straight line L₁: y=mx+12 which passes through (2, 4), has the same x-intercept

with another straight line L₂. It is given that the y-intercept of the straight line L₂ is 1.

(a) Find the value of m.

Find the equation of L₂.

(c) Mary claims that L, is not perpendicular to L₂.

Do you agree? Explain your answer.

L₁ passes through (2, 4).

... By substituting (2, 4) into y = mx + 12, we have....

4 = m(2) + 12

(b) By substituting y = 0 into y = -4x + 12, we have

The x-intercept of L₁ is 3.

L₁ has the same x-intercept with L₂.

L₂ passes through (3, 0).

The equation of L₂ is

Slope of L₁ = -4.

Slope of L2

y-0= -(x-3)

Slope of L₁ × slope of L2

Therefore the two lines are not perpendicular.

Mary's claim is agreed.

1M either one

1 for correct explanation

10. Figure 2 shows the graph of y = -x² + bx + c. Find

the values of b and c,

the axis of symmetry of the graph,

the maximum value of y.

(a) From the graph,

(b) Let A and B be the points (-1, 2) and (0, 2) respectively.

Coordinates of the mid-point of AB =

y-intercept = 2

The graph passes through (−1, 2).

2 = −(−1)² + b(-1) + 2

(c) By substituting x =

The axis of symmetry is x =

The axis of symmetry passes through the mid-point of AB.

x=-(-)²-(--)) + ²

into y = -x² - x + 2, we have

The maximum value of y is

The coordinates of the vertex of the graph are

y=-x² + bx + c

1M (or use completing square)

11. Figure 3 is the graph of y=1.5*.

(c) Plot y =

(a) Using the graph above to find the approximate values of the following expressions.

(Give your answers correct to the nearest 0.1.)

(b) Solve the following equation graphically. (Give your answers correct to the nearest 0.2.)

for -4≤ x ≤ 4 on the Figure 3.

√1.58 = 1.55

From the graph, when x=1.6, y = 1.9 (corr. to the nearest 0.1).

√1.58 = 1.9

اليدين

From the graph, when x = -0.6, y = 0.8 (corr. to the nearest 0.1).

(c) correct shape

From the graph, the solution for 1.5* = 4 is 3.4 (corr. to the nearest 0.2)

With correct y-intercept

(a) Solve the equation 9²x-

(b) Solve the exponential equation. 5-2 +5* - 130=0.

(a) 9²x-¹=27x+4

32(2x-1)= 33(x +

4x - 2 = 3x + 12

(b) 5-2 +5 -130 = 0.

5.52 +5 = 130

5* (5−²+1)=130

13. If a and are the roots of the equation 2x² – 3x + 1 = 0, find

(a) the value of a² + ß²,

(b) a quadratic equation in x whose roots are a² + 2 and ß² + 2.

13. From the equation 2x²-3x+1=0, we have

a² + B² = (a + B)² - 2aß

Sum of roots = (a² + 2) + (ß² + 2)

=a² + B² +4

Product of roots = (a² +2)(ß² +2)

= (aß)² +2(a² + ß²) + 4

..The required equation is

i.e. 4x² -21x+27= 0.

Answers written in the margins will not be marked.

3. Factorize

(a) x²-x²y + 18x²,

(b) _x³−x²y + 18x² − x +y −18.

4. The marked price of a vase is $280. The vase is sold at a discount of 40% of the marked price.

(a) Find the selling price of the vase.

If the marked price of the vase is 75% above its cost, determine whether there will be a gain

or a loss after selling the vase. Explain your answer.

Answers written in the margins will not be marked.

SECTION B (35 marks)

Answer ALL questions in this section and write your answers in the spaces provided.

14. Let z =

Express z in the form of a + bi, where a, b are real numbers.

If z is a real number, find

(i) the value of k,

where k is a real number.

5+ki+10i+2ki²

1² - (2i)²

5+ (10+ k)i +2k(-1)

z is a real number.

15. In Figure 4, L₁ is the straight line passing through A(-3, 2) and

B(2, 7). L₂ is perpendicular to L₁ and passes through B and C.

L3 is a horizontal line which intersects L₁ and L2 at A and C

respectively.

(a) Find the equation of L₂ in the general form, and the

coordinates of C

(b) D is a point on the line segment AC such that

area of AABD = 4 × area of ACBD.

(i) Find the coordinates of D.

(ii) Find the equation of BD in the general form.

15(a) Slope of L₁

Slope of L₂

The equation of L₂ is

Let (a, 2) be the coordinates of C.

By substituting (a, 2) into x+y-9=0, we have

The coordinates of C are (7, 2).

Let (d, 2) be the coordinates of D.

