Elective Mathematics — 2006

\(P = {x : 1 \leq x \leq 6}\) and \(Q = {x : 2 < x < 9}\) where \(x \in R\), find \(P \cap Q\).

Solve the inequality \(2x^{2} + 5x - 3 \geq 0\).

Simplify \(\sqrt{(\frac{-1}{64})^{\frac{-2}{3}}}\).

A binary operation ♦ is defined on the set R, of real numbers by \(a ♦ b = \frac{ab}{4}\). Find the value of \(\sqrt{2} ♦ \sqrt{6}\).

If \((x - 3)\) is a factor of \(2x^{3} + 3x^{2} - 17x - 30\), find the remaining factors.

Two functions f and g are defined by \(f : x \to 3x - 1\) and \(g : x \to 2x^{3}\), evaluate \(fg(-2)\).

Given that \(\frac{1}{8^{2y - 3y}} = 2^{y + 2}\).

Given that \((\sqrt{3} - 5\sqrt{2})(\sqrt{3} + \sqrt{2}) = p + q\sqrt{6}\), find q.

If \(f(x) = \frac{1}{2 - x}, x \neq 2\), find \(f^{-1}(-\frac{1}{2})\).

Find the coefficient of \(x^{4}\) in the binomial expansion of \((1 - 2x)^{6}\).

Find the equation of the line passing through (0, -1) and parallel to the y- axis.

The roots of the equation \(2x^{2} + kx + 5 = 0\) are \(\alpha\) and \(\beta\), where k is a constant. If \(\alpha^{2} + \beta^{2} = -1\), find the values of k.

Find the sum of the exponential series \(96 + 24 + 6 +...\)

Evaluate \(\log_{0.25} 8\)

Evaluate \(\lim \limits_{x \to 1} \frac{1 - x}{x^{2} - 3x + 2}\)

The mean age of n men in a club is 50 years. Two men aged 55 and 63 years left the club, and the mean age reduced by 1 year. Find the value of n.

A committee of 4 is to be selected from a group of 5 men and 3 women. In how many ways can this be done if the chairman of the committee must be a man?

Simplify \(\frac{^{n}P_{4}}{^{n}C_{4}}\)

Which of the following is a singular matrix?

Simplify \(8^{n} \times 2^{2n} \div 4^{3n}\)

The area of a sector of a circle is 3\(cm^{2}\). If the sector subtends an angle of 1.5 radians at the centre, calculate the radius of the circle.

A particle of mass 2.5 kg is moving at a speed of 12 m/s. If a force of magnitude 10 N acts against it, find how long it takes to come to rest.

Age(in years) 1 - 5 6 - 10 11 - 15 Frequency 3 5 2 Calculate the standard deviation of the distribution.

In a firing contest, the probabilities that Kojo and Kwame hit the target are \(\frac{2}{5}\) and \(\frac{1}{3}\) respectively. What is the probability that none of them hit the target?

The equation of the line of best fit for variables x and y is \(y = 19.33 + 0.42x\), where x is the independent variable. Estimate the value of y when x = 15.

Find the coordinates of the point on the curve \(y = x^{2} + 4x - 2\), where the gradient is zero.

Find the least value of the function \(f(x) = 3x^{2} + 18x + 32\).

A force of 32 N is applied to an object of mass m kg which is at rest on a smooth horizontal surface. If the acceleration produced is 8\(ms^{-2}\), find the value of m.

Given that \(\begin{pmatrix} 1 & -3 \\ 1 & 4 \end{pmatrix} \begin{pmatrix} -6 \\ P \end{pmatrix} = \begin{pmatrix} 3 \\ -26 \end{pmatrix}\), find the value of P.

Find the coordinates of the centre of the circle \(4x^{2} + 4y^{2} - 5x + 3y - 2 = 0\).

A and B are two independent events such that \(P(A) = \frac{2}{5}\) and \(P(A \cap B) = \frac{1}{15}\). Find \(P(B)\).

The parallelogram PQRS has vertices P(-2, 3), Q(1, 4), R(2, 6) and S(-1,5). Find the coordinates of the point of intersection of the diagonals.

Find, in surd form, the value of \(\cos 165\).

The mean and median of integers x, y, z and t are 5 and z respectively. If x < y < z < t and y = 4, find (x + t).

If \(a = \begin{pmatrix} 3 \\ 2 \end{pmatrix}\) and \(b = \begin{pmatrix} -3 \\ 5 \end{pmatrix}\), find a vector c such that \(4a + 3c = b\).

A lift moving upwards with a uniform acceleration of 5\(ms^{-2}\) carries a body of mass p kg. If the reaction on the floor is 480 N, find the value of p. [Take g = \(10 ms^{-2}\)].

Calculate, correct to one decimal place, the angle between 5i + 12j and -2i + 3j.

A particle is projected vertically upwards from a height 45 metres above the ground with a velocity of 40 m/s. How long does it take it to hit the ground? [Take g = \(10 ms^{-2}\)].

Two forces, each of magnitude 16 N, are inclined to each other at an angle of 60°. Calculate the magnitude of their resultant.

ABCD is a square. Forces of magnitude 14N, 4N, 2N and \(2\sqrt{2} N\) act along the sides AB, BC, CD and DA respectively. Find in Newtons, the magnitude of the resultant of the forces.