SSS Two Mathematics Objective Exam Questions

SSS 2 Mathematics Objective Questions

Mathematics Objective Questions for SSS 2

1. Find the range of values of 𝑥 for which $5𝑥 − 4 > 11$.

$𝑥 > 3$
$𝑥 < 3$
$𝑥 < 5$
$𝑥 < 1$

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Correct Answer: A. $𝑥 > 3$

Explanation:
$5𝑥 − 4 > 11$
$5𝑥 > 11 + 4$
$5𝑥 > 15$
$𝑥 > 15/5$
$𝑥 > 3$

2. If 𝑥 is an integer, find the three lowest possible values of 𝑥 in $2𝑥 − 1 > 5$.

3, 4, 5
4, 5, 6
1, 2, 3
3, 2, 1

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Correct Answer: B. 4, 5, 6

Explanation:
$2𝑥 − 1 > 5$
$2𝑥 > 5 + 1$
$2𝑥 > 6$
$𝑥 > 6/2$
$𝑥 > 3$
Since 𝑥 is an integer and must be greater than 3, the lowest possible integer values are 4, 5, and 6.

3. The statement 'there are less than 30 people on a bus’ can be represented mathematically as

$𝑥 < 30$
$𝑥 > 30$
$𝑥 ≤ 30$
$𝑥 ≥ 30$

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Correct Answer: A. $𝑥 < 30$

4. Solve for 𝑥 if $3 − 𝑥 < 1$.

$𝑥 < −2$
$𝑥 < 4$
$𝑥 > 2$
$𝑥 > −4$

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Correct Answer: C. $𝑥 > 2$

Explanation:
$3 − 𝑥 < 1$
$-𝑥 < 1 − 3$
$-𝑥 < -2$
Multiply both sides by -1 and reverse the inequality sign:
$𝑥 > 2$

5. Simplify $\frac{8x^2z}{10xyz}$

$\frac{4xyz}{5}$
$\frac{4z}{5}$
$\frac{4x}{5y}$
$\frac{4yz}{5}$

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Correct Answer: C. $\frac{4x}{5y}$

Explanation:
$\frac{8x^2z}{10xyz}$
Cancel common factors:
$8/10 = 4/5$
$x^2/x = x$
$z/z = 1$
The $y$ in the denominator remains.
So, $\frac{8x^2z}{10xyz} = \frac{4x}{5y}$

6. Simplify $\frac{18ab}{15bc} \times \frac{20cd}{24de}$

$\frac{a}{e}$
$\frac{3a}{5b}$
1
$\frac{4abcd}{2ac}$

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Correct Answer: A. $\frac{a}{e}$

Explanation:
First, simplify each fraction:
$\frac{18ab}{15bc} = \frac{18}{15} \cdot \frac{a}{1} \cdot \frac{b}{b} \cdot \frac{1}{c} = \frac{6}{5} \cdot a \cdot 1 \cdot \frac{1}{c} = \frac{6a}{5c}$
$\frac{20cd}{24de} = \frac{20}{24} \cdot \frac{c}{1} \cdot \frac{d}{d} \cdot \frac{1}{e} = \frac{5}{6} \cdot c \cdot 1 \cdot \frac{1}{e} = \frac{5c}{6e}$
Now multiply the simplified fractions:
$\frac{6a}{5c} \times \frac{5c}{6e}$
Cancel out common terms (6, 5, and c):
$\frac{\cancel{6}a}{\cancel{5}\cancel{c}} \times \frac{\cancel{5}\cancel{c}}{\cancel{6}e} = \frac{a}{e}$

7. For what value of 𝑥 is the expression $\frac{2x+1}{3x-12}$ not defined?

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Correct Answer: B. 4

Explanation:
A rational expression is not defined when its denominator is equal to zero. So, set the denominator to zero and solve for $x$:
$3x - 12 = 0$
$3x = 12$
$x = \frac{12}{3}$
$x = 4$

8. Solve for 𝑎 if $\frac{3}{a} = a - 2$.

−1, 2
5, 6
3, −1
2, 3

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Correct Answer: C. 3, −1

Explanation:
Multiply both sides by $a$ to clear the denominator (assuming $a \neq 0$):
$3 = a(a - 2)$
$3 = a^2 - 2a$
Rearrange into a standard quadratic equation form ($ax^2 + bx + c = 0$):
$a^2 - 2a - 3 = 0$
Factor the quadratic equation. We need two numbers that multiply to -3 and add to -2. These numbers are -3 and 1:
$(a - 3)(a + 1) = 0$
Set each factor to zero and solve for $a$:
$a - 3 = 0 \implies a = 3$
$a + 1 = 0 \implies a = -1$
So, the solutions are $a = 3$ and $a = -1$.

9. Simplify $\frac{h^2 - k^2}{(h - k)^2}$

$\frac{h+k}{h-k}$
$\frac{h-k}{h+k}$
$ℎ + 𝑘$
$ℎ − 𝑘$

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Correct Answer: A. $\frac{h+k}{h-k}$

Explanation:
Recall the difference of squares formula: $h^2 - k^2 = (h - k)(h + k)$
Also, $(h - k)^2 = (h - k)(h - k)$
Substitute these into the expression:
$\frac{(h - k)(h + k)}{(h - k)(h - k)}$
Cancel one common factor of $(h - k)$ from the numerator and denominator:
$\frac{h + k}{h - k}$

10. If $\frac{x}{y} = \frac{3}{4}$, evaluate $\frac{2x - y}{2x + y}$.

Click to reveal answer

Correct Answer: The correct answer is B. 1/5

Explanation:
Given $\frac{x}{y} = \frac{3}{4}$. This means we can let $x = 3k$ and $y = 4k$ for some non-zero constant $k$.
Substitute these into the expression we need to evaluate:
$\frac{2x - y}{2x + y} = \frac{2(3k) - (4k)}{2(3k) + (4k)}$
$= \frac{6k - 4k}{6k + 4k}$
$= \frac{2k}{10k}$
Cancel $k$ (since $k \neq 0$):
$= \frac{2}{10} = \frac{1}{5}$

11. Which of the following represents the statement ‘he is unhealthy only if he is neat’

$\sim p \to \sim q$
$\sim p \to q$
$p \to \sim p$
$\sim q \to \sim p$

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Correct Answer: B. $\sim p \to q$

Explanation:
Given statements:
$p$: He is healthy
$q$: He is neat
The statement "he is unhealthy" is the negation of $p$, written as $\sim p$.
The phrase "only if" indicates a conditional statement. "A only if B" means "If A, then B", which is $A \to B$.
So, "he is unhealthy only if he is neat" translates to "If he is unhealthy, then he is neat", which is $\sim p \to q$.

12. Which of the following represents ‘he is not neat only if he is unhealthy’

$\sim q \to \sim p$
$q \to p$
$\sim p \to \sim q$
$\sim q \to p$

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Correct Answer: A. $\sim q \to \sim p$

Explanation:
Given statements:
$p$: He is healthy
$q$: He is neat
The statement "he is not neat" is the negation of $q$, written as $\sim q$.
The statement "he is unhealthy" is the negation of $p$, written as $\sim p$.
Using "A only if B" as $A \to B$, "he is not neat only if he is unhealthy" translates to "If he is not neat, then he is unhealthy", which is $\sim q \to \sim p$.

13. If $p$ is the statement $3 + 4 = 8$ then negation $p$ written as $\sim p$ is the statement

$3 + 4 = 7$
$3 + 4 \neq 8$
$3 + 4 = 8$
$a \text{ and } b$

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Correct Answer: B. $3 + 4 \neq 8$

Explanation:
The negation of a statement asserts the opposite truth value. If $p$ is the statement "$3 + 4 = 8$", its negation $\sim p$ is "$3 + 4$ is not equal to $8$", which is written as $3 + 4 \neq 8$.

14. If $p \to q$, then the converse statement of $p \to q$ is

$p \to \sim q$
$\sim p \to q$
$\sim q \to \sim p$
$q \to p$

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Correct Answer: D. $q \to p$

Explanation:
For a conditional statement in the form "If P, then Q" (symbolically $P \to Q$), its converse is formed by interchanging the hypothesis (P) and the conclusion (Q). Therefore, the converse of $p \to q$ is $q \to p$.

15. A chord of length $24cm$ is $13cm$ from the centre of the circle. Calculate the radius of the circle

$17.69cm$
$15cm$
$7cm$
$12.57cm$

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Correct Answer: A. $17.69cm$

Explanation:
When a perpendicular line is drawn from the center of a circle to a chord, it bisects the chord. So, half the length of the chord is $\frac{24cm}{2} = 12cm$.
This forms a right-angled triangle where:
- One leg is the distance from the center to the chord ($13cm$).
- The other leg is half the chord length ($12cm$).
- The hypotenuse is the radius of the circle ($r$).
Using the Pythagorean theorem, $a^2 + b^2 = c^2$:
$r^2 = (13)^2 + (12)^2$
$r^2 = 169 + 144$
$r^2 = 313$
$r = \sqrt{313}$
$r \approx 17.6918cm$
Rounding to two decimal places, $r \approx 17.69cm$.

