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

57 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.Consider the data: 11, 13, 15, 17, 19, 21. Which one of the following is the mean deviation from the mean?
    • A. 4
    • B. 3.5
    • C. 4.5
    • D. 3
    Answer: D. Mean deviation is the average distance of the data values from their mean. The mean is $(11+13+15+17+19+21)/6=16$; the absolute deviations add to $5+3+1+1+3+5=18$, so the mean deviation is $18/6=3$. A gives $4$, not the average distance. B gives $3.5$, which is too large. C gives $4.5$, not the mean deviation. Textbook: Grade 12, unit 3 · section 3.1.
  2. 2.Consider the quadrilateral ABCD where $A=(0,0)$, $B=(-1,1)$, $C=(0,2)$ and $D=(2,2)$. Which one of the following is true?
    • A. CD is perpendicular to AB.
    • B. BC is parallel to AD.
    • C. CD is perpendicular to BE.
    • D. AB is parallel to CD.
    Answer: B. Parallel lines have equal slopes. The slope of $BC$ is $\frac{2-1}{0-(-1)}=1$, and the slope of $AD$ is $\frac{2-0}{2-0}=1$, so B is true. A and D are wrong because $CD$ is horizontal while $AB$ is neither horizontal nor vertical; C refers to point E, which is not defined. Textbook: Grade 10, unit 7 · section 7.4.
  3. 3.Consider the greatest integer function defined by $f(x)=[x-1]$. Which of the following sets is the domain of $f$?
    • A. The set of rational numbers.
    • B. The set of integers.
    • C. The set of real numbers.
    • D. The set of natural numbers.
    Answer: C. The key idea is that the greatest integer function is defined for every real input. Since $x-1$ is real for every real $x$, the domain is the set of real numbers. A is wrong because it leaves out irrational inputs. B is wrong because it leaves out non-integer inputs. D is wrong because it leaves out negative and non-natural inputs. Textbook: Grade 11, unit 1 · section 1.3.
  4. 4.Which one of the following relations is a function?
    • A. $R=\{(x,y): x,y\in\mathbb{R} \text{ and } y^2=x\}$
    • B. $R=\{(x,y): x,y\in\mathbb{R} \text{ and } |y|=x\}$
    • C. $R=\{(x,y): x,y\in\mathbb{R} \text{ and } y=x^2\}$
    • D. $R=\{(x,y): x,y\in\mathbb{R} \text{ and } y^4=x\}$
    Answer: C. A relation is a function if each input has exactly one output. In C, every real $x$ gives the single value $y=x^2$. A gives two possible $y$ values for positive $x$, and B gives two for positive $x$. D also gives two possible $y$ values for positive $x$. Textbook: Grade 10, unit 1 · section 1.2.
  5. 5.Which one of the following is a source of cash loan?
    • A. Non-government organization to alleviate public problems.
    • B. Investors involved in home construction for a community.
    • C. Group of investors committed to drilling out underground water.
    • D. A money line group investing in health improvement sectors.
  6. 6.What is the average rate of change of $y=f(x)$ with respect to $x$ on the interval $[a,b]$?
    • A. $\frac{f(b)+f(a)}{2}$
    • B. $\frac{b-a}{f(b)-f(a)}$
    • C. $\frac{f(b)-f(a)}{2}$
    • D. $\frac{f(b)-f(a)}{b-a}$
    Answer: D. The key idea is that average rate of change is the change in the function value divided by the change in input. On $[a,b]$, it is $\frac{f(b)-f(a)}{b-a}$. A gives an average of function values, not a rate of change. B reverses the numerator and denominator. C divides by $2$ instead of by the interval length. Textbook: Grade 12, unit 2 · section 2.1.
  7. 7.Which one of the following sets is the solution of $\log_3 x + \log_3 (x - 2) = 1$?
    • A. $\{-1,3\}$
    • B. $\{3\}$
    • C. $\{2\}$
    • D. $\{0,2\}$
    Answer, explanation and textbook page: open in the app.
  8. 8.Given the functions and , for and . Which of the following statements is true about the graph of and ?
    • A. The graph of f is an upward parabola, and the graph of g is obtained from that of f by shifting it downward by cc units.
    • B. The graph of f is a downward parabola, and the graph of g is obtained from that of f by shifting it to the right by cc units.
    • C. The graph of f is an upward parabola, and the graph of g is obtained from that of f by shifting it upward by cc units.
