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Grade 12 Mathematics (Social Science) national exam, 2014 E.C. (2022 G.C.)

50 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.Let A = {10, 2, 4, 6, 8, 10} and B = {1, 3, 5, 7, 9} Which one of the following is true?
    • A. B - A = B
    • B. A - B = A
    • C. B - A = {1, 3, 5, 7}
    • D. A - B = {0, 2, 4, 6, 8, 10}
  2. 2.Consider the polynomial function f(x) = ax 4 - b, where a and b are real numbers. Which one of the following is true about f?
    • A. The graph of f opens upward for any b > 0
    • B. The graph of f opens upward for any a ≥ 0
    • C. The graph of f opens downward for any b ≤ 0
    • D. The graph of f opens downward for any a ≤ 0
  3. 3.The solution set of the inequality 3x² + 35x - 12 ≤ 0 is
    • A. (-12, 1/3)
    • B. (-∞, 1/3)
    • C. (-12, 1)
    • D. (-∞, -12)
  4. 4.Which one of the following is the equation of a circle with center (2, -1) and radius 3 units?
    • A. x² + y² + 2x + 2y - 4 = 0
    • B. x² + y² - 4x + 2y - 4 = 0
    • C. x² + y² + 4x - 2y - 5 = 0
    • D. x² + y² - 4x + 2y - 5 = 0
    Answer: B. A circle with center $(h,k)$ and radius $r$ has equation $(x-h)^2+(y-k)^2=r^2$. Here, $(x-2)^2+(y+1)^2=9$ expands to $x^2+y^2-4x+2y-4=0$, which is B. A and C have the wrong center; D has the right center but the wrong radius. Textbook: Grade 10, unit 7 · section 7.5.
  5. 5.Consider the ellipse given by the equation 4(x + 1)² + (y - 1)² = 16. Which one of the following is the standard form of the equation of the ellipse? + 42 (x+1)2 = 12 (y−1)2 1 + 42 (x+1)2 = 42 (y−1)2 1
    • B. + 22 (x+1)2 = 42 (y−1)2 1 + 22 (x+1)2 = 12 (y−1)2 1
    Answer: C. The key idea is to write the ellipse with $1$ on the right and squared terms divided by their denominators. Divide $4(x+1)^2+(y-1)^2=16$ by $16$: $(x+1)^2/4+(y-1)^2/16=1$, which is option C. A has the wrong first denominator. B has the wrong first denominator. D has the wrong second denominator. Textbook: Grade 12, unit 5 · section 5.2.
  6. 6.Which of the following is true? A hyperbola is a locus of points in the plane so that:
    • A. the sum of the distances from two fixed points is constant
    • B. the difference between the distances from two fixed points is constant
    • C. the points have the same distance from a given point and a given line
    • D. the points have the same distance from any given two points
    Answer: B. The key idea is that a hyperbola is defined using the distances from two fixed points, called its foci. The absolute difference of those two distances is constant. A describes an ellipse. C describes a parabola, where distance from a focus equals distance from a fixed line. D describes points equally distant from two fixed points. Textbook: Grade 10, unit 7 · section 7.1.
  7. 7.Which of the following is the converse of: "If wages increase, then inflation decreases."?
    • A. If inflation decreases, then wages increase
    • B. If inflation decreases, then wages decrease
    • C. If inflation increases, then wages decrease
    • D. If inflation increases, then wages increase
    Answer: A. The key idea is that the converse of an if-then statement swaps its condition and result. Swapping the parts gives: “If inflation decreases, then wages increase,” which is A. B is wrong because it says wages decrease; C is wrong because it says inflation increases; D is wrong because it says inflation increases. Textbook: Grade 10, unit 3 · section 3.2.
  8. 8.Let A = Which one of the following is equal to A+A T ? 5 ( 0 −4 2 ) (0 4 2)
    • B. −4 4 (0 5 1) ( 0 −5 1 )
    • D. 5 −5
    Answer: B. The key idea is that a matrix transpose swaps its rows and columns. Here, $A=\begin{pmatrix}0&-1\\5&1\end{pmatrix}$ and $A^T=\begin{pmatrix}0&5\\-1&1\end{pmatrix}$, so $A+A^T=\begin{pmatrix}0&4\\4&2\end{pmatrix}$, option B. A has negative off-diagonal entries. C has incorrect entries in the second column. D has negative off-diagonal entries and the wrong bottom-right entry. Textbook: Grade 11, unit 3 · section 3.2.
