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4. Prove :
(1) sin 3A = 3 sin A - 4 (sin A)3.
(2) sin 54°=V5+1
sin (A - C) sin (B-A) sin (C-B) (3)
cos A cos C cos B cos A cos C cos B 5. Prove that in every circle the angle subtended by an arc equal to the radius is an invariable angle. Express that angle in degrees and decimals of degrees, and find the circular measure of one minute. On a circle 10 feet in radius it was found that an angle of 22° 30' was subtended by an arc 3 feet 115 inches in length; hence calculate, to four decimal places, the numerical ratio of the circumference of a circle to its diameter.
6. If a, b, c be the sides subtending respectively the angles A, B, C of a plane triangle, Prove :
: 45° when C is a right angle.
7. If (8) be the radius of the circle inscribed in the triangle ABC, prove;
b.sin sin (1) r=
(2) r= a+b+c
8. In a right-angled triangle ABC, C being the right angle, find AB, if B=30° and BC= 100 feet.
A'B'C' is also a right-angled triangle, C' the right angle and B'=300, find A'C' when the area of the triangle A'B'C' is three times the area of ABC.
9. When two sides and an included angle of a triangle are given, investigate a formula for determining the other two angles. Shew that for this determination it is not necessary to know the absolute lengths of the two sides, provided their ratio is given.
Ex. The included angle is 70° 30', the ratio of the containing sides is 5 : 3, find the other angles. Given
log10 2 = *3010300, log cot 35°15'=10'1507464, log tan 19° 28' 50" = 10'5486864.
A tower stood at the foot of an inclined plane whose inclination to the horizon was 9°; a line was measured straight up the incline from the foot of the tower of 100 feet in length, and at the upper extremity of this line the tower subtended an angle of 54°; find the height of the tower. Note. For sin 54° see question 4.
Shew that the logarithms of proper fractions are negative. Express the true value of the logarithm of ato to the base 10. How would it be expressed with a negative characteristic?—Since sin 30°=, explain why the logarithm of sin 30 is tabulated 9•6989700. Given that log tan 38° 16' is tabulated 9.8969714, determine log cotan 38° 16'. Given
log tan 38° 16' 10"=9*8970147,
log tan 38° 16' 20"=9.8970580, find the angle whose logarithmic tangent is 9.8970365.
Note. —Log10 2 is given in question 9.
V. PURE MATHEMATICS. (1.) 1. How do you measure the inclination of a straight line to a plane, and of a plane to a plane?
If two planes cut one another, their common section is a straight line.
3. Name and define the five regular solid figures; and shew that there cannot be more than five.
4. If two straight lines be cut by three parallel planes, they shall be cut in the same ratio. 5. Find the roots of the equation
373 + 3x + 36=0, and shew that every equation of the form 28+qx+rro has two impossible roots and one negative root. 6. Draw the straight lines
2x + 3y=4, and
3x - 2y=5; and prove that they are at right angles to each other.
7. Prove that the subnormal in the parabola is constant; and shew how to draw a normal to the curve at any given point.
8. A quadrilateral is inscribed in a circle, one of its diagonals coinciding with the diameter of the circle; find in terms of its sides (1) the radius of the circle, (2) the other diagonal, (3) the area of the quadrilateral. 9. Draw the circle represented by the equation
x2 = zay - ya, and transform the equation into polar co-ordinates.
Prove (1) geometrically, (2) analytically, that the perpendiculars dropped from the foci upon the tangent to any point of an ellipse intersect the tangent on the circumference of the circle described on the axis major as diameter.
1. Find the equation to the ellipse referred to a pair of conjugate diameters as axis, and shew that equal conjugate diameters are parallel to the lines joining the extremities of the major and minor axes.
12. Find the equation to the tangent to an hyperbola, and the locus of its intersection with the perpendicular upon it from the centre.
13. Prove the following trigonometrical formulæ (n being a positive integer):
14. Explain the method of computing the value of a=3:1415926..., and calculate e=
VI. PURE MATHEMATICS. (2.) [N.B.-Great importance will be attached to accuracy in numerical results.]
Find the algebraical expression which, when divided by x*+*-1, gives x3 – 3x2 + 4x – 7 for the quotient and 11x – 7 for the remainder.
3. Having given log102,= '301030 and log107 = '845098, find log1014, log1098,
7 4 3 5
to 25 terms,
1 I and I+ +
36 Prove that
I I.3 I 1.3.5 T 126.96.36.199 I
3 • 4
64 5. On a certain road the number of telegraph posts per mile is such that if there were one less in each mile the interval between the posts would be increased by 214 yards. Find the number of the posts per mile. 6. Prove that
cos 30° + cos 60° +cos 210° + cos 270°=1
sin 30 + 2 sin 50+ sin 70 and that
sin 50+ 2 sin 70+ sin 90 sin 70' 7. If A+B+C=90°, prove that
tan B tan C+tan Ctan A +tan A tan B=1. Hence shew that 2 (tan’A+tan B+ tanC) – 2 =(tan B - tan C)2
+(tan C -tan A)2 + (tan A - tan B)?, and that the expression tan’ A + tan’B+tanPC is never less than unity. 8. Expand each of the fractions 4.X
and (1 - 2) (1 + 2x)' (1 - x) (1 + x)' in a series of ascending powers of x, and find in each case the coefficient of x".
9. A length of 300 yards of paper, the thickness of which is the hundred and fiftieth part of an inch, is rolled up into a solid cylinder ; find approximately the diameter of the cylinder.
Find the centre of a circle cutting off three equal chords from the sides of a triangle.
In any triangle the straight line bisecting an angle, and the straight line passing through the middle point of the opposite side, perpendicular to it, meet on the circumscribing circle.
12. Having given the base of a triangle, the vertical angle, and the ratio of the sides, construct the triangle.
13. A man, walking along a straight road at the rate of three miles an hour, sees in front of him, at an elevation of 60', in the vertical plane through his path, a balloon which is travelling horizontally in the same direction at the rate of six miles an hour; ten minutes after he observes that the elevation is 30°; prove that the height of the balloon above the road is 4400 3 yards.
14. A quadrilateral ABCD can be inscribed in a circle; if E, F, G, H te the centres of the circles which circumscribe the triangles ABC, BCD, CDA, DAB, prove that the quadrilateral EFGH is also inscribable in a circle.
VII. PURE MATHEMATICS. (3.) Find the relation between the coefficients and roots of an equation. One root of the equation
*3 – 13x2 +15x+189=0 exceeds another root by 2; solve the equation.
Prove that incommensurable roots enter by pairs into an equation with rational coefficients.
If one root of such a biquadratic equation be vã+ No, where Va and Nē are dissimilar quadratic surds, prove that the other roots are
- Na+do, da-do-sa-do. 3. Prove that an odd number of roots of the equation f'(x)=0 lies between every two adjacent real roots of the equation f(x)=0.
Prove that the equation aya +23 – bx2 +o has always one positive value of y
for every value of x, and that it has two negative or two impossible values of y according as x is numerically greater than
3 4. Find the differential coefficient of xn for all values of n. Differentiate
I - X
1 + x2
a + bx' prove that 70
dy 2 5. If y=x log
dx) 6. State, without proving, the method of finding the limiting value of f(x)
for values of x which make f (x) and F(x) zero. F(x)