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Grade 12 Mathematics (Social Science) national exam, 2018 E.C. (2026 G.C.)

40 questions · 120 minutes. The first 5 answers are shown with explanations; sit the whole paper timed in the app to get your score and the textbook page behind every question.

1. 1.What is the solution set of $\sqrt{5^{x^2}} = 25^x$?
   • A. ∅
   • B. 0
   • C. $\sqrt{2}$
   • D. 0, 4
   Answer: D. The key idea is to rewrite both sides using the same base. Since $\sqrt{5^{x^2}}=5^{x^2/2}$ and $25^x=5^{2x}$, we get $x^2/2=2x$, or $x(x-4)=0$, so $x=0$ or $x=4$. Option A leaves out both solutions; option B leaves out $4$; option C gives values that do not solve the equation. Textbook: Grade 9, unit 3 · section 3.2.
2. 2.In an experiment of rolling a die, what is the probability of getting 1 or 5?
   • A. $\frac{1}{6}$
   • B. $\frac{1}{3}$
   • C. $\frac{2}{3}$
   • D. $\frac{1}{4}$
   Answer: B. Probability is the number of wanted outcomes divided by the number of equally likely outcomes. A die has $6$ outcomes, and $1$ or $5$ gives $2$ wanted outcomes, so the probability is $\frac{2}{6}=\frac13$. $\frac16$ counts only one outcome; $\frac23$ is too large; $\frac14$ does not match the count. Textbook: Grade 9, unit 9 · section 9.2.
3. 3.Which one of the following pairs of lines are perpendicular to each other?
   • A. $l_1:y=2x-1$ and $l_2:y=\frac{1}{2}x+1$
   • B. $l_1:y=1$ and $l_2:y=-2$
   • C. $l_1:y=3x+1$ and $l_2:y=-3x+4$
   • D. $l_1:x=3$ and $l_2:y=1$
   Answer: D. A vertical line and a horizontal line are perpendicular. In D, $x=3$ is vertical and $y=1$ is horizontal. A has slopes $2$ and $\frac12$; B is parallel lines; C has slopes $3$ and $-3$, so none of A, B, or C is perpendicular. Textbook: Grade 11, unit 2 · section 2.3.
4. 4.Suppose that a person has 4 jackets and 5 trousers. In how many different ways can the person dress?
   • A. 20
   • B. 16
   • C. 9
   • D. 25
   Answer: A. The fundamental counting principle says to multiply the choices for each clothing item. There are $4\times5=20$ jacket-and-trouser combinations, so option A is correct. Option B does not count all combinations; option C adds the choices instead of multiplying; option D uses the wrong number of choices. Textbook: Grade 11, unit 8 · section 8.2.
5. 5.Consider the following grouped data as shown in the table below. Which one of the following is the variance of the above data? Value | Frequency 1-5 | 4 6-10 | 5 11-15 | 6 16-20 | 5
   • A. 60
   • B. 29.5
   • C. 59
   • D. 28.5
   Answer: D. For grouped data, use each class midpoint as its representative value. The midpoints are $3,8,13,18$; the mean is $220/20=11$, and the variance is $[4(3-11)^2+5(8-11)^2+6(13-11)^2+5(18-11)^2]/20=28.5$. A is wrong because $60$ is not the variance. B is wrong because $29.5$ does not result from the calculation. C is wrong because $59$ is not the variance. Textbook: Grade 11, unit 7 · section 7.2.
6. 6.Consider the graph of a rational function f(x) as shown below. Which one of the following statements is true about the graph of the function?
   
   • A. The line y = 0 is a horizontal asymptote.
   • B. The range of f is R \ {0}.
   • C. The line y = x is an oblique asymptote.
   • D. The domain is the set of all real numbers.
   Answer, explanation and textbook page: open in the app.
7. 7.A rectangular metal sheet of dimension 12 m by 18 m is cut into six congruent rectangular sheets (or pieces). Then what is the area of each piece in m²?
   • A. 36
   • B. 24
   • C. 28
   • D. 32
   Answer, explanation and textbook page: open in the app.
8. 8.What is the solution set of the inequality ?
   • A. {x: x ≥ -2 or x ≤ 5}
   • B. {x: -2 < x < 5}
   • C. {x: -2 ≤ x ≤ 5}
   • D. {x: x ≤ -2 or x ≥ 5}
   Answer, explanation and textbook page: open in the app.
9. 9.What are the domain and range of the function $f(x)=x+|x|$ respectively?
