2023 - JAMB Mathematics Past Questions and Answers - page 5

 

1. Let \(y\) be the total number of items in the man's shop.
2. \(^1/_9\) of the items are from Brand A: \(\frac{1}{9}y\).
3. The remainder is \(\frac{8}{9}y\).
4. \(^5/_8\) of the remainder are from Brand B: \(\frac{5}{8} \times \frac{8}{9}y = \frac{5}{9}y\).
5. Total items from Brand A and Brand B: \(\frac{1}{9}y + \frac{5}{9}y = \frac{2}{3}y\).
6. Remaining items (Brand C): \(1 - \frac{2}{3}y = \frac{1}{3}y\).
7. Given that the number of Brand C items is 81: \(\frac{1}{3}y = 81\).
8. Solve for \(y\): \(y = 81 \times 3 = 243\).
9. Number of Brand B items: \(\frac{5}{9} \times 243 = 135\).
10. The number of more Brand B items than Brand C: \(135 - 81 = 54\).

 

Let's denote the third number as \(x\).

The first number is 35% more than the third number, which means it is \(x + 0.35x = 1.35x\).

The second number is 80% more than the third number, which means it is \(x + 0.80x = 1.80x\).

Now, the ratio of the two numbers is \(\frac{1.35x}{1.80x}\).

To simplify the ratio, we can divide both terms by \(0.45x\):

\[
\frac{1.35x}{1.80x} = \frac{1.35}{1.80} = \frac{135}{180} = \frac{27}{36} = \frac{3}{4}
\]

So, the correct ratio is \(3 : 4\)

Let's denote the exterior angle of an \(n\)-sided regular polygon as \(E_n\). The exterior angle of an \(n\)-sided regular polygon is given by:

\[ E_n = \frac{360^\circ}{n} \]

Now, according to the given information:

\[ E_{n-1} - E_{n+2} = 6^\circ \]

Substitute the expressions for \(E_{n-1}\) and \(E_{n+2}\) into the equation:

\[ \frac{360^\circ}{n-1} - \frac{360^\circ}{n+2} = 6^\circ \]

To solve this equation, we can find a common denominator:

\[ \frac{360(n+2) - 360(n-1)}{(n-1)(n+2)} = 6 \]

Simplify the numerator:

\[ \frac{360n + 720 - 360n + 360}{(n-1)(n+2)} = 6 \]

Combine like terms:

\[ \frac{1080}{(n-1)(n+2)} = 6 \]

Now, cross-multiply and simplify:

\[ 1080 = 6(n-1)(n+2) \]

Divide both sides by 6:

\[ 180 = (n-1)(n+2) \]

Expand the right side:

\[ 180 = n^2 + 2n - n - 2 \]

Combine like terms:

\[ 180 = n^2 + n - 2 \]

Rewrite the equation in standard form:

\[ n^2 + n - 182 = 0 \]

Factor the quadratic expression:

\[ (n - 13)(n + 14) = 0 \]

This gives two possible solutions:

\[ n - 13 = 0 \quad \text{or} \quad n + 14 = 0 \]

\[ n = 13 \quad \text{or} \quad n = -14 \]

Since the number of sides (\(n\)) cannot be negative, we discard the solution \(n = -14\). Therefore, the correct value of \(n\) is:

\[ n = 13 \]

 

To evaluate the expression \(\frac{5}{8} - \frac{3}{4} ÷ \frac{5}{12} \times \frac{1}{4}\), follow the order of operations (PEMDAS/BODMAS):

1. Perform the division: \(\frac{3}{4} ÷ \frac{5}{12}\)
2. Multiply the result by \(\frac{1}{4}\)
3. Subtract the product from \(\frac{5}{8}\)

Let's proceed with the calculations:

1. \(\frac{3}{4} ÷ \frac{5}{12} = \frac{3}{4} \times \frac{12}{5} = \frac{9}{5}\)
2. Multiply the result by \(\frac{1}{4}\): \(\frac{9}{5} \times \frac{1}{4} = \frac{9}{20}\)
3. Subtract the product from \(\frac{5}{8}\): \(\frac{5}{8} - \frac{9}{20} = \frac{25}{40} - \frac{9}{20} = \frac{16}{40} = \frac{2}{5}\)

Therefore, the correct option is \(\frac{1}{4}\)

Given polynomial: \( p(x) = -2x^3 + 6x^2 + 17x - 21 \)

We want to find the remainder when \( p(x) \) is divided by \( (x + 1) \). According to the Remainder Theorem, if you substitute \( x = -1 \) into \( p(x) \), you get the remainder.

So, substitute \( x = -1 \) into \( p(x) \):

\[ p(-1) = -2(-1)^3 + 6(-1)^2 + 17(-1) - 21 \]

Simplify each term:

\[ p(-1) = -2(-1) + 6(1) - 17 - 21 \]

\[ p(-1) = 2 + 6 - 17 - 21 \]

\[ p(-1) = -30 \]

Therefore, the remainder is \(-30\). 

