SAMPLE 2

General Certificate of Education (GCE) Examination Papers "Advanced" (A) and "Scholarship" (S) Levels

Set by Cambridge University examining body for mathematics; by Oxford University examining body for physics and chemistry. Age: 18.

Answers to not more than eight questions are to be given up. A pass mark can be obtained by good answers to about four questions, or their equivalent.

Begin each answer on a fresh sheet of paper and arrange your answers in numerical order.

Mathematical tables and squared paper are provided.

4

1. (i) If x=3+/5 find the exact values of x+- and x2+

2. (i) The least value of the function x2+px+q is 3, and this occurs when x=-2. Find the values of p and q.

(ii) Find for what values of k the equation

x2+(3k-7)x+(2k+6)=0

will have real roots in x.

3. (i) Find the coefficients of 25 and 210 in the expansion of (1−x)a (1+x)2 in powers of x.

4. Find the sum of n terms of the geometric progression

[If a general formula is used, it must first be proved.]

Deduce the sum to infinity of this series.

Find the least number of terms of the series which must be taken for their sum to exceed 1888.

2999

5. How many five-letter arrangements can be made from the letters of the word ROOMS? How many four-letter arrangements can be made from a selection of the same letters, (a) if both o's are included, (b) if no restriction is imposed on the choice of letters?

By taking a particular value for 0, or otherwise, prove that 4 cos 36° cos 72° =1, and deduce that cos 36° is a root of the equation

8c-4c-1=0.

By removing the factor 2c+1, obtain the value of cos 36° in surd form.

7. Assuming the sine rule for any triangle ABC, prove that its area is given by the formula c2 sin A sin B cosec C.

The point A is 4 miles due west of the point B. From a point C in the same horizontal plane the bearings of A and B are S 18° 21′ W and S 49° 38′ E respectively. Calculate the area of triangle ABC and the perpendicular distance of C from AB, each correct to three significant figures.

8. (i) Find all possible pairs of angles A and B between 0° and 360° such that A-B=240° and

cos A+cos B=0·825.

(ii) Find all angles x between 0° and 360° such that

2 tan x=4 cot x+cosec x.

9. By drawing graphs of the functions 2 cos 30+3 and 4 sin 20 for values of 0 from 0° to 90°, estimate as accurately as you can the roots, in this range, of the equation

2 cos 30+3=4 sin 20.

10. A geographical globe is a sphere with centre O and of radius R. Through two points A and B on the equator, whose · difference in longitude is 2a, meridians (i.e. great half-circles through the poles) are drawn, intersecting the circle of latitude λ (North) at P and Q respectively. (Thus the angle contained between the planes of these meridians is 2x, and the angles POA, QOB are each λ.) If the angle contained between the planes OPQ, OAB is o, prove that tan tan λ sec x.

=

If the angle POQ is 20, prove that sin 0=cos à sin «.

Answers to not more than eight questions are to be given up. A pass

mark can be obtained by good answers to about four questions, or their equivalent.

Begin each answer on a fresh sheet of paper and arrange your answers in numerical order.

Mathematical tables and squared paper are provided.

1. The vertices of a triangle are A (x1, y1), B (xq, Y2), C (X3, Ys). The mid-point of the side BC is M, and G is the point on AM

AG such that =2. Calculate the coordinates of G, and deduce GM

that the three medians of the triangle are concurrent.

The sides BC, CA, AB of the triangle are produced to P, Q, R respectively so that

BP CQ AR

k.

Calculate the coordinates of P, Q, and R and show that G is also

the point of concurrence of the medians of the triangle PQR.

2. Show by calculation that the point P (-9, -3) lies inside the circle

x2+ y2+4x-8y-80=0.

Find the equations of the two circles which pass through P, touch the given circle, and have radii of 5 units.

3. The point P in the first quadrant lies on the ellipse x2/a2+y2/b2=1. The points A' (—a, 0) and A (a, 0) are the extremities of the major axis of the ellipse, and O is the origin. The tangent to the ellipse at P meets the axis of y at Q and meets the line x=a at T. The chord A'P meets the axis of y at M, and when produced meets the line xa at R. Prove that (i) AT = TR, (ii) OQ3- MQ2= b2.

4. The point P on the hyperbola xy=c2 is such that the tangent to the hyperbola at P passes through the focus of the parabola y2 = 4ax. Find the coordinates of P in terms of a and c.

If P also lies on the parabola, prove that a=2c4, and calculate the acute angle between the tangents to the two curves at P. 5. (i) Differentiate with respect to x

(a) x3 (1-2x2), (b) √ sec (px+q).

(ii) Using the method of small increments, find the value of logo tan 45° 1' correct to five decimal places.

[Take π=3·142 and log10 e=0·4343.]

6. (i) If x=a+b sin 0 and y=a-b cos 0, where a and b are constants, find

dy

dx

in terms of 0, and prove that

(ii) The formula 0=27+36e-t/6 gives the temperature 0 degrees of a body t minutes after being placed in a certain room. Find after how many minutes the rate of cooling of the body will have fallen below one degree per minute, giving the answer correct to one decimal place.

7. Find the real root of the equation +3x-6=0, correct to two decimal places.