NCEA Level 1 Mathematics 91947 free practice questions with answers

Question 1 (12 marks) [Algebra, graphs and measurement]

(a) The diagram shows a cuboid with side lengths 3x + 3 cm, 2x + 5 cm and 5x + 4 cm. Find the total surface area of the cuboid, given that x = 6. 3x + 3 cm 5x + 4 cm 2x + 5 cm Diagram is NOT to scale

(a) Answer: 3298 cm²

• Evidence level: Achieved (substitute into an algebraic expression, then use the surface area of a cuboid).

• Substitute x = 6 into each side: 21 cm, 17 cm and 34 cm.

• A cuboid has three pairs of identical faces, so surface area = 2(lw + lh + wh).

• = 2(21 × 17 + 21 × 34 + 17 × 34) = 2(357 + 714 + 578) = 2 × 1649

• Total surface area = 3298 cm².

(b) The graph below shows two straight lines with equations x − 2y = −11 and 2x + y = 13 The lines intersect at the point Q. Find the coordinates of Q, using an algebraic method. Clearly show all steps of your working. -2 -1 1 2 3 4 5 6 7 8 9 10 -2 -1 1 2 3 4 5 6 7 8 9 10 x y Q Diagram is NOT to scale

(b) Answer: Q = (3, 7)

• Evidence level: Achieved (solve a pair of simultaneous equations by an algebraic method; do not read Q from the graph).

• Label the equations: (1) x − 2y = −11 and (2) 2x + y = 13.

• To eliminate y, multiply (1) by 1 and (2) by −2: x − 2y = −11 and −4x − 2y = −26.

• Subtract the second from the first: (5)x = 15, so x = 3.

• Substitute x = 3 into (1): 3 − 2y = −11, so y = 7.

• Check in (2): 2 × 3 + 1 × 7 = 13 ✓.

• Q = (3, 7).

(c) The diagram shows a rectangle ABCE. The point D lies on AB. Length AE = 70 cm, length AB = 1.3 m, and angle AED = 28°. Find the perimeter of triangle CDE. Give your answer in centimetres, correct to one decimal place. 70 cm 1.3 m A B C E D 28° Diagram is NOT to scale

(c) Answer: 325.5 cm

• Evidence level: Merit (relate the triangles in the diagram; work in one unit; choose the correct trigonometric ratio).

• Change to one unit: AB = 1.3 m = 130 cm.

• Triangle AED is right-angled at A. cos 28° = AE/ED, so ED = 70 ÷ cos 28° = 79.28 cm.

• tan 28° = AD/AE, so AD = 70 × tan 28° = 37.22 cm. Then DB = AB − AD = 130 − 37.22 = 92.78 cm.

• Triangle DBC is right-angled at B with BC = AE = 70 cm: DC² = 92.78² + 70², so DC = 116.22 cm (Pythagoras).

• EC is the top side of the rectangle, so EC = AB = 130 cm.

• Perimeter = ED + DC + EC = 79.28 + 116.22 + 130.00 = 325.5 cm (1 d.p.).

(d) The relationship between the x and y values in a pattern is shown in the table below. Find an equation that gives y for any value of x. Show working to support your answer.

(d) Answer: y = 2^x + 2

• Evidence level: Merit (spot the multiplying pattern in the differences and build the equation).

• The differences between consecutive y values are 2, 4, 8, 16.

• Each difference is 2 times the one before, so the pattern is exponential with base b = 2: y = a × 2^x + c.

• The first difference is a(2^2 − 2^1) = 2, so a = 1.

• Use the first point: 4 = 1 × 2^1 + c, so c = 2.

• Check on the last point: 1 × 2^5 + 2 = 34 ✓.

• Equation: y = 2^x + 2.

(e) The graph of y = 2^x + 2 could be drawn for all values of x. Identify THREE different features of the graph, using the equation.

(e) Answer: y-intercept (0, 3); asymptote y = 2; increasing and curving upwards

• Evidence level: Excellence (link the structure of the equation to features of its graph).