3.4.2 Sets, SPM Paper 1 (Short Questions 5 – 7)

Posted on May 25, 2020 by Myhometuition

3.4.2 Sets, SPM Practice Paper 1 (Short Questions)

Question 5:

Diagram below shows a Venn diagram with the number of elements of set P, set Q and set R.

It is given that the universal set,

$ξ = P \cup Q \cup R and n (Q') = n (Q \cap R) .$

Find the value of x.

Solution:

n(Q)’ = n(Q ∩ R)

3 + 8 + 5 = x– 3 + 9

16 = x + 6

x = 10

Question 6:

Diagram below is a Venn diagram showing the number of quiz participants in set P, set Q and set R.

It is given that the universal set,

$ξ = P \cup Q \cup R$ , set P = {Science quiz participants}, set Q = {Mathematics quiz participants} and set R = {History quiz participants}.

If the number of participants who participate in only one quiz is 76, find the total number of the participants.

Solution:

Number of participants who participate in only one quiz = 76

(5x – 2) + (x + 6) + (2x + 8) = 76

8x + 12 = 76
8x = 64
x = 8

Total number of the participants

= 76 + 7 + 4 + 5 + 3(8)

= 116

Question 7:

Diagram below is a Venn diagram showing the number of students in set K, set L and set M.

It is given that the universal set,

$ξ = K \cup L \cup M$ , set K = {Karate Club}, set L = {Life Guards Club} and set M = {Martial Arts Club}.

If the number of students who join both the Life Guards Club and the Martial Arts Club is 8, find the number of students who join only two clubs.

Solution:

Number of students who join both the Life Guards Club and the Martial Arts Club = n(L ∩ M) = 2 + 2x

2 + 2x = 8

2x = 6

x = 3

Number of students who join only two clubs

= x + 4 + 2x

= 3 + 4 + 2(3)

= 13

3.4.1 Sets, SPM Paper 1 (Short Questions 1 – 4)

Posted on May 25, 2020 by Myhometuition

Question 1:

List all the subsets of set P = {r, s}.

Solution:

There are 2 elements, so the number of subsets of set P is 2ⁿ = 2²= 4.

Set P = {r, s}

Therefore subsets of set P = {r}, {s}, {r, s}, { }.

Question 2:

Diagram above shows a Venn diagram with the universal set, ξ = Q υ P.

List all the subset of set P.

Solution:

Set P has 3 elements, so the number of subsets of set P is 2ⁿ = 2³ = 8.

Set P = {2, 3, 5}

Therefore subsets of set P = { }, {2}, {3}, {5}, {2, 3}, {2, 5}, {3, 5}, {2, 3, 5}.

Question 3:

It is given that the universal set, ξ = {x: 30 ≤ x < 42, x is an integer} and set P = {x: x is a number such that the sum of it its two digits is an even number}.

Find set P’.

Solution:

ξ = {30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41}

P = {31, 33, 35, 37, 39, 40}

Therefore P’ = {30, 32, 34, 36, 38, 41}.

Question 4:

Given that universal set ξ = {x : 3 < x ≤ 16, x is an integer},

Set A = {4, 11, 13, 16},

Set B = {x : x is an odd number} and

Set C = {x : x is a multiple of 3}.

The elements of the set (A υ C)’ ∩ B are

Solution:

ξ = {4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16}

A = {4, 11, 13, 16}

B = {5, 7, 9, 11, 13, 15}

C = {6, 9, 12, 15}

(A υ C)’ = {5, 7, 8, 10, 14}

Therefore (A υ C)’ ∩ B = {5, 7}.

3.3 Operations on Sets

Posted on May 25, 2020 by Myhometuition

3.3b Union of Sets (Part 2)

Example 1:

The Venn diagram below shows the number of elements in the universal set, ξ, set P, set Q and R.

Given n(Q) = n(P υ R)’, find n(ξ).

Solution:

n(Q) = n (P υ R)’

2x + 6 + 1 + 5 = 2x + 2x

2x + 12 = 4x

2x = 12

x = 6

n(ξ) = 2x + 2x + x + 7 + 6 + 1 + 5

= 5x+ 19

= 5(6) + 19

= 30 + 19

= 49

Example 2:

Diagram below is a Venn diagram showing the universal set, ξ = {Form 3 students}, set A = {Students who play piano} and set B = {Students who play violin}.

Given n(ξ) = 60, n(A) = 25, n(B) = 12 and n(A ∩ B) = 8, find the number of students who do not play the two instruments.

Solution:

The students who do not play the two instruments are represented by the shaded region, (A υ B)’.

Number of students who do not play the two instruments

= n (A υ B)’

= 60 – 17 – 8 – 4

= 31

3.3a Intersection of Sets (Part 2)

Posted on May 25, 2020 by Myhometuition

3.3a Intersection of Sets

Example 1:

Given that A= {3, 4, 5, 6, 7}, B = {4, 5, 7, 8, 9, 12} and C = {3, 5, 7, 8, 9, 10}.

