Mathematics 2022 Jamb Examination

The nth term of an AP is given by \(U_n = a + (n - 1)d\).
Given \(U_{10} = 32\) and \(a = 1.5\):\[32 = 1.5 + (10 - 1)d\]\[32 - 1.5 = 9d \implies 30.5 = 9d \implies d = \frac{30.5}{9} = \frac{61}{18}\]Now, find the 4th term (\(U_4\)):\[U_4 = a + 3d = 1.5 + 3\left(\frac{61}{18}\right)\]\[U_4 = \frac{3}{2} + \frac{61}{6} = \frac{9 + 61}{6} = \frac{70}{6} = \frac{35}{3}\]

The angles on a straight line (LNZ) sum up to 180°. From the diagram, the three angles \(\angle LNM\), \(\angle MNY\), and \(\angle YNZ\) are adjacent angles on the line LNZ.\[\angle LNM + \angle MNY + \angle YNZ = 180^o\]Substituting the given variables g and h:\[g + \angle MNY + h = 180^o\]Isolating \(\angle MNY\):\[\angle MNY = 180^o - g^o - h^o\]

The word ZOOLOGY has 7 letters in total. However, some letters repeat: 'O' appears 3 times. The number of permutations of n objects where p are of one kind is given by \(\frac{n!}{p!}\).\[\text{Ways} = \frac{7!}{3!} = \frac{7 \times 6 \times 5 \times 4 \times 3!}{3!}\]\[= 7 \times 6 \times 5 \times 4 = 840\]

To integrate \(\int (2x+1)^3 dx\), we use the power rule for a linear function \(\int (ax+b)^n dx = \frac{(ax+b)^{n+1}}{a(n+1)} + C\).
Here, a = 2, b = 1, and n = 3:\[\int (2x+1)^3 dx = \frac{(2x+1)^{3+1}}{2(3+1)} + C\]\[= \frac{(2x+1)^4}{2(4)} + C = \frac{(2x+1)^4}{8} + C\]

Multiply each row of A by each column of B:
Row 1: \((2 \times 3) + (1 \times 4) = 10\); \((2 \times 2) + (1 \times 2) = 6\)
Row 2: \((2 \times 3) + (3 \times 4) = 18\); \((2 \times 2) + (3 \times 2) = 10\)
Row 3: \((1 \times 3) + (2 \times 4) = 11\); \((1 \times 2) + (2 \times 2) = 6\)
The resulting matrix is \(\begin{pmatrix} 10 & 6 \\ 18 & 10 \\ 11 & 6 \end{pmatrix}\).

Square both sides to remove the square root:\[r^2 = \frac{3v}{\pi h}\]Multiply both sides by \(\pi h\):\[\pi r^2 h = 3v\]Divide both sides by 3 to isolate v:\[v = \frac{\pi r^2 h}{3}\]

The determinant of a matrix \(\begin{pmatrix} a & b \\ c & d \end{pmatrix}\) is \(ad - bc\).\[|A| = (2 \times 3) - (3 \times 1)\]\[|A| = 6 - 3 = 3\]

Express \(8\sqrt{2}\) as a power of 2:\[8\sqrt{2} = 2^3 \times 2^{1/2} = 2^{3 + 0.5} = 2^{3.5}\]Using the property \(\log_b(b^x) = x\):\[\log_2(2^{3.5}) = 3.5\]

A circle is defined geometrically as the locus of all points in a plane that are at a constant distance (the radius) from a fixed point (the center).

New Numerator: \(96 - (0.375 \times 96) = 96 - 36 = 60\)
New Denominator: \(50 - (0.20 \times 50) = 50 - 10 = 40\)
The new fraction is \(\frac{60}{40} = \frac{3}{2} = 1.5\).