   • A. ℝ and [2, ∞)
   • B. ℝ and ℝ
   • C. ℝ and [0, ∞)
   • D. [0, ∞) and ℝ
   Answer, explanation and textbook page: open in the app.
10. 10.Given below the ages of 40 people taken to form five football teams in a certain sport field. Ages | 10-18 | 18-26 | 26-34 | 34-42 | 42-50 No. of Peoples | 10 | 4 | 8 | 12 | 6 What is the class midpoint of the third class?
    • A. 34.5
    • B. 26
    • C. 30.5
    • D. 30
    Answer, explanation and textbook page: open in the app.
11. 11.Let a function, $f(x)=\begin{cases}2x^2-4, & \text{if } x\geq 1\ x^3-4x+1, & \text{if } x<1\end{cases}$. Then what is the equation of the tangent line to the graph of $f$, at $x=-2$?
    • A. $y=-4x+14$
    • B. $y=-8x-12$
    • C. $y=8x+17$
    • D. $y=4x-2$
    Answer, explanation and textbook page: open in the app.
12. 12.Let a function, $f(x) = \begin{cases} 2x^2 - 4, & \text{if } x \ge 1 \\ x^3 - 4x + 1, & \text{if } x < 1 \end{cases}$. Then what is the equation of the tangent line to the graph of $f$, at $x = -2$?
    • A. $y = -4x + 14$
    • B. $y = -8x - 12$
    • C. $y = 8x + 17$
    • D. $y = 4x - 2$
    Answer, explanation and textbook page: open in the app.
13. 13.Let A and B be two sets such that $A = \{1, 4, 9\}$ and $B = \{n \in \mathbb{N} : 3 < n \le 5\}$. Which of the following sets represents $A \times B$?
    • A. $\{(4,1), (5,1), (5,4), (5,5), (4,9), (5,9)\}$
    • B. $\{(1,4), (1,5), (4,4), (4,5), (9,4), (9,5)\}$
    • C. $\{(1,4), (1,5), (4,5), (5,5), (9,4), (9,5)\}$
    • D. $\{(4,1), (5,1), (4,4), (5,4), (9,4), (9,5)\}$
    Answer, explanation and textbook page: open in the app.
14. 14.If $\cos\theta=-\frac{5}{13}$ and $\theta$ is an angle in the second quadrant, then what is the value of $\cot\theta$?
    • A. $\frac{12}{13}$
    • B. $-\frac{5}{12}$
    • C. $-\frac{12}{13}$
    • D. $\frac{5}{12}$
    Answer, explanation and textbook page: open in the app.
15. 15.If $\cos \theta = -\frac{5}{13}$ and $\theta$ is an angle in the second quadrant, then what is the value of $\cot \theta$?
    • A. $\frac{12}{13}$
    • B. $-\frac{5}{12}$
    • C. $-\frac{12}{13}$
    • D. $\frac{5}{12}$
    Answer, explanation and textbook page: open in the app.
16. 16.Let a matrix $A = \begin{pmatrix} 1 & -3 & 2 \\ -3 & 3 & -1 \\ 2 & -1 & 0 \end{pmatrix}$. Then what is the inverse of A?
    • A. $\begin{pmatrix} 1 & 2 & 3 \\ 2 & 4 & 5 \\ 3 & 5 & 6 \end{pmatrix}$
    • B. $\begin{pmatrix} -1 & 2 & 4 \\ 3 & -3 & 1 \\ -2 & 1 & 0 \end{pmatrix}$
    • C. $\begin{pmatrix} 1 & \frac{1}{2} & \frac{1}{3} \\ \frac{1}{2} & \frac{1}{4} & \frac{1}{5} \\ \frac{1}{3} & \frac{1}{5} & \frac{1}{6} \end{pmatrix}$
    • D. $\begin{pmatrix} 1 & \frac{-1}{3} & \frac{1}{2} \\ \frac{-1}{3} & \frac{1}{3} & -1 \\ \frac{1}{2} & -1 & 0 \end{pmatrix}$
    Answer, explanation and textbook page: open in the app.
17. 17.A line is passing through the points $(-2, 5)$ and $(3, -5)$. What is the gradient of the line?
    • A. 0
    • B. $-\frac{1}{2}$
    • C. -2
    • D. $\frac{2}{3}$
    Answer, explanation and textbook page: open in the app.
18. 18.What is the simplified form of the expression $$\frac{(\sqrt[3]{64})^2 \times (\sqrt{8})^3}{(\sqrt{2})^3 \times (\sqrt[3]{32})^2}$$
    • A. $2\sqrt{2}$
    • B. $8\sqrt{2}$
    • C. $\frac{2}{\sqrt{2}}$
    • D. $\frac{8}{\sqrt{2}}$
    Answer, explanation and textbook page: open in the app.
