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UNIVERSITY OF LONDON
GENERAL CERTIFICATE OF EDUCATION
EXAMINATION
JANUARY 1968
Ordinary Level
PURE MATHEMATICS II
Syllabus B
Two and a half hours
Answer ALL questions in Section A and any FOUR questions in Section B.
Credit will be given for the orderly presentation of material; candidates who neglect this essential will be penalised.
All necessary working must be shown.
Section A
1. (i) Factorise 3x2 16x 12.
(ii) Solve the equation EMBED Equation.3 EMBED Equation.3 = EMBED Equation.3 .
(iii) Use tables to evaluate (2(068)3 EMBED Equation.3 .
TE&S 65/1320 11/2/100/17800
( 1968 University of London
2. (i) Evaluate (a) 26 24, (b) 26 ( 23, (c) 26 ( 20.
(ii) The scale of a map is 2 cm to represent 1 km. Express this as a fraction in the form EMBED Equation.3 where n is a whole number between 1 and 10.
(iii) In a triangle ABC the point D on AC is such that the angle ABD is equal to the angle ACB. Given that AB = 6 in., AC = 9 in. and the area of the triangle ABD is 10 sq in., calculate the area of the triangle ACB.
3. (i) From the top of a vertical cliff, 450 ft above sea level, the angle of depression of a marker buoy at sea is 17( 33(. Calculate, correct to the nearest 10 ft, the distance of the buoy from the foot of the cliff.
(ii) Solve the equations
2(2x 1(7y = 17(3
x + 3y = 11
(iii) In a triangle PQR the external bisector of the angle PQR meets PR produced at Y. If PQ= 8 in., QR = 3 in. and PY = 12 in., calculate the length of PR.
4. (i) Solve the equation 2x2 11x = 3, giving the roots correct to one decimal place.
(ii) A motorist on the Continent mistakenly asked to have his tyres inflated to a pressure of 2(10 kg per sq cm, when the correct pressure should have been 2(01 kg per sq cm. Calculate, correct to one decimal place, his percentage error.
5. (i) A circle with centre O and radius a is inscribed in a square of side 2a. One side PQ of the square touches the circle at A and the line OP cuts the circle at K.
Calculate, in terms of a,
(a) the area of the triangle OAP,
(b) the area of the triangle OAK,
(c) the area of the sector OAK of the circle.
[Take ( as 3 EMBED Equation.3 .]
(ii) Cylindrical cans of diameter 3 EMBED Equation.3 in. and height 4 in. are filled from a drum containing 1cu. ft. of oil. Calculate the number of cans filled and the quantity of oil left over.
[Take ( as 3 EMBED Equation.3 .]
6. Ruler and compasses only may be used in this question.
Construct a triangle ABC in which AB = 4 in., AC = 3 in. and the angle BAC = 60(. Construct two points P and Q which lie on the inscribed circle of the triangle ABC and which are equidistant from the two points B and C. Measure the length of PQ.
Section B
Answer FOUR questions in this section.
7. The base ABCD of a rectangular prism is a square of side 9 in. and lies in a horizontal plane. The four vertical edges of the prism AP, BQ, CR and DS are each of length 4 in.
Calculate
(a) the length of AR,
(b) the angle which AR makes with the base ABCD,
(c) the angle between the two planes AQC and ABC.
8. In the figure, not drawn to scale, PQ = 17 cm, QR = 21 cm and RP = 10 cm. PN is perpendicular to QR and RS is drawn parallel to NP to intersect QP produced at S.
Without using tables,
(a) calculate cos PQR and sin PQR, leaving your answers in ratio form,
(b) calculate the length of RS,
(c) show that the area of the trapezium PNRS is 57(6 sq cm.
9. The velocity v ft per sec of a body moving in a straight line is at any instant given by the sum of two terms, one of which is proportional to the time t seconds which has elapsed since the body started moving, and the other is proportional to the square of the time t.
Given that v = 68 when t = 1 and that v = 104 when t = 2 form an equation for v in terms of t. Use your equation to calculate
(a) the value of t when the body comes to rest,
(b) the acceleration of the body 2 seconds after the start.
10. ABCD is a cyclic quadrilateral. If S is the midpoint of the minor arc CD and if SA and SB intersect BD and AC respectively at P and Q, prove that
(a) SA and SB bisect the angles DAC and DBC respectively,
(b) the quadrilateral APQB is cyclic,
(c) PQ is parallel to DC.
11. The sum of the radii of two circles is 6 cm. If the radius of one of the circles is x cm form an equation for A, the sum of the areas of the two circles.
(a) Prove that A is a minimum when x = 3 cm.
(b) When the value of A is 20( calculate the radius of each circle.
(c) Calculate the rate of increase of A with respect to x when x = 3(5 cm, leaving ( as a factor of your answer.
12. Calculate
(a) the circumference of the circle of latitude 25(N on the earths surface, giving your answer to the nearest 100 miles,
(b) the latitude, to the nearest degree, of a circle of latitude whose circumference is 17,600 miles,
(c) the distance, correct to 3 significant figures, along a meridian between the two circles of latitude 25(N and 15(S respectively.
[Take ( as 3(142 and the earth to be a sphere of radius 3,960 miles.]
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