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Grade 12 Mathematics (Natural Science) national exam, 2017 E.C. (2025 G.C.)
51 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.Which one of the following function is NOT a power function?
- A. $f(x)=x^{-\frac{3}{5}}$
- B. $f(x)=\sqrt{2}x^{\frac{3}{2}}$
- C. $f(x)=\pi x^5$
- D. $f(x)=(\sqrt{3})^x$
Answer: D. The key idea is that a power function has the variable raised to a fixed power, as in $a x^n$. A, B, and C all have this form, even though their constants or powers differ. D has the variable in the exponent, so it is an exponential function, not a power function. Textbook: Grade 11, unit 1 · section 1.3. - 2.Let $f(x)=(1-2x)^{\frac{1}{2}}$. Then what is the values of $f'(x)$?
- A. $\frac{1}{\sqrt{1-2x}}$
- B. $-\frac{1}{2}\sqrt{1-2x}$
- C. $\frac{1}{2}\sqrt{1-2x}$
- D. $-\frac{1}{\sqrt{1-2x}}$
Answer: D. The chain rule is used when a function is inside another function. For $f(x)=(1-2x)^{1/2}$, $f'(x)=\frac12(1-2x)^{-1/2}(-2)=-\frac{1}{\sqrt{1-2x}}$. D matches this derivative. A has the wrong sign. B has the power in the numerator instead of the denominator. C has the wrong sign and power. Textbook: Grade 12, unit 2 · section 2.1. - 3.Which of the following statement describes the difference between scalar and vector quantities?
- A. Physical quantities that only have direction are scalars, while that have only magnitude are vectors.
- B. Physical quantities that only have magnitude are scalars, while that have both magnitude and direction are vectors.
- C. Physical quantities that only have magnitude are vectors, while that have both magnitude and direction are scalars.
- D. Physical quantities that only have direction are vectors, while that have both magnitude and direction are scalars.
Answer: B. By definition, scalar quantities possess only magnitude, whereas vector quantities possess both magnitude and a specific direction in space. Textbook: Grade 11, unit 5 · section 5.1. - 4.Consider the following data: 1, 3, 7, 9, 11, 13, 18, 28. Then the coefficient of mean deviation about the median is equal to:
- A. $\frac{1}{4}$
- B. $\frac{8}{5}$
- C. $\frac{5}{8}$
- D. $\frac{1}{2}$
Answer: C. The coefficient of mean deviation about the median is the mean of the absolute distances from the median, divided by the median. The median is $\frac{9+11}{2}=10$, and the absolute deviations total $50$, so the coefficient is $\frac{50/8}{10}=\frac{5}{8}$. A, B, and D are wrong because they are not this value. Textbook: Grade 12, unit 3 · section 3.1. - 5.Which one of the following is an irrational number?
- A. 3.4000
- B. $\sqrt{0.04}$
- C. 1.16222...
- D. $\sqrt{29}$
Answer: D. The key idea is that an irrational number cannot be written as a ratio of whole numbers. Since $29$ is not a perfect square, $\sqrt{29}$ is irrational. Option A is a terminating decimal, so it is rational. Option B equals $0.2$, so it is rational. Option C has repeating digits, so it is rational. Textbook: Grade 9, unit 2 · section 2.3. - 6.Consider the graph of a certain trigonometric function given below. Which one of the following functions represents the above graph for $-\frac{\pi}{2} \leq x \leq \frac{5\pi}{2}$?
- A. $y = \sec x$
- B. $y = \sin x$
- C. $y = \csc x$
- D. $y = \cos x$
Answer, explanation and textbook page: open in the app. - 7.Given that $\log 4.8 = 0.6812$, $\log 6.93 = 0.8406$ and $\log 6.84 = 0.8351$, then what is the value of $\sqrt{48}$?
- A. 6.83
- B. 6.71
- C. 6.93
- D. 6.84
Answer, explanation and textbook page: open in the app. - 8.Let $f(x)=x^3+2x+1$ and $g(x)=x^2+1$ be given two polynomial functions. Then which of the following is true?
