Linear Law: Additional Mathematics Tingkatan 4 questions

Section A: objective questions

12 marks

Choose the correct answer. Each question carries 1 mark.

Which equation gives a straight line when y is plotted against x?

y = x² + 1
y = 2x − 3
y = 4/x
y = √x

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A straight line has the form y = mx + c, and y = 2x − 3 has gradient 2 and y-intercept −3. The other three give curves.

For the teacher: Remembering, low difficulty, standard 6.1.1

Textbook p. 154

In the linear equation Y = mX + c, what does c represent?

The gradient of the line
The value of X when Y = 0
The Y-intercept
The value of Y when X = 1

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c is the value of Y where the line crosses the Y-axis (X = 0). The gradient is m.

For the teacher: Remembering, low difficulty, standard 6.1.1

Textbook p. 154

Which statement describes a line of best fit?

A straight line with the points that are not on it spread evenly on both sides
A line that joins every point in turn
A curve that passes through all the points
A line that starts from the origin and passes through the first point

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A line of best fit is straight and passes through or near most points, with the others spread evenly on both sides. It does not have to start at the origin.

For the teacher: Understanding, low difficulty, standard 6.1.2

Textbook p. 156

A line of best fit passes through (0, 3) and (4, 11). What is its equation?

y = 3x + 2
y = 4x + 3
y = x + 7
y = 2x + 3

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m = (11 − 3) ÷ (4 − 0) = 2 and the y-intercept is 3, so y = 2x + 3. y = 4x + 3 uses 4 (the x-value) instead of the gradient.

For the teacher: Applying, low difficulty, standard 6.1.3

Textbook p. 157

The equation y = 3x² + 5x is changed to the linear form Y = mX + c. Which of the following is correct?

Y = y/x, X = x, m = 3, c = 5
Y = y, X = x², m = 3, c = 5
Y = xy, X = x, m = 5, c = 3
Y = y/x, X = x², m = 5, c = 3

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Divide both sides by x: y/x = 3x + 5. So Y = y/x, X = x, m = 3 and c = 5. Plotting y against x² does not work because of the 5x term.

For the teacher: Applying, medium difficulty, standard 6.2.1

Textbook p. 162

The equation y = p/x + q, where p and q are constants, gives a straight line when you plot

y against x
y against p/x
y against 1/x
xy against x

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y = p(1/x) + q, so Y = y, X = 1/x, gradient p and intercept q. X must not contain the constant p.

For the teacher: Understanding, low difficulty, standard 6.2.1

Textbook p. 162

The variables are related by y = pq^x. A graph of log₁₀ y against x is a straight line. What is its gradient?

log₁₀ p
log₁₀ q
q
p

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log₁₀ y = x log₁₀ q + log₁₀ p. The gradient is log₁₀ q and the intercept is log₁₀ p. The gradient is not q itself because logarithms were taken.

For the teacher: Understanding, medium difficulty, standard 6.2.1

Textbook p. 162

A straight line graph of log₁₀ y against log₁₀ x has gradient 3 and intercept 0.5 on the vertical axis. Which relation between y and x does this show?

y = 0.5x³
y = 3x^0.5
y = 10^0.5 x + 3
y = 3.16x³

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log₁₀ y = 3 log₁₀ x + 0.5 gives y = 10^0.5 × x³ = 3.16x³. The gradient is the power of x and the intercept is log₁₀ of the coefficient.

For the teacher: Analysing, high difficulty, standard 6.2.1

Textbook p. 165

Diagram 1 shows the line of best fit when y is plotted against x². What is the value of y when x = 3?

Straight line on a graph of y against x squared passing through the points (0, 3) and (4, 11) — Diagram 1

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Gradient = (11 − 3) ÷ 4 = 2, intercept 3, so y = 2x² + 3. When x = 3, y = 2(9) + 3 = 21. Using x = 3 in place of x² gives 9.

For the teacher: Applying, low difficulty, standard 6.1.4

Textbook p. 159

A graph of xy against x² is a straight line for the equation y = 4x + 3/x. What is its gradient?

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Multiply by x: xy = 4x² + 3. So Y = xy, X = x², gradient 4 and intercept 3.

For the teacher: Applying, medium difficulty, standard 6.2.1

Textbook p. 163

A graph of log₁₀ y against x cuts the vertical axis at 0.6. What is the value of y when x = 0?

0.6
6
0.25
3.98

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When x = 0, log₁₀ y = 0.6, so y = 10^0.6 = 3.98. 0.6 is log₁₀ y, not y.

For the teacher: Understanding, low difficulty, standard 6.1.4

Textbook p. 160

The variables are related by y = px² + qx. The line of best fit of y/x against x has gradient 2 and y/x-intercept −3. What is the value of y when x = 2?

2
1
−2
−1

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y/x = 2x − 3. When x = 2, y/x = 1, so y = 2 × 1 = 2. The answer 1 is y/x, not y.

For the teacher: Evaluating, medium difficulty, standard 6.2.1

Textbook p. 163

Section B: structured questions

31 marks

Answer all questions. Show your working.

