Mathematics Harder Topics

Harder 2 Unit Topics

1. Write 8 + 27 + + n3 using notation.

2. Evaluate .

3. Factorise 2n-1 + 2n, hence write 21001 + 21000 as a power of 2.

3

4. Given that loga b = 2.8 and loga c = 4.1, find

loga ().

5. Given that log27 = 2.807 (to three decimal places, find log2 14.

6. Find the perpendicular distance from the point (1,2) to the line y

= 3x + 4.

7. Consider the circle x2 + y2 – 2x – 14y + 25 = 0.

(i) Show that if the line y = mx intersects the circle in

two distinct points, then

(1 + 7m)2 – 25 (1 + m2) > 0.

(ii) For what values of m is the line y = mx a tangent to

the circle?

8.

The figure shows the net of an oblique pyramid with a rectangu-

lar base. In this figure, PXZY R is a straight line, PX = 15 cm,

RY = 20 cm, AB = 25 cm and BC = 10 cm. Further, AP = PD

and BR = RC. When the net is folded, points P, Q, R and S all

meet at the apex T, which lies vertically above the point Z in the

hori-zontal base, as shown below.

(i) Show that the triangle TXY is right-angled.

(ii) Hence show that T is 12 cm above the base.

(iii) Hence find the angle that the face DCT makes with the

base.

9.

The figure shows a bottle-storage rack. It consists of n rows

of ‘bins’ stacked in such a way that the number of bins in

the rth row is r, counting from the top.

(i) Show that the total number of bins in the storage

rack is .

(ii) Each bin in the rth row contains c + r bottles, where c

is a constant. (For example, each bin in the third row

contains c + 3 bottles). Find an expression for the

total number of bottles in the storage rack. [You may

as-sume that 12 + 22 + + n2 + n (n + 1) (2n + 1).]

(iii) Enzo notices that c = 5 and that the average number

of bottles per bin in the storage rack is 10. Calculate

the number of rows in the storage rack.

10. Let f(x) = x

x2 – 1

(i) For what values of x is f(x) undefined?

(ii) Show that y = f(x) is an odd function.

(iii) Show that f’ (x) < 0 at all values of x for which the function

is defined.

(iv) Hence sketch y = f(x).

11. Consider y = ekx where k is a constant.

(i) Find d2y and dy

dx2 dx

(ii) Determine the values of k for which y = ekx

satisfies the equation

12. If y = 10x, find when x = 1.

13. Differentiate .

14. Consider the function y = f (), where

f () = cos - .

(i) Verify that f ’ () = 0

(ii) Sketch the curve y = f for 0 < ≤ given that

f”() < 0.

15. If y = cos(In x) find:

(i)

(ii) .

16. Differentiate e3xcos x.

17. Differentiate sin3 x.

18. Suppose the cubic f (x) = x3 + ax2 + bx + c has a relative

maximum at x = and a relative minimum at x = .

(i) Prove that + = - a,

(ii) Deduce that the point of inflexion occurs at

x = .

19.

The shaded area is bounded by the curve xy = 3, the lines x = 1 and

y = 6, and the two axes.

A solid is formed by rotating the shaded area about the y=axis.

Find the volume of this solid by considering separately the

regions above and below y = 3.

20. (i) By equating the coefficients of sin x and cos x, or otherwise,

find constants A and B satisfying the identity

A(2 sin x + cos x) + B(2 cos x – sin x) = sin x + 8 cos x

(iii) Hence evaluate dx.

21. (i) Evaluate .

(ii) Use Simpson’s rule with 3 function values to

approximate

.

(iii) Use your results to parts (i) and (ii) to obtain an

approx-imation for e. Give your answer correct to 3

decimal places.

22.

Points A, B and C lie on a circle. The length of the chord AB

is a constant k. The radian measures of ABC and BCA are and

respectively.

(i) Let l equal the sum of the lengths of chords CA and

CB. Show t hat l is given by

l = [sin + sin ( + )]

(ii) Why is a constant?

