Elective Mathematics — 2009

Solve \(x^{2} - 2x - 8 > 0\).

If (x + 3) is a factor of the polynomial \(x^{3} + 3x^{2} + nx - 12\), where n is a constant, find the value of n.

The line \(y = mx - 3\) is a tangent to the curve \(y = 1 - 3x + 2x^{3}\) at (1, 0). Find the value of the constant m.

The coordinates of the centre of a circle is (-2, 3). If its area is \(25\pi cm^{2}\), find its equation.

Given \(\sin \theta =  \frac{\sqrt{3}}{2}, 0° \leq \theta \leq 90°\), find \(\tan 2\theta\) in surd form.

Find the coefficient of \(x^{4}\) in the binomial expansion of \((2 + x)^{6}\).

Which of the following binary operations is not commutative?

Express \(\frac{2}{3 - \sqrt{7}} \text{ in the form} a + \sqrt{b}\), where a and b are integers.

The roots of the quadratic equation \(2x^{2} - 5x + m = 0\) are \(\alpha\) and \(\beta\), where m is a constant. Find \(\alpha^{2} + \beta^{2}\) in terms of m.

Given that \(2^{x} = 0.125\), find the value of x.

The gradient of point P on the curve \(y = 3x^{2} - x + 3\) is 5. Find the coordinates of P.

An arc of length 10.8 cm subtends an angle of 1.2 radians at the centre of a circle. Calculate the radius of the circle.

The first term of a geometric progression is 350. If the sum to infinity is 250, find the common ratio.

p and q are statements such that \(p \implies q\). Which of the following is a valid conclusion from the implication?

The roots of a quadratic equation are -3 and 1. Find its equation.

The derivative of a function f with respect to x is given by \(f'(x) = 3x^{2} - \frac{4}{x^{5}}\). If \(f(1) = 4\), find f(x).

Simplify \((216)^{-\frac{2}{3}} \times (0.16)^{-\frac{3}{2}}\)

Given that \(\log_{3}(x - y) = 1\) and \(\log_{3}(2x + y) = 2\), find the value of x.

If \(\frac{^{8}P_{x}}{^{8}C_{x}} = 6\), find the value of x.

Evaluate \(\int_{1}^{2} [\frac{x^{3} - 1}{x^{2}}] \mathrm {d} x\).

If \(P = \begin{vmatrix} 1 & 1 \\ 2 & 1 \end{vmatrix}\), find \((P^{2} + P)\).

Which of the following is the semi- interquartile range of a distribution?

A stone is projected vertically with a speed of 10 m/s from a point 8 metres above the ground. Find the maximum height reached. \([g = 10 ms^{-2}]\).

The velocity \(v ms^{-1}\) of a particle moving in a straight line is given by \(v = 3t^{2} - 2t + 1\) at time t secs. Find the acceleration of the particle after 3 seconds.

Three men, P, Q and R aim at a target, the probabilities that P, Q and R hit the target are \(\frac{1}{2}\), \(\frac{1}{3}\) and \(\frac{3}{4}\) respectively. Find the probability that exactly 2 of them hit the target.

The position vectors of A and B are (2i + j) and (-i + 4j) respectively; find |AB|.

Two fair dices, each numbered 1, 2, ..., 6, are tossed together. Find the probability that they both show even numbers.

Calculate, correct to the nearest degree, the angle between the vectors \(\begin{pmatrix} 13 \\ 1 \end{pmatrix}\) and \(\begin{pmatrix} 1 \\ 4 \end{pmatrix}\).

Simplify \(2\log_{3} 8 - 3\log_{3} 2\)

Evaluate \(\begin{pmatrix} 2 & 3 \\ 4 & 1 \end{pmatrix} \begin{pmatrix} 2 \\ 3 \end{pmatrix}\).

If the mean of -1, 0, 9, 3, k, 5 is 2, where k is a constant, find the median of the set of numbers.

Eight football clubs are to play in a league on home and away basis. How many matches are possible?

Two balls are drawn, from a bag containing 3 red, 4 white and 5 black identical balls. Find the probability that they are all of the same colour.

A force F acts on a body of mass 12kg increases its speed from 5 m/s to 35 m/s in 5 seconds. Find the value of F.

Express the force F = (8 N, 150°) in the form (a i + b j) where a and b are constants.

Three defective bulbs got mixed up with seven good ones. If two bulbs are selected at random, what is the probability that both are good?

The ages, in years, of 5 boys are 5, 6, 6, 8 and 10. Calculate, correct to one decimal place, the standard deviation of their ages.

A body is acted upon by forces \(F_{1} = (10 N, 090°)\) and \(F_{2} = (6 N, 180°)\). Find the magnitude of the resultant force.

In the diagram, a ladder PS leaning against a vertical wall PR makes angle x° with the horizontal floor. The ladder slides down to a point QT such that angle QTR = 30° and SNT = y°. Find the relation between x and y.

In the diagram, a ladder PS leaning against a vertical wall PR makes angle x° with the horizontal floor. The ladder slides down to a point QT such that angle QTR = 30° and SNT = y°. Find an expression for tan y.