(a) Find A∩B∩C.

(b) Draw a Venn diagram to represent A∩B∩C.

Solution:

(a) A∩B∩C= {5, 7}

(b)

4. The complement of the intersection of two sets, P and Q, represented by (P ∩ Q)’, is a set that consists of all the elements of the universal set, ξ, but not the elements of P ∩ Q.

5. The complement of set (P ∩ Q)’ is represented by the shaded region as shown in the Venn diagram.

Quadratic Equations Long Questions (Question 11 – 13)

Posted on May 25, 2020 by Myhometuition

Question 11:
Solve the following quadratic equation:

$\frac{5 x + 3 x^{2}}{1 - 2 x} = 6$

Solution:

$\begin{array}{l} \frac{5 x + 3 x^{2}}{1 - 2 x} = 6 \\ 5 x + 3 x^{2} = 6 - 12 x \\ 3 x^{2} + 5 x + 12 x - 6 = 0 \\ 3 x^{2} + 17 x - 6 = 0 \\ (3 x - 1) (x + 6) = 0 \\ 3 x - 1 = 0 or x + 6 = 0 \\ 3 x = 1 x = - 6 \\ x = \frac{1}{3} x = - 6 \end{array}$

Question 12:

Diagram above shows a rectangle ABCD.

(a) Express the area of ABCD in terms of n.

(b) Given the area of ABCD is 60 cm², find the length of AB.

Solution:

(a)

Area of ABCD

= (n + 7) × n

= (n²+ 7n) cm²

(b)

Given the area of ABCD = 60

n²+ 7n = 60

n²+ 7n – 60 = 0

(n – 5) (n + 12) = 0

n = 5 or n = – 12 (not accepted)

When n = 5,

Length of AB = 5 + 7 = 12 cm

Question 13 (4 marks):
Solve the following quadratic equation:

$- \frac{2}{3 x - 5} = \frac{x}{3 x - 1}$

Solution:

$\begin{array}{l} - \frac{2}{3 x - 5} = \frac{x}{3 x - 1} \\ - 2 (3 x - 1) = x (3 x - 5) \\ - 6 x + 2 = 3 x^{2} - 5 x \\ 3 x^{2} - 5 x + 6 x - 2 = 0 \\ 3 x^{2} + x - 2 = 0 \\ (3 x - 2) (x + 1) = 0 \\ 3 x - 2 = 0 or x + 1 = 0 \\ x = \frac{2}{3} or x = - 1 \end{array}$

SPM Practice 2 (Question 8 – 10)

Posted on May 25, 2020 by Myhometuition

Question 8:
The variables x and y are related by the equation $y = \frac{p}{3^{x}}$ , where k is a constant.
Diagram below shows the straight line graph obtained by plotting $\log_{10} y$ against x.

Express the equation $y = \frac{p}{3^{x}}$ in its linear form used to obtain the straight line graph shown in Diagram above.
Find the value of p.

Solution:

Question 9:
Variable x and y are related by the equation $y^{2} = p x^{q}$ . When the graph lg y against lg x is drawn, the resulting straight line has a gradient of -2 and an vertical intercept of 0.5 . Calculate the value of p and of q.

Solution:

Question 10:
Variable x and y are related by the equation $y = \frac{c}{d - x}$ . When the graph y against xy is drawn the resulting line has gradient 0.25 and an intercept on the y-axis of 1.25. Calculate the value of c and of d.

Solution:

SPM Practice 2 (Question 6 & 7)

Posted on May 25, 2020 by Myhometuition

Question 6:
The variables x and y are related by the equation $y = p x^{3}$ , where p is a constant. Find the value of p and n.

Solution:

Question 7:
Diagram A shows part of the curve $y = a x^{2} + b x$ . Diagram B shows part of the straight line obtained when the equation is reduced to the linear form. Find
(a) the values of a and b,
(b) the values of p and q.

Diagram A

Diagram B

Solution:

SPM Practice 2 (Question 4 & 5)

Posted on May 25, 2020 by Myhometuition

Question 4:
The diagram shows part of the straight line graph obtained by plotting $\frac{y}{x}$ against $\sqrt{x}$ .

Given its original non-linear equation is $y = p x + q x^{\frac{3}{2}}$ . Calculate the values of p and q.

Solution:

Question 5:
The diagram below shows the graph of the straight line that is related by the equation $\frac{x}{y} = \frac{2}{x} + 3 x$ .

Find the values of p and k.

Solution:

(G) Sum to Infinity of Geometric Progressions (Part 2)

Posted on May 24, 2020 by Myhometuition

(H) Recurring Decimal

Example of recurring decimal:

$\begin{array}{l} \frac{2}{9} = 0.2222222222222..... \\ \frac{8}{33} = 0.242424242424..... \\ \frac{41}{333} = 0.123123123123..... \end{array}$

Recurring decimal can be changed to fraction using the sum to infinity formula:

$S \infty = \frac{a}{1 - r}$

Example (Change recurring decimal to fraction)

Express each of the following recurring decimals as a fraction in its lowest terms.