  9. 9.Consider a line with equation y = mx + b. Which of the following is true?
    • A. The line intersects the y axis, if m ≠ 0 and b > 0
    • B. The line intersects the x axis, if m = 0 and b > 0
    • C. The line has no slope
    • D. The line passes through the origin
    Answer: A. The key idea is that a line $y=mx+b$ meets the $y$-axis when $x=0$. Then $y=b$, so it meets the axis at $(0,b)$; when $b>0$, this point is above the origin, and A is true. B is wrong because when $m=0$ and $b>0$, the line is horizontal and does not meet the $x$-axis. C is wrong because the line has slope $m$. D is wrong because the line passes through the origin only when $b=0$. Textbook: Grade 11, unit 6 · section 6.3.
  10. 10.Which one of the following statements is NOT true about the graph of the given linear inequality?
    • A. The graph of the linear inequality y > 2x - 1 contains all points of the line y = 2x - 1
    • B. The graph of the linear inequality y ≤ 2x + 1 is below the line y = x on the interval (-∞, -1)
    • C. The graph of the linear inequality y > 2x + 1 y > 2x + 1 contains no points of the line y = 2x - 3
    • D. The graph of the linear inequality y ≤ x - 1 is above the line y = 2x on the interval (-∞, -1)
    Answer, explanation and textbook page: open in the app.
  11. 11.Let A = $A=\begin{pmatrix}0&-1\\5&1\end{pmatrix}$ Which one of the following is equal to A+A T ?
    • A. $\begin{pmatrix}0&-4\\-4&2\end{pmatrix}$
    • B. $\begin{pmatrix}0&4\\4&2\end{pmatrix}$
    • C. $\begin{pmatrix}0&5\\5&1\end{pmatrix}$
    • D. $\begin{pmatrix}0&-5\\-5&1\end{pmatrix}$
    Answer, explanation and textbook page: open in the app.
  12. 12.You happened to notice that the 8% sales tax that your father paid on the new Television Set came to Birr 3,800. How much did the Television Set cost him?
    • A. Birr 47,500
    • B. Birr 45,800
    • C. Birr 35,400
    • D. Birr 30,600
    Answer, explanation and textbook page: open in the app.
  13. 13.Which one of the following numbers is equal to $\frac{\sqrt{3}+\sqrt{2}}{\sqrt{3}-\sqrt{2}}$?
    • A. $6 + 2\sqrt{2}$
    • B. $5 + 2\sqrt{6}$
    • C. $1 + 2\sqrt{3}$
    • D. $3 - 2\sqrt{3}$
    Answer, explanation and textbook page: open in the app.
  14. 14.What is the solution set of the equation 6 - 4x - 2x 2 = 0?
    • A. {-6, 2}
    • B. {-3, 1}
    • C. {-1, 3}
    • D. {1, 3}
    Answer, explanation and textbook page: open in the app.
  15. 15.What is the area of an equilateral triangle inscribed in a circle of radius 4 units?
    • A. 6√3 sq. units
    • B. 12√3 sq. units
    • C. 12 sq. units
    • D. 36 sq. units
    Answer, explanation and textbook page: open in the app.
  16. 16.What is the simplified form of $\frac{\sqrt[3]{32}}{16^2}$?
    • A. 2
    • B. 1/2
    • C. 8
    • D. 1/8
    Answer, explanation and textbook page: open in the app.
  17. 17.The base of a right prism is a rectangle with sides 3 cm and 4 cm and the altitude of the prism is 5 cm. What is the total surface area of the prism?
    • A. 70 cm²
    • B. 82 cm²
    • C. 94 cm²
    • D. 102 cm²
    Answer, explanation and textbook page: open in the app.
  18. 18.Consider the graphs of the functions y = a x and y = log a x for a > 0 and a ≠ 1. Which of the following is true about the functions?
    • A. The graph of y = a x is a reflection of the graph of y = log a x about the origin.
    • B. The graph of y = a x is a reflection of the graph of y = log a x along y = -x.
    • C. For a > 1, the graphs of both y = a x and y = log a x are increasing on (0, ∞).
