Mathematics
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3 Unit Exams Questions Paper 1
Western Region Trial 1996
QUESTION 1 |
Marks |
|
| a. | Find the exact value of
 |
3 |
|---|---|---|
| b. | Find  |
3 |
| c. | Solve the inequality
 |
3 |
| d. | Find the first derivative
of  |
3 |
QUESTION 2 |
Marks |
|
| a. |  AB is a tangent at B and AD || BC. Prove that  |
4 |
| b. | Find  using the substitution u
= x - 1 |
4 |
| c. | Prove by the method of Mathematical
Induction that |
4 |
QUESTION 3 |
Marks |
|
| a. | If 12Pr = 120.12 Cr find r. | 3 |
| b. | The velocity of a particle moving in a
straight line is given by v2 = 8x - 2x2 m/s |
5 |
| i. | Show that the particle is moving in simple harmonic motion. | |
| ii. | Find the centre of the motion. | |
| iii. | Determine the two end points between which the particle is oscillating. | |
| iv. | Find the maximum speed of the particle. | |
| c. |
A formula for the rate of change in population of a
colony of bacteria, is given by P = 3200 + 400
ekt
If the population doubles after 20 hours, how long would it take to triple the original population? |
4 |
QUESTION 4 |
Marks |
|
| a. | At what points on the curve y = cos
-1x is the gradient equal to  ? |
3 |
| b. | Find the middle term in the expansion of
 |
4 |
| c. | A capsule is in the shape of a cylinder with hemispherical ends. The radius of the cylindrical section is r cm and the volume of the capsule is 6 cm3 | 5 |
| i. | If the height of the
cylinder is 4 cm show that  |
|
| ii. | Show that one solution of
the equation lies between 0 and 1. |
|
| iii. | The equation has a root
close to . Use one application of Newton's method to
give a better approximation. |
|
QUESTION 5 |
Marks |
|
| a. | Solve the equation 3x3 - 17x2 - 8x + 12 = 0 given that the product of two of the roots is 4 | 3 |
| b. | The probability that a vaccine succeeds
is  An
experiment is conducted m times with
white mice. |
4 |
| i. | What is the probability that the experiment will fail at least once? | |
| ii. | Show that if the probability that the
experiment will fail at least once in m trials,
is greater than than  then  |
|
| c. | For a particular vessel, the rate of
increase of the volume with respect to its depth, is
given by  where V cm3 is the volume and h is the depth of the water. |
5 |
| i. | If the container is initially empty, show
that the volume as a function of the depth is  |
|
| ii. | Find the volume when the depth is 6 cm. | |
| iii. | If water is being poured into the vessel at a constant rate of 8 cm3 /s, find an expression for the rate of increase in the depth of the water. | |
| iv. | At what rate is the depth increasing when the water level is 6 cm, and how long will it take to the nearest second to reach this level? | |
QUESTION 6 |
Marks |
|
| a. | The letters of the word REPETITION are arranged at random in a row. | 3 |
| i. | How many different arrangements are possible? | |
| ii. | What is the probability that one particular arrangement will have vowels and consonants alternating? | |
| b. | ||
| i. | Write the general expansion of (1 + x)n | 3 |
| ii. | Hence or otherwise prove that  |
|
| c. | The curve y = sin -1x intersects the curve y = cos -1x at P, and the latter intersects the x axis at Q. | 6 |
| i. | Draw a neat sketch of this information. | |
| ii. | Verify that P has co-ordinates
 |
|
| iii. | Prove  |
|
| iv. | If O is the origin, find the
area enclosed by the arcs OP and PQ and
the x axis using the results in (iii) and the
fact that  |
|
QUESTION 7 |
Marks |
|
| A projectile fired with velocity V and at an angle 45 degrees to the horizontal, just clears the tops of two vertical posts of height 8a 2, and the posts are 2a 2 apart. There is no air resistance, and the acceleration due to gravity is g. | ||
| a. | If the projectile is at the point ( x,y ) at time t, derive expressions for x and y in terms of t. | 3 |
| b. | Hence show that the equation of the path
of the projectile is  |
2 |
| c. | Using the information in (b) show that
the range of the projectile is  |
2 |
| d. | If the first post is b units from the origin, show | 2 |
| i. |  |
|
| ii. |  |
|
| e. | Hence or otherwise prove that  |
3 |
|