- A. $\frac{f(x)}{g(x)}=\frac{x^2+1}{x-1}$
- B. $f(x)+g(x)=x^3+x^2$
- C. $f(x)-g(x)=x^3-x^2+2x$
- D. $f(x)g(x)=x^5+x^3+x^2-2x-1$
Answer, explanation and textbook page: open in the app. - 9.Let and be two sets of real numbers. What is the relative complement of with respect to ?
- B. {x∣−1 ≤x ≤0} ∅ {x∣−1 ≤x ≤1} {x∣−1 ≤x < 1}
Answer, explanation and textbook page: open in the app. - 10.Let $A = \{x|1 \leq x \leq 10\}$ and $B = \{x|-1 \leq x \leq 10\}$ be two sets of real numbers. What is the relative complement of $A$ with respect to $B$?
- A. $\{x|-1 \leq x \leq 0\}$
- B. $\emptyset$
- C. $\{x|-1 \leq x \leq 1\}$
- D. $\{x|-1 \leq x < 1\}$
Answer, explanation and textbook page: open in the app. - 11.The table below shows a grouped frequency distribution.
X | 0–9 | 10–19 | 20–29 | 30–39 | 40–49
f | 4 | 3 | 6 | 5 | 2
What is the class midpoint of the fourth class?
- A. 34
- B. 34.5
- C. 35
- D. 39
Answer, explanation and textbook page: open in the app. - 12.Given the data set: 29, 44, 46, 52, 35, 35, 25, 15, 10, 55, 56 What is the coefficient of quartile deviation?
- A. 0.25
- B. 0.35
- C. 0.45
- D. 0.55
Answer, explanation and textbook page: open in the app. - 13.Which of the following is the simplified form of the expression $\frac{81^{1/5}}{27^{3/5}}$?
- A. 3
- B. $\sqrt[5]{9}$
- C. $\sqrt[5]{3}$
- D. $\frac{1}{3}$
Answer, explanation and textbook page: open in the app. - 14.What is the total surface area (in $\text{cm}^2$) of the cylinder whose radius is 2 cm and height is 7 cm?
- A. $49\pi$
- B. $28\pi$
- C. $36\pi$
- D. $98\pi$
Answer, explanation and textbook page: open in the app. - 15.Suppose that we have 4 objects chosen 3 at a time. Then which of the following statement is true?
- A. The number of combinations is 6.
- B. The number of combinations is 24.
- C. The number of permutations is 24.
- D. The number of permutations is 6.
Answer, explanation and textbook page: open in the app. - 16.Consider the equation $\log x + \log(x - 3) = 1$. Then what is its solution set?
- A. $\{5\}$
- B. $\{2, -5\}$
- C. $\{-2, 5\}$
- D. $\{6\}$
Answer, explanation and textbook page: open in the app. - 17.If a fair six-sided die is tossed once, then what is the probability of getting a factor of 4?
- A. $\frac{1}{2}$
- B. $\frac{1}{6}$
- C. $\frac{5}{6}$
- D. $\frac{1}{3}$
Answer, explanation and textbook page: open in the app. - 18.Which of the following statement is true about a relation?
- A. The domain and the range of a relation are equal sets.
- B. The range of a relation is the set of all its first coordinates.
- C. The domain of a relation is the set of all its second coordinates.
- D. A relation from A to B is any subset of A × B for non-empty sets A and B.
Answer, explanation and textbook page: open in the app. - 19.Which one of the following expression is the correct decomposition of the rational expression $\frac{2x-3}{x^2+x-2}$?
- A. $\frac{5}{3x+2}+\frac{1}{x+1}$
- B. $\frac{1}{3-3x}+\frac{7}{3x+6}$
- C. $\frac{1}{1-x}+\frac{5}{3x+6}$
- D. $\frac{7}{3x+2}-\frac{1}{x+1}$
Answer, explanation and textbook page: open in the app. - 20.What is the average rate of change of $g(x)=x^3-2x$ over the interval $1\le x\le 6$?
- A. $\frac{203}{5}$
- B. 41
- C. $\frac{37}{5}$
- D. 46
Answer, explanation and textbook page: open in the app. - 21.Which of the following matrix is the inverse of the matrix $\begin{pmatrix}1&-1&2\1&0&1\3&1&3\end{pmatrix}$?