    • D. For 0 < a < 1, the y-axis is an asymptote of to the graphs of y = a x and y = log a x.
    Answer, explanation and textbook page: open in the app.
  19. 19.What is the solution set of the equation $3^{(x-x^2)}=\frac{1}{9}$?
    • A. {0, 1}
    • B. {-1, 0}
    • C. {1, -2}
    • D. {-1, 2}
    Answer, explanation and textbook page: open in the app.
  20. 20.Which of the following is true about the trigonometric values of $855^\circ$?
    • A. $\cos 855^\circ=-\frac{\sqrt{2}}{2}$
    • B. $\sin 855^\circ=\frac{\sqrt{3}}{2}$
    • C. $\tan 855^\circ=1$
    • D. $\cot 855^\circ=-\sqrt{2}$
    Answer, explanation and textbook page: open in the app.
  21. 21.Which one of the following is true about the function y = sin x?
    • A. Its domain is [-1, 1]
    • B. Its period is π
    • C. Its range is [-1, 1]
    • D. Its domain is (0, ∞)
    Answer, explanation and textbook page: open in the app.
  22. 22.Let . Which one of the following is the inverse of f? x−3 2x−1 f (x) = −1 x−3 −2x+1 f (x) = −1
    • B. x−2 f (x) = −1 x+2 f (x) = −1
    • D. −3x−1 3x+1
    Answer, explanation and textbook page: open in the app.
  23. 23.Let $\vec{u}$ be a vector of length 3.5, cm directed to the East and $\vec{v}$ be a vector of length 3cm directed to the north.Then $\vec{u} + \vec{v}$ is equals to a vector of length:
    • A. 3 cm in the direction East 60° North
    • B. 3 cm in the direction East 30° North
    • C. 6 cm in the direction East 60° North
    • D. 6 cm in the direction East 30° North
    Answer, explanation and textbook page: open in the app.
  24. 24.Below is a histogram showing the time taken to complete a test. Based on the above histogram, which of the following is true?
    • A. 30% of the students completed the test with less than 40 minutes.
    • B. 25% of the students completed the test with less than 40 minutes.
    • C. 50% of the students completed the test with less than 50 minutes.
    • D. 40% students completed the test with more than 40 minutes.
    Answer, explanation and textbook page: open in the app.
  25. 25.A bag contains 2 red, 3 white and 1 black balls. We draw two balls one after the other without replacement. What is the probability that the first ball is red and the second ball is black?
    • A. 1/10
    • B. 1/12
    • C. 1/15
    • D. 1/18
    Answer, explanation and textbook page: open in the app.
  26. 26.Consider the grouped frequency table for students' weight in a certain school. Weight (kg) Frequency 31 – 35 3 36 – 40 5 41 – 45 6 46 – 50 7 51 – 55 4 What is the mean weight of the student?
    • A. 41.5kg
    • B. 43kg
    • C. 43.8kg
    • D. 45.5kg
    Answer, explanation and textbook page: open in the app.
  27. 27.Which of the following is NOT true?
    • A. There is no single rule for determining the size of a sample.
    • B. Random sampling requires time to plan and implement.
    • C. Random sampling is free from personal bias of the investigator.
    • D. Any person can implement random sampling from a given population.
    Answer, explanation and textbook page: open in the app.
  28. 28.One special perfume costs a cosmetic shop Birr 2,100. If the shop desires a 25% markup rate based on the selling price, then which one of the following should be set as a selling price for this perfume?
    • A. Birr 2,800
    • B. Birr 2,600
    • C. Birr 2,400
    • D. Birr 2,300
    Answer, explanation and textbook page: open in the app.
  29. 29.Let $f(x) = \frac{3x + 1}{x + 2}$. Which one of the following is the inverse of f?
    • A. $f^{-1}(x) = \frac{2x - 1}{x - 3}$
    • B. $f^{-1}(x) = \frac{-2x + 1}{x - 3}$
    • C. $f^{-1}(x) = \frac{x - 2}{-3x - 1}$
    • D. $f^{-1}(x) = \frac{x + 2}{3x + 1}$
    Answer, explanation and textbook page: open in the app.