16. Find the value of 𝑥 in the figure below (O is the centre of the circle)

$49^\circ$
$82^\circ$
$108^\circ$
$98^\circ$

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Correct Answer: B. $82^\circ$

Explanation:
The angle at the center subtended by an arc is twice the angle the same arc subtends at any point on the circumference.

17. Find the value of y in the figure below (O is the centre of the circle)

$180^\circ$
$45^\circ$
$90^\circ$
$60^\circ$

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Correct Answer: C. $90^\circ$

Explanation:
The figure shows an angle ($y$) inscribed in a semicircle (an angle whose vertex lies on the circle and whose sides pass through the endpoints of a diameter).
Theorem: The angle subtended by a diameter at any point on the circumference is always a right angle ($90^\circ$).
Therefore, $y = 90^\circ$.

18. For what value of 𝑥 is the expression $\frac{x^2+3x+2}{x+4}$ equals to zero?

2, 3
−1, −2
3, −4
3 − 5

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Correct Answer: B. −1, −2

Explanation:
A rational expression (a fraction) is equal to zero if and only if its numerator is zero AND its denominator is non-zero.
First, set the numerator to zero and solve for $x$:
$x^2 + 3x + 2 = 0$
Factor the quadratic equation: Find two numbers that multiply to 2 and add to 3. These are 1 and 2.
$(x + 1)(x + 2) = 0$
This gives two possible values for $x$:
$x + 1 = 0 \implies x = -1$
$x + 2 = 0 \implies x = -2$
Next, check that these values do not make the denominator zero:
For $x = -1$, denominator is $(-1 + 4) = 3 \neq 0$.
For $x = -2$, denominator is $(-2 + 4) = 2 \neq 0$.
Both values are valid solutions.

19. Solve for 𝑥 and 𝑦, $2𝑥 + 5𝑦 = 1$, $3𝑥 − 2𝑦 = 30$

$𝑥 = 8, 𝑦 = −3$
$𝑥 = 2, 𝑦 = 5$
$𝑥 = 4, 𝑦 = 2$
$𝑥 = 6, 𝑦 = 2$

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Correct Answer: A. $𝑥 = 8, 𝑦 = −3$

Explanation:
We have a system of two linear equations:
1) $2x + 5y = 1$
2) $3x - 2y = 30$
Using the elimination method, multiply equation (1) by 2 and equation (2) by 5 to eliminate $y$:
$(2x + 5y = 1) \times 2 \implies 4x + 10y = 2$ (Equation 3)
$(3x - 2y = 30) \times 5 \implies 15x - 10y = 150$ (Equation 4)
Add Equation 3 and Equation 4:
$(4x + 10y) + (15x - 10y) = 2 + 150$
$19x = 152$
$x = \frac{152}{19}$
$x = 8$
Now substitute the value of $x$ into equation (1) to find $y$:
$2(8) + 5y = 1$
$16 + 5y = 1$
$5y = 1 - 16$
$5y = -15$
$y = \frac{-15}{5}$
$y = -3$
Thus, the solution is $x = 8$ and $y = -3$.

20. What is the range of value of 𝑥 represented by the number line?

$-4 < 𝑥 < 5$
$-4 \le 𝑥 \le 5$
$5 > 𝑥 < -4$
$-4 > 𝑥 > 5$

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Correct Answer: B. $-4 \le 𝑥 \le 5$

Explanation:
The number line shows a shaded segment between -4 and 5, with solid circles at both -4 and 5.
- A solid circle indicates that the endpoint is included in the range.
- The shaded region indicates all values between these endpoints are also included.
Therefore, the range of values for $x$ is from -4 to 5, inclusive, which is written as $-4 \le x \le 5$.

21. Find the range of value of 𝑥 for which $2 − 𝑥 < 8 < 9 − 𝑥$

$-6 < 𝑥 < 1$
$67 < 𝑥 < 2$
$2 < 𝑥 < 9$
$2 < 𝑥 < 4$

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Correct Answer: A. $-6 < 𝑥 < 1$

Explanation:
This is a compound inequality, which can be split into two separate inequalities:
1) $2 - x < 8$
2) $8 < 9 - x$

Solve the first inequality:
$2 - x < 8$
$-x < 8 - 2$
$-x < 6$
Multiply by -1 and reverse the inequality sign:
$x > -6$

Solve the second inequality:
$8 < 9 - x$
$x < 9 - 8$
$x < 1$

Combine both solutions. We need $x$ to be greater than -6 AND less than 1. This is written as:
$-6 < x < 1$

22. Which equation represents the graph of inequality below,

$𝑦 < 4$
$𝑥 < 4$
$𝑥 + 𝑦 \le 4$
$𝑥 − 𝑦 < 4$

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Correct Answer: C. $𝑥 + 𝑦 \le 4$)

Explanation:
First, find the equation of the boundary line. The line passes through the points $(4, 0)$ on the x-axis and $(0, 4)$ on the y-axis. The equation of a line with x-intercept $a$ and y-intercept $b$ is $\frac{x}{a} + \frac{y}{b} = 1$.
Substituting the intercepts: $\frac{x}{4} + \frac{y}{4} = 1$
Multiply the entire equation by 4 to eliminate denominators: $x + y = 4$.
Next, determine the inequality sign. The line is dashed, which typically means the inequality is strict (either $<$ or $>$), not inclusive ($\le$ or $\ge$). The shaded region is below the line. To confirm the inequality direction, pick a test point not on the line, for example, $(0,0)$.
Substitute $(0,0)$ into $x + y = 4$: $0 + 0 = 0$. Since $0$ is less than $4$, the inequality that represents the shaded region is $x + y < 4$.
Among the given options, $x + y \le 4$ (Option C) has the correct linear expression ($x+y$) and constant ($4$). The only discrepancy is the inequality symbol (dashed line implies strict inequality `<` rather than inclusive `$\le$`). However, it is the only option that correctly represents the relationship between $x$ and $y$ for the boundary line.

23. What is the length of a chord of a circle of radius $26cm$ if the chord is $10cm$ from the centre of the circle.

$24cm$
$48cm$
$12 cm$
$7cm$.

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Correct Answer: B. $48cm$

Explanation:
When a perpendicular line is drawn from the center of a circle to a chord, it bisects the chord. This forms a right-angled triangle where:
- The hypotenuse is the radius of the circle ($r = 26cm$).
- One leg is the distance from the center to the chord ($d = 10cm$).
- The other leg is half the length of the chord ($\frac{l}{2}$).
Using the Pythagorean theorem, $r^2 = d^2 + \left(\frac{l}{2}\right)^2$:
$(26)^2 = (10)^2 + \left(\frac{l}{2}\right)^2$
$676 = 100 + \left(\frac{l}{2}\right)^2$
$\left(\frac{l}{2}\right)^2 = 676 - 100$
$\left(\frac{l}{2}\right)^2 = 576$
$\frac{l}{2} = \sqrt{576}$
$\frac{l}{2} = 24$
To find the full length of the chord, multiply by 2:
$l = 24 \times 2 = 48cm$

24. If $P, q$ and $r$ are logical statements such that if $p \to q$ and $q \to r$ then $p \to r$ this is called

chain rule
logical reasoning
conditional statement
equivalent statement

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Correct Answer: A. chain rule

Explanation:
This principle is a fundamental rule of inference in logic, formally known as the **Law of Syllogism** or **Transitivity of Implication**. It states that if one statement implies a second, and the second implies a third, then the first statement implies the third. Informally, in some contexts, this is referred to as a "chain rule" because the implications link together like a chain ($p$ implies $q$, and $q$ implies $r$, forming a chain from $p$ to $r$).

25. The Venn diagram below represents…

$D \lor G$
$D \land G$
$D \to G$
$D \varepsilon G$

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Correct Answer: B. $D \land G$

Explanation:
The Venn diagram shows two overlapping circles, $D$ and $G$. The shaded region is the area where the two circles overlap. This overlapping region represents the elements that are common to both set $D$ and set $G$. In set theory, this is known as the **intersection** of sets $D$ and $G$.
The logical symbol for "AND" ($\land$) is often used to represent set intersection, especially when thinking of elements satisfying both conditions. $D \land G$ means "elements that are in D AND in G".
Let's review other options:
- $D \lor G$ (D OR G) represents the union of the sets, which would be the entire area covered by both circles.
- $D \to G$ represents a conditional statement ("If D, then G"), not a set operation.
- $D \varepsilon G$ is not standard set notation; the symbol for "element of" is $\in$, and "subset of" is $\subseteq$.

