2024 - JAMB Mathematics Past Questions and Answers - page 1

Step-by-Step Solution:

Count the total number of letters:

The word "MATHEMATICIAN" has 13 letters.

Identify the frequency of each letter:

• M: 2
• A: 3
• T: 2
• H: 1
• E: 1
• I: 2
• C: 1
• N: 1

Use the formula for permutations of a multiset:

Total permutations = \(\frac{13!}{2! \cdot 3! \cdot 2! \cdot 1! \cdot 1! \cdot 2! \cdot 1!}\) 

Plug in the values:

Total permutations = \(\frac{13!}{2! \cdot 3! \cdot 2! \cdot 1! \cdot 1! \cdot 2! \cdot 1!}\) 

Total permutations = \(\frac{6227020800}{48} = 129729600\) 

The number of ways to arrange the letters in the word MATHEMATICIAN is 129729600.

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.

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.

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.