  30. 30.Suppose that the lines $\overline{AB}$ and \overline{BC} intersect at B outside the circle as shown in the figure below. If m(∠ABC) = 30° and m(∠BCD) = 45°, then what is $m(\overset{\triangle}{AC})$
    • A. 90°
    • B. 105°
    • C. 120°
    • D. 150°
    Answer, explanation and textbook page: open in the app.
  31. 31.Let f be a function that is continuous at x o . Which of the following is true about f?
    • A. x o does not belong to the domain of f.
    • B. lim x→x o f(x) does not exists.
    • C. lim x→x o + f(x) ≠ lim x→x o - f(x)
    • D. lim x→x o f(x o )
    Answer, explanation and textbook page: open in the app.
  32. 32.Let $f(x)=2\sqrt{x+1}$. Which one of the following indicates the interval and its derivative of f respectively?
    • A. $[0,\infty),\quad (x+1)^{\frac{3}{2}}$
    • B. $[-1,\infty),\quad \frac{1}{\sqrt{x+1}}$
    • C. $(-1,\infty),\quad \frac{1}{\sqrt{x+1}}$
    • D. $(-\infty,-1),\quad (x+1)^{-\frac{3}{2}}$
    Answer, explanation and textbook page: open in the app.
  33. 33.Let $f(x)=x^2+\sqrt{x}$ and $g(x)=e^{2x}$. Which of the following is true?
    • A. $(f+g)'(x)=2x+\frac{1}{2\sqrt{x}}+2e^{2x}$
    • B. $(f+g)'(x)=2x+\frac{1}{2}\cdot\sqrt{x}+2e^{2x}$
    • C. $(f+g)'(x)=2x-\frac{1}{2}\cdot\sqrt{x}-2e^{2x}$
    • D. $(f+g)'(x)=2x-\frac{1}{2\sqrt{x}}-2e^{2x}$
    Answer, explanation and textbook page: open in the app.
  34. 34.What is the equation of the tangent line to the graph of the function $f(x)=e^x$ at $(1,e)$?
    • A. $y=x+\frac{1}{e}$
    • B. $y=\frac{1}{e}x+\frac{1}{e}$
    • C. $y=ex+e$
    • D. $y=ex$
    Answer, explanation and textbook page: open in the app.
  35. 35.Let $A=\begin{pmatrix}a_1&b_1&c_1\\a_2&b_2&c_2\\a_3&b_3&c_3\end{pmatrix}$ and det(A)=3. Which one of the following is NOT true?
    • A. $\left| \begin{array}{ccc} a_1 & b_1 & c_1 \\ a_3 & b_3 & c_3 \\ a_2 & b_2 & c_2 \end{array} \right| = 3$
    • B. $\left| \begin{array}{ccc} 2a_1 & 2b_1 & 2c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3 \end{array} \right| = 6$
    • C. $\left| \begin{array}{ccc} -a_1 & b_1 & -c_1 \\ a_2 & b_2 & -c_2 \\ a_3 & b_3 & -c_3 \end{array} \right| = 3$
    • D. $\left| \begin{array}{ccc} a_1 & b_1 & b_1 \\ a_2 & b_2 & b_2 \\ a_3 & b_3 & b_3 \end{array} \right| = -3$
    Answer, explanation and textbook page: open in the app.
  36. 36.Which one of the following is equal to $\int (e^{2x}+\cos x)\,dx$?
    • A. $\frac{1}{2}e^{2x}+\sin x+C$
    • B. $\frac{1}{2}e^{2x}-\sin x+C$
    • C. $e^{2x}+\cos x+C$
    • D. $e^{2x}+\sin x+C$
    Answer, explanation and textbook page: open in the app.
  37. 37.Which one of the following is equal to $\int_1^3 2x\sqrt{x^2+1}\,dx$?
    • A. $\frac{4(10\sqrt{10}-2\sqrt{2})}{3}$
    • B. $\frac{2(5\sqrt{10}-\sqrt{2})}{3}$
    • C. $\frac{2(5\sqrt{10}+2\sqrt{2})}{3}$
    • D. $\frac{4(5\sqrt{10}-\sqrt{2})}{3}$
    Answer, explanation and textbook page: open in the app.
  38. 38.Consider the following data: 6, 6, 7, 7, 7, 8, 8. Then which of the following is true?