26. Simplify $\frac{4}{x} - \frac{6}{x+2}$

$\frac{8-2x}{x^2+2x}$
$\frac{5x-6}{2x+4}$
$\frac{2x-5}{x^2+2x}$
$\frac{6x+8}{5x}$

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Correct Answer: A. $\frac{8-2x}{x^2+2x}$

Explanation:
To subtract algebraic fractions, find a common denominator, which is the product of the individual denominators: $x(x+2)$.
$\frac{4}{x} - \frac{6}{x+2}$
Convert each fraction to have the common denominator:
$= \frac{4(x+2)}{x(x+2)} - \frac{6x}{x(x+2)}$
Combine the numerators over the common denominator:
$= \frac{4(x+2) - 6x}{x(x+2)}$
Distribute and simplify the numerator:
$= \frac{4x + 8 - 6x}{x^2 + 2x}$
Combine like terms in the numerator:
$= \frac{(4x - 6x) + 8}{x^2 + 2x}$
$= \frac{-2x + 8}{x^2 + 2x}$
This can also be written as $\frac{8 - 2x}{x^2 + 2x}$.

27. Let $p$: I Study hard, $q$: I pass mathematics. Translate $\sim p \to \sim q$ into a statement.

if I study hard, then I will pass mathematics
if I pass mathematics, then I study hard
if I don’t study hard, then I will fail mathematics
if I fail mathematics then, I won’t study hard

Click to reveal answer

Correct Answer: C. if I don’t study hard, then I will fail mathematics

Explanation:
Given:
$p$: I study hard
$q$: I pass mathematics

The symbol $\sim$ denotes negation. So:
$\sim p$: I do not study hard
$\sim q$: I do not pass mathematics (which means I fail mathematics)

The symbol $\to$ denotes a conditional statement ("if...then..."). So, $\sim p \to \sim q$ translates to:
"If I do not study hard, then I do not pass mathematics."
This is equivalent to: "If I don’t study hard, then I will fail mathematics."

28. Solve the inequality $3𝑥 − 5 \ge 20 − 2𝑥$

$𝑥 > 4$
$𝑥 \ge 4$
$𝑥 \ge 5$
$𝑥 \le 6$

Click to reveal answer

Correct Answer: C. $𝑥 \ge 5$

Explanation:
To solve the inequality, we want to isolate $x$ on one side. We treat it much like solving an equation, with the key difference being that multiplying or dividing by a negative number reverses the inequality sign.
Given: $3x - 5 \ge 20 - 2x$

Add $2x$ to both sides:
$3x + 2x - 5 \ge 20$
$5x - 5 \ge 20$

Add 5 to both sides:
$5x \ge 20 + 5$
$5x \ge 25$

Divide both sides by 5 (a positive number, so the inequality sign does not change):
$x \ge \frac{25}{5}$
$x \ge 5$

29. Simplify $\frac{m^2-9}{m^2-m-6} \times \frac{m^2+2m}{m^2}$

$\frac{m+1}{m+3}$
$\frac{m+3}{m}$
$\frac{m+2}{m(m-1)}$
$\frac{m^2+9}{m^2}$

Click to reveal answer

Correct Answer: B. $\frac{m+3}{m}$

Explanation:
To simplify the expression, factorize each numerator and denominator:
1. Numerator of the first fraction: $m^2 - 9$. This is a difference of squares ($a^2 - b^2 = (a-b)(a+b)$).
$m^2 - 9 = (m - 3)(m + 3)$

2. Denominator of the first fraction: $m^2 - m - 6$. Factor this quadratic. We need two numbers that multiply to -6 and add to -1. These are -3 and 2.
$m^2 - m - 6 = (m - 3)(m + 2)$

3. Numerator of the second fraction: $m^2 + 2m$. Factor out the common term $m$.
$m^2 + 2m = m(m + 2)$

4. Denominator of the second fraction: $m^2$. This is already in simplest factored form ($m \times m$).

Now substitute the factored forms back into the expression:
$\frac{(m - 3)(m + 3)}{(m - 3)(m + 2)} \times \frac{m(m + 2)}{m \cdot m}$

Cancel out common factors from the numerator and denominator:
- $(m-3)$ cancels from the first fraction's numerator and denominator.
- $(m+2)$ cancels from the first fraction's denominator and the second fraction's numerator.
- One $m$ cancels from the second fraction's numerator and denominator.
After cancellation, the expression becomes:
$\frac{\cancel{(m - 3)}(m + 3)}{\cancel{(m - 3)}\cancel{(m + 2)}} \times \frac{\cancel{m}\cancel{(m + 2)}}{m \cdot \cancel{m}} = \frac{m + 3}{m}$

30. Evaluate $\frac{x^2+x-1}{2x^2+x-1}$ when $x = 1$

2
−1
1/2
1

Click to reveal answer

Correct Answer: C. 1/2

Explanation:
To evaluate the expression when $x = 1$, substitute $x = 1$ into both the numerator and the denominator.

Numerator: $x^2 + x - 1$
Substitute $x = 1$: $(1)^2 + (1) - 1 = 1 + 1 - 1 = 1$

Denominator: $2x^2 + x - 1$
Substitute $x = 1$: $2(1)^2 + (1) - 1 = 2(1) + 1 - 1 = 2 + 1 - 1 = 2$

Now, form the fraction with the evaluated numerator and denominator:
$\frac{\text{Numerator}}{\text{Denominator}} = \frac{1}{2}$

31. What is the arithmetic mean of 8, 5, 7, 2, and 3?

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Correct Answer: C. 5

Explanation:
The arithmetic mean (average) is calculated by summing all the values and then dividing by the number of values.
Sum of values $= 8 + 5 + 7 + 2 + 3 = 25$
Number of values $= 5$
Arithmetic Mean $= \frac{\text{Sum of values}}{\text{Number of values}} = \frac{25}{5} = 5$

32. The bearing of $320^\circ$ is equivalent to ………

N40W
S50E
N10W
N54E

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Correct Answer: A. N40W

Explanation:
Bearings are measured clockwise from North ($0^\circ$ or $360^\circ$).
A bearing of $320^\circ$ is in the fourth quadrant (between $270^\circ$ and $360^\circ$). This direction is generally North-West.
To find the equivalent bearing using cardinal points (N, S, E, W), we determine the angle from the nearest North or South line.
Since $320^\circ$ is closer to $360^\circ$ (North), we calculate the angle from North:
Angle from North (counter-clockwise towards West) $= 360^\circ - 320^\circ = 40^\circ$.
Therefore, the equivalent bearing is N$40^\circ$W (North 40 degrees West).

33. Given that $\cos\theta = 0.2874$. What is the value of $\theta$ lying between $0^\circ$ to $360^\circ$?

$73.3^\circ$ and $286.7^\circ$
$73.3^\circ$ and $324.5^\circ$
$23.4^\circ$ and $82.5^\circ$
$33.4^\circ$ and $73.4^\circ$

Click to reveal answer

Correct Answer: A. $73.3^\circ$ and $286.7^\circ$

Explanation:
First, find the principal value of $\theta$ using the inverse cosine function:
$\theta_1 = \arccos(0.2874)$
Using a calculator, $\theta_1 \approx 73.3^\circ$ (rounded to one decimal place).

The cosine function is positive in the first and fourth quadrants. So, there will be two solutions between $0^\circ$ and $360^\circ$.
The first solution is the principal value: $\theta_1 = 73.3^\circ$.
The second solution is in the fourth quadrant, found by subtracting the reference angle from $360^\circ$:
$\theta_2 = 360^\circ - \theta_1 = 360^\circ - 73.3^\circ = 286.7^\circ$.
Thus, the values of $\theta$ are $73.3^\circ$ and $286.7^\circ$.

34. $\sin 118^\circ$ is equivalent to

$\sin (-42^\circ)$
$\sin 345^\circ$
$\sin 62^\circ$
$\sin 240^\circ$

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Correct Answer: C. $\sin 62^\circ$

Explanation:
The sine function has a property where $\sin \theta = \sin(180^\circ - \theta)$ for angles in the first and second quadrants. This is because sine values are positive in both quadrants.
The angle $118^\circ$ lies in the second quadrant ($90^\circ < 118^\circ < 180^\circ$).
To find its equivalent acute angle (reference angle) in the first quadrant, we subtract it from $180^\circ$:
$\sin 118^\circ = \sin(180^\circ - 118^\circ)$
$\sin 118^\circ = \sin 62^\circ$.

35. What is the range of the data? 10, 30, 25, 40, 50.

Click to reveal answer

Correct Answer: A. 40

Explanation:
The range of a set of data is the difference between the highest (maximum) value and the lowest (minimum) value in the set.
Given data set: $10, 30, 25, 40, 50$
Maximum value $= 50$
Minimum value $= 10$
Range $= \text{Maximum value} - \text{Minimum value} = 50 - 10 = 40$

36. Find the mean deviation of 20, 30, 25, 40, 35, 50, 45, 40, 20, and 45

Click to reveal answer

Correct Answer: B. 9

Explanation:
To find the mean deviation, follow these steps:
1. Calculate the arithmetic mean ($\bar{x}$) of the data.
2. Find the absolute deviation of each data point from the mean ($|x_i - \bar{x}|$).
3. Calculate the mean of these absolute deviations.