- A. $\begin{pmatrix}1&5&-1\-3&0&1\-4&-1&1\end{pmatrix}$
- B. $\begin{pmatrix}-1&0&1\5&-3&-4\-1&1&1\end{pmatrix}$
- C. $\begin{pmatrix}-1&0&1\5&3&4\1&-1&1\end{pmatrix}$
- D. $\begin{pmatrix}-1&5&-1\0&-3&1\1&-4&1\end{pmatrix}$
Answer, explanation and textbook page: open in the app. - 22.When the number $43\times10^8\times10^{-5}$ is written in standard form, it equals
- A. $4.3\times10^4$
- B. $43\times10^3$
- C. $4.3\times10^3$
- D. $0.43\times10^5$
Answer, explanation and textbook page: open in the app. - 23.If a square pyramid has a base side of 6 cm long and its lateral faces are equilateral triangles, then what is its volume (in $\text{cm}^3$)?
- A. 72
- B. $3\sqrt{2}$
- C. 48
- D. $36\sqrt{2}$
Answer, explanation and textbook page: open in the app.
A television company assembles flat screen and curved screen televisions. The company has to invest Birr 15,000 to assemble one flat screen and Birr 20,000 to assemble one curved screen television. The company allocates a maximum of Birr 900,000 in a week for assembling. The company gets Birr 1800 profit from a sale of one flat screen television and Birr 2000 profit from a sale of one curved screen television. Moreover, the company has a store of maximum capacity of storing 50 televisions. Let x be the number of flat screens and y be the number of curved screens.
- 24.Then which one of the following inequalities should model the profit for the objective function Z=1800x+2000y? ⎩ ⎨ ⎧20, 000x + 15, 000y ≤360, 000 ⎩ ⎨ ⎧15, 000x + 20, 000y ≤900, 000
- B. x + y ≤50 x, y ≥0 x + y ≤50 x, y ≥0 ⎩ ⎨ ⎧15, 000x + 20, 000y ≤360, 000 ⎩ ⎨ ⎧20, 000x + 15, 000y ≤900, 000
- D. x + y ≤50 x, y ≥0 x + y ≤50 x, y ≥0
Answer, explanation and textbook page: open in the app. - 25.Which one of the following distribution is multimodal?
- A. 18, 14, 15, 10, 11, 3, 18, 12
- B. 6, 10, 9, 6, 10, 4
- C. 8, 7, 11, 13, 17, 9, 8, 7, 6, 11
- D. 12, 13, 26, 15, 25, 38, 28, 27, 32
Answer, explanation and textbook page: open in the app. - 26.What is the equation of a line that passes through $(-2,5)$ and has a slope $\frac{1}{2}$?
- A. $2y-x-12=0$
- B. $y=2x+12$
- C. $2y+x=5$
- D. $y=\frac{1}{2}x+5$
Answer, explanation and textbook page: open in the app. - 27.What is the minimum value of the function defined by $f(x)=-3+4x+x^2$?
- A. $-7$
- B. 1
- C. 0
- D. $-2$
Answer, explanation and textbook page: open in the app. - 28.Which of the following statement is true about the given system of linear equations?
- A. $\begin{cases}x+3y=6\2x-5y=2\end{cases}$ is a homogeneous system.
- B. $\begin{cases}2x+y=0\x-4y=0\end{cases}$ is a non-homogeneous system.
- C. $\begin{cases}x+3y=2\2x-5y=1\end{cases}$ is a non-homogeneous system.
- D. $\begin{cases}3x-y=1\x+3y=0\end{cases}$ is a homogeneous system.
Answer, explanation and textbook page: open in the app. - 29.Suppose and be given two functions. Then which of the following is equal to ? 3x + − 2 1 4x + 3 2 x + 3 2 − 3 1 4x + 3 2x
- B. x + 3 2 − 3 1 4x + 3 2 x + 2 3 − 2 1 x + 3 4 3 2
Answer, explanation and textbook page: open in the app. - 30.The table given below shows the ages distribution of children in a small village in years.
Age interval | Frequency (f)
2-4 | 4
5-7 | 1
8-10 | 2
11-13 | 3
Then what is the mean age of the children’s (in years)?
