WAEC SSCE - Further Mathematics - 2007 (Objective)

The binary operation * is defined on the set of R, of real numbers by \(x * y = 3x + 3y - xy, \forall x, y \in R\). Determine, in terms of x, the identity element of the operation.

If \(f(x) = mx^{2} - 6x - 3\) and \(f'(1) = 12\), find the value of the constant m.

Given that \(R = (4, 180°)\) and \(S = (3, 300°)\), find the dot product.

The tangent to the curve \(y = 4x^{3} + kx^{2} - 6x + 4\) at the point P(1, m) is parallel to the x- axis, where k and m are constants. Find the value of k.

The equation of a circle is given by \(x^{2} + y^{2} - 4x - 2y - 3\). Find the radius and the coordinates of its centre.

Simplify \(\frac{\sqrt{3}}{\sqrt{3} - 1} + \frac{\sqrt{3}}{\sqrt{3} +1}\)

Given that \(P = \begin{pmatrix} 4 & 9 \end{pmatrix}\) and \(Q = \begin{pmatrix} -1 & -2 \\ 3 & 2 \end{pmatrix}\). Evaluate \(|Q|P\).

The tangent to the curve \(y = 4x^{3} + kx^{2} - 6x + 4\) at the point P(1, m) is parallel to the x- axis, where k and m are constants. Determine the coordinates of P.

X and Y are two independent event. If \(P(X) = \frac{1}{5}\) and \(P(X \cap Y) = \frac{2}{15}\), find \(P(Y)\).

Three students are working independently on a Mathematics problem. Their respective probabilities of solving the problem are 0.6, 0.7 and 0.8. What is the probability that at least one of them solves the problem?

Find the fourth term of the binomial expansion of \((x - k)^{5}\) in descending powers of x.

Given that \(P = \begin{pmatrix} 4 & 9 \end{pmatrix}\) and \(Q = \begin{pmatrix} -1 & -2 \\ 3 & 2 \end{pmatrix}\). Which of the following operations is possible?

Given that \(\log_{2} y^{\frac{1}{2}} = \log_{5} 125\), find the value of y.

A force of 30 N acts at an angle of 60° on a body of mass 6 kg initially at rest on a smooth horizontal plane. Find the distance covered in 10 seconds.

Simplify \(\frac{^{n}P_{3}}{^{n}C_{2}} + ^{n}P_{0}\)

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

Two bodies of masses 8 kg and 5 kg travelling in the same direction with speeds x m/s and 2 m/s respectively collide. If after collision, they move together with a speed of 3.85 m/s, find, correct to the nearest whole number, the value of x.

Two vectors m and n are defined by \(m = 3i + 4j\) and \(n = 2i - j\). Find the angle between m and n.

Calculate, correct to one decimal place, the standard deviation of the numbers: -1, 5, 0, 2 and 9.

Find the value of p for which \(x^{2} - x + p\) becomes a perfect square. 

Find the area of the circle whose equation is given as \(x^{2} + y^{2} - 4x + 8y + 11 = 0\).

Calculate in surd form, the value of \(\tan 15°\).

The sum of the first three terms of an Arithmetic Progression (A.P) is 18. If the first term is 4, find their product.

A group of 5 boys and 4 girls is to be chosen from a class of 8 boys and 6 girls. In how many ways can this be done?

The gradient of the line passing through the points P(4, 5) and Q(x, 9) is \(\frac{1}{2}\). Find the value of x.

Simplify \(\frac{\tan 80° - \tan 20°}{1 + \tan 80° \tan 20°}\)

A bag contains 2 red and 4 green sweets of the same size and shape. Two boys pick a sweet each from the box, one after the other, without replacement. What is the probability that at least a sweet with green wrapper is picked?

Which of the following is the same as \(\sin (270 + x)°\)?

A particle is projected vertically upwards with a speed of 40 m/s. At what times will it be 35m above its point of projection? \(\text{Take g} = 10 ms^{-2}\)

The table above shows the marks obtained by 100 pupils in a test. Find the upper class boundary of the class containing the third quartile.

How many ways can 12 people be divided into three groups of 2, 7 and 3 in that order?

Given that \(p = 4i + 3j\), find the unit vector in the direction of p.

The equation of a curve is given by \(y = 2x^{2} - 5x + k\). If the curve has two intercepts on the x- axis, find the value(s) of constant k.

Given that \(\alpha\) and \(\beta\) are the roots of an equation such that \(\alpha + \beta = 3\) and \(\alpha \beta = 2\), find the equation.

A body is acted upon by two forces \(F_{1} = (5 N, 060°)\) and \(F_{2} = (10 N, 180°)\). Find the magnitude of the resultant force.

Given that the straight lines \(kx - 5y + 6 = 0\) and \(mx + ny - 1 = 0\) are parallel, find a relationship connecting the constants m, n and k.

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

The table above shows the marks obtained by 100 pupils in a test. Find the probability that a student picked at random scored at least 14 marks.

The polynomial \(2x^{3} + 3x^{2} + qx - 1\) has the same reminder when divided by \((x + 2)\) and \((x - 1)\). Find the value of the constant q.

Three forces \(F_{1} = (8 N, 300°), F_{2} = (6 N, 090°)\) and \(F_{3} = (4 N, 180°)\) act on a particle. Find the vertical component of the resultant force.