(iii) Evaluate when = -

(iv) Hence show that l is a maximum when

= -

23.

The diagram shows a cylindrical barrel of length l and radius r.

The point A is at one end of the barrel, at the very bottom of

the rim. The point B is at the very top of the barrel, halfway

along its length. The length of AB is d.

(i) Show that the volume of the barrel is

(ii) Find l is terms of d if the barrel has a maximum volume

for the given d.

24.

In the diagram, the x-axis represents a major blood vessel whilst

the line PA represents a minor blood vessel that joins the major

blood vessel at P. The point A has coordinates (t, d) and PA

makes an angle < with the normal to the x-axis a P, of chords

as shown in the dagram.

It is known that the resistance to flow in a blood vessel is pro-

portional to its length, where the constant of proportionality

de-pends upon the particular blood vessel.

Let R be the sum of the resistances to flow in OP and PA.

(i) Show that R = c1 (l – d tan ) + c2d sec , where c1 and

c2 are constants.

(ii) The blood vessel PA is joined to the blood vessel 0x in

such a way that R is minimized. If = 2, find the angle

that minimizes R. (You may assume that l is large

com-pared to d.)

25. (i) Prove that the graph of y = In x is concave down for all x >

0.

(ii) Sketch the graph of y = In x.

(iii) Suppose 0 < a < b and consider the points barrel A (a,

In a) and B (b, In b) the graph of at the In x. Find the

co-ordinates of the point P that divides the line segment

AB in the ratio 2:1.

(iv) By using (ii) and (iii) deduce that

In a + In b < In ( a + b).

26. Write down the equation of the vertical asymptote of

y = .

27.

The triangle ABC is isosceles, with AB = BC, and BD is per-

pendicular to AC. Let ∠ABD = ∠CBD = and ∠BAD = as

shown in the diagram.

(i) Show that sin = cos

(ii) By applying the sine rule in ∆ABC, show that sin 2 =

2 sin cos .

(iii) Given that 0 < < , show that the limiting sum of the geo-

metric series

Sin 2 + sin 2 cos2 + sin 2 cos4 + sin 2 cos6 + = 2 cot .

28.

The triangle ABC has sides of length a, b and c, as shown in

the diagram. The point D lies on AR and CD is perpendicular

to AB.

(i) Show that a sin = b sin A.

(ii) Show that c = a cos B + b cos A

(iii) Given that c2 = 4ab cos A cos B, show that a = b.

29.

The diagram shows a hanging basket in the shape of a

hemisphere with radius 10 cm. Let 0 be the centre of the

sphere and let 0A be the central axis. Two vertical wire

supports, AB and AC, are shown on the diagram. The

length of the arc BC is 4 cm.

A horizontal wire support is placed around the surface of

the basket. This wire meets AB at D and AC at E.

The plane through DE parallel to the plane OBC cuts OA at

F. The length OF is 6 cm. Note that ∠BOC = ∠DFE.

(i) Show that the length of FD is 8 cm.

(ii) Find ∠DFE in radians

(iii) Find the size of ∠DOE in radians , correct to 3

decimal places

30. An amount $A is borrowed at r% per annum reducible interest,

calculated monthly. The loan is to be repaid in equal monthly

instalments of $M. Let R = (1 + ) and let $Bn be the amount

owing after n monthly repayments have been made.

(i) Show that Bn = ARn – M ().

Pat borrow $30 000 at 9% per annum reducible interest,

calcu-lated monthly. The loan is to be repaid in 60 equal

monthly in-stalments.

(ii) Show that the monthly repayments should be $622.75.

(iii) With the twelfth repayment, Pat pays an additional

$5000, so this payment is $5622.75. After this, repayments

continue at $622.75 per month. How many more repayments

will be needed?

31. Paul plans to contribute to a retirement fund. He will invest

$500 on each birthday from age 25 to 64 inclusive. That is,

he will make 40 contributions to the fund. The retirement

funds pay interest on the investments at the rate of 8% per

annum, compounded annually.

How much money will be in Paul’s fund on his 65th birthday?