Convert each equation to the linear form Y = mX + c, where h, k, a, b are constants. Identify Y, X, m and c.

[2 marks] (a)y = hx² + kx

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y/x = hx + k [1]1
Y = y/x, X = x, m = h, c = k [1]1

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[2 marks] (b)y = 5/x + k

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y = 5(1/x) + k [1]1
Y = y, X = 1/x, m = 5, c = k [1]1

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[2 marks] (c)y = ab^x

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log₁₀ y = x log₁₀ b + log₁₀ a [1]1
Y = log₁₀ y, X = x, m = log₁₀ b, c = log₁₀ a [1]1

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For the teacher: Applying, low difficulty, standard 6.2.1

Textbook p. 162

The variables x and y are related by y = px² + qx, where p and q are constants. The table shows values from an experiment.

x	1	2	3	4	5
y	−1	2	9	20	35

Five points on a straight line of y/x against x, rising from −1 at x = 1 to 7 at x = 5 — Diagram 2: y/x against x

[1 mark] (a)Write the equation in the form Y = mX + c with Y = y/x and X = x.

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y/x = px + q [1]1

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[2 marks] (b)Diagram 2 shows the line of best fit of y/x against x. Find the gradient and the y/x-intercept.

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Gradient = (7 − (−1)) ÷ (5 − 1) = 2 [1]1
Intercept: −1 = 2(1) + c, c = −3 [1]1

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[1 mark] (c)Hence state the value of p and of q.

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p = 2, q = −3 [1]1

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[1 mark] (d)Find y when x = 6.

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y = 2(36) − 3(6) = 54 [1]1

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For the teacher: Applying, medium difficulty, standard 6.2.1

Textbook p. 163

The variables x and y are related by y = pq^x, where p and q are constants. The table shows an experiment.

x	1	2	3	4	5
y	6	12	24	48	96

[2 marks] (a)Express the equation in the form Y = mX + c. State Y, X, m and c.

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log₁₀ y = (log₁₀ q)x + log₁₀ p [1]1
Y = log₁₀ y, X = x, m = log₁₀ q, c = log₁₀ p [1]1

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[2 marks] (b)Complete the values of log₁₀ y (2 decimal places).

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0.78, 1.08, 1.38, 1.68, 1.98

Any three correct [1]1
All five: 0.78, 1.08, 1.38, 1.68, 1.98 [1]1

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[2 marks] (c)Use the points (1, 0.78) and (5, 1.98) to find the value of q.

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Gradient = (1.98 − 0.78) ÷ 4 = 0.30 [1]1
q = 10^0.30 = 2 [1]1

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[2 marks] (d)Find the value of p.

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log₁₀ p = 0.78 − 0.30 = 0.48 [1]1
p = 10^0.48 = 3 [1]1

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For the teacher: Analysing, high difficulty, standard 6.2.1

Textbook p. 164

A ball is thrown upwards. Its height h metres at time t seconds is given by h = at − bt², where a and b are constants.

t	1	2	3	4	5
h	20	32	36	32	20

[1 mark] (a)Express the equation in the form Y = mX + c, where Y = h/t and X = t.

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h/t = −bt + a [1]1

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[1 mark] (b)Complete the table of h/t against t.

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20, 16, 12, 8, 4

20, 16, 12, 8, 4 [1]1

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[3 marks] (c)From the values in (b), find a and b.

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Gradient = (4 − 20) ÷ 4 = −4 [1]1
−b = −4, so b = 4 [1]1
Intercept: 20 = −4(1) + a, a = 24 [1]1

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[1 mark] (d)Find h when t = 6. What does your answer mean?

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h = 24(6) − 4(36) = 0: the ball is back on the ground [1]1

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For the teacher: Applying, low difficulty, standard 6.3.1

Textbook p. 166

The variables x and y are related by y = 5 + 3x². Aini says that plotting y against x gives a straight line. Bala says that plotting y against x² gives a straight line.

[1 mark] (a)Who is correct?

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Bala [1]1

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[2 marks] (b)Explain your answer by comparing the equation with Y = mX + c.

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y = 3(x²) + 5, so Y = y, X = x², m = 3, c = 5 [1]1
y against x gives a curve because y is not linear in x [1]1

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[1 mark] (c)Using Bala's graph, find y when x² = 4.

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y = 3(4) + 5 = 17 [1]1

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[2 marks] (d)Create a non-linear equation relating y and x for which a graph of log₁₀ y against x is a straight line with gradient log₁₀ 3 and intercept log₁₀ 2.

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log₁₀ y = x log₁₀ 3 + log₁₀ 2 [1]1
y = 2 × 3^x [1]1

Tick each point you got.

For the teacher: Evaluating, medium difficulty, standard 6.2.1

Textbook p. 162

Section A: objective questions

Question 1

Question 2

Question 3

Question 4

Question 5

Question 6

Question 7

Question 8

Question 9

Question 10

Question 11

Question 12

Section B: structured questions

Question 13

Question 14

Question 15

Question 16

Question 17