ST.

JOHN’S GRAMMAR SCHOOL

3
SECOND MOCK EXAMINATION, June, 2023 b) Given that 𝑐𝑜𝑠𝑥 = , 0° ≤ 𝑥 ≤ 90°, without using
𝟏 5
CORE MATHEMATICS TIME: 𝟐 𝟐 hours 3𝑡𝑎𝑛𝑥
mathematical tables or calculator, calculate
2𝑠𝑖𝑛𝑥+3𝑐𝑜𝑠𝑥

2 5. (a) In an event, a fair die is rolled once and two

unbiased coins are tossed once. What is the probability
of obtaining
Write your name and index number in ink in the space (i) 3 and a tail?
provided above. (ii) exactly one head and a 1 or 2?
Answer ten questions in all: all the five questions in Section (iii) at least one head and not a two?
A and five questions from Section B. (b) A bag contains 10 marbles, 7 of which are black and 3
are red. Two marbles are drawn one after the other
In each question, all the necessary details of working,
without replacement. Find the probability of getting
including rough work, must be shown with the answer. Give
answers as accurately as data and tables allow. Graph papers (i) a red, then a black marble
are provided for your use in the examination. The use of
non-programmable, silent and cordless calculator is (ii) two black marbles
allowed.
SECTION B
SECTION A
[60marks]
[40 Marks] Answer five questions only from this section. All questions
Answer all the questions in this section. All questions carry carry equal marks
equal marks
6. (a) Draw the table of values for 𝑦 = 𝑠𝑖𝑛𝑥 + 2𝑐𝑜𝑠𝑥, for
1. (a) The electrical resistance R, ohms of a wire varies 0° ≤ 𝑥 ≤ 360° using intervals at 60°
directly as the length l cm and inversely as twice the (b) Using a scale of 2 cm to 30° on the 𝑥 − 𝑎𝑥𝑖𝑠 and 2
square root of the decimeter d cm. Given that R = 5, cm to 0.5 on the 𝑦 − 𝑎𝑥𝑖𝑠, draw the graph of 𝑦 =
when l = 2 and d = 4, find 𝑠𝑖𝑛𝑥 + 2𝑐𝑜𝑠𝑥, for 0° ≤ 𝑥 ≤ 360°.
(i) the relation for l in terms of R and d (c) Use the graph to
(ii) R when l = 4 and d = 16 (i) solve the equation 2sin𝑥 = 3 − 4𝑐𝑜𝑠𝑥
81𝑥−2 (ii.) find the minimum point of the graph
(b) If 92𝑥+1 = , find the value of 𝑥.
3𝑥
(iii.) value of 𝑥 when 𝑦 = −1
2. (a) A cube and cuboid have the same volume. The length
of the cuboid is 3𝑐𝑚 bigger, the breadth 2𝑐𝑚 bigger and 7. a) 𝐴 and 𝐵 are two observation posts on the same
the height 3𝑐𝑚 smaller than the edge of the cube. Find horizontal ground at the foot 𝑀 of a vertical tower 𝑀𝑁.
the length of the edge of the cube. The tower is 18m due north of 𝐵 and 24m east of 𝐴. The
3√1.376
(b) Find the value of 4 0.007 using logarithm angle of elevation of 𝑁 from 𝐵 is 35ᵒ. Calculate, correct
√ to three significant figures the:
3. (a) The angle of a sector of a circle of radius 7 cm (i) height 𝑀𝑁
is 1080 . Calculate the perimeter of the sector. (Take π= (ii) angle of elevation of 𝑁 from 𝐴
22
7
) (iii) distance 𝐴𝐵
(b) The sum of the 9 terms of an A.P is 72 and the sum (iv) bearing of 𝐴 from 𝐵
of the next 4 terms is 71. Find the sequence
b) Simplify: 3[4𝑥 − 5(3𝑥 − 5)] − 1
4. (a) In the diagram, ̅̅̅̅
𝐶𝐷 is a tangent to the circle 8. (a) Using a ruler and a pair of compasses only,
𝑃𝑄𝑀 𝑎𝑡 𝑀, 𝑃𝑄𝑅 is a straight line and 𝑀𝑅 bisects construct:
∠𝑄𝑀𝐷. Find the value of the angle 𝑄𝑅𝑀. (i) ∆𝐴𝐵𝐶 such that |𝐴𝐵| = 5𝑐𝑚, |𝐴𝐶| = 7.5𝑐𝑚
and ∠𝐶𝐴𝐵 = 135°
𝑄 𝑅 (ii) the locus 𝑙1 of points equidistant from A and B
(iii) the locus 𝑙2 of points equidistant from AB and
𝑃
60° AC which passes through a ∆𝐴𝐵𝐶
(b) (i) Locate the point of intersection, 𝑃, of 𝑙1 and 𝑙2 .
(ii) With the point 𝑃 as the center, draw a circle to pass
through the vertices 𝐴 and 𝐵

