H
Bearsden Academy
Mathematics Department
Mathematics Higher Paper 2 Practice Prelim B Time allowed 1 hour 10 minutes NATIONAL
QUALIFICATIONS
Read carefully 1 Calculators may be used in this paper. 2 Full credit will be given only where the solution contains appropriate working. 3 Answers obtained by readings from scale drawings will not receive any credit. FORMULAE LIST Circle: The equation x 2 + y 2 + 2 gx + 2 fy + c = 0 represents a circle centre ( − g , − f ) and radius g 2 + f 2 − c . The equation ( x − a)2 + ( y − b)2 = r 2 represents a circle centre ( a , b) and radius r. Trigonometric formulae: sin(A ± B) = sin A cos B ± cos A sin B cos(A ± B) = cos A cos B m sin A sin B
sin 2A = 2 sin A cos A
cos 2A = cos 2 A − sin 2 A
= 2 cos 2 A − 1
= 1 − 2 sin 2 A
ALL questions should be attempted. Marks 16
, x > 0. x
1. A curve has equation y = x −
Find the equation of the tangent at the point where x = 4. 6 2. Triangles ACD and BCD are right angled at D with angles p and q and lengths as shown in the diagram. (a) Show that the exact value of sin( p + q) is (b) Calculate the exact values of: (i) cos( p + q); (ii) tan( p + q). 84
. 85
4 3 3. (a) A chord joins the points A(1, 0) and B(5, 4) on the circle as shown in the diagram. Show that the equation of the perpendicular bisector of chord AB is x + y = 5. 4 (b) The point C is the centre of the circle. The tangent at the point A on the circle has equation x + 3 y = 1. Find the equation of the radius CA. 4 (c) (i) Determine the coordinates of the point C. (ii) Find the equation of the circle. 4 4. The curves with equations y = x 2 and y = 2 x 2 − 9 intersect at K and L as shown. 5. Calculate the area enclosed between the curves. Marks 8 A zookeeper wants to fence off six individual animal pens. Each pen is a rectangle measuring x metres by y metres, as shown in the diagram. (a) (i) Express the total length of fencing in terms of x and y. (b) Find the values of x and y which give the maximum area and write down this maximum area. (ii) Given that the total length of fencing is 360 m, show that the total 16
area, A m2, of the six pens is given by A( x) = 240 x − x 2 . 3
3 6 6. (a) Write x 2 − 10 x + 27 in the form ( x + a)2 + b. (b) Hence show that the function g( x) = x 3 − 5x 2 + 27 x − 2 is always increasing. 1
3
2 4 Marks 7. Two functions f and g, are defined by f ( x) = k sin 2 x and g( x) = sin x where k > 1. The diagram shows the graphs of y = f ( x) and y = g( x) intersecting at 0, A, B, C and D. Show that, at A and C, cos x =
1
. 2k
6 8. The diagram shows a sketch of a parabola passing through ( −1, 0), (0, p) and ( p , 0). (a) Show that the equation of the parabola is y = p + ( p − 1)x − x 2 . (b) For what value of p will the line y = x + p be a tangent to 3 3 this curve? End of Question Paper