Save as PDF or print — the answers and worked solutions live online, where your work gets marked free.

Examinations Council of Zambia — past paper

ECZ 2019 O-Level Paper 2 — Mathematics

12 questions · downloaded from g12titan.com, where this paper is marked free online

(a)

Given that matrix A = 6x23,

(a(i))

find the value of x for which the determinant of A is 36, [2]

(a(ii))

hence, find the inverse of A. [2]

(b)

A box contains 4 red pens and 5 green pens. A pen is picked at random from the box without replacement and a second pen is then picked.

(b(i))

Draw a tree diagram to illustrate the outcomes. [3]

(b(ii))

What is the probability of picking one red pen and one green pen? [2]

(a)

Simplify 6x² - 24x - 2. [2]

(b)

Given the geometric progression 4, 8, 16, ..., find

(b(i))

the geometric mean of 256 and 1 024, [2]

(b(ii))

the 11th term, [2]

(b(iii))

the sum of the first 11 terms. [3]

(a)

Solve the equation p² - p = 4, giving your answers correct to 2 decimal places. [5]

(b)

The Venn diagram below shows the number of elements in sets A, B and C. Find Diagram for part b

(b(i))

x, such that n(B) = n(B union C)', [2]

(b(ii))

y, such that n(C) = n(A), [1]

(b(iii))

n(E), [1]

(b(iv))

n(B'). [1]

(a)

Express 52x - 4 - 43x + 5 as a single fraction in its simplest form. [3]

(b)

The program below is given in the form of a pseudocode. Begin Enter length If length < 0 Then display "error message" and re-enter positive length Else enter height If height < 0 Then display "error message" and re-enter positive height Else volume = 13 * l * l * h End if Display volume End Draw the corresponding flowchart for the information given above. [5]

(a(i))

Construct triangle ABC in which AB = 8 cm, angle BAC = 110° and angle ABC = 35°. [1]

(a(ii))

Measure and write the length of BC. [1]

(b)

Within the triangle ABC, construct the locus of points which are

(b(i))

3 cm from AB, [1]

(b(ii))

equidistant from AC and BC, [2]

(b(iii))

4 cm from A. [1]

(c)

A point Q inside triangle ABC is greater than or equal to 3 cm from AB, less than or equal to 4 cm from A and nearer to AC than BC. Indicate clearly, by shading, the region in which Q must lie. [2]

(a)

In the diagram below, OB = 4OA and AC = 5AX. M is the midpoint of BC, OA = a and BM = b. Diagram for part a

(a(i))

Express in terms of a and/or b.

(a(i)(a))

AB, [1]

(a(i)(b))

AC, [1]

(a(i)(c))

OM. [1]

(a(ii))

Show that OX = (25)(4a + b). [2]

(b)

Find the equation of the normal to the curve y = 5x³ - 6x² + 2x + 5 at the point (1, 2). [3]

The vertices of triangle ABC are A(1, 1), B(1, 3) and C(3, 3). The vertices of triangle A1B1C1 are A1(-1, 1), B1(-3, 1) and C1(-3, 3).

(a)

Using a scale of 1 cm to represent 1 unit on each axis, draw the x and y axes for -6 ≤ x ≤ 6 and -6 ≤ y ≤ 6. Draw and label triangles ABC and A1B1C1. [2]

(b)

Describe fully a single transformation that maps triangle ABC onto triangle A1B1C1. [2]

(c)

An enlargement maps triangle ABC onto triangle A2B2C2 with vertices A2(-2, -2), B2(-2, -6) and C2(-6, -6).

(c(i))

Draw and label triangle A2B2C2. [1]

(c(ii))

Find the scale factor. [1]

(d)

The transformation represented by the matrix 2001 maps triangle ABC onto triangle A3B3C3.

(d(i))

Find the coordinates of the vertices A3, B3 and C3. [2]

(d(ii))

Draw and label triangle A3B3C3. [1]

(e)

Triangle ABC is mapped onto triangle A4B4C4 with vertices A4(1, -2), B4(1, 0) and C4(3, -6).

(e(i))

Draw and label triangle A4B4C4. [1]

(e(ii))

Find the matrix representing this transformation. [2]

The table below shows the expenditure of 90 farmers in a particular farming season. Amount in K0<x≤100100<x≤200200<x≤300300<x≤400400<x≤500500<x≤600600<x≤700700<x≤800Number of farmers5161717141272

(a)

Calculate the standard deviation. [6]

(b(i))

Using the table above, copy and complete the cumulative frequency table below. Amount in K≤0≤100≤200≤300≤400≤500≤600≤700≤800Frequency0521385569 [1]

(b(ii))

Using a scale of 2 cm to represent 100 units on the horizontal axis and 2 cm to represent 10 units on the vertical axis, draw a smooth cumulative frequency curve. [3]

(b(iii))

Showing your method clearly, use your graph to estimate the interquartile range. [2]

Kuunika wishes to build a lodge with single and double rooms. He needs to decide the number of each room type he should build to maximize profit. Let x represent the number of single rooms and y the number of double rooms.

(a)

Write the inequalities which represent each of the following conditions:

(a(i))

There must be at least one single room. [1]

(a(ii))

There must be at least 10 rooms altogether. [1]

(a(iii))

The total number of rooms should not exceed 15. [1]

(a(iv))

The number of double rooms must be at least twice the number of single rooms. [1]

(a(v))

The number of double rooms should not be more than 12. [1]

(b)

Using a scale of 2 cm to 5 units on both axes, draw x and y axes for 0 ≤ x ≤ 16 and 0 ≤ y ≤ 16 respectively and shade the unwanted region to indicate clearly the region where the solution of the inequalities lie. [5]

(c)

The rate for a single room is K600.00 and K900.00 for a double room. How many rooms of each type should Kuunika build to maximize the income? [2]

10.

(a)

The diagram below shows part of the graphs of y = x³ + 2x - 1 and y = 10x. Diagram for part a

(a(i))

Use the graphs to solve the equations

(a(i)(a))

x³ + 2x = 6, [2]

(a(i)(b))

x³ + 2x - 1 = 10x. [2]

(a(ii))

Calculate an estimate of

(a(ii)(a))

the gradient of the curve at the point (2, 11), [2]

(a(ii)(b))

the area bounded by the curve, y = 10x, y = 0 and x = 2. [3]

(b)

Evaluate ∫31 (3x² + 4x) dx. [3]

11.

(a)

The points P, Q, R and T are on the surface of the earth as shown in the diagram below. [Take π as 3.142 and R = 3437 nm] Diagram for part a

(a(i))

Find the difference in longitude between the points T and R. [2]

(a(ii))

Find, in nautical miles, the distance between

(a(ii)(a))

P and Q along the latitude 65° N, [2]

(a(ii)(b))

P and T along the longitude 90° W. [2]

(b)

The figure below shows a right pyramid with a vertex O and a square base ABCD of side 8 cm. Angle CPD = angle OPC = 90°. Given that OA = OB = OC = OD = 10 cm, calculate Diagram for part b

(b(i))

the height OP, [4]

(b(ii))

the angle between the edge OC and the base PC. [2]

12.

(a)

The diagram below shows a triangle KMN in which KM = 8 km, MN = 10 km and angle KMN = 92°. Calculate Diagram for part a

(a(i))

KN, [5]

(a(ii))

the area of triangle KMN, [2]

(a(iii))

the shortest distance from M to KN. [2]

(b)

Solve the equation 2 tan θ = -3 for 0° ≤ θ ≤ 180°. [1]

(c)

Simplify 25p⁴7q² ÷ 5p⁶21q⁴ × p15q. [2]