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Examinations Council of Zambia — past paper
ECZ 2023 GCE Paper 2 — Mathematics
12 questions · downloaded from g12titan.com, where this paper is marked free online
1.
(a)
Simplify 13a328a2 b2 ÷ 65a4 b56a2 b4. [2]
(b)
Learners in a Grade 10 class were asked the types of drinks they liked. The Venn diagram shows their responses.
(b(i))
Given that 40 learners liked Fanta, find the value of x. [2]
(b(ii))
Find the total number of learners in the class. [1]
(b(iii))
How many learners
(b(iii)(a))
did not like Fanta, [1]
(b(iii)(b))
liked two types of drinks only? [1]
2.
(a)
In a geometric progression, the second term is 21 and the fourth term is 189. Calculate the
(a(i))
first term and the common ratio, [3]
(a(ii))
sixth term, [2]
(a(iii))
sum of the first 5 terms of the progression. [2]
(b)
Given that matrix P = 7x-2-x1,
(b(i))
find the value of x for which the determinant of P is -10, [2]
(b(ii))
hence, find the inverse of P. [2]
3.
(a)
Express 91 - 4k - 81 - 3k as a single fraction in its simplest form. [3]
(b)
A girl has 7 green apples and 6 red apples in a bag. She picks one apple at random from the bag and eats it. She picks another apple and eats it.
(b(i))
Draw a tree diagram to illustrate all the possible outcomes. [2]
(b(ii))
What is the probability that the first apple eaten was green? [3]
4.
(a(i))
Construct a triangle KLM in which KL = 10 cm, LM = 7 cm and angle KLM = 120°. [1]
(a(ii))
Measure and write the length KM. [1]
(b)
Within triangle KLM, draw the locus of points which are
(b(i))
5.5 cm from M, [1]
(b(ii))
1 cm from LM, [1]
(b(iii))
equidistant from M and L. [2]
(c)
A point P, within triangle KLM, is such that it is less than or equal to 5.5 cm from M, greater than or equal to 1 cm from LM and nearer to L than M. Indicate clearly, by shading, the region in which P must lie. [2]
5.
(a)
The program below is given in the form of a pseudocode.
Begin
Enter n
IF n < 0 THEN
Display 'error n must be positive'
ELSE sum = n2 * [2*a + (n - 1)*d]
ENDIF
Display sum
End
Draw a corresponding flowchart for the information given above. [5]
(b)
Solve the equation 8x2 - 9x + 2 = 0, giving your answers correct to 2 decimal places. [5]
6.
(a)
In the diagram, OA = 2a, AB = b, OC = 3b, M is the midpoint of OB and OC is parallel to AB.
Express in terms of a and/or b
(a(i))
OB, [1]
(a(ii))
CB, [1]
(a(iii))
AC, [1]
(a(iv))
CM. [2]
(b)
The equation of a curve is y = x3 - 27x. Find the coordinates of the turning points of the curve. [3]
7.
(a)
The diagram shows the positions of three towns P, Q and R. PQ = 7.1 km, PR = 23.8 km and angle RPQ = 92.7°.
Calculate the
(a(i))
distance RQ to 2 decimal places, [5]
(a(ii))
area of the triangle PQR, [2]
(a(iii))
shortest distance of P from RQ giving your answer to 2 decimal places. [2]
(b)
Solve the equation 3 tan θ = 89 for 0° ≤ θ ≤ 180°. [1]
(c)
Simplify 9x2 - 19x + 3. [2]
8.
(a)
The diagram shows the frustum of a cone. The perpendicular height is 40 cm. The top and bottom radii are 10 cm and 30 cm respectively. [Take π as 3.142]
Calculate the volume of the frustum. [6]
(b)
Four towns A(70° N, 65° W), B(70° N, 65° E), C(70° S, 65° E) and D(70° S, 65° W) are on the surface of the earth. Take π as 3.142 and R = 3 437 nm.
(b(i))
Sketch the surface of the earth showing all the four towns A, B, C and D. [2]
(b(ii))
Calculate, in nautical miles, the distance
(b(ii)(a))
AB along latitude 70° N, [2]
(b(ii)(b))
BC along longitude 65° E. [2]
9.
Answer the whole of this question on a sheet of graph paper.
A lady intends to bake two types of cakes, type A and type B for sale. She intends to bake at least 30 cakes of type A and at least 20 cakes of type B. The number of cakes of type A must be equal to or more than the number of cakes of type B. The total number of cakes must not exceed 90.
(a)
Taking x to represent the number of cakes of type A and y to represent the number of type B cakes, write four inequalities to represent the information above. [4]
(b)
Using a scale of 2 cm to represent 10 cakes on each axis, draw x and y axes for 0 ≤ x ≤ 90 and 0 ≤ y ≤ 90 respectively and shade the unwanted region to indicate clearly the region where the solution of the inequalities lie. [5]
(c)
The profit on the sale of a type A cake is K30.00 and the profit on a type B cake is K50.00. How many cakes of each type can be baked to make maximum profit? [2]
(d)
Calculate the maximum profit. [1]
10.
Answer the whole of this question on a sheet of graph paper.
The vertices of triangle ABC are A(0, 4), B(0, 6) and C(-4, 6) while the vertices of triangle A1B1C1 are A1(4, 0), B1(6, 0), and C1(6, 4).
(a)
Using a scale of 1 cm to represent 1 unit on each axis, draw x and y axes for -4 ≤ x ≤ 8 and -3 ≤ y ≤ 12. Draw and label triangle ABC and triangle A1B1C1. [2]
(b)
Describe fully a single transformation that maps triangle ABC onto triangle A1B1C1. [3]
(c)
The 1002 maps triangle ABC onto triangle A2B2C2.
(c(i))
Find the coordinates of A2, B2 and C2. [3]
(c(ii))
Draw and label triangle A2B2C2. [1]
(d)
Triangle A1B1C1 is mapped onto triangle A3B3C3 with vertices A3(4, 0), B3(6, 0) and C3(-2, 4).
(d(i))
Draw and label triangle A3B3C3. [1]
(d(ii))
Find the matrix representing this transformation. [2]
11.
The ages (in years) of 100 patients treated at a certain health centre on a particular day are given in the table below.
Age in years0 < x ≤ 1010 < x ≤ 2020 < x ≤ 3030 < x ≤ 4040 < x ≤ 5050 < x ≤ 60Number of patients51025302010
(a)
Calculate the standard deviation. [6]
(b)
Answer this part of the question on a sheet of graph paper.
(b(i))
Using the table above, copy and complete the cumulative frequency table below.
Age in years≤ 0≤ 10≤ 20≤ 30≤ 40≤ 50≤ 60Number of patients0515100 [1]
(b(ii))
Using a scale of 2 cm to represent 10 units on each axis for 0 ≤ x ≤ 60 and 0 ≤ y ≤ 100, draw a smooth cumulative frequency curve. [3]
(b(iii))
Showing your method clearly, use your graph to estimate the semi-interquartile range. [2]
12.
(a)
Evaluate ∫1-1 (-1 - 3x2) dx. [3]
(b)
The diagram shows the graph of y = x3 - x2 - 5x - 3.
(b(i))
Use the graph to find the solution of the equations
(b(i)(a))
x3 - x2 - 5x - 3 = 0, [2]
(b(i)(b))
x3 - x2 - 5x = x - 3. [3]
(b(ii))
Find the
(b(ii)(a))
gradient of the curve at the point (-2, -5), [2]
(b(ii)(b))
area bounded by the curve, x = 1, y = 0 and x = 3. [2]