- A. 6.9
- B. 6.6
- C. 7.5
- D. 7.2
Answer, explanation and textbook page: open in the app. - 31.A worker of a company borrowed Birr 100,000 from the company's credit and saving association. She agreed to pay 10% of the loan during the agreement and on the remaining amount to pay Birr 3000 (monthly plus interest) in 36 monthly installments. What is the total amount of money she paid to the credit and saving association?
- A. Birr 106,000
- B. Birr 118,000
- C. Birr 108,000
- D. Birr 116,000
Answer, explanation and textbook page: open in the app. - 32.Given the system of linear equations $\begin{cases}a_1x+b_1y=c_1\a_2x+b_2y=c_2\end{cases}$, where $a_1, a_2, b_1, b_2, c_1, c_2\in\mathbb{R}$ and $a_2, b_2, c_2\ne 0$, then which of the following is true?
- A. If $\frac{a_1}{a_2}\ne\frac{b_1}{b_2}$, then the system has one solution.
- B. If $\frac{a_1}{a_2}=\frac{c_1}{c_2}$, then the system has no solution.
- C. If $\frac{b_1}{b_2}=\frac{c_1}{c_2}$, then the system has unique solution.
- D. If $\frac{a_1}{a_2}\ne\frac{c_1}{c_2}$, then the system has infinitely many solutions.
Answer, explanation and textbook page: open in the app. - 33.Which of the following is true about the cosecant value of the given angles?
- A. $\csc 135^\circ=-\frac{1}{\sqrt{2}}$
- B. $\csc 135^\circ=-\sqrt{2}$
- C. $\csc 45^\circ=\frac{1}{\sqrt{2}}$
- D. $\csc 45^\circ=\sqrt{2}$
Answer, explanation and textbook page: open in the app. - 34.Which of the following statement defines geometric progression?
- A. It is a sequence in which the ratio between consecutive terms is a non-zero constant.
- B. It is a sequence in which the difference between consecutive terms is a positive constant.
- C. It is a sequence in which the difference between consecutive terms is a non-zero constant.
- D. It is a sequence in which the ratio between consecutive terms is a positive constant.
Answer, explanation and textbook page: open in the app. - 35.Consider the line segment $PQ$ with $P=(1,-1)$ and $Q=(4,2)$. Then which of the following point divides $PQ$ internally in the ratio of $2:1$?
- A. $(3,1)$
- B. $(2,0)$
- C. $(2,1)$
- D. $(3,0)$
Answer, explanation and textbook page: open in the app. - 36.What is the determinant of the matrix $\begin{pmatrix}\frac{1}{2} & 1 & \frac{3}{2} \ 2 & \frac{5}{2} & 3 \ \frac{7}{2} & 4 & \frac{9}{2}\end{pmatrix}$?
- A. 0
- B. $\frac{1}{2}$
- C. $\frac{1}{8}$
- D. 1
Answer, explanation and textbook page: open in the app. - 37.Given the objective function $Z=-x+2y$ that is subjected to the constraints $\begin{cases}x\geq 3\x+y\geq 5\x+2y\leq 6\y\geq 0\end{cases}$, what are the maximum and minimum values of $Z$ respectively?
- A. -2 and -6
- B. -2 and 1
- C. 1 and -2
- D. -6 and -2
Answer, explanation and textbook page: open in the app. - 38.Let such that and be a polynomial function. Then which one of the following statement states the location theorem?
- A. If and have opposite signs, then there is at least one zero of between and . f(a) f(b) f a b
- B. If and have the same signs, then there is at least one zero of between and . f(a) f(b) f a b f(a) f(b) f a b
- C. If and have the same signs, then there exist a unique zeros of between and .
- D. If and have opposite signs, then there is a unique zeros of between and . f(a) f(b) f a b
Answer, explanation and textbook page: open in the app. - 39.Suppose two triangles are similar by SSS similarity theorem. If the side lengths of one triangle are 2 cm, 4 cm and 6 cm, and the shorter side of the other triangle is 9 cm long, then what are the lengths (in cm) of the other two sides of the triangle?
