Elective Mathematics — 2008

Simplify \(\frac{\sqrt{3} + \sqrt{48}}{\sqrt{6}}\)

Find the range of values of x for which \(2x^{2} + 7x - 15 > 0\).

A function f is defined on R, the set of real numbers, by: \(f : x \to \frac{x + 3}{x - 2}, x \neq 2\), find \(f^{-1}\).

The sum of the first n terms of a linear sequence is \(S_{n} = n^{2} + 2n\). Find the common difference of the sequence.

The sum of the first n terms of a linear sequence is \(S_{n} = n^{2} + 2n\). Determine the general term of the sequence.

If \(f(x) = 2x^{2} - 3x - 1\), find the value of x for which f(x) is minimum.

The polynomial \(2x^{3} + x^{2} - 3x + p\) has a remainder of 20 when divided by (x - 2). Find the value of constant p.

If \(2\log_{4} 2 = x + 1\), find the value of x.

Which of the following quadratic curves will not intersect with the x- axis?

What is the coordinate of the centre of the circle \(5x^{2} + 5y^{2} - 15x + 25y - 3 = 0\)?

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

A rectangle has a perimeter of 24m. If its area is to be maximum, find its dimension.

Express \(\frac{7\pi}{6}\) radians in degrees.

If \(P = \begin{pmatrix} 1 & -2 \\ 3 & 4 \end{pmatrix}\) and \(Q = \begin{pmatrix} -2 & 3 \\ 1 & 0 \end{pmatrix}\), find PQ.

Two statements are represented by p and q as follows: p : He is brilliant; q : He is regular in class Which of the following symbols represent "He is regular in class but dull"?

Find the locus of points which is equidistant from P(4, 5) and Q(-6, -1).

A binary operation ,*, is defined on the set R, of real numbers by \(a * b = a^{2} + b + ab\). Find the value of x for which \(5 * x = 37\).

Find the derivative of \(3x^{2} + \frac{1}{x^{2}}\)

The coefficient of the 5th term in the binomial expansion of \((1 + kx)^{8}\), in ascending powers of x is \(\frac{35}{8}\). Find the value of the constant k.

Given that \(f '(x) = 3x^{2} - 6x + 1\) and f(3) = 5, find f(x).

Express \(\frac{1}{1 - \sin 45°}\) in surd form.

If \(\begin{vmatrix} 4 & x \\ 5 & 3 \end{vmatrix} = 32\), find the value of x.

If events A and B are independent and \(P(A) = \frac{7}{12}\) and \(P(A \cap B) = \frac{1}{4}\), find P(B).

Given that \(\overrightarrow{AB} = 5i + 3j\) and \(\overrightarrow{AC} = 2i + 5j\), find \(\overrightarrow{BC}\).

The probability of Jide, Atu and Obu solving a given problem are \(\frac{1}{12}\), \(\frac{1}{6}\) and \(\frac{1}{8}\) respectively. Calculate the probability that only one solves the problem.

Two forces \(F_{1} = (10N, 020°)\) and \(F_{2} = (7N, 200°)\) act on a particle. Find the resultant force.

Marks 2 3 4 5 6 7 8 No of students 5 7 9 6 3 6 4 The table above shows the distribution of marks by some candidates in a test. What is the median score?

Marks 2 3 4 5 6 7 8 No of students 5 7 9 6 3 6 4 The table above shows the distribution of marks by some candidates in a test. Find, correct to one decimal place, the mean of the distribution.

Marks 2 3 4 5 6 7 8 No of students 5 7 9 6 3 6 4 The table above shows the distribution of marks by some candidates in a test. If a student is selected at random, what is the probability that she scored at least 6 marks?

Express \(r = (12, 210°)\) in the form \(a i + b j\).

A test consists of 12 questions out of which candidates are to answer 10. If the first 6 are compulsory, in how many ways can each candidate select her questions?

A body starts from rest and moves in a straight line with uniform acceleration of \(5 ms^{-2}\). How far, in metres, does it go in 10 seconds?

If n items are arranged two at a time, the number obtained is 20. Find the value of n.

If \(p = \begin{pmatrix} 2 \\ -2 \end{pmatrix} \) and \(q = \begin{pmatrix} 3 \\ 4 \end{pmatrix}\), find \(|q - \frac{1}{2}p|\).

Find the value of the constant k for which \(a = 4 i - k j\) and \(b = 3 i + 8 j\) are perpendicular.

The initial and final velocities of an object of mass 5 kg are \(u = \begin{pmatrix} 1 \\ 3 \end{pmatrix}\) and \(v = \begin{pmatrix} 4 \\ 7 \end{pmatrix}\) respectively. Find the magnitude of its change in momentum.

If \(y = x^{2} - 6x + 11\) is written in the form \(y = a(x - h)^{2} + k\), find the value of \((a + h + k)\).

The distance between P(x, 7) and Q(6, 19) is 13 units. Find the values of x.

In the diagram above, forces P, Q and 50N are acting on a body at M. If the system is in equilibrium, calculate, in terms of \(\theta\), the magnitude of P.