    • D. The graph of f is a downward parabola, and the graph of g is obtained from that of f by shifting it upward by cc units.
    Answer, explanation and textbook page: open in the app.
  9. 9.Consider the test results of 20 students in a given class. 13 14 14 15 19 18 13 15 16 16 13 14 14 15 19 18 13 15 16 16 17 18 15 13 14 16 14 18 16 16 Which one of the following numbers is the mode of the above data?
    • A. 16
    • B. 15
    • C. 14
    • D. 18
    Answer, explanation and textbook page: open in the app.
  10. 11.Which one of the following equations represents a line that passes through the points $(2, -1)$ and $(-3, 4)$?
    • A. $x + y - 1 = 0$
    • B. $x + y + 1 = 0$
    • C. $x - y - 1 = 0$
    • D. $x - y + 1 = 0$
    Answer, explanation and textbook page: open in the app.
  11. 12.What is the coefficient of in the expansion of
    • A. 35
    • B. 70
    • C. 560
    • D. -280
    Answer, explanation and textbook page: open in the app.
  12. 13.Which one of the following is NOT a power function?
    • A. $f(x)=x^3$
    • B. $f(x)=3x$
    • C. $f(x)=4x^{-6}$
    • D. $f(x)=7x^{-1/3}$
    Answer, explanation and textbook page: open in the app.
  13. 14.Given the problem: Maximize $Z=3x+y$ subject to: \[\begin{cases}x+y\ge 2\y+2x\le 4\y\ge 1,\ x\ge 0\end{cases}\] Which one of the following is the maximum value of $Z$?
    • A. 2
    • B. 5.5
    • C. 4
    • D. 5.8
    Answer, explanation and textbook page: open in the app.
  14. 15.Let $r_1$ and $r_2$ be the roots of the quadratic equation $2x^2+3x-4=0$. Which one of the following is equal to $\frac{1}{r_1^2}+\frac{1}{r_2^2}$?
    • A. 3/4
    • B. 25/16
    • C. 3/16
    • D. 1/4
    Answer, explanation and textbook page: open in the app.
  15. 16.Consider the matrix $A=\begin{pmatrix}-1&2&1\3&4&2\1&1&5\end{pmatrix}$. Which of the following statements is true about the minor of the entries of A?
    • A. The cofactor of 3 is 9.
    • B. The cofactor of 4 is -6.
    • C. The minor of 4 is 6.
    • D. The minor of 3 is -9.
    Answer, explanation and textbook page: open in the app.
  16. 17.In the figure below, let m ( )=15° and m(arc BC)=70°. What is the measure of arc CD? [Figure: A circle with center O is shown. Point A lies outside the circle to the right. A secant line passes through point A, intersects the circle at point D, and continues to point B on the upper left of the circle. A tangent line starts from point A and touches the bottom of the circle at point C. Angle DAC at vertex A is marked as 15 degrees. The arc between B and C contains points D along the secant line.]
    • A. 42.5°
    • B. 55°
    • C. 30°
    • D. 40°
    Answer, explanation and textbook page: open in the app.
Consider the right-angled triangle below, where and . Which one of the following is the length of BD? [Figure: A right-angled triangle labeled ΔABC with the right angle at vertex B. The vertices are labeled A at the top left, B at the bottom left, and C at the bottom right. A line segment is drawn from vertex B perpendicularly to the hypotenuse AC, meeting AC at a point labeled D.
  1. 18.There is a right-angle symbol at vertex B and another right- angle symbol where the line segment BD meets the hypotenuse AC at point D.]
    • A. 5.76cm
    • B. 2.4cm
    • C. 5cm
    • D. 7cm
    Answer, explanation and textbook page: open in the app.
  2. 19.Which one of the following is true about definite integrals?
    • A. $\int_a^b f(x)\,dx=0,\ a\ne0$
    • B. $\int_a^b(f(x)+g(x))\,dx=\int_a^b f(x)\,dx+\int_a^b g(x)\,dx$
    • C. $\int_a^b f(x)\,dx=\int_a^b g(x)\,dx$
    • D. $\int_a^b(f(x)-g(x))\,dx=\int_a^b(g(x)-f(x))\,dx$
    Answer, explanation and textbook page: open in the app.
  3. 20.Which of the following statements defines a circle?
    • A. It is the set of points in a plane formed by the intersection of two secants in the plane.