To factorize the given expression \(16x^4 - y^4\), we can use the difference of squares formula, which states that \(a^2 - b^2 = (a - b)(a + b)\).

Let \(a = 2x^2\) and \(b = y^2\). Then, \(a^2 - b^2 = (2x^2 - y^2)(2x^2 + y^2)\).

Now, \(2x^2 - y^2\) is itself a difference of squares, where \(c = 2x\) and \(d = y\). Applying the difference of squares formula again, we get \(2x^2 - y^2 = (2x - y)(2x + y)\).

Substitute this back into the original expression:

\[16x^4 - y^4 = (2x - y)(2x + y)(2x^2 + y^2)\]

So, the correct factorization is:

\[16x^4 - y^4 = (2x - y)(2x + y)(2x^2 + y^2)\]

 

To evaluate the limit \(\lim_{{x \to 2}} \frac{{x^2 + 4x - 12}}{{x^2 - 2x}}\), let's substitute \(x = 2\) into the expression:

\[ \lim_{{x \to 2}} \frac{{x^2 + 4x - 12}}{{x^2 - 2x}} = \frac{{2^2 + 4(2) - 12}}{{2^2 - 2(2)}} \]

\[ = \frac{{4 + 8 - 12}}{{4 - 4}} \]

\[ = \frac{0}{0} \]

Since the expression results in an indeterminate form (\(\frac{0}{0}\)), we can simplify it further by factoring the numerator and denominator:

\[ \frac{{x^2 + 4x - 12}}{{x^2 - 2x}} = \frac{{(x - 2)(x + 6)}}{{x(x - 2)}} \]

Now, we can cancel the common factor \((x - 2)\) from the numerator and denominator:

\[ \lim_{{x \to 2}} \frac{{x^2 + 4x - 12}}{{x^2 - 2x}} = \lim_{{x \to 2}} \frac{{x + 6}}{{x}} \]

Now, substitute \(x = 2\) into the simplified expression:

\[ \lim_{{x \to 2}} \frac{{x + 6}}{{x}} = \frac{{2 + 6}}{{2}} = \frac{8}{2} = 4 \]

 

The relationship between the interior angle (\(I\)) and the exterior angle (\(E\)) of a regular polygon is given by the formula:

\[I = 180^\circ - E\]

Given that the interior angle is five times the size of the exterior angle, we can set up the equation:

\[I = 5E\]

Now, substitute \(180^\circ - E\) for \(I\) in the equation:

\[180^\circ - E = 5E\]

Combine like terms:

\[180^\circ = 6E\]

Solve for \(E\):

\[E = \frac{180^\circ}{6} = 30^\circ\]

So, the exterior angle of the regular polygon is \(30^\circ\).

Now, we know that the exterior angle of a regular polygon is given by \(360^\circ/n\), where \(n\) is the number of sides. Set up the equation:

\[30^\circ = \frac{360^\circ}{n}\]

Solve for \(n\):

\[n = \frac{360^\circ}{30^\circ} = 12\]

So, the polygon has 12 sides, and it is a dodecagon. 

The gain percent can be calculated using the following formula:

\[ \text{Gain Percent} = \left( \frac{\text{Profit}}{\text{Cost Price}} \right) \times 100 \]

First, let's calculate the profit. The profit is the difference between the selling price (SP) and the total cost price (CP), where the total cost price is the sum of the original cost price and any additional expenses.

\[ \text{Profit} = \text{SP} - \text{Total CP} \]

Given:
- Cost Price (CP) = N9,200.00 (original cost of the bicycle)
- Additional Expenses = N1,500.00 (repairs)

\[ \text{Total CP} = \text{CP} + \text{Additional Expenses} = 9,200 + 1,500 = N10,700.00 \]

\[ \text{Profit} = \text{SP} - \text{Total CP} = 13,400 - 10,700 = N2,700.00 \]

Now, plug these values into the gain percent formula:

\[ \text{Gain Percent} = \left( \frac{2,700}{10,700} \right) \times 100 \]

\[ \text{Gain Percent} \approx 25.23\% \]

Therefore, Bello's gain percent is approximately 25.23%. 

The percentage decrease in population can be calculated using the formula:

\[ \text{Percentage Decrease} = \left( \frac{\text{Decrease in Population}}{\text{Original Population}} \right) \times 100 \]

Given that the original population is 1,230 and it decreased to 1,040:

\[ \text{Decrease in Population} = \text{Original Population} - \text{Final Population} = 1,230 - 1,040 = 190 \]

\[ \text{Percentage Decrease} = \left( \frac{190}{1,230} \right) \times 100 \]

\[ \text{Percentage Decrease} \approx 15.45\% \]