(a) Given that \(m = (6i + 8j)\) and \(n = (-8i + \frac{7}{3}j)\), find the :
(i) magnitudes and direction of m and n ; (ii) angle between m and n.
(b) The position vectors of points P, Q, R and S are \(\begin{pmatrix} -2 \\ 3 \end{pmatrix}, \begin{pmatrix} 10 \\ 4 \end{pmatrix}, \begin{pmatrix} 3 \\ 12 \end{pmatrix}\) and \(\begin{pmatrix} 4 \\ 0 \end{pmatrix}\) respectively. Show that \(\overrightarrow{PQ}\) is perpendicular to \(\overrightarrow{RS}\).
(a) A body P of mass q kg is suspended by two light inextensible strings AB and DB attached to a horizontal table. The strings are inclined at 30° and 60° respectively to the horizontal and the tension in AB is 48N. If the system is in equilibrium :
(i) sketch a diagram to represent the information ; (ii) calculate the tension in DB ;
The table shows the frequency distribution of the ages of patients in a clinic.
| Ages (years) | 17 – 19 | 20 – 22 | 23 – 28 | 29 – 34 | 35 – 43 |
| No. of patients | 6 | 9 | 12 | 18 | 18 |
(a) Draw a histogram for the distribution
(b) Find, correct to two decimal places, the mean age of the patients.
(a) In a school, the ratio of those who passed to those who failed in a History test is 4 : 1. If 7 students are selected at random from the school, find, correct to two decimal places, the probability that :
(i) at least 3 passed the test ; (ii) between 3 and 6 students failed the test.
(b) A fair die is thrown five times; find the probability of obtaining a six three times.
(a) If \(f(x) = \int (4x – x^{2}) \mathrm {d} x\) and f(3) = 21, find f(x).
(b) The second, fourth and eigth terms of an Arithmetic Progression (A.P) form the first three consecutive terms of a Geometric Progression (G.P). The sum of the third and fifth terms of the A.P is 20, find the :
(i) first four terms of the A.P
(ii) sum of the first ten terms of the A.P
(a) If \(f(x) = \frac{2x – 3}{(x^{2} – 1)(x + 2)}\)
(i) find the values of x for which f(x) is undefined.
(ii) express f(x) in partial fractions.
(b) A circle with centre (-3, 1) passes through the point (3, 1). Find its equation.
(a) Simplify : \(\frac{1}{1 – \cos \theta} + \frac{1}{1 + \cos \theta}\) and leave your answer in terms of \(\sin \theta\).
(b) Find the equation of the line joining the stationary points of \(y = x^{2} (x – 3)\) and the distance between them.
Forces \(F_{1} (18N, 330°), F_{2} (10N, 090°)\) and \(F_{3} (25N, 180°)\) act on a body at rest. Find, correct to one decimal place, the magnitude and direction of the resultant force.
A parallelogram MNQR has vertices M(4, -6), N(10, 2), Q(8, 16) and R(x, y). Find the coordinates of R.
The table shows the heights in cm of some seedlings in a certain garden.
| Height (cm) | 36-40 | 41-45 | 46-50 | 51-55 | 56-60 |
| Frequency | 3 | 9 | 21 | 12 | 5 |
(a) Draw the cumulative frequency curve for the distribution.
(b) Using the curve in (a), find thesemi-interquartile range.
Bottles of the same sizes produced in a factory are packed in boxes. Each box contains 10 bottles. If 8% of the bottles are defective, find, correct to two decimal places, the probability that box chosen at random contains at least 3 defective bottles.
If (x + 1) and (x – 2) are factors of the polynomial \(g(x) = x^{4} + ax^{3} + bx^{2} – 16x – 12\), find the values of a and b.
(a) Given that \(\log_{10} p = a, \log_{10} q = b\) and \(\log_{10} s = c\), express \(\log_{10} (\frac{p^{\frac{1}{3}}q^{4}}{s^{2}}\) in terms of a, b and c.
(b) The radius of a circle is 6cm. If the area is increasing at the rate of 20\(cm^{2}s^{-1}\), find, leaving the answer in terms of \(\pi\), the rate at which the radius is increasing.
Evaluate : \(\int_{1}^{3} (\frac{x – 1}{(x + 1)^{2}}) \mathrm {d} x\).
(a) If \(f(x) = \frac{4 – 5x}{2}\), and \(g(x) = x + 6, x \in R\), find \(f \circ g^{-1}\).
(b) P(x, y) divides the line joining (7, -5) and (-2, 7) internally in 5 : 4. Find the coordinates of P.
If \((2x^{2} – x – 3)\) is a factor of \(f(x) = 2x^{3} – 5x^{2} – x + 6\), find the other factor
- A. (x - 2)
- B. (x - 1)
- C. (x + 1)
- D. (x + \(\frac{3}{2}\))
If P = \({n^{2} + 1: n = 0,2,3}\) and Q = \({n + 1: n = 2,3,5}\), find P\(\cap\) Q.
- A. {5, 10}
- B. {4, 6}
- C. {1, 3}
- D. { }
Given that \(f(x) = 2x^{2} – 3\) and \(g(x) = x + 1\) where \(x \in R\). Find g o f(x).
- A. \(2(x^{2} - 1)\)
- B. \(2x^{2} + 4x - 1\)
- C. \(2x^{2} + 6x - 1\)
- D. \(3(x^{2} - 1)\)
The velocity, V, of a particle after t seconds, is \(V = 3t^{2} + 2t – 1\). Find the acceleration of the particle after 2 seconds.
- A. 10\(ms^{-2}\)
- B. 12\(ms^{-2}\)
- C. 14\(ms^{-2}\)
- D. 17\(ms^{-2}\)
Find the magnitude and direction of the vector \(p = (5i – 12j)\)
- A. (13, 113.38°)
- B. (13, 067.38°)
- C. (13, 025.38°)
- D. (13, 157.38°)
Solve \(3^{2x} – 3^{x+2} = 3^{x+1} – 27\)
- A. 1 or 0
- B. 1 or 2
- C. 1 or -2
- D. -1 or 2