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Grade 12 Mathematics (Social Science) national exam, 2016 E.C. (2024 G.C.)

37 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.

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  1. 1.In the set of natural numbers, which one of the following defines a composite number?
    • A. A number that has only two distinct factors.
    • B. A number that is divisible by numbers from 2 to 9.
    • C. A number that has two or more different factors.
    • D. A number that has three or more different divisors.
    Answer: D. The key idea is that a composite natural number has more than two distinct divisors. D gives this definition: a composite number has at least three distinct divisors. A describes a prime number, B is not a definition of composite, and C also includes prime numbers, which have two factors. Textbook: Grade 9, unit 2 · section 2.1.
  2. 2.Which of the following is true about rationalizing the denominator?
    • A. Only denominators can be multiplied by an irrational number.
    • B. Only numerators can be rationalized when there is a radical.
    • C. It is a method of changing the denominator into a rational number.
    • D. It is a technique of multiplying a given number by an irrational expression.
    Answer: C. Rationalizing the denominator means changing a denominator with a radical into a rational number. C gives this meaning. A and B are wrong because rationalizing is not limited to one part of a fraction. D is wrong because it does not describe the goal of the method. Textbook: Grade 9, unit 2 · section 2.4.
  3. 3.When dividing 142 by 13 and expressed the result by applying the division algorithm, we get?
    • A. $142 = 10(13) + 2$
    • B. $142 = 11(13) - 1$
    • C. $142 = 11(13) + 1$
    • D. $142 = 10(14) + 2$
  4. 4.The solution set of the system $\begin{cases} 2x - y = 5 \\ 3y + 15 = 6x \end{cases}$ is:
    • A. $\emptyset$
    • B. $\{(1, 2t - 15): t\in\mathbb{R}\}$
    • C. $\{(1, 2), (1,3), (2,2), \dots\}$
    • D. $\{(0, 5), (1, 3), (2, 2), \dots\}$
  5. 5.Let $f$ be a function whose graph is the given below Which one of the following is equal to $f(x)$?
    • A. $-2x^2 + 4x - 8$
    • B. $2x^2 - 8x + 8$
    • C. $8x - 2x^2 + 8$
    • D. $8x - 2x^2 - 8$
  6. 6.Let $\bar{x}$ denote the arithmetic mean of a numerical data. If each observation is multiplied by 0.5, which of the following is the new mean?
    • A. $\bar{x}$
    • B. $\frac{1}{2}\bar{x}$
    • C. $2\bar{x}$
    • D. $\bar{x}-0.5$
    Answer: B. The key idea is that multiplying every observation by a number multiplies the mean by that number. Therefore, the new mean is $0.5\bar{x}=\frac12\bar{x}$. Option A leaves the mean unchanged; option C doubles it; option D subtracts a number instead of scaling the mean. Textbook: Grade 9, unit 9 · section 9.2.
  7. 7.A regular triangular pyramid labeled 1, 2 and 3, is thrown twice and what comes down on a flat surface is observed. What is the probability that the sum of the numbers faced down from this experiment less than 5?
    • A. $\frac{1}{3}$
    • B. $\frac{2}{3}$
    • C. $\frac{4}{9}$
    • D. $\frac{5}{9}$
    Answer: B. The key idea is to count equally likely pairs of results and find what fraction have a sum less than $5$. There are $3\times3=9$ pairs, and $6$ give sums $2$, $3$, or $4$, so the probability is $\frac{6}{9}=\frac{2}{3}$. A and C count too few successful pairs; D counts too many. Textbook: Grade 11, unit 8 · section 8.6.
  8. 8.Which one of the following is the definition of a polynomial function? f(x) = a x + n n a x + n−1 n−1 ... + a x + 1 a 0 n an 0 a , a , ..., a ∈ 0 1 n R
    • A. A function where is a non-negative integer, ≠ and . f(x) = a x + n n a x + n−1 n−1 ... + a x + 1 n+1 a 0 n a n 0 a
    • B. A function where is a positive integer, ≠ and ∈ 0 1 n R f(x) = a x + n n a x + n−1 n−1 ... + a x + 1 n−1 a 0 n a n 0 a , a , ..., a ∈ 0 1 n R , a , ..., a .
    • C. A function where is an integer, ≠ and . f(x) = a x − n n a x + n−1 n−1 ... + a x + 1 a 0 n a n 0 a
    • D. A function where is a rational number, ≠ and ∈ 0 1 n R , a , ..., a .
