1.       (a)     Expand and simplify (7 + √5)(3 – √5).

(3)

(b)     Express

7 +   3 +

in the form a + b √5, where a and b are integers.

(3) (Total 6 marks)

Show that x2 + 6x + 11 can be written as

(x + p)2 + q

where p and q are integers to be found.

(2)

(b)     In the space at the top of page 7, sketch the curve with equation y = x2 + 6x + 11, showing

clearly any intersections with the coordinate axes.

(2)

(c)     Find the value of the discriminant of x2 + 6x + 11

(2) (Total 6 marks)

3.       Solve the simultaneous equations

y – 3x + 2 = 0

y2 – x – 6x2 = 0

(Total 7 marks)

4.       Find the set of values of x for which

(a)     3(x – 2) < 8 – 2x

(2)

(b)     (2x – 7)(1 + x) < 0

(3)

(c)     both 3(x – 2) < 8 – 2x and (2x – 7)(1 + x) < 0

(1) (Total 6 marks)

5.       (a)     Find an equation of the line joining A (7, 4) and B (2, 0), giving your answer in the form ax + by + c = 0, where a, b and c are integers.

(3)

(b)     Find the length of AB, leaving your answer in surd form.

(2)

The point C has coordinates (2, t), where t > 0, and AC = AB.

(c)     Find the value of t.

(1)

(d)     Find the area of triangle ABC.

(2) (Total 8 marks)

6.

An emblem, as shown in the diagram above, consists of a triangle ABC joined to a sector CBD of a circle with radius 4 cm and centre B. The points A, B and D lie on a straight line with AB = 5 cm and BD = 4 cm. Angle BAC = 0.6 radians and AC is the longest side of the triangle ABC.

(a)      Show that angle ABC = 1.76 radians, correct to 3 significant figures.

(4)

(b)     Find the area of the emblem.

(3) (Total 7 marks)

7.       An experiment consists of selecting a ball from a bag and spinning a coin. The bag contains 5   red balls and 7 blue balls. A ball is selected at random from the bag, its colour is noted and then the ball is returned to the bag.

When a red ball is selected, a biased coin with probability    of landing heads is spun.

When a blue ball is selected a fair coin is spun.

(a)     Complete the tree diagram below to show the possible outcomes and associated

probabilities.

(2

Shivani selects a ball and spins the appropriate coin.

(b)     Find the probability that she obtains a head.

(2)

Given that Tom selected a ball at random and obtained a head when he spun the appropriate coin,

(c)     find the probability that Tom selected a red ball.

(3

Shivani and Tom each repeat this experiment.

(d)     Find the probability that the colour of the ball Shivani selects is the same as the colour of the ball Tom selects.

(3) (Total 10 marks)

8.       A car of mass 800 kg pulls a trailer of mass 200 kg along a straight horizontal road using a light towbar which is parallel to the road. The horizontal resistances to motion of the car and the        trailer have magnitudes 400 N and 200 N respectively. The engine of the car produces a             constant horizontal driving force on the car of magnitude 1200 N. Find

(a)     the acceleration of the car and trailer,

(3)

(b)     the magnitude of the tension in the towbar.

(3)

The car is moving along the road when the driver sees a hazard ahead. He reduces the force       produced by the engine to zero and applies the brakes. The brakes produce a force on the car of magnitude F newtons and the car and trailer decelerate. Given that the resistances to motion are unchanged and the magnitude of the thrust in the towbar is 100 N,

(c)     find the value of F.

(7) (Total 13 marks)

9.       A ball is projected vertically upwards with a speed of 14.7 ms– 1 from a point which is 49 m above horizontal ground. Modelling the ball as a particle moving freely under gravity, find

(a)     the greatest height, above the ground, reached by the ball,

(4)

(b)     the speed with which the ball first strikes the ground,

(3)

(c)     the total time from when the ball is projected to when it first strikes the ground.

(3)

(Total 10 marks)