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Midterm Examination for Mathematics

1. (a) Simplify (i)

x? y 8x ? 8 y

(ii)

x(3x- 2) - (3x2 - 5).

(b) Solve the equations (i) 3t ? 4 ? 7 ? 2(t

? 3) (ii)

18 16 ? ?1 q q?2

(iii)

5 x 2 ? x ? 7 ? 0 , giving each solution correct to 2 decimal places.

(c) Factorise 5px ¨C 7qx + 10py ¨C 14qy.

(d) (i) When x = ¨C2, which of the two expressions, 3x + 4 and 2 ¨C x, has the greater value? You must show your working.

(ii) Solve the inequality 3x + 4 < 2 ¨C x.

(e) It is given that m ? 2.1?10 and n ? 3?10 . Express your answers in standard form,

7 4

find (i) m? n

(ii) n ? m

2

2. (a) The rate of exchange between pounds (? and dollars ($) is ? = $1.87. ) 1 The rate of exchange between pounds (? and euros (€) is ? = €1.21. ) 1 (i) Catherine changes ? into dollars. 500 Calculate how many dollars she receives.

(ii) Esther changes €726 into pounds. Calculate how many pounds she receives.

(iii) Rose changes $850 into euros. Calculate how many euros she receives.

(b) Matthew changes $770 into rupees. He receives 40 000 rupees. How many rupees did he receive for each dollar?

(c) (i) Lily bought a car for $13 500. She paid for it in 36 equal monthly payments. Calculate the amount she paid each month.

(ii) George bought a car for $27 000. He borrowed the $27 000 at 15% per year simple interest for 3 years. He repaid the total amount in 36 equal monthly payments. Calculate the amount he paid each month.

3. (a)

? ? ? ,2,3,4,5,6,7,8,9,10,11,12,13,14,15? 1

L = {x : x is an odd number} M = {x : x is a multiple of 3} (i) Write down (a) L ? M (b) L ? M

'

(ii)

A number n is chosen at random from Find the probability that n ? L ? M .

?

(b) In a survey, a number of people were asked ¡°Do you own a car?¡± and ¡°Do you own a bicycle?¡±. The Venn diagram shows the set C of car owners and the set B of bicycle owners. The letters p, q and x are the numbers of people in each subset. 11 people owned neither a car nor a bicycle.

A total of 66 people owned a car. 4 times as many people owned a car only as owned a bicycle only. (i) Write down expressions, in terms of x, for (a) p, (b) q.

(ii) A total of 27 people owned a bicycle. Calculate (a) x, (b) the total number of people who were in the survey.

4. (a) Evaluate

? 2? ? 1 ? ? ? ? ? (i) 3? 4 ? ? 2? 6 ? ? 0 ? ? ? 3? ? ? ? ?

?0 4? ? ? (ii) ?1 3 4 ?? 3 1 ? ?5 0? ? ?

(b) Given that A ? ? ? (i) Find A

?1

? 3 2? ? ? 2 1? ? and B ? ? ? ? ? 3 2? , ? ?5 4? ? ?

(ii) Find the matrix C if 4 A ? 2 BC ? O where O is the zero matrix.

?1

5. Two shops sell the same type of CD-R and photocpy paper. A box of 5 CD-R cost $3.70 at shop A while a ream of photocopy paper costs $6.50 at shop B. Peter and Mary plan to buy the Following quantities: CD-R (in boxes) Peter Mary 6 4 photocopy paper (in reams) 2 3

Peter needs to pay $35.80 for his supply regardless of whether he buys from Shop A or Shop B. It is given that P ? ? ?

? 6 2? ? 3.70 b ? ? 35.80 35.80? ?, Q ? ? ? and R ? ? ? ? a ? ? 35.20 34.70? . ? 6.50? ? 4 3? ? ? ?

(a) Write down an equation connecting P, Q and R.

(b) What is the significance of the elements in the second row of the matrix R?

(c) Find the values of a and b and explain the significance of these values.

6. The numbers of 225 and 540, written as the products of their prime factors, are

225 ? 32 ? 52 , 540 ? 22 ? 33 ? 5

(i) Write 2250 as the products of their prime factors.

(ii)

Find the smallest positive integer value of n for which 540n is a multiple of 225.

7. A light aircraft flew from Maseru to Nata and returned to Maseru. (a) The distance from Maseru to Nata is 1080 km. (i) On the outward flight, the average speed of the aircraft was x kilometres per hour. Write down an expression, in terms of x, for the time taken in hours.

(ii) On the return flight, the average speed was 30 km/h greater than the average speed on the outward flight. Write down an expression, in terms of x, for the time taken, in hours, on the return flight.

(b) The time taken on the return flight was half an hour less than the time taken on the outward flight. Form an equation in x and show that it reduces to x2+30x-64800=0.

(c) Solve the equation x2 + 30x¨C64800 = 0.

(d) Calculate (i) the time taken, in hours, on the outward flight,

(ii) the average speed for the whole flight from Maseru to Nata and back to Maseru.

8. The diagram is the speed-time graph for the first 20 seconds of a Cyclist's journey.

(a) Calculate the distance travelled in the first 16 seconds.

(b) By drawing a tangent, find the acceleration of the cyclist when t=18.

(c) On the grid in the answer space, sketch the distance-time graph for the first 16 seconds of the cyclist's journey.

9.

The diagram shows the speed-time graphs of two objects, A and B, for the first 10 seconds of their motion. Object A travelled at a constant speed of 12 m/s throughout the 10 seconds. Object B started from rest, and accelerated at a constant rate, attaining a speed of 20 m/s after 5 seconds. It then travelled at a constant speed of 20 m/s. (a) Calculate (i) the distance travelled by object B during the first 5 seconds of its motion,

(ii) the average speed of object B for the first 10 seconds of its motion,

(iii) the value of t when both objects were travelling at the same speed,

(iv) the value of t when both objects had travelled the same distance.

(b) The diagram below shows the distance-time graph for object B.

In the diagram, OP is a curve and PQ is a straight line.

(i) State the values of d1 and d2.

(ii) What does the gradient of the straight line PQ represent?

(iii) Write down the gradient of the tangent to the curve at t = 2.5.

(c) After 10 seconds, both objects slowed down at the same constant rate. Object A came to rest after a further 9 seconds. After how many seconds from the start of its motion did object B come to rest?

10. The table below shows some values of x and the corresponding values of y for

y?

2x 4

(a) Calculate the values of m and n.

(b) Using a scale of 2 cm to represent 1 unit, draw a horizontal x-axis for ? 1 ? x ? 5 . Using a scale of 2 cm to represent 1 unit, draw a vertical y-axis for 0 ? y ? 8 . On your axes, plot the points given in the table and join them with a smooth curve. (c) Use your graph to solve the equations (i)

2x ? 3, 4

x

(ii) 2 ? 6 . (d) The equation y ?

2x t can be written in the form y ? 2 . 4

(i) Find an expression for t in terms of x. (ii) Hence, find the equation of the line that can be drawn on your graph to evaluate y when

3 t?? . 4

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