Mathematics 2016 Jamb Examination

Parallel lines have identical gradients (slopes). For any line in the form \( y = mx + c \), the coefficient \( m \) represents the gradient. Comparing both equations: \[ m_1 = 5 \] \[ m_2 = w \] Since the lines are parallel, \( m_1 = m_2 \), therefore \( w = 5 \).

To evaluate the integral, we use the standard trigonometric integration formulas: \[ \int \cos(ax) dx = \dfrac{1}{a} \sin(ax) + k \] \[ \int \sin(ax) dx = -\dfrac{1}{a} \cos(ax) + k \] Applying these to the expression: \[ \int \cos 4x dx + \int \sin 3x dx = \dfrac{1}{4} \sin 4x - \dfrac{1}{3} \cos 3x + k \]

To find the value of \( x \) in base 10, we expand the number in base 5 using powers of 5: \[ x = (2 \times 5^1) + (3 \times 5^0) \] \[ x = 10 + 3 \] \[ x = 13 \] Thus, the decimal equivalent of \( 23_5 \) is 13.

x^o + 36^o + 54^o + 28^o + 90^o + 120^o = 360^o

x + 328^o = 360^o
x = 360^o - 328^o
x = 32^o

The range of a set of data is the difference between the highest value and the lowest value in the set: \[ \text{Range} = \text{Maximum} - \text{Minimum} \] In the given data: \[ \text{Maximum} = 34 \] \[ \text{Minimum} = -6 \] \[ \text{Range} = 34 - (-6) = 34 + 6 = 40 \]

By observing the heights of the bars for the specified days on the vertical axis:
No of visitors = 10 + 5 + 1 = 16

The expression is a perfect square trinomial following the algebraic identity: \[ a^2 - 2ab + b^2 = (a - b)^2 \] By substituting \( a = k \) and \( b = p \), we get: \[ k^2 - 2kp + p^2 = (k - p)^2 \]

The perimeter of a sector consists of two radii and the arc length. First, calculate the arc length \( L \): \[ L = \dfrac{\theta}{360} \times 2\pi r = \dfrac{60}{360} \times 2\pi(12) = \dfrac{1}{6} \times 24\pi = 4\pi \] Now, add the lengths of the two radii to the arc length: \[ \text{Perimeter} = 2r + L = 2(12) + 4\pi = (24 + 4\pi) \; \text{cm} \]

The total mark is the sum of the products of each mark and its corresponding frequency: \[ \sum (\text{Mark} \times \text{Frequency}) = 48 \] \[ (2 \times 4) + (3 \times 4) + (4 \times y) = 48 \] \[ 8 + 12 + 4y = 48 \] \[ 20 + 4y = 48 \] \[ 4y = 28 \Rightarrow y = 7 \]

List the elements of the Universal set \( U \) and set \( P \): \[ U = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14\} \] \[ P = \{2, 4, 6, 8, 10, 12, 14\} \] The complement \( P' \) consists of elements in \( U \) that are not in \( P \): \[ P' = \{1, 3, 5, 7, 9, 11, 13\} \]