Mathematics 2018 Jamb Examination

The angle of depression is equal to the angle of elevation from the boat to the man. Let \( \displaystyle d \) be the distance from the base. Using trigonometry: \[ \displaystyle \tan 74^\circ = \frac{150}{d} \] \[ \displaystyle d = \frac{150}{\tan 74^\circ} = \frac{150}{3.487} \approx 43\text{cm} \]

Apply the power rule for integration \( \displaystyle \int x^n dx = \frac{x^{n+1}}{n+1} \):
\( \displaystyle \int 6x^2 dx = \frac{6x^3}{3} = 2x^3 \)
\( \displaystyle \int -2x dx = -\frac{2x^2}{2} = -x^2 \)
\( \displaystyle \int 1 dx = x \) \[ \displaystyle \int (6x^2 - 2x + 1) dx = 2x^3 - x^2 + x + c \]

The word LEADER has 6 letters. We check for repeated letters: - E appears 2 times. The number of unique arrangements is calculated as: \[ \displaystyle \frac{6!}{2!} = \frac{720}{2} = 360 \]

Since \( \displaystyle \angle XMN = \angle XZY \) and \( \displaystyle \angle X \) is common, triangles XMN and XZY are similar. Using the ratio of corresponding sides: \[ \displaystyle \frac{MX}{XZ} = \frac{XN}{XY} \] Given \( \displaystyle XZ = XN + NZ = 12 + 4 = 16\text{cm} \): \[ \displaystyle \frac{8}{16} = \frac{12}{XY} \implies \frac{1}{2} = \frac{12}{XY} \implies XY = 24\text{cm} \] \( \displaystyle YM = XY - MX = 24 - 8 = 16\text{cm} \)

First, convert both values to the same unit (grams): \[ \displaystyle 16.5\text{kg} = 16.5 \times 1000 = 16500\text{g} \] Now calculate the percentage: \[ \displaystyle \frac{495}{16500} \times 100 = \frac{495}{165} = 3\% \]

This is a difference of two squares in the form \( \displaystyle (a - b)(a + b) = a^2 - b^2 \). \[ \displaystyle (2\sqrt{3})^2 - 4^2 \] \[ \displaystyle (4 \times 3) - 16 \] \[ \displaystyle 12 - 16 = -4 \]

First, find the gradient of the curve by differentiating: \[ \displaystyle \frac{dy}{dx} = 2x - 2 \] At the point (2, 0), the gradient \( \displaystyle m \) is: \[ \displaystyle m = 2(2) - 2 = 2 \] Using the equation of a straight line \( \displaystyle y - y_1 = m(x - x_1) \): \[ \displaystyle y - 0 = 2(x - 2) \implies y = 2x - 4 \]

Solve each logarithm separately:
\( \displaystyle \log_2 8 = \log_2 2^3 = 3 \)
\( \displaystyle \log_3 \frac{1}{9} = \log_3 3^{-2} = -2 \)
Subtracting the two values: \[ \displaystyle 3 - (-2) = 3 + 2 = 5 \]

We use the CAST rule: - 1st Quadrant: All (Sin, Cos, Tan) are positive. - 2nd Quadrant: Only Sin is positive. - 3rd Quadrant: Only Tan is positive (meaning Sin and Cos are negative). - 4th Quadrant: Only Cos is positive. Since Tanθ is positive and Sinθ is negative, the angle must be in the Third Quadrant.

The minimum value of a quadratic graph (parabola) is the y-coordinate of its lowest point, also known as the vertex. Looking at the provided graph, the curve dips to its lowest point at approximately \( \displaystyle y = -5.3 \) on the vertical axis.