    • B. It is the set of points in a space formed by the intersection of two secants in the space.
    • C. It is the set of points in a space each of which is equidistant from a fixed point in the space.
    • D. It is the set of points in a plane each of which is equidistant from a fixed point in the plane.
    Answer, explanation and textbook page: open in the app.
  4. 21.What is the transpose of the matrix $M=\begin{pmatrix}1&2&1\0&-1&3\2&1&5\end{pmatrix}$?
    • A. $\begin{pmatrix}1&0&2\2&-1&1\1&3&5\end{pmatrix}$
    • B. $\begin{pmatrix}1&2&1\2&-1&3\0&1&5\end{pmatrix}$
    • C. $\begin{pmatrix}2&1&5\0&-1&3\1&2&1\end{pmatrix}$
    • D. $\begin{pmatrix}1&2&2\0&-1&1\1&3&5\end{pmatrix}$
    Answer, explanation and textbook page: open in the app.
  5. 22.Which one of the following sets is the domain of the rational function $f(x)=\dfrac{x^3+x^2}{x^4+2x^2-3}$?
    • A. $\mathbb{R}\setminus\{1\}$
    • B. $\mathbb{R}\setminus\{-1,1\}$
    • C. $\mathbb{R}\setminus\{-3,1\}$
    • D. $\mathbb{R}\setminus\{-1,1,-3\}$
    Answer, explanation and textbook page: open in the app.
  6. 23.Which of the following statements is true in borrowing money?
    • A. The borrower only pays back the principal amount.
    • B. The lender assumes no risk when giving out a loan.
    • C. Borrowing money involves an agreement to repay the principal amount along with accrued interestz
    • D. Collateral is always required for any type of financial borrowing.
    Answer, explanation and textbook page: open in the app.
  7. 24.Let f(x)=2x+1/x–1. Which of the following equations is the equation of the tangent line to the graph of f at (2,f(2))?
    • A. 3x+y=11
    • B. y–3x=–1
    • C. x+3y=17
    • D. 3x–y=13
    Answer, explanation and textbook page: open in the app.
  8. 25.What is the measure of each interior angle of a regular polygon with 15 sides?
    • A. 144°
    • B. 150°
    • C. 156°
    • D. 162°
    Answer, explanation and textbook page: open in the app.
  9. 26.Let and . What is the derivative of (f+g)(x)? 12x − 2 2x + 4 4x − 3 x + 2 4x 12x − 2 x + 4 9x − 2 2x + 4
      Answer, explanation and textbook page: open in the app.
    • 27.Let $A=\{x\in\mathbb{R}:x^3-x=0\}$and $B=\{x\in\mathbb{Z}:-1\leq x\}$. What is the relative complement of B with respect to A (i.e., $A\backslash B$)?
      • A. $\{2\}$
      • B. $\{0,1\}$
      • C. $\emptyset$
      • D. $\{-1,0,1\}$
      Answer, explanation and textbook page: open in the app.
    • 28.Given the set B={ ,{–2,0}}, what is the power set of B?
      • A. { ,{–2,0}}
      • B. { ,{ },{{–2,0}},{ ,{–2,0}}} ∅ ∅∅ ∅
      • C. {{ },{{–2,0}}}
      • D. { ,{–2},{0},{–2,0}} ∅ ∅
      Answer, explanation and textbook page: open in the app.
    • 29.What is the simplified form of $\log_{16}(4)-\log_2(\sqrt{12.5})+\log_{0.5}(2)$?
      • A. $\log_2(5)$
      • B. $1-\log_2(5)$
      • C. $-\log_2(5)$
      • D. $0$
      Answer, explanation and textbook page: open in the app.
    • 30.Which statement is true about the graphs of the exponential functions f(x)= and g(x)= where a>1?
      • A. They are reflections of each other across the x-axis.
      • B. They are reflections of each other across the y-axis.
      • C. They never intersect each other.
      • D. Both functions are monotonically increasing.
      Answer, explanation and textbook page: open in the app.
    • 31.Which of the following statements is NOT true about the mean of a given data set?
      • A. It is highly affected by extreme values (outliers).
      • B. The sum of the deviations of the items from the mean is always zero.
      • C. It must always be one of the actual data values present in the set.
      • D. It uses every value in the data set for its calculation.
      Answer, explanation and textbook page: open in the app.
    • 32.Which one of the following numbers is an irrational number?
