JAMB 1999 · UME

Find the value of $x$ if $\dfrac{\sqrt2}{x + \sqrt2} = \dfrac{1}{x - \sqrt2}$.

A trader bought 100 oranges at 5 for ₦1.20; 20 oranges got spoilt and the remaining were sold at 4 for ₦1.50. Find the percentage gain or loss.

If $U = \{1, 2, 3, 4, 5, 6\}$, $P = \{3, 4, 5\}$, $Q = \{2, 4, 6\}$ and $R = \{1, 2, 3, 4\}$, list the elements of $(P \cup Q)' \cap R$.

Divide $2434_6$ by $42_6$.

Simplify $\sqrt{\dfrac{0.0023 \times 750}{0.00345 \times 1.25}}$.

If $\log_8 10 = x$, evaluate $\log_8 5$ in terms of $x$.

A group of market women sell at least one of yam, plantain and maize. 12 of them sell maize, 10 sell yam and 14 sell plantain; 5 sell plantain and maize, 4 sell yam and maize, 2 sell yam and plantain only, while 3 sell all three items. How many women are in the group?

The first term of a geometric progression is twice its common ratio. Find the sum of the first two terms of the progression if its sum to infinity is 8.

If $m * n = \frac mn - \frac nm$ for real $m$, $n$, evaluate $-3 * 4$.

Find the matrix $T$ if $ST = I$, where $S = \begin{pmatrix} -1 & 1 \\ 1 & -2 \end{pmatrix}$ and $I$ is the identity matrix.

Divide $4x^3 - 3x + 1$ by $2x - 1$.

Three consecutive positive integers $k$, $l$ and $m$ are such that $l^2 = 3(k + m)$. Find the value of $m$.

Factorize completely $x^2 + 2xy + y^2 + 3x + 3y - 18$.

The sum of two numbers is twice their difference. If the difference of the numbers is $p$, find the larger of the two numbers.

Find the equation of the locus of a point $P(x, y)$ such that $PV = PW$, where $V = (1, 1)$ and $W = (3, 5)$.

A frustum has top radius 3 cm, bottom radius 6 cm and vertical height 4 cm. Find its slant height $l$.

In triangle $XYZ$, $XY = 2$ m, $YZ = 1$ m and $\angle XYZ = 120^\circ$. Find the length $XZ$.

Find a positive value of $a$ if the centre of the circle $x^2 + y^2 - 2ax + 4y - a = 0$ is $(a, -2)$ and the radius is 4 units.

A man 1.7 m tall observes a bird on top of a tree at an angle of elevation of $30^\circ$. If the distance between the man's head and the bird is 25 m, what is the height of the tree?

In the figure, $TZ$ is a tangent to the circle $QPZ$ and $TPQ$ is a straight line. Find $x = TP$ if $TZ = 6$ units and $PQ = 9$ units.

Find the tangent of the acute angle between the lines $2x + y = 3$ and $3x - 2y = 5$.

From a point $P$, the bearings of two points $Q$ and $R$ are N$67^\circ$W and N$23^\circ$E respectively. If the bearing of $R$ from $Q$ is N$68^\circ$E and $PQ = 150$ m, calculate $PR$.

In the figure, $PQRS$ is a circle with $ST \parallel RQ$ ($T$ on $PQ$) and $PT = PS$. If $\angle QRS = 110^\circ$, find the value of $x = \angle PQR$.

In the diagram, $EFGH$ is a cyclic quadrilateral in which $EH \parallel FG$, and $FH$ and $EG$ are chords. If $\angle FHG = 42^\circ$ and $\angle EFH = 34^\circ$, calculate $\angle HEG$.

If the maximum value of $y = 1 + hx - 3x^2$ is 13, find $h$.

Evaluate $\displaystyle\int_{-2}^{1} (x - 1)^2\,dx$.

Find the area bounded by the curve $y = x(2 - x)$, the $x$-axis, $x = 0$ and $x = 2$.

If $y = 3x^2(x^3 + 1)^{\frac12}$, find $\frac{dy}{dx}$.

Find the volume of the solid generated when the area enclosed by $y = 0$, $y = 2x$ and $x = 3$ is rotated about the $x$-axis.

What is the derivative of $t^2\sin(3t - 5)$ with respect to $t$?

Find the value of $x$ for which the function $y = x^3 - x$ has a minimum value.

Three boys play a game of luck in which their respective chances of winning are $\frac12$, $\frac13$ and $\frac14$. What is the probability that one and only one of the boys wins the game?

A number is selected at random from 0 to 20. What is the probability that the number is an odd prime?

If $\dfrac{^6C_r}{^6P_r} = \frac16$, find the value of $r$.

If the standard deviation of the numbers 3, 6, $x$, 7, 5 is $\sqrt2$, find the least possible value of $x$.

The grades of 36 students in a class test are shown in the pie chart. How many students had excellent?

The marks scored by students in a test are given below. Find the median.

A student calculated the mean of 5 numbers as 45.3. While rechecking his working, he discovered that his total was short by 20.5. What is the correct mean of the 5 numbers?

The allocations to various ministries in a state budget are: Agriculture ₦25,000,000; Education ₦20,000,000; Women Affairs ₦35,000,000; Commerce and Industries ₦20,000,000. In a pie chart of this information, the angle for agriculture is