Given data: $20, 30, 25, 40, 35, 50, 45, 40, 20, 45$
Number of values ($n$) $= 10$

Step 1: Calculate the Mean ($\bar{x}$)
Sum of values $= 20+30+25+40+35+50+45+40+20+45 = 350$
$\bar{x} = \frac{350}{10} = 35$

Step 2: Calculate the Absolute Deviations ($|x_i - \bar{x}|$)
$|20 - 35| = 15$
$|30 - 35| = 5$
$|25 - 35| = 10$
$|40 - 35| = 5$
$|35 - 35| = 0$
$|50 - 35| = 15$
$|45 - 35| = 10$
$|40 - 35| = 5$
$|20 - 35| = 15$
$|45 - 35| = 10$

Step 3: Sum the Absolute Deviations
Sum $= 15+5+10+5+0+15+10+5+15+10 = 90$

Step 4: Calculate the Mean Deviation
Mean Deviation $= \frac{\text{Sum of absolute deviations}}{\text{Number of values}} = \frac{90}{10} = 9$

37. The mean of 1, $𝑥$, 5, 7, and 3 is 4. Find the value of $𝑥$

Click to reveal answer

Correct Answer: B. 4

Explanation:
The formula for the mean is: $\text{Mean} = \frac{\text{Sum of values}}{\text{Number of values}}$
Given values: $1, x, 5, 7, 3$
Number of values $= 5$
Given Mean $= 4$

First, find the sum of the values:
Sum $= 1 + x + 5 + 7 + 3 = 16 + x$

Now, substitute the values into the mean formula:
$4 = \frac{16 + x}{5}$

Multiply both sides by 5:
$4 \times 5 = 16 + x$
$20 = 16 + x$

Subtract 16 from both sides to find $x$:
$x = 20 - 16$
$x = 4$

38. Find the median of 2, 1, 0, 3, 1, 1, 4, 0, 1 and 2.

Click to reveal answer

Correct Answer: C. 1.0

Explanation:
To find the median, first arrange the data set in ascending order.
Given data: $2, 1, 0, 3, 1, 1, 4, 0, 1, 2$
Sorted data: $0, 0, 1, 1, 1, 1, 2, 2, 3, 4$

Count the number of data points ($n$). In this set, $n = 10$.
Since $n$ is an even number, the median is the average of the two middle values.
The positions of the middle values are $\frac{n}{2}$ and $\left(\frac{n}{2}\right) + 1$.
$\frac{10}{2} = 5^\text{th}$ value
$\frac{10}{2} + 1 = 6^\text{th}$ value

From the sorted list:
The $5^\text{th}$ value is $1$.
The $6^\text{th}$ value is $1$.

Median $= \frac{\text{5th value} + \text{6th value}}{2} = \frac{1 + 1}{2} = \frac{2}{2} = 1$
So the median is $1.0$.

39.

From the diagram above, find the value of 𝑥.

$20cm$
$\sqrt{304}cm$
$\sqrt{108}cm$
$10cm$

Click to reveal answer

Correct Answer: B. $\sqrt{304}cm$

Explanation:
Use Cosine Rule

40. From the top of a building $10m$ high, the angle of depression of a stone lying on the horizontal ground is $69^\circ$. Calculate the distance of the stone from the foot of the building.

$3.6m$
$3.8m$
$8.0m$
$9.3m$

Click to reveal answer

Correct Answer: B. $3.8m$

Explanation:
Imagine a right-angled triangle formed by the building, the horizontal ground, and the line of sight from the top of the building to the stone.
- The height of the building is the opposite side to the angle of elevation from the stone ($h = 10m$).
- The distance of the stone from the foot of the building is the adjacent side ($d$).
- The angle of depression from the top of the building is $69^\circ$. By alternate interior angles, the angle of elevation from the stone to the top of the building is also $69^\circ$.
We can use the tangent function, which relates the opposite and adjacent sides to the angle:
$\tan(\text{angle}) = \frac{\text{Opposite}}{\text{Adjacent}}$
$\tan(69^\circ) = \frac{10}{d}$
Now, solve for $d$:
$d = \frac{10}{\tan(69^\circ)}$
Using a calculator, $\tan(69^\circ) \approx 2.60509$
$d = \frac{10}{2.60509}$
$d \approx 3.8386m$
Rounding to one decimal place, the distance is approximately $3.8m$.

41. The table below shows the frequency distribution of the number of chairs in each 40 rooms of various houses.

Number of chairs ($x$)	1	2	3	4	5	6	7
Frequency ($f$)	2	7	5	4	9	7	6

Use the table above to answer Questions 42 – 44.

42. Find the mean of the distribution

Click to reveal answer

Correct Answer: C. 4.4

Explanation:
To find the mean ($\bar{x}$) of a frequency distribution, use the formula: $\bar{x} = \frac{\sum fx}{\sum f}$
First, calculate $fx$ for each category and sum them:
$ (1 \times 2) = 2 $
$ (2 \times 7) = 14 $
$ (3 \times 5) = 15 $
$ (4 \times 4) = 16 $
$ (5 \times 9) = 45 $
$ (6 \times 7) = 42 $
$ (7 \times 6) = 42 $
$\sum fx = 2 + 14 + 15 + 16 + 45 + 42 + 42 = 176$

Next, find the sum of the frequencies ($\sum f$):
$\sum f = 2 + 7 + 5 + 4 + 9 + 7 + 6 = 40$

Now, calculate the mean:
$\bar{x} = \frac{176}{40} = 4.4$

43. Find the mode of the distribution

Click to reveal answer

Correct Answer: C. 5

Explanation:
The mode of a frequency distribution is the value (or class interval) that has the highest frequency.
From the given table:
Number of chairs ($x$): 1, 2, 3, 4, 5, 6, 7
Frequency ($f$): 2, 7, 5, 4, 9, 7, 6
The highest frequency is 9, which corresponds to the 'Number of chairs' value of 5.
Therefore, the mode is 5.

44. Find the median of the distribution

Click to reveal answer

Correct Answer: D. 5

Explanation:
To find the median of a frequency distribution, first determine the total number of data points ($\sum f$) and then find the position of the median.
Total number of rooms ($N$) = $\sum f = 40$.
Since $N$ is an even number, the median is the average of the values at the $\left(\frac{N}{2}\right)^\text{th}$ and $\left(\frac{N}{2} + 1\right)^\text{th}$ positions.
The middle positions are: $\frac{40}{2} = 20^\text{th}$ and $\left(\frac{40}{2} + 1\right) = 21^\text{st}$.

Now, find these values using the cumulative frequency:
Number of chairs | Frequency | Cumulative Frequency
1 | 2 | 2
2 | 7 | $2+7 = 9$
3 | 5 | $9+5 = 14$
4 | 4 | $14+4 = 18$
5 | 9 | $18+9 = 27$ (The 20th and 21st values fall within this category)
6 | 7 | $27+7 = 34$
7 | 6 | $34+6 = 40$

Both the $20^\text{th}$ and $21^\text{st}$ values fall into the 'Number of chairs' category of 5.
Therefore, the median is $\frac{5+5}{2} = 5$.

45. Express the true bearing of $250^\circ$ as a compass bearing

N20°E
S20°E
N20°W
S70°W

Click to reveal answer

Correct Answer: D. S70°W

Explanation:
True bearings are measured clockwise from North ($0^\circ$).
A bearing of $250^\circ$ falls in the third quadrant, between $180^\circ$ (South) and $270^\circ$ (West). This corresponds to a South-West direction.
To convert to a compass bearing (e.g., S $\theta^\circ$ W), we measure the angle from the South line ($180^\circ$).
Angle from South $= 250^\circ - 180^\circ = 70^\circ$.
Since it's towards the West from South, the compass bearing is S$70^\circ$W.

46. A bag contains 4 red, 5 white and 8 blue identical balls. What is the probability of picking a blue ball?

$\frac{4}{17}$
$\frac{5}{17}$
$\frac{8}{17}$
$\frac{17}{5}$

Click to reveal answer

Correct Answer: C. $\frac{8}{17}$

Explanation:
Probability is calculated as: $\text{Probability} = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}$

Number of red balls $= 4$
Number of white balls $= 5$
Number of blue balls $= 8$

Total number of balls in the bag $= 4 + 5 + 8 = 17$

Number of favorable outcomes (picking a blue ball) $= 8$
Probability of picking a blue ball $= \frac{8}{17}$

47. Express $\frac{4}{5}$ as a percentage

Click to reveal answer

Correct Answer: C. 80%

Explanation:
To express a fraction as a percentage, multiply the fraction by $100\%$.
Percentage $= \frac{4}{5} \times 100\%$
Percentage $= \frac{400}{5}\%$
Percentage $= 80\%$

48. There are $m$ girls and $12$ boys in a class, what is the probability of selecting at random a boy from the class?

$\frac{m}{12}$
$\frac{12}{m}$
$\frac{12}{12+m}$
$\frac{m}{m+12}$

Click to reveal answer

Correct Answer: C. $\frac{12}{12+m}$

Explanation:
Probability is calculated as: $\text{Probability} = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}$

Number of girls $= m$
Number of boys $= 12$

Total number of students in the class $= \text{Number of girls} + \text{Number of boys} = m + 12$

Number of favorable outcomes (selecting a boy) $= 12$
Probability of selecting a boy $= \frac{12}{m + 12}$

49. Find the mean deviation of 6, 7, 8, 9, 10

Click to reveal answer

Correct Answer: A. 1.2

Explanation:
To find the mean deviation, follow these steps:
1. Calculate the arithmetic mean ($\bar{x}$) of the data.
2. Find the absolute deviation of each data point from the mean ($|x_i - \bar{x}|$).
3. Calculate the mean of these absolute deviations.