    Answer: A. A polynomial function has whole-number powers of $x$ starting from $0$, and its highest power has a non-zero coefficient. A gives this form, with $n$ a non-negative integer, $a_n\ne0$, and all coefficients real. B has an incorrect power of $x$; C allows negative values of $n$ and has an incorrect power; D allows a rational value of $n$ instead of a non-negative integer. Textbook: Grade 10, unit 2 · section 2.1.
  9. 9.Which of the following is the approximate surface area of the bases of a regular pentagonal prism of radius 6 cm? (Given values: sin72° ≈ 0.3, cos72° ≈ 0.95)
    • A. 54 sq. cm
    • B. 85.5 sq. cm
    • C. 27 sq. cm
    • D. 27.5 sq. cm
    Answer, explanation and textbook page: open in the app.
  10. 10.What is the solution set of the equation $\log_{4}(x+1)-2\log_{4}x=0.5$?
    • A. $\{1,-\frac{1}{2}\}$
    • B. $\{\frac{1}{2},-1\}$
    • C. $\{1\}$
    • D. $\{\frac{1}{2}\}$
    Answer, explanation and textbook page: open in the app.
  11. 11.Which of the following is the characteristic of $\log(20 + \frac{1}{5})$?
    • A. -1
    • B. 1
    • C. 2.02
    • D. 0.2
    Answer, explanation and textbook page: open in the app.
  12. 12.Given vectors $\vec{u}$ having initial point $(-1, 0)$ and terminal point $(2, 3)$ and $\vec{v}$ with initial point $(5, -2)$ and terminal point $(2, 3)$, then $\vec{v} - \vec{u}$ is equal to:
    • A. $(-6, 3)$
    • B. $(2, -6)$
    • C. $(0, -2)$
    • D. $(-6, -2)$
    Answer, explanation and textbook page: open in the app.
  13. 13.Which one of the following is the definition of a polynomial function?
    • A. A function $f(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0$ where $n$ is a non-negative integer, $a_n$ ≠ $0$ and $a_0, a_1, ..., a_n \in \mathbb{R}$.
    • B. A function $f(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x^{n+1} + a_0$ where $n$ is a positive integer, $a_n$ ≠ $0$ and $a_0, a_1, ..., a_n \in \mathbb{R}$.
    • C. A function $f(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x^{n-1} + a_0$ where $n$ is an integer, $a_n$ ≠ $0$ and $a_0, a_1, ..., a_n \in \mathbb{R}$.
    • D. A function $f(x) = a_n x^n - a_{n-1} x^{n-1} + ... + a_1 x + a_0$ where $n$ is a rational number, $a_n$ ≠ $0$ and $a_0, a_1, ..., a_n \in \mathbb{R}$.
    Answer, explanation and textbook page: open in the app.
  14. 14.For , which of the following identity is correct? csc (90 − 2 ∘ θ) = 1 + tan θ 2 cot (90 − 2 ∘ θ) = 1 + csc θ 2
    • B. sec θ + 2 tan θ = 2 1 cot θ + 2 csc θ = 2 1
    Answer, explanation and textbook page: open in the app.
  15. 15.What is the solution set of the equation $\log_4(x + 1) - 2\log_4 x = 0.5$?
    • A. $\left\{1, -\frac{1}{2}\right\}$
    • B. $\left\{\frac{1}{2}, -1\right\}$
    • C. $\{1\}$
    • D. $\left\{\frac{1}{2}\right\}$
    Answer, explanation and textbook page: open in the app.
  16. 16.A person would like to fence all sides of a regular hexagonal polygonal park of radius 12 meter. What is the length of a fencing material needed in meter?
    • A. $72\sqrt{3}$
    • B. 72
    • C. $36\sqrt{3}$
    • D. 36
    Answer, explanation and textbook page: open in the app.
  17. 17.Which one of the following is the solution set of the equation $(\frac{1}{25})^x -5(0.04)^x = 30$?
    • A. $\{5\}$
    • B. $\left\{\frac{1}{5}\right\}$
    • C. $\left\{-\frac{1}{2}\right\}$
    • D. $\left\{\frac{1}{2}\right\}$
    Answer, explanation and textbook page: open in the app.
