JAMB UTME - Mathematics - 1992

\(p = \begin{vmatrix} x & 3 & 0 \\ 2 & y & 3 \\ 4 & 2 & 4 \end{vmatrix}\)

\(q = \begin{vmatrix} x & 2 & z \\ 3 & y & 2 \\ 0 & 3 & z \end{vmatrix}\) Where \(p^T\) is the transpose p calculate \(/p^T/\) when x = 0, y = 1 and z = 2

Solve the equation \(y^2 - 11y + 24 = 0\)

If the binary operation \( \ast \) is defined by \( m \ast n = mn + m + n \) for any real number m and n, find the identity of the elements under this operation

Make t the subject of formula \(S = ut + \frac{1}{2}at^2\)

Evaluate \( \frac{(x-2)(x^2+3x-2)}{x^2-4} \)

In the figure, if XZ is 10cm, calculate RY in cm.

a student blows a balloon and its volume increases at a rate of \( \pi(20 - t^2)\text{ cm}^3\text{ s}^{-1} \) after t seconds. If the initial volume is 0 cm³, find the volume of the balloon after 2 seconds

Find the sum to infinity to the following series \(3 + 2 + \frac{4}{3} + \frac{8}{9} + \frac{16}{17} + .....\)

The angle between latitudes 30oS and 13oN is

If in the diagram above, FG is parallel to KM, find the value of x

Simplify \(\frac{(1.25 \times 10^{-4}) \times (2.0 \times 10^{-1})}{(6.25 \times 10^{5})}\)

Simplify \(5\sqrt{18} - 3\sqrt{72} + 4\sqrt{50}\)

The probability of an event P is \( \frac{3}{4} \) while that of another event Q is \( \frac{1}{6} \). If the probability of both P and Q is \( \frac{1}{2} \). What is the probability of either P or Q.

If the mean of the above frequency distribution is 5.2, find y

\(p = \begin{vmatrix} x & 3 & 0 \\ 2 & y & 3 \\ 4 & 2 & 4 \end{vmatrix}\)

\(q = \begin{vmatrix} x & 2 & z \\ 3 & y & 2 \\ 0 & 3 & z \end{vmatrix}\)

pq is equivalent to

The diagram is a circle with centre O. Find the area of the shaded portion.

If \(y = x \sin x\), Find \(\dfrac{d^2y}{d^2x}\)

Given that \( \theta \) is an acute angle and \( \sin \theta = \frac{m}{n} \), find \( \cos \theta \)

If \(x = 3 - \sqrt{3}\), find \(x^2 + \frac{36}{x^2}\)

Obtain a maximum value of the function \(f(x)=x^3-12x+11\).

What is the value of x satisfying the equation \( \frac{4^{2x}}{4^{3x}} = 2 \)?

Resolve \( \frac{3}{x^2+x-2} \) into partial fractions

The sketch is the curve of y = ax2 + bx + c. Find a, b and c respectively

Find all values of x satisfying the inequality \( -11 \le 4 - 3x \le 28 \)

Find the mode and median respectively of the distribution above

A cliff on the bank of a river is 300 meter high. if the angle of depression of a point on the opposite side of the river is \(60^\circ\), find the width of the river.

Evaluate the integral \( \int_{\frac{\pi}{12}}^{\frac{\pi}{4}} 2 \cos 2x\, dx \)

In the figure, PQr is a semicircle while PQ and Qr are chords. QS is the perpendicular from Q to the diameter PR. What is the expression for QS?