If the trains were travelling in the same direction, what would be the time occupied by the faster train in passing the slower train? 6. A clockmaker was asked how many clocks he had, and replied: I have more than 500 but less than 1000; if I count them four by four, or five by five, or six by six, there remains one over, if I count them by sevens none remain over. How many clocks had he? 7. Show that (ax+by+cz)2+bc ( y − z)2+ca ( z − x)2 + ab (x − y)2 8. Determine c so that the equation in x has zero for a root. With this value of c solve the equation completely. 9. Solve the simultaneous equations x (b −c)+by − c =0, y(c-a)-ax+c=0. Show that the values which satisfy the above equations will also satisfy the equation ax - by + a − b = 0. 10. Prove the Index Law, namely that am × an = am+n, when m and n are positive integers. m If the Index Law be assumed to hold good for all values of the indices, find what meanings must be assigned to ao, a ̄m, and añ. -a Multiply a3 – a3¿13 – 263 by 2a3 – a3¿3 – 63, and divide the product by 2a3 – 3a3b3 – 2b3. II. Find the arithmetic mean of x and y. -36, 26 - 3c, and If a, b, c, are in arithmetical progression, show that 2a – 36-4c will also be in arithmetical progression. Show that 3(1 + 3+ 5+ ... + 99) = 101 + 103 + 105+ + 199. ... one of them valid when Use one of the expansions to find its value to the nearest millionth when x = 19. II. PART I. SECOND PAPER. N.B.-In questions on Geometry ordinary abbreviations may be employed, but the method of proof must be geometrical. Proofs other than Euclid's must not violate Euclid's sequence of propositions. In the absence of special directions to Candidates, any of the propositions within the limits prescribed for Examination may be used in the solution of deductions. I. (i.) Define a parallelogram. (ii.) If, of the sides of a quadrilateral, one pair is parallel and the other pair equal, prove that any angle is either equal or supplementary to the opposite angle. 2. (i.) Prove that the exterior angle of a triangle equals the sum of the two interior and opposite angles. (ii.) In a certain quadrilateral the sum of two internal adjacent angles is 200°. Find, in degrees, the size of the obtuse angle contained by the bisectors of the two remaining internal angles of the quadrilateral. 3. (i.) Show, with accompanying proof, how to construct a square equal to a given rectilinear figure. (ii.) By means of a figure carefully drawn to scale, apply the construction in (i.) to find, as accurately as you can, the lengths of the sides of a rectangle whose area is 9 sq. in. and whose perimeter (the sum of the sides) is 14 inches. A proof is not required. 4. Prove that the sum of two opposite angles of a quadrilateral inscribed in a circle equals two right angles. 5. (i.) Give, without the proof, the construction for drawing a tangent to a circle from an external point. (ii.) P, C and L denote respectively a given point, circle and unlimited straight line, P and C lying on the same side of L. Find, with proof, a point Q in Z such that PQ and a tangent from Q to C (not in the same line with PQ) make equal angles with L. Only one solution is required. 6. Show how to circumscribe a circle about a given triangle ABC. If O is the centre of this circle, and the lines AO, BO, CO produced meet the circumference in A', B', C', show that the triangle A'B'C' is in every respect equal to ABC. 7. Show how to find a mean proportional between two given right lines. O is a point outside a given circle; through O is drawn any right line meeting the circle in A and B ; state (without proof) why the mean proportional between OA and OB is the same for all lines drawn through 0, and exhibit this mean proportional. 8. Prove that the area of a trapezium is equal to half the sum of the parallel sides multiplied by the perpendicular distance between them. The parallel sides of a trapezium are 9 and 30 feet long, and the other sides are 17 and 10 feet long; find its area. 9. On the base of a hemisphere of radius is constructed a right cone whose volume is equal to that of the hemisphere; what is the height of the cone? IO. A spherical shell I foot in external diameter and 2 inches thick is made of metal having a mass of 504 pounds per cubic foot; find (to the nearest pound) the mass of metal in the shell. (The circumference of a circle is 3.14159 times the diameter.) III. PART I. THIRD PAPER. I. State the most approximate value you know for the numerical quantity π, and compare it with the value of the fraction Calculate the value of a radian expressed in seconds. 2. expressed as Express all the other trigonometric functions of A in terms of sin A. Why must the double sign ± be prefixed to all the square roots occurring in your expressions? Write down a general formula for all angles whose sine is equal to expressed either in degrees or in radians. 3. Simplify (sec- cose) (cosec @ – sin @) (tan+cot 0) and find a positive angle @ less than 180° satisfying the equation 4. I - cos 0=√√√3 sin 0. Write down and prove the formula giving tan (A+B) in terms of tan A and tan B. The perpendicular from the vertex of a triangle on the base is 6 inches long, and it divides the base into segments which are 2 and 3 inches long respectively. Find the tangent of the vertical angle of the triangle. 5. An observer wishing to determine the length of an object in the horizontal plane through his eye, finds that the object subtends the angle a at his eye when he is in a certain position A. He then finds two other positions B and C where the object subtends the same angle a. Express the length of the object in terms of the sides of the triangle ABC and of the angle a. 6. What is meant by the supplement of an angle? What is the supplement of an angle of 330°? Draw a figure showing the angle and its supplement, and prove that the sines of these two angles are algebraically equal. Find cosec 630°. 7. Prove the formula a2 = (b + c − 2 cos + √bc)(b+c+2 cos 2 2 Apply it to find the side a of a triangle when 6=132.5 feet, c=97.32 feet, A = 37° 46', as accurately as the tables permit. 8. Find an expression for the difference in area of the two triangles which can be drawn having two given sides a, c and a given angle A opposite the smaller side a. 9. Explain the meaning of logam. Prove that logam × logma = 1. Without using the tables find the characteristics of (1) log715914, (2) logo 00187. IO. Throw the expression 5 sin 0+3584 cos 0 5 sin 0-3 584 cos 0 into a form suitable to logarithmic calculation when different values of 0 are introduced, and use your form to evaluate the expression when ℗ = 71° 59′ correct to three places of decimals. |