55° (c) Measure |𝐵𝐶|

𝐶 𝐷
𝑀
 (d) Calculate, correct to 3 significant figures the area of (i) ∠ 𝑄𝑃𝑅 (ii) ∠ 𝑀𝑄𝑂
the circle in b)(ii). [Take 𝜋 = 3.142] 𝑃
9. (a) The diagram below represents a playing field in the
form of a trapezium. If |𝑃𝑄| = 40𝑚, |𝑄𝑅| = 𝑆
70𝑚, |𝑆𝑅| = 120𝑚, |𝑃𝑆| = 50𝑚 and 𝑃𝑄//𝑆𝑅. 15°
Calculate
(i) angle 𝑃𝑆𝑅 𝑀 𝑂
(ii) the length of PR, correct to three significant
figures
40𝑚 𝑄 32°
𝑅
𝑄

12. (a) The hire purchase price of a machine is $4045.00. If

20% is paid as deposit and the rest is paid in 18 equal
𝑅 monthly installments, calculate
𝑆 (i) the amount of deposit
b) The universal set 𝐸 is the set of all integers A, B and C
(ii) the amount of each monthly installment
are subsets of E defined as follows:A=
(b) In the diagram below, BA is parallel to DE. Find the
{… 6, −4, −2, 0, 2, 4, 6, … }, B= {𝑥: 0 ≤ 𝑥 ≤ 9} and C=
value of 𝑥
{𝑥: −4 ≤ 𝑥 ≤ 10} 𝐴
(i) Write down set 𝐴′ where 𝐴′ , the compliment of A with
respect to 𝐸
(ii) find the members of the sets 𝐵 ∪ 𝐶, 𝐴 ∩ 𝐵 and 𝐵 ∩ 𝐶 𝐸
and hence show that 𝐴 ∩ (𝐵 ∪ 𝐶) = (𝐴 ∩ 𝐵) ∪ (𝐴 ∩ 𝐶) 52ᵒ 𝑥ᵒ
𝐵
10. (a) Using a scale of 2 cm to 2 unit on both axes, draw on
a sheet of graph paper two perpendicular
axes𝑂𝑥 𝑎𝑛𝑑 𝑂𝑦, for the intervals −10 ≤ 𝑥 ≤ 10 and 𝐷
−12 ≤ 𝑥 ≤ 12.
(b) Draw on the same graph sheet, showing clearly the 312ᵒ
coordinates of all vertices
(i) 𝐵𝐴 = (−2
the triangle 𝐴𝐵𝐶 with 𝐴(4,8), ⃗⃗⃗⃗⃗ 2
) and
2
⃗⃗⃗⃗⃗ = ( ) 13. (a)Two functions 𝑔 and ℎ defined by 𝑔: 𝑥 → 2𝑥² −
𝐶𝐴 4
(ii) ′ ′ ′
the image ∆ 𝐴 𝐵 𝐶 of ∆ 𝐴𝐵𝐶 under a reflection 1 and ℎ: 𝑥 → 3𝑥 + 2, where 𝑥 is a real number
in the line 𝑦 = −2, where 𝐴 → 𝐴′ , 𝐵 → 𝐵′ and (i) if 𝑔(𝑥 − 1) − 7 = 0, find the value of 𝑥
𝐶 → 𝐶′ 𝑔(−12)∙× ℎ(3)
(ii) Evaluate:
(iii) the image ∆ 𝐴′′ 𝐵′′ 𝐶 ′′ of ∆ 𝐴𝐵𝐶 under a 𝑔(4)−ℎ(5)

translation by a vector (−8 2

) where
b) Consider the statements:
𝐴 → 𝐴 , 𝐵 → 𝐵 and 𝐶 → 𝐶 ′′
′′ ′′
𝑋 ∶ Emmanuel trains hard 𝑌 ∶ Emmanuel wins the race
(iv) the image ∆ 𝐴′′′ 𝐵′′′ 𝐶 ′′′ of ∆ 𝐴′𝐵′𝐶′ under a (i) Illustrate the information above on a Venn diagram
rotation through 180ᵒ about the origin If 𝑋 ⇒ 𝑌, state whether or not the following
where A ′ → 𝐴′′′ , 𝐵′ → 𝐵′′′ and 𝐶′ → 𝐶 ′′′ statements are valid:
⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗
c) Calculate 𝐵′′𝐵′′′ (𝛼) If Emmanuel wins the race, then he has trained hard
d) Find the equation of the line joining points 𝐵′ and 𝐴
(𝛽) If Emmanuel does not train hard then he will not win the
11. (a) Three towns A, B and C are such that the distance race
between A and B is 50km and the distance between A (𝛾) If Emmanuel does not win the race then he has not train
and C is 90km. If the bearing B from A is 075ᵒ and the hard
bearing of C from A is 310ᵒ, find:
(i) distance between B and C
(ii) bearing of C from B
END OF PAPER
(b) In the diagram below, O is the Centre of the circle.
∠ OQR=32ᵒ and angle 𝑀𝑃𝑄 = 15ᵒ Calculate