Elective Mathematics — 2011

A binary operation * is defined on the set of real numbers R, by a* b = -1. Find the identity element under the operation *.

Express 75° in radians, leaving your answer in terms of \(\pi\).

If \(\log_{9} 3 + 2x = 1\), find x.

Evaluate \(\cos (\frac{\pi}{2} + \frac{\pi}{3})\)

Find the remainder when \(5x^{3} + 2x^{2} - 7x - 5\) is divided by (x - 2).

A function is defined by \(f(x) = \frac{3x + 1}{x^{2} - 1}, x \neq \pm 1\). Find f(-3).

Simplify \(\sqrt[3]{\frac{8}{27}} - (\frac{4}{9})^{-\frac{1}{2}}\)

Solve \(3x^{2} + 4x + 1 > 0\)

The equation of a circle is \(3x^{2} + 3y^{2} + 6x - 12y + 6 = 0\). Find its radius

\(f(x) = p + qx\), where p and q are constants. If f(1) = 7 and f(5) = 19, find f(3).

The sum and product of the roots of a quadratic equation are \(\frac{4}{7}\) and \(\frac{5}{7}\) respectively. Find its equation.

\(f(x) = (x^{2} + 3)^{2}\) is defines on the set of real numbers, R. Find the gradient of f(x) at x = \(\frac{1}{2}\).

Find \(\lim \limits_{x \to 3} \frac{x + 3}{x^{2} - x - 12}\)

If \(y^{2} + xy - x = 0\), find \(\frac{\mathrm d y}{\mathrm d x}\).

A line is perpendicular to \(3x - y + 11 = 0\) and passes through the point (1, -5). Find its equation.

Solve \(9^{2x + 1} = 81^{3x + 2}\)

The inverse of a function is given by \(f^{-1} : x \to \frac{x + 1}{4}\).

If \(\begin{pmatrix} 3 & 2 \\ 7 & x \end{pmatrix} \begin{pmatrix} 2 \\ 3 \end{pmatrix} = \begin{pmatrix} 12 \\ 29 \end{pmatrix} \), find x.

The fourth term of a geometric sequence is 2 and the sixth term is 8. Find the common ratio.

What percentage increase in the radius of a sphere will cause its volume to increase by 45%?

Evaluate \(\frac{1}{1 - \sin 60°}\), leaving your answer in surd form.

Find the equation of a circle with centre (-3, -8) and radius \(4\sqrt{6}\).

Determine the coefficient of \(x^{2}\) in the expansion of \((a + 3x)^{6}\).

The mean of 2, 5, (x + 2), 7 and 9 is 6. Find the median.

The probability that Kofi and Ama hit a target in a shooting competition are \(\frac{1}{6}\) and \(\frac{1}{9}\) respectively. What is the probability that only one of them hit the target?

In how many ways can 3 prefects be chosen out of 8 prefects?

Find the standard deviation of the numbers 3,6,2,1,7 and 5.

Marks 5-7 8-10 11-13 14-16 17-19 20-22 No of students 4 7 26 41 14 8 The table above shows the distribution of marks of students in a class. Find the upper class boundary of the modal class.

If \(^{3x}C_{2} = 15\), find the value of x?

Four doctors and two nurses are to sit round a circular table. In how many ways can this be done if the nurses are to sit together?

A basket contains 3 red and 1 white identical balls. A ball is drawn from the basket at random. Calculate the probability that it is either white or red.

A force of 200N acting on a body of mass 20kg initially at rest causes it to move a distance of 320m along a straight line for t secs. Find the value of t.

Two forces 10N and 15N act on an object at an angle of 120° to each other. Find the magnitude of the resultant.

A body of mass 25kg changes its speed from 15m/s to 35m/s in 5 seconds by the action of an applied force F. Find the value of F.

A particle starts from rest and moves in a straight line such that its velocity, v, at time t seconds is given by \(v = (3t^{2} - 2t) ms^{-1}\). Calculate the distance covered in the first 2 seconds.

A particle starts from rest and moves in a straight line such that its velocity, v, at time t seconds is given by \(v = (3t^{2} - 2t) ms^{-1}\). Determine the acceleration when t = 2 secs.

Given that \(q = 9i + 6j\) and \(r = 4i - 6j\), which of the following statements is true?

The functions f and g are defined on the set, R, of real numbers by \(f : x \to x^{2} - x - 6\) and \(g : x \to x - 1\). Find \(f \circ g(3)\).

Find the unit vector in the direction of (-5i + 12j).

Find, correct to two decimal places, the acute angle between \(p = \begin{pmatrix} 13 \\ 14 \end{pmatrix}\) and \(q = \begin{pmatrix} 12 \\ 5 \end{pmatrix}\).