5 Marks Questions

Question 15 Marks

Two separate figures are given below. Each figure shows the first few positions in a sequence of arrangements made with sticks. Identify the pattern and answer the following questions for each figure:
(a) How many squares are in position number 11 of the sequence?
(b) How many sticks are needed to arrange position number 11 of the sequence?
(c) Can an arrangement in this sequence be made using exactly 85 sticks? If yes, which position number will it correspond to?
(d) Can an arrangement in this sequence be made using exactly 150 sticks? If yes, which position number will it correspond to?

Answer

So, no. of sticks needed in 11th position will be 13 + 9 × 10 = 103
(c) Let the nth arrangement have 85 sticks.
13 + (n – 1) × 9 = 85
⇒ 13 + 9n – 9 = 85
⇒ 4 + 9n = 85
⇒ 9n = 85 – 4
⇒ 9n = 81
⇒ n = 9
Yes, the arrangement will have 85 sticks.
(d) Let the nth arrangement have 150 sticks.
13 + (n – 1) × 9 = 150
⇒ 13 + 9n – 9 = 150
⇒ 4 + 9n = 150
⇒ 9n = 150 – 4
⇒ 9n = 146
$\Rightarrow n =\frac{146}{9}$
$\Rightarrow n=16 \frac{2}{9}$
It is not a whole number.
So, no arrangement can be made with 150 sticks.

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Question 25 Marks

Write 4 equations whose solution is u = 6.

Answer

(i) u = 6
Multiply both sides by $\frac{2}{3}$
$\begin{array}{l}\Rightarrow \frac{2}{3} u=\frac{2}{3} \times 6 \\ \Rightarrow \frac{2}{3} u=4\end{array}$
(ii) u = 6
Add 7 to both sides u + 7 = 6 + 7
⇒ u + 1 = 13
(iii) u = 6
Multiply both sides by 2
2u = 12
Add 3 to both sides
⇒ 2u + 3 = 12 + 3
⇒ 2u + 3 = 15
(iv) u = 6
Multiply both sides by 3
3u = 18
Subtract 5 from both sides
3u – 5 = 18 – 5
⇒ 3u – 5 = 13

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Question 35 Marks

Here are machines that take an input, perform an operation on it, and send out the result as an output.

Find the inputs in the following cases:

Answer

(i) Let the unknown number be x.
x × 3 – (x + 3) = 63
⇒ 3x – x – 3 = 63
⇒ 2x – 3 = 63
⇒ 2x = 66
⇒ x = 33

(ii) Let the unknown number be y.
y × 3 – (y + 3) = 227
⇒ 3y – y – 3 = 227
⇒ 2y = 227 + 3
⇒ 2y = 230
⇒ y = 115
The unknown number is 115.

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Question 45 Marks

Given 4k + 1 = 13, find the values of:
(a) 8k+ 2
(b) 4k
(c) k
(d) 4k – 1
(e) -k – 2

Answer

4k + 1 = 13
Subtract 1 from both sides
4k + 1 – 1 = 13 – 1
⇒ 4k = 12
Divide both sides by 4
4k ÷ 4 = 12 ÷ 4
⇒ k = 3
(a) 8k + 2 = 8(3)+ 2
= 24 + 2
= 26
(b) 4k = 4 × 3 = 12
(c) k = 3
(d) 4k – 1 = 4(3) – 1
= 12 – 1
= 11
(e) -k – 2 = -3 – 2 = -5

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Question 55 Marks

Write 5 equations whose solution is x = -2.

Answer

(i) x = -2
Add 3 on both sides
x + 3 = -2 + 3
⇒ x + 3 = 1
Multiply both sides by 2
2(x + 3) = 2
(ii) x = -2
Multiply both sides by 3
3x = -6
Add 2 on both sides
3x + 2 = -6 + 2
⇒ 3x + 2 = -4
(iii) x = -2
Divide both sides by 4
$\begin{array}{l}\frac{x}{4}=-\frac{2}{4} \\ \frac{x}{4}=-\frac{1}{2}\end{array}$
(iv) x = -2
Multiply both sides by 5
5x = -10
Add 12 on both sides
5x + 12 = -10 + 12
⇒ 5x + 12 = 2
(v) x = -2
Subtract 7 from both sides
x – 7 = -2 – 7
x – 7 = -9

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Question 65 Marks

Solve these equation and check the solution.
4(m + 6) – 8 = 2m – 4

Answer

4(m + 6) – 8 = 2m – 4
Apply the distributive property
4m + 24 – 8 = 2m – 4
⇒ 4m + 16 = 2m – 4
Subtract 2m from both sides
4m + 16 – 2m = 2m – 4 – 2m
⇒ 2m + 16 = -4
Subtract 16 from both sides
2m + 16 – 16 = -4 – 16
⇒ 2m = -20
Divide by 2 on both sides
2m ÷ 2 = -20 ÷ 2
⇒ m = -10
Check:
LHS = 4(m + 6) – 8
= 4(-10 + 6) – 8
= 4(-4) – 8
= -16 – 8
= -24
RHS = 2m – 4
= 2(-10) – 4
= -20 – 4
= -24
LHS = RHS
Hence checked.

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