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Junior, Senior, and Higher Local.
Junior Work, Nos. 1-410 inclusive.
1. Prove that a -(6-c)=a-b+c, and simplify
If a= $, b=2, x=1', and y=$, find the value of (a + b) 50 – 6)y? – a Vy (x – 6) + x.
4. Find the H.C.F. of a3 – 2a- 4 and 23 - Q2 _-4.
7. A boat's crew can row 9 miles an hour. What is the speed of the current if it takes them 24 hours to row 9 miles up stream and 9 miles down ?
1 8. Define a logarithm. Prove that logab =
log, a logarithm of 32 to base 8, and log 8 to basé 3/2.
9. The 2nd, 4th, and 7th terms of an A.P. are in G.P. Find the common ratio of this latter series.
10. Sum to n terms and to infinity, when possible, the series :
4 +31 + 2 +
11. Find for what other value of x the expression
the same value as it has when x=a.
12. Prove that the number of all possible combinations of n things is 2n – 1.
14. Show that 1 + 2x4 is never less than x2 + 2x3.
ANSWERS TO APRIL PAPER.
1. 208 + x^yt + y8 ; (cr2 + 2xy + 4y?) (* - 4x*y* + 16y). 2. 4.2-1. 3. 6x4 – 7x3 + x. 4. 2, 5, and 13. 5. (i) x = - 6 ; (ii)
+1, y = +2. 7. 1+
9. (a+b+c) (b+c-a) (c+a-b) (a+b-c). 11. n = 15," = = 6. 14. 1
; i, 4, ils 2-5 3+
TRIGONOMETRY; STATICS ; HYDROSTATICS.
Junior and Senior.
1. If a be the circular measure of a positive angle less than 90°, prove (i) O is greater than sin 0, and less than tan@ ; (ii) sin e
= 1 when 0 is very small. If a contain n degrees find the
A 2. Show that 2 sin
= + v1+ sin A + -sin A. Which
signs must be taken when A lies between 0° and 90° ?
A 3. In a triangle show that cos
= 3(s—a) where 28=a+ s
2 +c. If the sides are 32 + x + 1, 2 x + 1, and 22-1, show that the greatest angle is 120°.
4. Find the centre of gravity of the perimeter of a triangle.
5. Having given the C.G. of a body and of a portion of it, find that of the remainder. ABCD is a rectangle; A is joined to E, the middle point of CD. Find the C.G. of the quadrilateral АВСЕ.
6. When a weight is supported on a smooth inclined plane by a force along the plane, the force is to the weight as the height of the plane to its length.
7. Define centre of pressure, and distinguish between whole and resultant pressure.
8. A rectangle has one side in the surface of a liquid, divide it by a horizontal line into two parts on which the pressures are equal.
9. Find the resultant pressure of a liqnid on the surface of a solid either wholly or partially immersed. Hence find the conditions of equilibrium of a floating body.
STATICS, DYNAMICS, ASTRONOMY.
1. An inclined plane makes an angle of 45° with the horizon, a weight W is supported by a force P, such that 2 W?= 3 P?; find the direction in which P acts.
2. In a false balance, the arms being unequal, a weight is measured in one scale by P lbs. and in the other by Q lbs. Show that the arms are as P to ✓ Q.
3. If a power P balance a weight W in a combination of n movable pullies, each of weight w, show that
W=(P + w) (2n +1 -1)-(n + 1) no the strings being parallel and each attached to the weight.
4. If a weight of 10 lbs. be placed on a plane which descends with a uniform acceleration of 10 feet per second, what is the pressure on the plane ?
5. A particle is projected up a rough inclined plane. It te= time of ascending, tz= time of descending show that
2_sin (a - 0)
sin (a + 0) tan & being the co-efficient of friction.
6. When a heavy particle falls down a smooth curve, the velocity at any point is that due to the vertical height through which it has fallen.
7. Describe the diurnal and annual motions of the sun as seen from the North Pole.
8. Explain how the attractions of the Sun and Moon produce the tides, and account for spring and neap tides.
. 9. If the earth's orbit were circular, find when the equation of time would vanish.
TRIGONOMETRY AND CONICS.
1. The top of a pole 32 feet high is just seen by a man 6 feet in height, at a distance of 10 miles ; find the earth's radius.
2. A regular polygon is described in a circle, and the tangent of half the acute angle subtended by a side at the circumference =t, show that a side is to the diameter as 2t to 1+ t.
3. The sides of a triangle are as 3, 5, 6; compare the radii of
; the inscribed and circumscribed circles.
5. The perpendiculars from the foci on the tangent to an ellipse intersect the tangent in the circumference of a circle on the axis major as diameter.
Deduce from this an analogous proposition for the parabola.
6. In the ellipse of the conjugate diameter meet either focal distance in E, PE will be equal to AC.
7. If a tangent be drawn to a hyperbola, and be terminated by the asymptotes, it will be bisected at the point of contact.
Apply this to prove that the area of the triangle contained by the tangent and the asymptotes is constant.
8. A and B are fixed points ; draw through B any line, and let fall on it a perpendicular AP. Produce AP so that AP. AQ. may be constant; find the locus of Q, using polar co-ordinates.
9. What are the centres of similitude of two circles? Find the co-ordinates of those of the circles (x – a)? + (y - 3)2=r?, (x – a')? +(y -B)?=r'?.
10. Find the equation of an ellipse referred to its vertex as origin. Hence find the equation of a parabola.