2023 - JAMB Mathematics Past Questions and Answers - page 6

The determinant (|D|) of a 3x3 matrix \(D = \begin{bmatrix}a&b&c\\d&e&f\\g&h&i\\\end{bmatrix}\) is given by the formula:

\[ |D| = a(ei - fh) - b(di - fg) + c(dh - eg) \]

For the given matrix \(D = \begin{bmatrix}2& -1&3\\4&1&2\\1&-3&1\\\end{bmatrix}\):

\[ |D| = 2(1 \cdot 1 - (-3) \cdot 2) - (-1)(4 \cdot 1 - 2 \cdot 1) + 3(4 \cdot (-3) - 1 \cdot 2) \]

\[ |D| = 2(1 + 6) + (4 - 2) + 3(-12 - 2) \]

\[ |D| = 2(7) + 2 - 3(14) \]

\[ |D| = 14 + 2 - 42 \]

\[ |D| = -26 \]

To find the equation of a line perpendicular to the given line \(3x + 2y + 4 = 0\) and passing through the point (2, 3), we need to follow these steps:

1. Find the slope (m) of the given line. The slope of the given line \(ax + by + c = 0\) is given by \(-\frac{a}{b}\).
2. Find the negative reciprocal of the slope found in step 1. This will be the slope of the line perpendicular to the given line.
3. Use the point-slope form of the equation of a line: \((y - y_1) = m(x - x_1)\), where \((x_1, y_1)\) is the given point and \(m\) is the slope found in step 2.
4. Simplify the equation into the desired form.

Let's go through the steps:

1. Given line: \(3x + 2y + 4 = 0\)
   Slope (m) = \(-\frac{3}{2}\)

2. Slope of the line perpendicular to the given line:
   Perpendicular slope = \(\frac{2}{3}\)

3. Point-slope form using the point (2, 3):
   \((y - 3) = \frac{2}{3}(x - 2)\)

4. Simplify the equation:
   \[3(y - 3) = 2(x - 2)\]
   \[3y - 9 = 2x - 4\]
   \[3y = 2x + 5\]

The nth term \(a_n\) of an arithmetic progression (A.P.) is given by the formula:

\[a_n = a + (n-1)d\]

where:
- \(a\) is the first term,
- \(n\) is the term number,
- \(d\) is the common difference.

Given that the third term \(a_3 = 6\) and the fifth term \(a_5 = 12\), we can set up a system of equations to solve for the first term \(a\) and the common difference \(d\):

1. For the third term:
\[a_3 = a + 2d = 6\]

2. For the fifth term:
\[a_5 = a + 4d = 12\]

Solving this system will give us the values of \(a\) and \(d\). Once we have those, we can use the formula for the sum of the first \(n\) terms of an A.P.:

\[S_n = \frac{n}{2}[2a + (n-1)d]\]

to find the sum of the first twelve terms.

Let's solve for \(a\) and \(d\):

From equation (1):
\[a + 2d = 6\]

From equation (2):
\[a + 4d = 12\]

Subtracting equation (1) from equation (2):
\[(a + 4d) - (a + 2d) = 12 - 6\]
\[2d = 6\]

Dividing both sides by 2:
\[d = 3\]

Now substitute \(d = 3\) back into equation (1):
\[a + 2(3) = 6\]
\[a + 6 = 6\]
\[a = 0\]

So, \(a = 0\) and \(d = 3\).

Now, we can find the sum of the first twelve terms using the formula:
\[S_{12} = \frac{12}{2}[2(0) + (12-1)(3)]\]

Calculating this expression will give us the sum.

\[S_{12} = \frac{12}{2}[2(0) + 11(3)] = 6 \times 33 = 198\]

The formula for the volume (\(V\)) of a cone is given by:

\[V = \frac{1}{3} \pi r^2 h\]

where:
- \(\pi\) is a mathematical constant approximately equal to 3.14159,
- \(r\) is the radius of the base of the cone,
- \(h\) is the height of the cone.

Given that the base radius (\(r\)) is 5 cm and the slant height is the hypotenuse of the right triangle formed by the radius, and the height (\(h\)) can be found using the Pythagorean theorem:

\[h = \sqrt{\text{{slant height}}^2 - \text{{radius}}^2}\]

Substitute the values into the formula for the volume:

\[V = \frac{1}{3} \pi (5)^2 \sqrt{13^2 - 5^2}\]

Let's calculate this to find the correct volume.

\[h = \sqrt{13^2 - 5^2} = \sqrt{169 - 25} = \sqrt{144} = 12 \, \text{cm}\]

Now, substitute the values into the formula for the volume:

\[V = \frac{1}{3} \pi (5)^2 \times 12 = \frac{1}{3} \pi \times 25 \times 12 = 100 \pi \, \text{cm}^3\]

Let's solve for \(x\) in the inequality \(3(x - 1) \leq 2(x - 3)\):

\[3(x - 1) \leq 2(x - 3)\]

Distribute the terms:

\[3x - 3 \leq 2x - 6\]

Subtract \(2x\) from both sides:

\[x - 3 \leq -6\]

Add 3 to both sides:

\[x \leq -3\]

Let's find the combinations that result in a prime number when two dice are tossed. There are 6 faces on each die, so there are \(6 \times 6 = 36\) possible outcomes.

Prime numbers less than or equal to 12 are: 2, 3, 5, 7, 11.

Now, let's count the combinations that give each prime number:

- For 2: (1, 1)
- For 3: (1, 2), (2, 1)
- For 5: (1, 4), (2, 3), (3, 2), (4, 1)
- For 7: (1, 6), (2, 5), (3, 4), (4, 3), (5, 2), (6, 1)
- For 11: (5, 6), (6, 5)

There are 15 combinations that result in a prime number.

Therefore, the probability of getting a prime number is \(\frac{15}{36}\), which simplifies to \(\frac{5}{12}\).

Let's denote the length of the side of the square as \(s\) and the length of the diagonal as \(d\). For a square, the length of the diagonal is related to the side length by \(d = s\sqrt{2}\).

Given that \(d = 20\sqrt{2}\), we can equate this to the diagonal formula:

\[s\sqrt{2} = 20\sqrt{2}\]

Now, solving for \(s\):

\[s = \frac{20\sqrt{2}}{\sqrt{2}}\]

\[s = 20\]

Now, we can find the area (\(A\)) and perimeter (\(P\)) of the square:

Area of a square: \(A = s^2\)

Perimeter of a square: \(P = 4s\)

Substitute the value of \(s\) into these formulas:

\[A = 20^2 = 400 \, \text{cm}^2\]

\[P = 4 \times 20 = 80 \, \text{cm}\]