    • A. The mean deviation from the mode is 4/7
    • B. The mean deviation from the median is 4/7
    • C. The mean deviation from the mean is 4/7
    • D. The mean deviation from the mean is 3/2
    Answer, explanation and textbook page: open in the app.
  39. 39.What is the solution set of $\frac{2}{x-1}+\frac{4}{x}=\frac{6}{x-2}$?
    • A. $\left\{\frac{4}{11}\right\}$
    • B. $\left\{\frac{11}{4}\right\}$
    • C. $\left\{\frac{4}{5}\right\}$
    • D. $\left\{\frac{5}{4}\right\}$
    Answer, explanation and textbook page: open in the app.
  40. 40.A couple own a house that values Birr 2,000,000. The house added Birr 10,000 in the first year, Birr 10,800 in the second year, and so on. What is the value of the house at the end of 6 years?
    • A. Birr 2,012,000
    • B. Birr 2,054,000
    • C. Birr 2,060,000
    • D. Birr 2,064,000
    Answer, explanation and textbook page: open in the app.
  41. 41.Let p: 2 is a prime number and q: 3 is an odd number. Then the contrapositive and its truth value of the statement p ⇒ q are respectively:
    • A. ┐q ⇒ -p and True
    • B. ┐q ⇒ -p and False
    • C. ┐p ⇒ q and True
    • D. q ⇒ -p and False
    Answer, explanation and textbook page: open in the app.
  42. 42.The following table shows the test scores of 20 students in a class. Test Scores Frequency 0 - 2 3 3 - 5 5 6 - 8 5 9 - 11 3 12 - 14 4 What is the variance of the given data?
    • A. 16.2
    • B. 17.2
    • C. 18.45
    • D. 19.45
    Answer, explanation and textbook page: open in the app.
  43. 43.What is the solution of the system of linear equations $\begin{cases}2x+7y=-5\5x+11y=2\end{cases}$
    • A. $x=-\frac{5}{3}$ and $y=\frac{25}{21}$
    • B. $x=\frac{5}{3}$ and $y=-\frac{25}{21}$
    • C. $x=\frac{69}{13}$ and $y=-\frac{29}{13}$
    • D. $x=-\frac{69}{13}$ and $y=\frac{29}{13}$
    Answer, explanation and textbook page: open in the app.
  44. 44.What is the simplified form of ? − 8 7 i 8 3 − 2 5 i 2 5
    • B. + 2 5 i 4 5 − 2 5 i 4 5
    Answer, explanation and textbook page: open in the app.
  45. 45.Consider the sequence Which one of the following is true about the sequence?
    • A. 2 is its lower bound
    • B. 3/2 is its lower bound
    • C. 4 is its upper bound
    • D. 6/5 is its upper bound
    Answer, explanation and textbook page: open in the app.
  46. 46.Which one of the following is equal to $\lim_{x\to 0}\frac{x}{1-\sqrt{x+1}}$
    • A. 0
    • B. -2
    • C. -1
    • D. -1/2
    Answer, explanation and textbook page: open in the app.
  47. 47.Consider the function f(x) = x 2 - 3x on [-1, 2]. What is the value x that satisfies the Mean Value Theorem?
    • A. 0
    • B. 1
    • C. 1/2
    • D. 3/2
    Answer, explanation and textbook page: open in the app.
  48. 48.A carpenter wants to design a closed box that has a square base and total surface area of 54 cm 2 . What is the maximum volume that can be constructed with the given total surface area?
    • A. 12 cm 3
    • B. 13.5 cm 3
    • C. 18 cm 3
    • D. 27 cm 3
    Answer, explanation and textbook page: open in the app.
  49. 49.What is the area of the region bounded by the line $y=x$ and the parabola $y=x^2-2x$ on the interval $[0,3]$?
    • A. 9 square units
    • B. $9/2$ square units
    • C. 3 square units
    • D. 2 square units
    Answer, explanation and textbook page: open in the app.
  50. 50.A painter buys 5 liters of paint for Birr 36 and 6 liters of the same brand paint for Birr 43. Which of the following is true?
    • A. Buying 5 liters is more economical than buying 6 liters.
    • B. Buying 6 liters is more economical than buying 5 liters.
    • C. Both have the same economical purchase.
    • D. We cannot compare.
    Answer, explanation and textbook page: open in the app.
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