Given data: $6, 7, 8, 9, 10$
Number of values ($n$) $= 5$

Step 1: Calculate the Mean ($\bar{x}$)
Sum of values $= 6 + 7 + 8 + 9 + 10 = 40$
$\bar{x} = \frac{40}{5} = 8$

Step 2: Calculate the Absolute Deviations ($|x_i - \bar{x}|$)
$|6 - 8| = |-2| = 2$
$|7 - 8| = |-1| = 1$
$|8 - 8| = |0| = 0$
$|9 - 8| = |1| = 1$
$|10 - 8| = |2| = 2$

Step 3: Sum the Absolute Deviations
Sum of absolute deviations $= 2 + 1 + 0 + 1 + 2 = 6$

Step 4: Calculate the Mean Deviation
Mean Deviation $= \frac{\text{Sum of absolute deviations}}{\text{Number of values}} = \frac{6}{5} = 1.2$

50. express 302.10495 correct to five significant figures

302.10
302.11
302.105
302.1049

Click to reveal answer

Correct Answer: A. 302.10

Explanation:
To express a number to a certain number of significant figures, start counting from the first non-zero digit.
Given number: $302.10495$
1st significant figure: $3$
2nd significant figure: $0$
3rd significant figure: $2$
4th significant figure: $1$
5th significant figure: $0$

The digit immediately after the 5th significant figure is $4$ (from $302.10\underline{4}95$).
Since this digit ($4$) is less than $5$, we do not round up the preceding digit ($0$). We simply truncate the number at the 5th significant figure.
Therefore, $302.10495$ correct to five significant figures is $302.10$.

51. The Probability that Kebba, Ebou and Omar will hit the target are $\frac{2}{3}, \frac{3}{4}, \frac{4}{5}$ respectively. Find the probability that only Kebba will hit the target.

$\frac{2}{5}$
$\frac{7}{3}$
$\frac{1}{30}$
$\frac{1}{60}$

Click to reveal answer

Correct Answer: C. $\frac{1}{30}$

Explanation:
Let $P(K)$, $P(E)$, and $P(O)$ be the probabilities that Kebba, Ebou, and Omar hit the target, respectively.
$P(K) = \frac{2}{3}$
$P(E) = \frac{3}{4}$
$P(O) = \frac{4}{5}$

The probability that a person does NOT hit the target is $1$ minus the probability that they do hit the target:
$P(K') = 1 - P(K) = 1 - \frac{2}{3} = \frac{1}{3}$ (Kebba misses)
$P(E') = 1 - P(E) = 1 - \frac{3}{4} = \frac{1}{4}$ (Ebou misses)
$P(O') = 1 - P(O) = 1 - \frac{4}{5} = \frac{1}{5}$ (Omar misses)

We want the probability that only Kebba hits the target. This means Kebba hits, AND Ebou misses, AND Omar misses. Since these are independent events, we multiply their probabilities:
$P(\text{only Kebba}) = P(K) \times P(E') \times P(O')$
$P(\text{only Kebba}) = \frac{2}{3} \times \frac{1}{4} \times \frac{1}{5}$
$P(\text{only Kebba}) = \frac{2 \times 1 \times 1}{3 \times 4 \times 5} = \frac{2}{60} = \frac{1}{30}$

The table below shows the marks scored by several pupils in a mathematics test.
Use the information to answer questions 52 - 54

Marks	30	40	50	60	80
Number of Pupils	11	8	4	5	2

52. How many pupils are there?

Click to reveal answer

Correct Answer: B. 30

Explanation:
The total number of pupils is the sum of the frequencies (Number of Pupils) from the table.
Total Pupils $= 11 + 8 + 4 + 5 + 2 = 30$

53. What is the probability that a pupil chosen at random from the class scored 60%?

$\frac{1}{6}$
$\frac{11}{30}$
$\frac{3}{40}$
$\frac{7}{20}$

Click to reveal answer

Correct Answer: A. $\frac{1}{6}$

Explanation:
Total number of pupils $= 30$ (from Question 52).
From the table, the number of pupils who scored 60% is 5.
Probability (score 60%) $= \frac{\text{Number of pupils who scored 60%}}{\text{Total number of pupils}}$
Probability (score 60%) $= \frac{5}{30} = \frac{1}{6}$

54. What is the probability that a pupil chosen at random from the class will pass the test if the pass mark is 50%?

$\frac{4}{30}$
$\frac{11}{30}$
$\frac{4}{20}$
$\frac{11}{20}$

Click to reveal answer

Correct Answer: B. $\frac{11}{30}$

Explanation:
Total number of pupils $= 30$ (from Question 52).
The pass mark is 50%. This means pupils who scored 50% or above have passed.
From the table:
- Number of pupils who scored 50% = 4
- Number of pupils who scored 60% = 5
- Number of pupils who scored 80% = 2
Total number of pupils who passed $= 4 + 5 + 2 = 11$
Probability (pass) $= \frac{\text{Number of pupils who passed}}{\text{Total number of pupils}}$
Probability (pass) $= \frac{11}{30}$

55. The bearing of point X from point Y is $074^\circ$. What is the bearing of Y from X?

$106^\circ$
$148^\circ$
$164^\circ$
$254^\circ$

Click to reveal answer

Correct Answer: D. $254^\circ$

Explanation:
When you have the bearing of point X from point Y, and you want to find the bearing of Y from X (back bearing), you add or subtract $180^\circ$.
Rule:
- If the given bearing is less than $180^\circ$, add $180^\circ$.
- If the given bearing is $180^\circ$ or more, subtract $180^\circ$.

Given bearing of X from Y $= 074^\circ$.
Since $074^\circ < 180^\circ$, we add $180^\circ$:
Bearing of Y from X $= 074^\circ + 180^\circ = 254^\circ$

56. Express 0.0462 in standard form.

$0.462 \times 10^{-1}$
$0.462 \times 10^{-2}$
$4.62 \times 10^{-1}$
$4.62 \times 10^{-2}$

Click to reveal answer

Correct Answer: D. $4.62 \times 10^{-2}$

Explanation:
Standard form (or scientific notation) requires a number to be written as $a \times 10^b$, where $1 \le |a| < 10$ and $b$ is an integer.
Given number: $0.0462$
To get 'a' between 1 and 10, move the decimal point two places to the right, from its current position to after the first non-zero digit (4).
$0.0462 \rightarrow 4.62$
Since the decimal point moved 2 places to the right, the exponent $b$ will be negative, equal to the number of places moved.
So, $0.0462 = 4.62 \times 10^{-2}$.

57. Town A is $4km$ due north of Town B and Town C is $3km$ due east of Town A. What is the distance between Town B and Town C

$7km$
$8km$
$5km$
$4km$

Click to reveal answer

Correct Answer: C. $5km$

Explanation:
This problem forms a right-angled triangle.
1. Town A is $4km$ due north of Town B. This is one leg of the triangle (vertical side).
2. Town C is $3km$ due east of Town A. This is the other leg of the triangle (horizontal side).
The distance between Town B and Town C is the hypotenuse of this right-angled triangle.
Using the Pythagorean theorem, $a^2 + b^2 = c^2$:
Where $a = 4km$ (distance BA) and $b = 3km$ (distance AC), and $c$ is the distance BC.
$4^2 + 3^2 = c^2$
$16 + 9 = c^2$
$25 = c^2$
$c = \sqrt{25}$
$c = 5km$

58. A fair die is rolled once, What is the probability of obtaining a number less than 3?

$\frac{1}{6}$
$\frac{1}{3}$
$\frac{2}{3}$
$\frac{1}{2}$

Click to reveal answer

Correct Answer: B. $\frac{1}{3}$

Explanation:
When a fair die is rolled, the possible outcomes are the numbers from 1 to 6.
Total number of possible outcomes $= \{1, 2, 3, 4, 5, 6\}$, so there are 6 outcomes.

We want to find the probability of obtaining a number less than 3.
The numbers less than 3 are $\{1, 2\}$.
Number of favorable outcomes $= 2$.

Probability $= \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}$
Probability (number less than 3) $= \frac{2}{6} = \frac{1}{3}$

59. A ladder $6m$ long leans against a vertical wall so that it makes an angle of $60^\circ$ with the wall. Calculate the distance of the foot of the ladder from the wall.

$3m$
$6m$
$3.46m$
$5.20m$

Click to reveal answer

Correct Answer: D. $5.20m$

Explanation:
This scenario forms a right-angled triangle where:
- The ladder is the hypotenuse, with length $6m$.
- The wall is one leg of the triangle.
- The ground is the other leg of the triangle.
- The angle between the ladder and the wall is given as $60^\circ$. This is the angle at the top of the triangle.