      • A. $3.14159$ (terminating)
      • B. $\sqrt{16}$
      • C. $22/7$
      • D. $\pi$
      Answer, explanation and textbook page: open in the app.
    • 33.What is the area of the parallelogram spanned by the vectors $\vec{u}=i-2j+k$ and $\vec{v}=2i+2j-k$
      • A. $\sqrt{14}$
      • B. $3\sqrt{5}$
      • C. 9
      • D. $\sqrt{35}$
      Answer, explanation and textbook page: open in the app.
    • 34.What is the fundamental period of the function $f(x)=\sin(x)$?
      • A. $\pi/2$
      • B. $\pi$
      • C. $2\pi$
      • D. $4\pi$
      Answer, explanation and textbook page: open in the app.
    • 35.What is the simplified form of the expression ( ?
      • A. 16
      • B. -16
      • C. 8
      • D. -8
      Answer, explanation and textbook page: open in the app.
    • 36.In statistical sampling, if a population is divided into homogeneous groups and a random sample is drawn from each group, this method is known as:
      • A. Simple random sampling
      • B. Cluster sampling
      • C. Stratified sampling
      • D. Systematic sampling
      Answer, explanation and textbook page: open in the app.
    • 37.A mathematics exam contains 10 multiple-choice questions and 3 short-answer questions. According to the multiplication principle, how many ways can a student select exactly one multiple-choice question and exactly one short-answer question to solve first?
      • A. 13
      • B. 30
      • C. $10^3$
      • D. $\binom{13}{2}$
      Answer, explanation and textbook page: open in the app.
    • 38.A translation vector maps the origin (0,0) to the point (-2,4). Under this same translation, what is the image of the point (3,1)?
      • A. (5,-3)
      • B. (1,5)
      • C. (-6,4)
      • D. (-5,3)
      Answer, explanation and textbook page: open in the app.
    • 39.If the initial point of a vector is $A(2,-1,4)$ and the terminal point is $B(4,3,2)$, what is the position vector $\overrightarrow{AB}$?
      • A. $\langle -2,-4,2\rangle$
      • B. $\langle 6,2,6\rangle$
      • C. $\langle 2,4,-2\rangle$
      • D. $\langle 8,-3,8\rangle$
      Answer, explanation and textbook page: open in the app.
    • 40.Let vector have an initial point at (1,2)and a terminal point at (-3,6). What are the components of the vector ?
      • B. ⟨2, −2⟩ ⟨−2, 2⟩
      • D. \langle 1,-1\rangle ⟨4, −4⟩
      Answer, explanation and textbook page: open in the app.
    • 41.Consider the rational function f(x)= -4/x-2. The domain of f is the set of all real numbers EXCEPT:
      • A. -2
      • B. 2
      • C. 0
      • D. The domain is all real numbers.
      Answer, explanation and textbook page: open in the app.
    • 42.Which of the following properties of matrices and scalar multiplication r is TRUE for any two matrices A and B of the same dimension?
      • A. r(AB)=(rA)(rB)
      • B. r(A+B)=rA+rB
      • D. det(A+B)=det(A)+det(B) (A + B) = T A − T BT
      Answer, explanation and textbook page: open in the app.
    • 43.If AA is a 3×3 square matrix, which of the following is always true about its transpose AT? det(A ) = T −det(A) det(A ) = T 1/det(A)
      • B. det(A ) = T det(A) A = T A−1
      Answer, explanation and textbook page: open in the app.
    • 44.Let AA be a 3×3 invertible matrix with determinant |A|=4. What is the determinant of ?
      • A. 1/2
      • B. 2
      • C. 8
      • D. 1/4
      Answer, explanation and textbook page: open in the app.
    • 45.What is the solution set for the radical equation $\sqrt{x^2+3x}=2$?
      • A. {1,-4}
      • B. {1,4}
      • C. {-1,4}
      • D. {-1,-4}
      Answer, explanation and textbook page: open in the app.
    • 46.Given the exponential functions and , which statement describes their monotonicity?
      • A. Both functions are increasing.
      • B. Both functions are decreasing.
      • C. f(x) is decreasing and g(x) is increasing.
      • D. f(x)is increasing and g(x)is decreasing.
      Answer, explanation and textbook page: open in the app.
    • 47.In an arithmetic progression, the first term is 11 and the common difference is –13. What is the sum of the first 7 terms?