- A. 9 and 21
- B. 18 and 27
- C. 9 and 27
- D. 18 and 21
Answer, explanation and textbook page: open in the app. - 40.Let such that and be a polynomial function. Then which one of the following statement states the location theorem?
- A. If and have opposite signs, then there is at least one zero of between and . f(a) f(b) f a b f(a) f(b) f a b
- B. If and have the same signs, then there is at least one zero of between and .
- C. If and have the same signs, then there exist a unique zeros of between and . f(a) f(b) f a b f(a) f(b) f a b
- D. If and have opposite signs, then there is a unique zeros of between and .
Answer, explanation and textbook page: open in the app. - 41.If $\begin{pmatrix}x+y&2\5&3\end{pmatrix}=\begin{pmatrix}7&2\5&x-y\end{pmatrix}$, then what is the value of $x$?
- A. -2
- B. -5
- C. 5
- D. 2
Answer, explanation and textbook page: open in the app. - 42.Consider the following data: 13, 16, 19, 19, 19, 19, 24, 27 What is the mean deviation about the mode?
- A. 4
- B. 2.75
- C. 3.75
- D. 3
Answer, explanation and textbook page: open in the app. - 43.Which of the following statement is true about types of data:
- A. The age of students in a class is a quantitative data.
- B. The height of students in a class is a qualitative data.
- C. The weight of students in class is a qualitative data.
- D. The gender of students in a class is a quantitative data.
Answer, explanation and textbook page: open in the app. - 44.A company offers a discount of 25% on one of its product for promotion. What is a regular price of the product if it is purchased for Birr 99?
- A. 110
- B. 198
- C. 132
- D. 125
Answer, explanation and textbook page: open in the app. - 45.What is the total surface area (in cm ) of the cylinder whose radius is 2cm and height is 7cm?
- A. 49π
- B. 28π
- C. 36π
- D. 98π
Answer, explanation and textbook page: open in the app. - 46.Which of the following statement is true about the given infinite series?
- A. $\sum_{n=1}^{\infty}(-1)^n 2^n$ is convergent.
- B. $\sum_{n=1}^{\infty}\left(\frac{4}{3}\right)^n$ is convergent.
- C. $\sum_{n=1}^{\infty}2^n$ is divergent.
- D. $\sum_{n=1}^{\infty}(-1)^n\left(\frac{2}{5}\right)^n$ is divergent.
Answer, explanation and textbook page: open in the app. - 47.Let $\theta$ be an angle in standard position and $(x, y)$ be a point on the terminal side of $\theta$, which is different from the origin. Let $r$ be the distance from $(x, y)$ to the origin. Then which of the following defines $\cos \theta$?
- A. $\frac{r}{y}$
- B. $\frac{y}{r}$
- C. $\frac{r}{x}$
- D. $\frac{x}{r}$
Answer, explanation and textbook page: open in the app. - 48.Which one of the following is the solution set of $5^x = 11.2$? [Use $ ext{log } 5 = 0.6990$ and $ ext{log } 1.12 = 0.0485$]
- A. 1.12
- B. 1.5
- C. 1.2
- D. 1.69
Answer, explanation and textbook page: open in the app. - 49.A boy and girl are both standing on a sun light. The boy’s height is 1.6m and casts a shadow of 2m. If the girl’s height casts a shadow of 1.8m, then what is the height (in meter) of the girl?
- A. 1.5
- B. 1.54
- C. 1.68
- D. 1.44
Answer, explanation and textbook page: open in the app. - 50.If two vectors $\vec{V}=\overrightarrow{AB}$, and $\vec{U}=\overrightarrow{BC}$ are given as shown in the figure below. Then what is the vector $\vec{V}+\vec{U}$?
- A. $\overrightarrow{AC}$
- B. $\overrightarrow{BA}$
- C. $\overrightarrow{CA}$
- D. $\overrightarrow{CB}$
Answer, explanation and textbook page: open in the app. - None.Which one of the following function is NOT a power function?
- A. $f(x)=x^{-\frac{3}{5}}$
- B. $f(x)=\sqrt{2}x^{\frac{3}{2}}$
- C. $f(x)=\pi x^5$
- D. $f(x)=(\sqrt{3})^x$
Answer, explanation and textbook page: open in the app.