We want to find the distance of the foot of the ladder from the wall, which is the side opposite to the $60^\circ$ angle.
Using trigonometry, specifically the sine function:
$\sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}}$
$\sin(60^\circ) = \frac{\text{distance}}{6}$
Distance $= 6 \times \sin(60^\circ)$
We know that $\sin(60^\circ) = \frac{\sqrt{3}}{2} \approx 0.8660$
Distance $= 6 \times \frac{\sqrt{3}}{2} = 3\sqrt{3}$
Distance $\approx 3 \times 1.73205 = 5.19615m$
Rounding to two decimal places, the distance is approximately $5.20m$.

60.

What is the value of $y$ in the figure above?

$4\sqrt{3}$
$5\sqrt{3}$
$6\sqrt{2}$
$5\sqrt{5}$

Click to reveal answer

Correct Answer: B. $5\sqrt{3}$

61. Express 0.00045 in standard form.

$45 \times 10^{-5}$
$45000$
$4.5 \times 10^{-4}$
$4.5 \times 10^{4}$

Click to reveal answer

Correct Answer: C. $4.5 \times 10^{-4}$

Explanation:
Standard form (or scientific notation) expresses a number as $a \times 10^b$, where $1 \le |a| < 10$ and $b$ is an integer.
Given number: $0.00045$
To get 'a' between 1 and 10, move the decimal point to the right until it is after the first non-zero digit (4).
$0.00045 \rightarrow 4.5$
The decimal point moved 4 places to the right. When moving the decimal to the right for a number less than 1, the exponent $b$ is negative, and its value is equal to the number of places moved.
So, $0.00045 = 4.5 \times 10^{-4}$.

62. What is the logarithm of 0.00325?

$\bar{3}.5119$
$4.3521$
$\bar{3}.4371$
$0.3245$

Click to reveal answer

Correct Answer: A. $\bar{3}.5119$

Explanation:
To find the logarithm of a number (base 10), we first express the number in standard form (scientific notation):
$0.00325 = 3.25 \times 10^{-3}$

Using logarithm properties, $\log(a \times b) = \log a + \log b$ and $\log(10^n) = n$:
$\log_{10}(0.00325) = \log_{10}(3.25 \times 10^{-3})$
$= \log_{10}(3.25) + \log_{10}(10^{-3})$
$= \log_{10}(3.25) - 3$

Now, we find the value of $\log_{10}(3.25)$ using a calculator or logarithm table. The mantissa (decimal part) is positive, and the characteristic (integer part) comes from the power of 10.
$\log_{10}(3.25) \approx 0.5119$ (rounded to 4 decimal places)

So, $\log_{10}(0.00325) \approx 0.5119 - 3 = -2.4881$

In bar notation, a negative logarithm is expressed with a negative characteristic and a positive mantissa. We can write $-2.4881$ as $-3 + 0.5119$.
This is represented as $\bar{3}.5119$.

63. Correct 457.43 to the two significant figures.

457.43
457
460
500

Click to reveal answer

Correct Answer: C. 460

Explanation:
To correct a number to a specific number of significant figures, start counting from the first non-zero digit.
Given number: $457.43$
1. The first significant figure is 4.
2. The second significant figure is 5.
3. Look at the digit immediately after the second significant figure, which is 7.
4. Since 7 is 5 or greater, we round up the second significant figure. So, 5 becomes 6.
5. Replace any digits after the significant figures, up to the decimal point, with zeros to maintain the place value.
Thus, $457.43$ corrected to two significant figures is $460$.

64. Find the $n_{th}$ term of the sequence $3, 5, 7, ....$

$U_n = 2n + 1$
$U_n = n + 1$
$U_n = 2n - 1$
$U_n = 3n + 1$

Click to reveal answer

Correct Answer: A. $U_n = 2n + 1$

65. What is the fifth term of the sequence whose $n_{th}$ term $U_n=\frac{n+1}{n-1}$

$\frac{2}{3}$
$1\frac{1}{4}$
$1\frac{1}{2}$
$\frac{1}{4}$

Click to reveal answer

Correct Answer: C. $1\frac{1}{2}$

Explanation:
To find the fifth term ($U_5$), substitute $n=5$ into the given formula for the $n_{th}$ term, $U_n=\frac{n+1}{n-1}$.
$U_5 = \frac{5+1}{5-1}$
$U_5 = \frac{6}{4}$
Simplify the fraction:
$U_5 = \frac{3}{2}$
As a mixed number, this is $1\frac{1}{2}$.

66. The first term of a GP is 2 and the common ratio is 4.What is the $4^{th}$ term of the sequence?

Click to reveal answer

Correct Answer: A. 128

Explanation:
For a Geometric Progression (GP), the formula for the $n^{th}$ term is $U_n = ar^{n-1}$, where:
- $a$ is the first term
- $r$ is the common ratio
- $n$ is the term number

Given:
First term ($a$) $= 2$
Common ratio ($r$) $= 4$
We need to find the $4^{th}$ term, so $n = 4$.

Substitute these values into the formula:
$U_4 = 2 \times 4^{(4-1)}$
$U_4 = 2 \times 4^3$
$U_4 = 2 \times (4 \times 4 \times 4)$
$U_4 = 2 \times 64$
$U_4 = 128$

67. What are the roots of the quadratic equation, $𝑥^2 + 8𝑥 + 16$?

$-4$ twice
$2$ and $4$
$4$ twice
$3$ and $2$

Click to reveal answer

Correct Answer: A. $-4$ twice

Explanation:
The given quadratic equation is $x^2 + 8x + 16 = 0$.
This is a perfect square trinomial because it follows the pattern $(a+b)^2 = a^2 + 2ab + b^2$.
Here, $a=x$ and $b=4$, so $x^2 + 2(x)(4) + 4^2 = x^2 + 8x + 16$.
So, the equation can be factored as $(x + 4)^2 = 0$.
To find the roots, set the factor to zero:
$x + 4 = 0$
$x = -4$
Since it's a perfect square, the root is repeated. Therefore, the roots are $-4$ twice.

68. Find the quadratic equation whose roots are -3 and 2.

$𝑥^2 − 3𝑥 + 2 = 0$
$𝑥^2 + 4𝑥 + 5 = 0$
$𝑥^2 + 𝑥 − 6 = 0$
$𝑥^2 − 𝑥 + 1 = 0$

Click to reveal answer

Correct Answer: C. $𝑥^2 + 𝑥 − 6 = 0$

Explanation:
If $\alpha$ and $\beta$ are the roots of a quadratic equation, the equation can be written in the form:
$x^2 - (\alpha + \beta)x + \alpha\beta = 0$

Given roots: $\alpha = -3$ and $\beta = 2$.

1. Calculate the sum of the roots ($\alpha + \beta$):
$\alpha + \beta = -3 + 2 = -1$

2. Calculate the product of the roots ($\alpha\beta$):
$\alpha\beta = (-3) \times (2) = -6$

3. Substitute these values into the quadratic equation formula:
$x^2 - (-1)x + (-6) = 0$
$x^2 + x - 6 = 0$

Alternatively, using factors: If the roots are -3 and 2, then $(x - (-3))$ and $(x - 2)$ are the factors.
$(x + 3)(x - 2) = 0$
Expand the expression:
$x(x - 2) + 3(x - 2) = 0$
$x^2 - 2x + 3x - 6 = 0$
$x^2 + x - 6 = 0$

69. For what value of k is the quadratic equation $𝑥^2 + 5𝑥 + 𝑘$ a perfect square?

4
24/4
18
12/5

Click to reveal answer

Correct Answer: (The exact value is $25/4$ or $6.25$, which is not among the options. Option B is numerically closest if there's a typo.)

Explanation:
A quadratic expression of the form $ax^2 + bx + c$ is a perfect square if its discriminant, $D = b^2 - 4ac$, is equal to zero.
In the given equation $x^2 + 5x + k$, we have $a=1$, $b=5$, and $c=k$.
Set the discriminant to zero:
$b^2 - 4ac = 0$
$(5)^2 - 4(1)(k) = 0$
$25 - 4k = 0$
$25 = 4k$
$k = \frac{25}{4}$
As a decimal, $k = 6.25$.

Reviewing the given options:
A. 4
B. $24/4 = 6$
C. 18
D. $12/5 = 2.4$

The calculated value $k=6.25$ is not exactly present in the options. However, option B is $6$, which is numerically the closest integer value. It is highly probable that option B intended to be $\frac{25}{4}$.