AABD and ACBD have the same height.

Area of ABD AD

Area of CBD DC

.. The coordinates of D are (5,2).

The equation of BD is

y-2=-=(x-5)

3y-6=-5x+25

As shown in Figure 5, in AABC, ZABC = 90°, AB = 8 cm and BC = 6 cm. The rectangle BDEF is

inscribed in AABC. Let DB = EF = x cm. Let the area of rectangle BDEF be S cm².

(a) Express the length of DE in terms of x.

(b) Express S in terms of x.

(c) A student thinks that the area of BDEF is maximum when D is the mid-point of AB.

Do you agree? Explain your answer.

(c) S= 7x(8 - x)

ΔADE ~ ΔΕFC

(x²+6x) cm²

=-²/(x-4)² +12

(b) The area S of the rectangle BDEF = BD × DE

= -³2 (x² − 8x + 4²) + ²/(4²)

The maximum value of S is 12 when x = 4.

(corr. sides, As)

(x-4)² ≤0 for all x, ..S≤ 12 for all x.

The area of rectangle BDEF is maximum when D is the mid-point of AB

The student is correct.

17 In a city, the number N of babies born in the t-th year after 2008 can be represented by the

following formula:

N = a - b(0.9)', where a and b are constants.

The number of babies born in 2010 was 43 480. The number of babies born in 2011 was 44 452.

(a) Find the values of a and b.

(b) Find the number of babies born in 2013.

In 2016, the government of the city will give a subsidy of $5 000 to each baby born. Will the

total amount of subsidy in 2016 exceed $200 000 000? Explain your answer.

(Give the answers correct to the nearest integer if necessary.)

17(a) In 2010, t = 2010 - 2008 = 2, and N = 43 480.

43 480 = a - b(0.9)²

In 2011, t = 2011 - 2008 = 3, and N = 44 452.

44 452 = ab(0.9)³

44 452 a 0.729b............

972 = 0.081b

Substituting b= 12 000 into (i), we have

43 480 a 0.81(12 000)

(ii) - (i):

(b) From (a), N = 53 200 12 000(0.9)¹.

In 2013, t = 2013 - 2008 = 5, and

N = 53 200 12 000(0.9)³

= 46 114, cor. to the nearest integer

The required number of babies is 46 114.

In 2016, t = 2016 - 2008 = 8, and

N = 53 200 12 000(0.9)⁹

Total amount of subsidy

= $5 000(48 034)

= $240 170 000

= 48 034, cor. to the nearest integer

$240 170 000 > $200 000 000

The total amount of subsidy will exceed $200 000 000.

END OF PAPER

Answers written in the margins will not be marked.

Consider the graph of the quadratic function y = −3(x+5)² +7.

(a) State the direction of opening.

(b) Find coordinates of the vertex.

(c) Find the y-intercept of the graph.

6. Solve the following quadratic equations. (Leave your answers in surd form if necessary.)

(a) x² - 12x - 45 = 0

(b) 2(x + 1)(x − 1) = x

Answers written in the margins will not be marked.

Answers written in the margins will not be marked.

It is given that f(x) = ax² + 24x + 7 and ƒ(-1) = −13.

(a) Find the value of a.

(b) It is given that g(x) = f

Find the algebraic representation of g(x).

Determine the nature of the roots of the equation g(x) = 0.

Answers written in the margins will not be marked.

Answers written in the margins will not be marked.

It is given that the quadratic equation_x(x-7)= k-7 has real roots.

(a) Find the range of values of k.

(b) Solve the equation when k is the smallest positive integer in (a).

Answers written in the margins will not be marked.

Answers written in the margins will not be marked.

SECTION A(2) (35 marks)

In Figure 1, the straight line L₁: y=mx+12 which passes through (2, 4), has the same x-intercept

with another straight line L₂. It is given that the y-intercept of the straight line L2 is 1.

(a) Find the value of m.

Find the equation of L2.

Mary claims that L, is not perpendicular to L2.

Do you agree? Explain your answer.

Answers written in the margins will not be marked.

Answers written in the margins will not be marked.

10. Figure 2 shows the graph of y = -x²+bx+c. Find

the values of b and c.

the axis of symmetry of the graph,

the maximum value of y.

(-1,2)X 2 y = -x² + bx + C

Answers written in the margins will not be marked.

Answers written in the margins will not be marked.

11. Figure 3 is the graph of y = 1.5*.

(i) √√√₁.

(a) Using the graph above to find the approximate values of the following expressions.

(Give your answers correct to the nearest 0.1.)

(b) Solve the following equation graphically. (Give your answers correct to the nearest 0.2.)

for -4 ≤x≤4 on the Figure 3.

Answers written in the margins will not be marked.