2025 - JAMB Mathematics Past Questions and Answers - page 3

First rewrite the function in standard quadratic form:\[y = -2x^2 - 4x + 2\]Here, the coefficient of \(x^2\) is negative, meaning the graph opens downward.This tells us the function does not have a minimum value, because as x becomes very large or very small, the value of y decreases without bound.To show this clearly, consider the behavior when x becomes very large in magnitude.For example, if x = 100:\[y = -2(100)^2 - 4(100) + 2\]\[= -20000 - 400 + 2\]\[= -20398\]If x = 1000:\[y = -2(1000)^2 - 4(1000) + 2\]\[= -2000000 - 4000 + 2\]\[= -2003998\]We can see that y continues decreasing as x increases.Similarly, if x becomes very negative, y also decreases without limit.Therefore, the function has no minimum value.However, it does have a maximum value at the vertex.Hence, the minimum value is negative infinity, meaning there is no minimum.

In a geometric progression, each term is obtained by multiplying the previous term by the common ratio r.The general formula for the nth term is:\[T_n = ar^{n-1}\]Where a is the first term and r is the common ratio.The second term is:\[T_2 = ar = 1\]The fifth term is:\[T_5 = ar^4 = \frac{1}{8}\]Now divide the fifth term equation by the second term equation:\[\frac{ar^4}{ar} = \frac{1/8}{1}\]Simplify the left side:\[r^3 = \frac{1}{8}\]Take the cube root of both sides:\[r = \sqrt[3]{\frac{1}{8}}\]Since:\[\frac{1}{8} = \left(\frac{1}{2}\right)^3\]Therefore:\[r = \frac{1}{2}\]Hence, the common ratio is \(\frac{1}{2}\).

Draw a perpendicular from the centre of the circle to the chord. This perpendicular bisects the chord into two equal parts.Half the chord length is:\[\frac{30}{2} = 15 \text{ cm}\]Now we form a right triangle where:Hypotenuse = radius = 17 cmBase = 15 cmHeight = required distance dUsing the Pythagorean theorem:\[\text{hypotenuse}^2 = \text{base}^2 + \text{height}^2\]Substitute the values:\[17^2 = 15^2 + d^2\]Calculate the squares:\[289 = 225 + d^2\]Subtract 225 from both sides:\[d^2 = 64\]Take the square root:\[d = 8\]Therefore, the distance from the centre to the chord is 8 cm.

First factorize the quadratic expression.We look for two numbers whose product is -4 and whose sum is 3.These numbers are 4 and -1.So we write:\[x^2 + 3x - 4 = (x + 4)(x - 1)\]Now solve the inequality:\[(x + 4)(x - 1) \leq 0\]The critical values are obtained by setting each factor equal to zero:\[x + 4 = 0 \Rightarrow x = -4\]\[x - 1 = 0 \Rightarrow x = 1\]These values divide the number line into three intervals.Test values in each interval:Between -4 and 1, the expression is negative.Since we want values less than or equal to zero, we include this entire interval including endpoints.Therefore, the solution is:\[-4 \leq x \leq 1\]

The midpoint formula is:\[\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)\]Substitute the coordinates of P(-3, 4) and Q(5, 6):x-coordinate:\[\frac{-3 + 5}{2} = \frac{2}{2} = 1\]y-coordinate:\[\frac{4 + 6}{2} = \frac{10}{2} = 5\]Therefore, the midpoint is:\[(1, 5)\]

The mean of numbers is calculated using:\[\text{Mean} = \frac{\text{Sum of values}}{\text{Number of values}}\]There are four numbers, so:\[\frac{0 + (x + 2) + (3x + 6) + (4x + 8)}{4} = 4\]Simplify the numerator:\[0 + x + 2 + 3x + 6 + 4x + 8\]\[= 8x + 16\]So we have:\[\frac{8x + 16}{4} = 4\]Multiply both sides by 4:\[8x + 16 = 16\]Subtract 16 from both sides:\[8x = 0\]Divide by 8:\[x = 0\]Therefore, the value of x is 0.

First find the total cost price.This includes purchase price and repair cost:\[\text{Total cost} = 270000 + 70000\]\[= 340000\]Next find the profit:\[\text{Profit} = \text{Selling price} - \text{Cost price}\]\[= 490000 - 340000\]\[= 150000\]Now calculate percentage profit using:\[\text{Percentage profit} = \frac{\text{Profit}}{\text{Cost price}} \times 100\]Substitute values:\[= \frac{150000}{340000} \times 100\]Simplify fraction:\[= \frac{15}{34} \times 100\]Calculate:\[= 44.12\%\]The closest option is 45%.

A full pie chart represents 360°.First find the total number of people represented:\[16 + 8 + 24 + 21 + 27 = 96\]The number of doctors is 16.The fraction representing doctors is:\[\frac{16}{96}\]Now convert this fraction into degrees:\[\frac{16}{96} \times 360\]Simplify:\[= \frac{1}{6} \times 360\]Calculate:\[= 60^\circ\]Therefore, the angle for doctors is 60°.

Rewrite each square root in terms of \(\sqrt{3}\).\[\sqrt{75} = \sqrt{25 \times 3} = 5\sqrt{3}\]\[\sqrt{12} = \sqrt{4 \times 3} = 2\sqrt{3}\]\[\sqrt{27} = \sqrt{9 \times 3} = 3\sqrt{3}\]Substitute into the equation:\[5\sqrt{3} - 2\sqrt{3} + 3\sqrt{3}\]Combine like terms:\[(5 - 2 + 3)\sqrt{3}\]\[= 6\sqrt{3}\]Compare with:\[y\sqrt{3}\]Therefore:\[y = 6\]

The area of a triangle is calculated using the formula:\[\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}\]From the diagram, the base QR is given as 20 cm and the perpendicular height PC is 13 cm.Substitute these values into the formula:\[\text{Area} = \frac{1}{2} \times 20 \times 13\]First multiply 20 and 13:\[20 \times 13 = 260\]Now multiply by \(\frac{1}{2}\):\[\frac{1}{2} \times 260 = 130\]Therefore, the area of triangle PQR is 130 cm\(^2\).