70. What is the gradient of a straight line passing through the points $A(8,4)$ and $B(10,10)$

Click to reveal answer

Correct Answer: D. 3

Explanation:
The gradient ($m$) of a straight line passing through two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by the formula:
$m = \frac{y_2 - y_1}{x_2 - x_1}$

Given points: $A(x_1, y_1) = (8, 4)$ and $B(x_2, y_2) = (10, 10)$.
Substitute the coordinates into the formula:
$m = \frac{10 - 4}{10 - 8}$
$m = \frac{6}{2}$
$m = 3$

71. Correct 0.04945 to two significant figures.

0.040
0.049
0.050
0.49

Click to reveal answer

Correct Answer: B. 0.049

Explanation:
To correct a number to two significant figures, we start counting significant figures from the first non-zero digit.
Given number: $0.04945$
1. The first non-zero digit is 4, so it's the 1st significant figure.
2. The next digit is 9, so it's the 2nd significant figure.
3. Look at the digit immediately after the 2nd significant figure, which is 4.
4. Since this digit (4) is less than 5, we do not round up the preceding digit (9). The subsequent digits are simply dropped.
Therefore, $0.04945$ corrected to two significant figures is $0.049$.

72. Evaluate $\log_{10} 6 + \log_{10} 45 − \log_{10} 27$

Click to reveal answer

Correct Answer: A. 1

Explanation:
We use the properties of logarithms:
- $\log a + \log b = \log (ab)$ (Product Rule)
- $\log a - \log b = \log \left(\frac{a}{b}\right)$ (Quotient Rule)

Apply the product rule first:
$\log_{10} 6 + \log_{10} 45 = \log_{10} (6 \times 45) = \log_{10} (270)$

Now, apply the quotient rule:
$\log_{10} 270 - \log_{10} 27 = \log_{10} \left(\frac{270}{27}\right)$

Simplify the fraction inside the logarithm:
$\frac{270}{27} = 10$

So the expression becomes:
$\log_{10} 10$

Since $10^1 = 10$, $\log_{10} 10 = 1$.

73. If $\log x = \bar{2}.3675$ and $\log y = 0.9750$, what is the value of $x + y$? Correct to three significant figures.

1.18
1.31
9.03
9.46

Click to reveal answer

Correct Answer: D. 9.46

Explanation:
Given:
$\log x = \bar{2}.3675 = -2 + 0.3675$
$\log y = 0.9750$

To find $x$, we take the antilog of $\log x$:
$x = 10^{\log x} = 10^{\bar{2}.3675} = 10^{-2 + 0.3675} = 10^{0.3675} \times 10^{-2}$
Using a calculator, $10^{0.3675} \approx 2.3307$
So, $x \approx 2.3307 \times 10^{-2} = 0.023307$

To find $y$, we take the antilog of $\log y$:
$y = 10^{\log y} = 10^{0.9750}$
Using a calculator, $10^{0.9750} \approx 9.4406$

Now, calculate $x + y$:
$x + y \approx 0.023307 + 9.4406$
$x + y \approx 9.463907$

Finally, correct the result to three significant figures:
The first three significant figures are 9, 4, 6.
The digit after the third significant figure (6) is 3, which is less than 5, so we do not round up.
$x + y \approx 9.46$

74. Find the sum of the first five terms of the GP $2, 6, 18, \dots$

Click to reveal answer

Correct Answer: C. 242

Explanation:
For a Geometric Progression (GP), the sum of the first $n$ terms ($S_n$) is given by the formula:
$S_n = \frac{a(r^n - 1)}{r - 1}$, where $r \neq 1$
- $a$ is the first term
- $r$ is the common ratio
- $n$ is the number of terms

From the given GP $2, 6, 18, \dots$ :
First term ($a$) $= 2$
Common ratio ($r$) $= \frac{6}{2} = 3$
Number of terms ($n$) $= 5$

Substitute these values into the formula:
$S_5 = \frac{2(3^5 - 1)}{3 - 1}$
$S_5 = \frac{2(243 - 1)}{2}$
$S_5 = \frac{2(242)}{2}$
$S_5 = 242$

75. Simplify $\frac{\log \sqrt{8}}{\log 8}$

$\frac{1}{3}$
$\frac{1}{2}$
$\frac{1}{2}\log \sqrt{2}$
$\frac{1}{3}\sqrt{8}$

Click to reveal answer

Correct Answer: B. $\frac{1}{2}$

Explanation:
We can simplify this expression using the properties of logarithms and exponents.
Recall that $\sqrt{x} = x^{1/2}$. So, $\sqrt{8} = 8^{1/2}$.
The expression becomes: $\frac{\log (8^{1/2})}{\log 8}$

Now, use the logarithm property $\log a^b = b \log a$ (Power Rule). Apply this to the numerator:
$\log (8^{1/2}) = \frac{1}{2} \log 8$

Substitute this back into the expression:
$\frac{\frac{1}{2} \log 8}{\log 8}$

Since $\log 8$ is a common factor in both the numerator and the denominator, we can cancel it out (assuming $\log 8 \neq 0$, which is true).
The simplified expression is $\frac{1}{2}$.

76. The population of a State is 5846. Express this number to the nearest hundred.

5800
6000
5850
5900

Click to reveal answer

Correct Answer: A. 5800

Explanation:
To round a number to the nearest hundred, we look at the tens digit.
Given number: $5846$
The hundreds digit is 8.
The tens digit is 4.
Since the tens digit (4) is less than 5, we round down. This means the hundreds digit remains the same, and all digits to its right become zero.
Therefore, 5846 rounded to the nearest hundred is 5800.

77. Which of the following equations has its root as 4 and -5?

$x^2 + 4x + 10 = 0$
$x^2 + x - 20 = 0$
$x^2 + x + 20 = 0$
$x^2 - x - 20 = 0$

Click to reveal answer

Correct Answer: B. $x^2 + x - 20 = 0$

Explanation:
If $\alpha$ and $\beta$ are the roots of a quadratic equation, the equation can be written in the form:
$x^2 - (\alpha + \beta)x + \alpha\beta = 0$

Given roots: $\alpha = 4$ and $\beta = -5$.

1. Calculate the sum of the roots ($\alpha + \beta$):
$\alpha + \beta = 4 + (-5) = -1$

2. Calculate the product of the roots ($\alpha\beta$):
$\alpha\beta = (4) \times (-5) = -20$

3. Substitute these values into the quadratic equation formula:
$x^2 - (-1)x + (-20) = 0$
$x^2 + x - 20 = 0$

Alternatively, using factors: If the roots are 4 and -5, then $(x - 4)$ and $(x - (-5))$ are the factors.
$(x - 4)(x + 5) = 0$
Expand the expression:
$x(x + 5) - 4(x + 5) = 0$
$x^2 + 5x - 4x - 20 = 0$
$x^2 + x - 20 = 0$

78. Find $x$ if $\log_2 2x + \log_2 8 = 5$

Click to reveal answer

Correct Answer: A. 2

Explanation:
Given the logarithmic equation: $\log_2 2x + \log_2 8 = 5$

Use the logarithm property: $\log_b A + \log_b B = \log_b (A \times B)$
Combine the terms on the left side:
$\log_2 (2x \times 8) = 5$
$\log_2 (16x) = 5$

Convert the logarithmic equation to an exponential equation. The definition of a logarithm states that if $\log_b M = N$, then $M = b^N$.
In this case, $b=2$, $M=16x$, and $N=5$.
$16x = 2^5$
Calculate $2^5$:
$2^5 = 2 \times 2 \times 2 \times 2 \times 2 = 32$
So, the equation becomes:
$16x = 32$

Divide both sides by 16 to solve for $x$:
$x = \frac{32}{16}$
$x = 2$

79. The actual length of a table is $8.40cm$. A boy measured it to be $7.64cm$. What is the percentage error in the measurement?

9.0%
-9.0%
-8.6%
8.6%

Click to reveal answer

Correct Answer: B. -9.0%

Explanation:
The formula for percentage error is typically given as:
Percentage Error $= \frac{\text{Measured Value} - \text{Actual Value}}{\text{Actual Value}} \times 100\%$

Given:
Actual Length $= 8.40 \text{ cm}$
Measured Length $= 7.64 \text{ cm}$

Calculate the difference (error):
Error $= \text{Measured Value} - \text{Actual Value} = 7.64 - 8.40 = -0.76 \text{ cm}$

Calculate the percentage error:
Percentage Error $= \frac{-0.76}{8.40} \times 100\%$
Percentage Error $\approx -0.090476 \times 100\%$
Percentage Error $\approx -9.0476\% $

Rounding to one decimal place as suggested by the options:
Percentage Error $\approx -9.0\%$

The negative sign indicates that the measured value is less than the actual value (an underestimation). Since options include both positive and negative values, the signed percentage error is expected.

Use the graph below to answer question 80, 81 and 82

80. What are the roots of the equation?

2 and -3
-1 and 3
4 and 4
3 and 2

Click to reveal answer

Correct Answer: B. -1 and 3

Explanation:
The roots of a quadratic equation are the x-intercepts of its graph, which are the points where the curve crosses the x-axis (where $y=0$).
From the provided graph, the curve clearly intersects the x-axis at $x = -1$ and $x = 3$.
Therefore, the roots of the equation are $-1$ and $3$. This matches option B.