      • A. -28
      • B. -196
      • C. -67
      • D. 196
      Answer, explanation and textbook page: open in the app.
    • 48.In a geometric sequence, the first term is and the fifth term is (assume the common ratio is positive). What is the value of the 13th term ( )?
      • A. 32
      • B. 64
      • C. 128
      • D. 256
      Answer, explanation and textbook page: open in the app.
    • 49.The infinite repeating decimal 0.373737... can be expressed as which rational fraction in lowest terms?
      • A. 37/100
      • B. 37/99
      • C. 37/90
      • D. 1/3
      Answer, explanation and textbook page: open in the app.
    • 50.According to the properties of integration, which of the following is equivalent to $\int (3f(x)-g(x))\,dx$?
      • A. $3\int f(x)\,dx\cdot\int g(x)\,dx$
      • B. $\int 3f(x)\,dx-\int g(x)\,dx$
      • C. $3\left(\int f(x)\,dx-\int g(x)\,dx\right)$
      • D. $\int f(x)\,dx-3\int g(x)\,dx$
      Answer, explanation and textbook page: open in the app.
    • 51.Three vertices of a parallelogram ABCD are A(0,0), B(5,0), and C(7,4). If AD is parallel to BC, what are the coordinates of vertex D?
      • A. (2,4)
      • B. (-2,4)
      • C. (12,4)
      • D. (2,-4)
      Answer, explanation and textbook page: open in the app.
    • 52.If $\alpha$ and $\beta$ are complementary angles ($\alpha + \beta = 90^\circ$), which of the following trigonometric relations is always true?
      • A. $\sin(\alpha) = \sin(\beta)$
      • B. $\tan(\alpha) = \tan(\beta)$
      • C. $\sin(\alpha) = \cos(\beta)$
      • D. $\sec(\alpha) = \cos(\beta)$
      Answer, explanation and textbook page: open in the app.
    • 53.In descriptive statistics, the third decile ($D_3$) corresponds to which percentile?
      • A. 3rd percentile
      • B. 30th percentile
      • C. 33rd percentile
      • D. 70th percentile
      Answer, explanation and textbook page: open in the app.
    • 54.Which one of the following statements defines a matrix?
      • A. A matrix is a rectangular array of numbers.
      • B. A matrix is a vertical array of letters.
      • C. A matrix is a horizontal array of letters.
      • D. A matrix is a circular array of numbers.
      Answer, explanation and textbook page: open in the app.
    • 55.Which one of the following is the decomposition of the rational expression as a sum of partial fractions? 2 3 (x+1)2 1 3 (x+1)2 3
      • B. + x+2 + x+1 − x+2 1 2 (x+1)2 3 5 (x+1)2 4
      • D. + x+2 − x+1 + x+2
      Answer, explanation and textbook page: open in the app.
    • 56.A farmer having a total capital of Birr 400,000 bought oxen and sheep. An ox costs Birr 60,000 and a sheep Birr 5,000. He sold each ox at Birr 65,000 and each sheep at Birr 6,000. Which one of the following is a representation of the problem in order to maximize his profit, assuming uniform prices where x is the number of oxen and y is the number of sheep? x + y ≤ 35, 12x + y ≤80, x ≥0, y ≥0 x + y ≤ 35, 12x + y ≤80, x ≥0, y ≥0
      • A. Z(x,y)=6000x+5000y subject to
      • B. Z(x,y)=60000x+5000y subject to
      • C. Z(x,y)=5000x+1000ysubject to
      • D. x + y ≤35, 12x + y ≤800, x ≥0, y ≥0 x + y ≤ 35, 12x + y ≤80, x ≥0, y ≥0 Z(x,y)=5000x+1000ysubject to
      Answer, explanation and textbook page: open in the app.
    • 57.Which one of the following statements is true about the given numbers and points on the number line?
      • A. There is a one-to-one correspondence between the set of natural numbers and points on the number line.
      • B. There is a one-to-one correspondence between the set of rational numbers and points on the number line.
      • C. There is a one-to-one correspondence between the set of irrational numbers and points on the number line.
      • D. There is a one-to-one correspondence between the set of real numbers and points on the number line.
      Answer, explanation and textbook page: open in the app.
    • 58.A spherical ball has diameter 30cm. What is its surface area (in )?
      • A. 3600
      • B. 1000 π π
      • C. 900
      • D. 600 π π
      Answer, explanation and textbook page: open in the app.
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