12. In a plane triangle the two sides are 4 3+1 and √3 - 1, and the included angle is 60°, find the remaining angles. 13. If a, b, c be the sides of a triangle, and a, ß, y the perpendiculars drawn upon them from the centre of the circumscribing circle, prove that abc αβγ 14. Find the area of a triangle in terms of its perimeter, and the radius of the inscribed circle. The area of an equilateral triangle described about a circle is four times the area of an equilateral triangle inscribed in the same circle. 15. A person wanting to calculate the perpendicular height of a cliff, takes its angular altitude 12° 30', and then measures 950 yards in a direct line towards the base, when he is stopped by a river; he then takes a second altitude, and finds it 69° 30′. Find the height of the cliff. SET V. 1. Prove that x* MATHEMATICS. (1.) 1 is divisible by xa – 1 without remainder; and divide a3(b − c) + b3(c − a) + c3(a − b) by a + b + c. 3. If m and n are two whole numbers, each of which leaves the remainder 1 when divided by 4, show that m × n leaves the remainder 1 when divided by 4. Prove also that every uneven square number leaves either the remainder 1, or the remainder 9, when divided by 16. 4. Find the highest common divisor of 3x4 10x3 + 9x3 2x and 2x4 7x3 + 2x2+8x; and the least common multiple of ac2 - 1, x2 2, x2 + x − 2. ab Show that if d is the greatest common divisor of two numbers a and b, their least common multiple is d and prove that a quadratic surd cannot be equal to the sum or difference of two dissimilar quadratic surds. 7. What are the conditions that the roots of the equation ax2+2bx + c = 0 should be (1) real, (2) equal, (3) imaginary? What do you know about the roots of this equation, (1) if a = c, (2) if a and c have opposite signs? 8. If I pay one guinea for a cubical block of marble, of which the side is one foot, what ought I to pay for another cubical block of the same marble, of which the side is equal in length to the diagonal of the first block? 9. Determine to the nearest hundredth of an inch the radius (1) of a sphere whose volume is one cubic foot, (2) of a sphere whose surface is one square foot. [In this and the following question = 3.1416.] 10. If a room be 40 feet long by 20 feet broad, and contain 12,800 cubic feet; what addition will be made to its cubic contents by throwing out a semicircular bow at one end? 11. Given a circle, show how to find the radii of the n concentric circles which divide the area of the given circle into n + 1 equal parts. If the radius of the given circle is one foot, find to the nearest hundredth of an inch the radii of the two concentric circles which divide its area into three equal parts. 12. A field is in the form of a trapezoid; its parallel sides are respectively 10 chains 30 links, and 7 chains 70 links; the distance between them is 7 chains 50 links; find the acreage. 13. If is the logarithm of a to the base b, what is the logarithm of am to the base b"? 1 Write down the logarithms of 256 and 0625 to the base 2 = 14. Given log 18 = 1.2552725, log 125 2.0969100, log 21 = 1.3222193, find the logarithms of the numbers from 2 to 9, MATHEMATICS. (2.) 1. In any obtuse-angled triangle, the square on the side subtending the obtuse angle is greater than the sum of the squares on the other two sides, by twice the rectangle contained by one of the sides, and the distance between the obtuse angle and the foot of the perpendicular let fall on the side from opposite angle. 2. Prove that one circle cannot touch another in more than one point, whether it touches on the inside or outside. 3. Divide a straight line into two parts, so that the rectangle contained by them may equal the square of the difference. 4. Describe a circle (1.) Passing through three given points, which are not in the same straight line. (2.) Passing through two given points, and touching a given circle. 5. Explain the following terms: (1.) Homologous sides. (2.) Ex æquali. (3.) Duplicate ratio. (4.) Reciprocal figures. (5.) Alternando. 6. Divide a given straight line similarly to a given divided straight line. Hence show how a given straight line may be divided into any number of equal parts. 7. The rectangle contained by the figure inscribed in a circle is contained by its opposite sides. = diagonals of a quadrilateral equal to both the rectangles = 8. Define the tangent of an angle, find the value of tan 60°, and prove that tan (180 — A): tan A, and tan (90 + A) cot A. Also trace the changes in its sign and magnitude, as A varies from 0° to 180°. 9. If A and B be each less than a right angle, and A > B, prove that cos (AB) = cos A .cos B+ sin A. sin B. Draw the figure for a case (1) when A is greater than a right angle, (2) when A is < B. 10. If the sine of an angle is given, show how to find the sine of half the angle, and explain why the general expression obtained ought to include four values. Given sin 300° 3, find sin 150°. = 2 12. The angles in one regular polygon are twice as many as in another regular polygon; and an angle of the former is to an angle of the latter as 3 : 2. Find the number of sides. 13. In an oblique-angled triangle, given a = 145, b = 178, B = 41° 10′, find A. and show whether there is any ambiguity in the result. 14. In any triangle given a, b, and A, find an equation for determining c, and discuss its values, according as (a) is equal to, less than, or greater than b sin A. |