(a) 0.8888 ...

(b) 0.171717...

(c) 0.513513513 ….

Solution:

(a)

0.8888 = 0.8 + 0.08 + 0.008 +0.0008 + ….. (recurring decimal)

$\begin{array}{l} G P, a = 0.8, r = \frac{0.08}{0.8} = 0.1 \\ S_{\infty} = \frac{a}{1 - r} \\ S_{\infty} = \frac{0.8}{1 - 0.1} \\ S_{\infty} = \frac{0.8}{0.9} \\ S_{\infty} = \frac{8}{9} \to \begin{array}{l} check using calculator \\ \frac{8}{9} = 0.888888.... \end{array} \end{array}$

(b)

0.17171717 …..

= 0.17 + 0.0017 + 0.000017 + 0.00000017 + …..

$\begin{array}{l} G P, a = 0.17, r = \frac{0.0017}{0.17} = 0.01 \\ S_{\infty} = \frac{a}{1 - r} \\ S_{\infty} = \frac{0.17}{1 - 0.01} = \frac{0.17}{0.99} = \frac{17}{99} \to \begin{array}{l} remember to check the \\ answer using calculator \end{array} \end{array}$

(c)

0.513513513…..

= 0.513 + 0.000513 + 0.000000513 + …..

$\begin{array}{l} G P, a = 0.513, r = \frac{0.00513}{0.513} = 0.001 \\ S_{\infty} = \frac{a}{1 - r} \\ S_{\infty} = \frac{0.513}{1 - 0.001} = \frac{0.513}{0.999} = \frac{513}{999} = \frac{19}{37} \end{array}$

3. The nth Term of Geometric Progressions (Part 2)

Posted on May 24, 2020 by Myhometuition

(D) The Number of Term of a Geometric Progression

Smart TIPS: You can find the number of term in an arithmetic progression if you know the last term

Example 2:
Find the number of terms for each of the following geometric progressions.

(a) 2, 4, 8, ….., 8192

(b) $\frac{1}{4}, \frac{1}{6}, \frac{1}{9}, ....., \frac{16}{729}$

(c) $- \frac{1}{2}, 1, - 2, ....., 64$

Solution:
(a)

$\begin{array}{l} 2, 4, 8, ....., 8192 \leftarrow (Last term is given) \\ a = 2 \\ r = \frac{T_{2}}{T_{1}} = \frac{4}{2} = 2 \\ T_{n} = 8192 \\ a r^{n - 1} = 8192 \leftarrow (T h e n th term of GP, T_{n} = a r^{n - 1}) \\ (2) {(2)}^{n - 1} = 8192 \\ 2^{n - 1} = 4096 \\ 2^{n - 1} = 2^{12} \\ n - 1 = 12 \\ n = 13 \end{array}$

(b)

$\begin{array}{l} \frac{1}{4}, \frac{1}{6}, \frac{1}{9}, ....., \frac{16}{729} \\ a = \frac{1}{4}, r = \frac{\frac{1}{6}}{\frac{1}{4}} = \frac{2}{3} \\ T_{n} = \frac{16}{729} \\ a r^{n - 1} = \frac{16}{729} \\ (\frac{1}{4}) {(\frac{2}{3})}^{n - 1} = \frac{16}{729} \\ {(\frac{2}{3})}^{n - 1} = \frac{16}{729} \times 4 \\ {(\frac{2}{3})}^{n - 1} = \frac{64}{729} \\ {(\frac{2}{3})}^{n - 1} = {(\frac{2}{3})}^{6} \\ ∴ n - 1 = 6 \\ n = 7 \end{array}$

(c)

$\begin{array}{l} - \frac{1}{2}, 1, - 2, ....., 64 \\ a = - \frac{1}{2}, r = \frac{- 2}{1} = - 2 \\ T_{n} = 64 \\ a r^{n - 1} = 64 \\ (- \frac{1}{2}) {(- 2)}^{n - 1} = 64 \\ {(- 2)}^{n - 1} = 64 \times - 2 \\ {(- 2)}^{n - 1} = - 128 \\ {(- 2)}^{n - 1} = {(- 2)}^{7} \\ n - 1 = 7 \\ n = 8 \end{array}$

(E) Three consecutive terms of a geometric progression

If e, f and g are 3 consecutive terms of GP, then

$\frac{g}{f} = \frac{f}{e}$

Example 3:

If p + 20, p − 4, p −20 are three consecutive terms of a geometric progression, find the value of p.

Solution:

$\begin{array}{l} \frac{p - 20}{p - 4} = \frac{p - 4}{p + 20} \\ (p + 20) (p - 20) = (p - 4) (p - 4) \\ p^{2} - 400 = p^{2} - 8 p + 16 \\ 8 p = 416 \\ p = 52 \end{array}$