  18. 18.Given the set $A = {x \in \mathbb{R} : x \notin (\pi,\infty)}$, which of the following sets is equal to $A$?
    • A. $\{-\infty, \pi \}$
    • B. $\{-\pi, \pi\}$
    • C. $\{-\infty, -\pi \}$
    • D. $\{\pi , \infty \}$
    Answer, explanation and textbook page: open in the app.
  19. 19.Which one of the following is the solution set of $(2x - 1)^2 + 2 \leq 6x^2 - 3x$?
    • A. $[-1.5, 1]$
    • B. $[1.5, \infty]$
    • C. $(-\infty, -1.5] \cup [1, \infty)$
    • D. $(-\infty, -1.5] \cup [1, \infty)$
    Answer, explanation and textbook page: open in the app.
  20. 20.The line $y = x$ is intersected by the horizontal line $y = a$ at point $P$ and $y = b$ at point $Q$ for $a ≠ b$. Which of the following gives the distance from $Q$ to $P$?
    • A. $\sqrt{2}|b-a|$
    • B. $\sqrt{2}|b-a|$
    • C. $\sqrt{2}(b-a)$
    • D. $2\sqrt{|a-b|}$
    Answer, explanation and textbook page: open in the app.
  21. 21.Let $P(-\frac{1}{2}, -3)$ and $Q(4, -5)$ are points on the line $y = kx + b$. Which of the following is the value of $k$?
    • A. $-\frac{7}{16}$
    • B. $-\frac{9}{19}$
    • C. $-\frac{16}{7}$
    • D. $-\frac{9}{16}$
    Answer, explanation and textbook page: open in the app.
  22. 22.Which of the following is true about a trigonometric value of angles?
    • A. $\sin(270^\circ) = 0$
    • B. $\cos(-180^\circ) = -1$
    • C. $\tan(-90^\circ) = 0$
    • D. $\cot(540^\circ) = -1$
    Answer, explanation and textbook page: open in the app.
  23. 23.For $0^\circ \le \theta \le 90^\circ$, which of the following relationships describes the co-relationship of angle $\theta$?
    • A. $\cos \theta$ is equal to secant of its complementary angle
    • B. $\csc \theta$ is equal to sine of its supplementary angle
    • C. $\cot \theta$ is equal to tangent of its complementary angle
    • D. $\sec \theta$ is equal to sine of its supplementary angle
    Answer, explanation and textbook page: open in the app.
  24. 24.Given the linear system, $\begin{cases}a_1x+b_1y+c_1z=3\a_2x+b_2y+c_2z=-1\a_3x+b_3y+c_3z=2\end{cases}$, such that the determinant of the coefficient matrix is $\frac{1}{3}$. Which of the following is the value of y?
    • A. $\frac{1}{3}\begin{vmatrix}a_1&b_1&c_1\3&-1&2\a_3&b_3&c_3\end{vmatrix}$
    • B. $\frac{1}{3}\begin{vmatrix}a_1&3&c_1\a_2&-1&c_2\a_3&2&c_3\end{vmatrix}$
    • C. $-3\begin{vmatrix}a_1&b_1&c_1\-3&1&-2\a_3&b_3&c_3\end{vmatrix}$
    • D. $-3\begin{vmatrix}a_1&-3&c_1\a_2&1&c_2\a_3&-2&c_3\end{vmatrix}$
    Answer, explanation and textbook page: open in the app.
  25. 25.If $w=i-\frac{3+i}{5}$, then which of the following is equal to the modulus of $w$?
    • A. $\frac{3}{5}\sqrt{5}$
    • B. 1
    • C. $1-\frac{1}{5}\sqrt{10}$
    • D. $1+\frac{1}{5}\sqrt{10}$
    Answer, explanation and textbook page: open in the app.
  26. 26.Few terms of a sequence are given as follows: -2, 0, 4, 10, ... Which of the following can be a fifth term?
    • A. 14
    • B. 16
    • C. 18
    • D. 20
    Answer, explanation and textbook page: open in the app.
  27. 27.For the arithmetic sequence with $A_3=5$ and $A_5=10$, which of the following is the sum of the first 20 terms?
    • A. 475
    • B. 495
    • C. 570
    • D. 950
    Answer, explanation and textbook page: open in the app.
  28. 28.Suppose a and b are real numbers such that $a\ne 0$ and $b>0$. If $\sum_{n=1}^{\infty} b\left(\frac{a}{b}\right)^n$ converge, which of the following is true?