19. 19.Consider the following data: . Which of the following is true about the mode of the given data?
    • A. Multimodal
    • B. Unimodal
    • C. No mode
    • D. Bimodal
    Answer, explanation and textbook page: open in the app.
20. 20.Given the data: 10, 4, 2, 8, 2, 4. What is the standard deviation of the given data?
    • A. $\sqrt{3}$
    • B. 3
    • C. 5
    • D. $2\sqrt{5}$
    Answer, explanation and textbook page: open in the app.
21. 21.Consider the following system of linear equations. $\begin{cases}x-y=2\3x+y=4\2x-2y=3\end{cases}$ Then which one of the following matrices is the reduced form of the augmented matrix into triangular matrix?
    • A. $\begin{pmatrix}1&-1&2\0&-2&-2\0&0&-1\end{pmatrix}$
    • B. $\begin{pmatrix}-1&0&0\-2&4&0\1&-1&2\end{pmatrix}$
    • C. $\begin{pmatrix}-1&0&0\-2&-2&0\1&-1&2\end{pmatrix}$
    • D. $\begin{pmatrix}1&-1&2\0&4&-2\0&0&-1\end{pmatrix}$
    Answer, explanation and textbook page: open in the app.
22. 22.Let $MD$ refers to the mean deviation from the mean and $\bar{x}$ denotes the mean of a data set. Which of the following formula represents the coefficient of Mean deviation (CMD) from the mean of the data?
    • A. $CMD=\frac{MD+\bar{x}}{MD-\bar{x}}$
    • B. $CMD=\frac{\bar{x}}{MD}$
    • C. $CMD=\frac{MD-\bar{x}}{MD+\bar{x}}$
    • D. $CMD=\frac{MD}{\bar{x}}$
    Answer, explanation and textbook page: open in the app.
23. 23.Which one of the following statements is NOT true about regular polygon?
    • A. A circle can be circumscribed about the regular polygon.
    • B. It has n lines of symmetries.
    • C. Its apothem is equal to the side of the polygon.
    • D. A circle can be inscribed in the regular polygon.
    Answer, explanation and textbook page: open in the app.
24. 24.Let and . Consider and . Then which one of the following statements is true about their domains and ranges?
    • A. The domain of is equal to the range of . f(x) g(x) f(x) g(x)
    • B. The domain of is equal to the domain of . f(x) g(x)
    • C. The domain of is a subset of the domain of .
    • D. The range of is equal to the range of . f(x) g(x)
    Answer, explanation and textbook page: open in the app.
25. 25.What is the solution set of $\log_{2} x + \log_{2}(x + 3) = 2$?
    • A. {1}
    • B. {2}
    • C. {−4, 1}
    • D. {4}
    Answer, explanation and textbook page: open in the app.
26. 26.If $A = \begin{pmatrix}3 & 1 & 1 \ 2 & -1 & 1 \ 0 & 1 & 3\end{pmatrix}$, then what is the value of $\det(A)$?
    • A. -4
    • B. -16
    • C. 14
    • D. 2
    Answer, explanation and textbook page: open in the app.
27. 27.Consider the rhombus ABCD as shown in the figure below, where AC = 6 units and BD = 8 units. Then which one of the following statements is true about the above rhombus? m(∠DAB) = m(∠OBA) + m(∠ODA)
    • B. The side of the rhombus is 10 units.
    • C. The perimeter of the rhombus is 20 units.
    • D. m(∠OAB) = m(∠OBA)
    Answer, explanation and textbook page: open in the app.
28. 28.What is the fraction form of the decimal $0.343434...$?
    • A. $\frac{17}{50}$
    • B. $\frac{34}{99}$
    • C. $\frac{5}{9}$
    • D. $\frac{34}{90}$
    Answer, explanation and textbook page: open in the app.
29. 29.Let $P(a,b)$ be a point, other than the origin $(0,0)$, on the terminal side of an angle $\theta$ in standard position. Then what is the value of $\cos\theta$?
    • A. $\frac{a}{\sqrt{a^2+b^2}}$
    • B. $\frac{b}{a^2+b^2}$
    • C. $\frac{b}{\sqrt{a^2+b^2}}$
    • D. $\frac{a}{a^2+b^2}$
    Answer, explanation and textbook page: open in the app.
30. 30.Which one of the following is a skew symmetric matrix?