81. What is the minimum value of y?

-2
-3
-4
3

Click to reveal answer

Correct Answer: C. -4

Explanation:
The minimum value of $y$ for a parabola opening upwards (like the one in the graph) occurs at its vertex, which is the lowest point on the curve.
From the provided graph, the lowest point of the parabola is at the coordinates $(1, -4)$.
Therefore, the minimum value of $y$ is $-4$. This matches option C.

82. Which equation satisfies the curve?

$y = x^2 - x - 6$
$y = x^2 - 2x + 3$
$y = x^2 - 2x - 3$
$y = x^2 + 8x - 4$

Click to reveal answer

Correct Answer: C. $y = x^2 - 2x - 3$

Explanation:
To find the equation that satisfies the curve, we can use the x-intercepts (roots) and a known point on the graph.
1. Identify the roots: From the graph, the curve crosses the x-axis at $x = -1$ and $x = 3$. This means that if $y=0$, then $(x - (-1))(x - 3)=0$.
2. Form a basic equation: A quadratic equation with roots $r_1$ and $r_2$ can be written as $y = a(x - r_1)(x - r_2)$.
Substituting the roots $r_1 = -1$ and $r_2 = 3$:
$y = a(x + 1)(x - 3)$
3. Find the value of 'a': We can use another clear point from the graph, such as the y-intercept $(0, -3)$. Substitute $x=0$ and $y=-3$ into the equation:
$-3 = a(0 + 1)(0 - 3)$
$-3 = a(1)(-3)$
$-3 = -3a$
$a = 1$
4. Write the full equation: Substitute $a=1$ back into the equation:
$y = 1(x + 1)(x - 3)$
Expand the factors:
$y = x(x - 3) + 1(x - 3)$
$y = x^2 - 3x + x - 3$
$y = x^2 - 2x - 3$

This equation matches Option C.

83. Find the gradient of a straight line whose equation is $5x + 2y - 3 = 0$

5
$-\frac{5}{2}$
$\frac{5}{3}$
$-\frac{3}{2}$

Click to reveal answer

Correct Answer: B. $-\frac{5}{2}$

Explanation:
To find the gradient of a straight line from its general equation ($Ax + By + C = 0$), we need to rearrange it into the slope-intercept form, $y = mx + c$, where $m$ is the gradient and $c$ is the y-intercept.
Given equation: $5x + 2y - 3 = 0$
1. Isolate the $y$ term:
$2y = -5x + 3$
2. Divide the entire equation by the coefficient of $y$ (which is 2):
$y = \frac{-5}{2}x + \frac{3}{2}$
Comparing this to $y = mx + c$, the gradient $m$ is the coefficient of $x$.
Therefore, the gradient $m = -\frac{5}{2}$.

84. Find the equation of the straight line passing through the pair of points $(2, 4)$ and $(5, 6)$

$3y = 2x + 8$
$2x + 2y = 5$
$x + 4y = 10$
$x + y = 4$

Click to reveal answer

Correct Answer: A. $3y = 2x + 8$

Explanation:
1. Calculate the gradient (slope), $m$:
Using the formula $m = \frac{y_2 - y_1}{x_2 - x_1}$ with points $(x_1, y_1) = (2, 4)$ and $(x_2, y_2) = (5, 6)$:
$m = \frac{6 - 4}{5 - 2} = \frac{2}{3}$

2. Use the point-slope form of the equation of a line:
The point-slope form is $y - y_1 = m(x - x_1)$. We can use either point; let's use $(2, 4)$.
$y - 4 = \frac{2}{3}(x - 2)$

3. Convert to the general form or match options:
Multiply both sides by 3 to eliminate the fraction:
$3(y - 4) = 2(x - 2)$
$3y - 12 = 2x - 4$
Rearrange the terms to match the format of option A:
$3y = 2x - 4 + 12$
$3y = 2x + 8$

85. Correct 45.8472 to two decimal places

45.00
45.85
45.80
40.00

Click to reveal answer

Correct Answer: B. 45.85

Explanation:
To correct a number to two decimal places, we look at the digit in the third decimal place.
Given number: $45.8472$
1. The first decimal place is 8.
2. The second decimal place is 4.
3. The digit in the third decimal place is 7.
4. Since 7 is 5 or greater, we round up the digit in the second decimal place. So, 4 becomes 5.
5. All digits after the second decimal place are dropped.
Therefore, $45.8472$ corrected to two decimal places is $45.85$.

86. The first term of an AP is 3, if the common difference is 2, find the $10^{th}$ term.

Click to reveal answer

Correct Answer: A. 21

Explanation:
For an Arithmetic Progression (AP), the formula for the $n^{th}$ term ($U_n$) is:
$U_n = a + (n-1)d$
Where:
- $a$ is the first term
- $d$ is the common difference
- $n$ is the term number

Given:
First term ($a$) $= 3$
Common difference ($d$) $= 2$
We need to find the $10^{th}$ term, so $n = 10$.

Substitute these values into the formula:
$U_{10} = 3 + (10 - 1)2$
$U_{10} = 3 + (9) \times 2$
$U_{10} = 3 + 18$
$U_{10} = 21$

87. What is the $n^{th}$ term of the GP $6, 18, 54, 162, \dots$

$2 \times 4^n$
$5 \times 3^n$
$2 \times 3^n$
$2 \times 3^{n-1}$

Click to reveal answer

Correct Answer: C. $2 \times 3^n$

Explanation:
For a Geometric Progression (GP), the formula for the $n^{th}$ term ($U_n$) is $U_n = ar^{n-1}$, where $a$ is the first term and $r$ is the common ratio.

From the given GP $6, 18, 54, 162, \dots$ :
First term ($a$) $= 6$
Common ratio ($r$) $= \frac{18}{6} = 3$ (also $\frac{54}{18} = 3$, and so on).

So, the standard $n^{th}$ term formula gives $U_n = 6 \times 3^{n-1}$.
Now, let's examine the options and simplify our formula to match one of them. We know that $6 = 2 \times 3$.
So, we can rewrite $U_n$ as:
$U_n = (2 \times 3) \times 3^{n-1}$
Using the exponent rule $x^a \times x^b = x^{a+b}$:
$U_n = 2 \times 3^{1} \times 3^{n-1}$
$U_n = 2 \times 3^{(1 + n - 1)}$
$U_n = 2 \times 3^n$

Let's verify this formula with the given terms:
For $n=1, U_1 = 2 \times 3^1 = 6$ (Correct)
For $n=2, U_2 = 2 \times 3^2 = 2 \times 9 = 18$ (Correct)
For $n=3, U_3 = 2 \times 3^3 = 2 \times 27 = 54$ (Correct)

88. Factorize the quadratic expression $𝑥^2 + 4𝑥 + 4$

$(𝑥 + 2)(𝑥 + 2)$
$(𝑥 + 3)(𝑥 + 2)$
$(𝑥 + 1)(𝑥 + 3)$
$(𝑥 + 2)(𝑥 − 3)$

Click to reveal answer

Correct Answer: A. $(𝑥 + 2)(𝑥 + 2)$

Explanation:
The given quadratic expression is $x^2 + 4x + 4$.
We are looking for two numbers that multiply to give 4 (the constant term) and add up to give 4 (the coefficient of the $x$ term).
The numbers are 2 and 2 (since $2 \times 2 = 4$ and $2 + 2 = 4$).
So, the expression can be factored as $(x + 2)(x + 2)$.

This is also a perfect square trinomial, following the pattern $a^2 + 2ab + b^2 = (a+b)^2$. Here, $a=x$ and $b=2$, so $x^2 + 2(x)(2) + 2^2 = (x+2)^2$.

89. What must be added to $b^2 + 4b$ to make a perfect square?

Click to reveal answer

Correct Answer: A. 4

Explanation:
To make a quadratic expression of the form $x^2 + Kx$ (or $b^2 + Kb$) a perfect square, you need to add the square of half the coefficient of the linear term ($K$). That is, you add $\left(\frac{K}{2}\right)^2$.
In the expression $b^2 + 4b$, the coefficient of the linear term ($b$) is 4.
So, we need to add $\left(\frac{4}{2}\right)^2$:
$\left(\frac{4}{2}\right)^2 = (2)^2 = 4$

Adding 4 to the expression gives $b^2 + 4b + 4$, which is a perfect square: $(b+2)^2$.

90. For what value of x and y is the simultaneous equation $x + y = 12$ and $x - y = 6$ satisfied?

$x = 9, y = 3$
$x = 10, y = 2$
$x = 20, y = −8$
$x = 14, y = −2$

Click to reveal answer

Correct Answer: A. $(x=9, y=3)$

Explanation:
We have a system of two linear equations:
1. $x + y = 12$
2. $x - y = 6$

We can solve this system using the elimination method. Add Equation (1) and Equation (2) together:
$(x + y) + (x - y) = 12 + 6$
$x + y + x - y = 18$
$2x = 18$
$x = \frac{18}{2}$
$x = 9$

Now, substitute the value of $x=9$ into either Equation (1) or Equation (2) to find $y$. Let's use Equation (1):
$9 + y = 12$
$y = 12 - 9$
$y = 3$

So, the solution that satisfies both equations is $x=9$ and $y=3$

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