    • A. $|a|=b$
    • B. $|a|>b$
    • C. $a<b$
    • D. $|a|<b$
    Answer, explanation and textbook page: open in the app.
  29. 29.An enterprise found out its profit by the formula $f(t)=t^2-t+100$, t in days. Which of the following is the average rate of change of its profit on $0\le t\le 2$?
    • A. $\frac{f(0)-f(2)}{2}$
    • B. $\frac{f(2)-f(1)}{2}$
    • C. $\frac{f(2)-f(t)}{t-2}$
    • D. $\frac{1}{2}\left(f(2)-f(0)\right)$
    Answer, explanation and textbook page: open in the app.
  30. 30.For the purpose of road management, flow of vehicles needs to be regulated. On the average it was observed that the speed of vehicles in km/hr follow the formula $f(x)=10\left(x+\frac{50}{2x+1}\right)$ where $x$ is the speed in km/hr. Which of the following is the minimum speed in km/hr of a vehicle in the situation?
    • A. 45
    • B. 95
    • C. 105
    • D. 115
    Answer, explanation and textbook page: open in the app.
  31. 31.Which of the following gives the definition of integration for a function?
    • A. It is a process of reversing the derivative of the function.
    • B. It is the derivative of a function at a point.
    • C. It is the rate of change of a function.
    • D. It is the derivative of a function on the given interval
    Answer, explanation and textbook page: open in the app.
  32. 32.Given the grouped data: x | 8 - 12 | 13 - 17 | 18 - 22 | 23 - 27 | 28 - 32 f | 3 | 4 | 5 | 6 | 2 What is the standard deviation of the data?
    • A. 6
    • B. 6.5
    • C. $\sqrt{37}$
    • D. $\sqrt{37.5}$
    Answer, explanation and textbook page: open in the app.
  33. 33.Given two sets of data A and B with ranges 15 and 15 respectively, coefficient of range (CR) of A is 0.2 and CR of B is 0.21. Then which of the following is true?
    • A. The dispersion of A is slightly greater than that of B.
    • B. The dispersion of B is slightly greater than that of A.
    • C. Data A has the same dispersion as B even if CR varies.
    • D. No enough information to compare dispersion roughly.
    Answer, explanation and textbook page: open in the app.
  34. 34.In a homogeneous population where there is a categorical difference, items are selected randomly in each category. Which of the following describes the sampling method stated above?
    • A. Simple random sampling
    • B. Systematic sampling
    • C. Stratified sampling
    • D. Cluster sampling
    Answer, explanation and textbook page: open in the app.
  35. 35.Which one of the following is the minimum value of the following linear programming problem? Minimize: $Z = 3x + y$ Subject to: $y \ge -x + 4, y \ge 5x - 20, x \ge 1, y \le 5$
    • A. 12
    • B. 8
    • C. 4
    • D. 6
    Answer, explanation and textbook page: open in the app.
An agent planned to transport fertilizers in barrels and sacks from a port. The agent can transport 30 barrels and 50 sacks charging Birr 8,000 for a trip, and each car B can transport 45 barrels and 80 sacks charging Birr 12,000.
  1. 36.If there are at least 400 barrels and at most 1,500 sacks to be transported from the port, which of the z = 8, 000x + 12, 000y following is a constraint of the linear programming problem to minimize the cost of transportation given by ? ⎩ ⎨ ⎧6x + 9y ≥80 ⎩ ⎨ ⎧6x + 9y ≤80
    • B. 5x + 8y ≤150 x ≥0, y ≥0 5x + 8y ≥150 x ≥0, y ≥0 ⎩ ⎨ ⎧3x + 5y ≥40 ⎩ ⎨ ⎧3x + 5y ≤40
    • D. 9x + 16y ≤30 x ≥0, y ≥0 9x + 16y ≥30 x ≥0, y ≥0
    Answer, explanation and textbook page: open in the app.
  2. 37.A college graduate plans to deposit Birr 2000 each year starting from the first day of the year into a saving account that pays 8% interest compounded annually. How much will he/she have just after the first day of the third year?
    • A. Birr4492.2
    • B. Birr6332.8
    • C. Birr6492.8
    • D. Birr7012.2
    Answer, explanation and textbook page: open in the app.
Other Mathematics (Social Science) years: 2018 2017 2015 2014 2013