    • A. $\begin{pmatrix}0&-2&3\2&0&4\-3&4&0\end{pmatrix}$
    • B. $\begin{pmatrix}-1&0&0\2&2&0\1&-3&1\end{pmatrix}$
    • C. $\begin{pmatrix}0&-2&1\2&0&-3\-1&3&0\end{pmatrix}$
    • D. $\begin{pmatrix}1&-5&4\-5&2&0\4&0&3\end{pmatrix}$
    Answer, explanation and textbook page: open in the app.
31. 31.In a bank, there are 10 customers waiting for service. In how many different ways can an officer choose six customers in all, if three older customers are given priority?
    • A. 120
    • B. 35
    • C. 72
    • D. 210
    Answer, explanation and textbook page: open in the app.
32. 32.Given below the amount of time (in minutes) spent on running each day by a group of 16 students. Time spent on running | 1-4 | 5-8 | 9-12 | 13-16 | 17-20 No. of students | 3 | 2 | 4 | 4 | 3 What is the coefficient of quartile deviation of the given data?
    • A. $\frac{9}{22}$
    • B. $\frac{9}{2}$
    • C. 9
    • D. 11
    Answer, explanation and textbook page: open in the app.

1. 33.Which one of the following represents the linear objective function and constraints of the profit? (let S and M be the numbers of Samsung and Motorola mobile respectively)
   • B. Z = 1, 200S + 1, 500M Z = 1, 500S + 1, 200M subject to: ⎩ ⎨ ⎧16, 000S + 18, 000M ≤10, 000, 000 subject to: ⎩ ⎨ ⎧18, 000S + 16, 000M ≤10, 000, 000 S + M ≤600 S, M ≥0 S + M ≤600 S, M ≥0
   • D. Z = 1, 200S + 1, 500M Z = 1, 500S + 1, 200M subject to: ⎩ ⎨ ⎧18, 000S + 16, 000M ≤10, 000, 000 subject to: ⎩ ⎨ ⎧18, 000S + 16, 000M ≥10, 000, 000 S + M ≥600 S, M ≥0 S + M ≤600 S, M ≥0
   Answer, explanation and textbook page: open in the app.
2. 34.The two consecutive sides of a parallelogram are $3\,\text{m}$ and $4\,\text{m}$ long and the angle between them is $30^\circ$. Then what is the area of the parallelogram?
   • A. $7.5\,\text{m}^2$
   • B. $12\,\text{m}^2$
   • C. $6\,\text{m}^2$
   • D. $24\,\text{m}^2$
   Answer, explanation and textbook page: open in the app.
3. 35.What is the solution set of the quadratic inequality $3x^2 - 3x - 6 < 0$?
   • A. $(-3, 6)$
   • B. $(-\infty, -1) \cup (2, \infty)$
   • C. $(-1, 2)$
   • D. $(-\infty, -3) \cup (6, \infty)$
   Answer, explanation and textbook page: open in the app.
4. 36.There are three red, four yellow and three green balls in a bag. If two balls are picked randomly from the bag at a time, what is the probability of getting at least one green ball?
   • A. $\frac{1}{3}$
   • B. $\frac{7}{15}$
   • C. $\frac{8}{15}$
   • D. $\frac{7}{45}$
   Answer, explanation and textbook page: open in the app.
5. 37.A shop sale a shoe at a price of Birr 1225 with a 30% discount notice. How much was the original price of the shoe in Birr?
   • A. 1432.50
   • B. 1750
   • C. 3675
   • D. 1592.50
   Answer, explanation and textbook page: open in the app.
6. 38.Two six face dice are thrown at a time. What is the probability of getting the sum of the outcome is a prime number?
   • A. $\frac{5}{12}$
   • B. $\frac{5}{8}$
   • C. $\frac{7}{12}$
   • D. $\frac{2}{3}$
   Answer, explanation and textbook page: open in the app.

1. 39.A person is given the colors blue, red and green and asked to paint six houses in a row. The rules are: R1: No two neighboring houses can have the same color. R2: The second and the last houses must have same color. R3: The third and fifth houses cannot have the same color. Then which houses will always have the same color?
   • A. The second and the fifth
   • B. The first and the fourth
   • C. The first and the sixth
   • D. The fourth and the sixth
   Answer, explanation and textbook page: open in the app.
2. 40.In a household, there are three mothers, six daughters and four granddaughters living together. What is the smallest possible number of females in the house?
   • A. 13
   • B. 12
   • C. 7
   • D. 9
   Answer, explanation and textbook page: open in the app.