2024 - JAMB Mathematics Past Questions and Answers - page 1

To convert 27₁₀ to base 3, we divide 27 by 3 repeatedly. 27 ÷ 3 = 9 (remainder 0), 9 ÷ 3 = 3 (remainder 0), 3 ÷ 3 = 1 (remainder 0), 1 ÷ 3 = 0 (remainder 1). Therefore, 27₁₀ = 1000₃.

1. Count the letters: The word MATHEMATICIAN has 13 letters.

2. Identify repetitions:

   • M appears 2 times.
   • A appears 3 times.
   • T appears 2 times.
   • I appears 2 times.
   • C, E, and N appear once each.

3. Apply the formula for permutations with repetitions: The number of arrangements is given by: $$\frac{n!}{n_1!n_2!...n_k!}$$ where n is the total number of letters, and n₁, n₂, ..., n_k are the counts of each repeated letter.

4. **

The diagonal of the room forms a 3D triangle with the floor. Using the Pythagorean theorem, the length of the diagonal is (\sqrt{12^2 + 9^2 + 8^2} = 15). The cosine of the angle with the floor is (\frac{12}{15} = \frac{4}{5} = \frac{15}{17}).

We are given the equation of two 2x2 determinants:

\(\begin{vmatrix} 5 & 3 \\ x & 2 \end{vmatrix} = \begin{vmatrix} 3 & 5 \\ 4 & 5 \end{vmatrix}\)

First, we calculate the determinant of the left-hand side:

\(\begin{vmatrix} 5 & 3 \\ x & 2 \end{vmatrix} = (5 \times 2) - (3 \times x) = 10 - 3x\)

Now, calculate the determinant of the right-hand side:

\(\begin{vmatrix} 3 & 5 \\ 4 & 5 \end{vmatrix} = (3 \times 5) - (5 \times 4) = 15 - 20 = -5\)

Now, equate both determinants:

10 - 3x = -5

Solve for \(x\):

Subtract 10 from both sides:

-3x = -5 - 10

-3x = -15

Now, divide by -3:

x = \(\frac{-15}{-3} = 5\)

Thus, the value of \(x\) is 5.

The correct option is x = 5.

The total surface area of a cylindrical pipe with radius 7m and length 50cm (converted to 0.5m) and one end open is given by (A = \pi r^2 + 2\pi r h). Substituting the values, we get (A = \pi(7^2) + 2\pi(7)(0.5) = 49\pi + 7\pi = 56\pi) m².

(\frac{2 - \sqrt5}{3 - \sqrt5}) x (\frac{3 + \sqrt5}{3 + \sqrt5})

(\frac{(2 - \sqrt5)(3 + \sqrt5)}{(3 - \sqrt5)(3 + \sqrt5)}) = (\frac{6 +2\sqrt5 - 3\sqrt5 - \sqrt25}{9 + 3\sqrt5 - 3\sqrt5 - \sqrt25})

= (\frac{6 - \sqrt5 - 5}{9 - 5})

= (\frac{1 - \sqrt5}{4})

To find the standard deviation, we follow these steps:

1. Find the mean of the data: The data set is: 2, 3, 8, 10, 12.
2. Find the squared differences from the mean for each data point:
   • For 2: \( (2 - 7)^2 = (-5)^2 = 25 \)
   • For 3: \( (3 - 7)^2 = (-4)^2 = 16 \)
   • For 8: \( (8 - 7)^2 = (1)^2 = 1 \)
   • For 10: \( (10 - 7)^2 = (3)^2 = 9 \)
   • For 12: \( (12 - 7)^2 = (5)^2 = 25 \)
3. Find the variance: The variance \( \sigma^2 \) is the average of the squared differences:
4. Find the standard deviation: The standard deviation \( \sigma \) is the square root of the variance:

S.D = \(\sqrt{\frac{(x - \varkappa)^2}{n}}\)

S.D = \(\sqrt{\frac{76}{5}}\)

S.D = 3.9

Therefore, the standard deviation is approximately 3.9.

If y = (2x + 1)³, then

Let u = 2x + 1 so that, y = u³

(\frac{dy}{du}) = 3u² and (\frac{dy}{dx}) = 2

Hence by the chain rule,

(\frac{dy}{dx}) = (\frac{dy}{du}) x (\frac{du}{dx})

= 3u² x 2

= 6u²

= 6(2x + 1)²

To solve the quadratic equation \( 3x^2 - 4x - 5 = 0 \), we will use the quadratic formula:

The quadratic formula is given by:

\( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

In the equation \( 3x^2 - 4x - 5 = 0 \), the coefficients are:

• a = 3
• b = -4
• c = -5

Now, substitute the values of \( a \), \( b \), and \( c \) into the quadratic formula:

\( x = \frac{-(-4) \pm \sqrt{(-4)^2 - 4(3)(-5)}}{2(3)} \)

\( x = \frac{4 \pm \sqrt{16 + 60}}{6} \)

\( x = \frac{4 \pm \sqrt{76}}{6} \)

Now, calculate the square root of 76:

\( \sqrt{76} \approx 8.7178 \)

Thus, the equation becomes:

\( x = \frac{4 \pm 8.7178}{6} \)

Now, solve for both values of \( x \):

1) For \( x_1 = \frac{4 + 8.7178}{6} \):

\( x_1 = \frac{12.7178}{6} \approx 2.12 \)

2) For \( x_2 = \frac{4 - 8.7178}{6} \):

\( x_2 = \frac{-4.7178}{6} \approx -0.79 \)

Therefore, the solutions to the equation are \( x = 2.12 \) or \( x = -0.79 \)

To find the volume of a cuboid, we use the formula:

\(\text{Volume} = \text{Length} \times \text{Breadth} \times \text{Height}\)

Given:

• Length = 0.76 cm
• Breadth = 2.6 cm
• Height = 0.82 cm

Now, we calculate the volume:

\(\text{Volume} = 0.76 \times 2.6 \times 0.82\)

Multiplying the values step by step:

\(0.76 \times 2.6 = 1.976\)

\(1.976 \times 0.82 = 1.61952\)

Rounding to two decimal places, we get:

\(\text{Volume} \approx 1.62 \, \text{cm}^3\)

Thus, the correct option is a volume of 1.62 cm³.