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

53 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.What is the locus all points in the plane in which the sum of the distances from two fixed points is constant?
    • A. Hyperbola
    • B. Parabola
    • C. Ellipse
    • D. Circle
    Answer: C. The locus is the set of points that meet the stated distance rule. Points whose distances from two fixed points add to a constant form an ellipse. A hyperbola uses a constant difference of distances; a parabola uses a focus and a directrix; a circle uses a constant distance from one fixed point. Textbook: Grade 10, unit 7 · section 7.5.
  2. 2.The radian measure of an angle of 120° is equal to:
    • A. $\frac{2\pi}{3}$
    • B. $\frac{4\pi}{3}$
    • C. $\frac{\pi}{2}$
    • D. $\frac{\pi}{4}$
    Answer: A. The key idea is to convert degrees to radians using $180^\circ=\pi$ radians. Thus $120^\circ\times\frac{\pi}{180^\circ}=\frac{2\pi}{3}$. B, C, and D are not equal to this radian measure. Textbook: Grade 10, unit 4 · section 4.1.
  3. 3.What is the surface area of a sphere with radius 6 cm?
    • A. 24π cm²
    • B. 36π cm²
    • C. 48π cm²
    • D. 144π cm²
    Answer: D. The surface area of a sphere depends on its radius. Using $S=4\pi r^2$, we get $S=4\pi(6)^2=144\pi\,\mathrm{cm}^2$. A, B, and C do not equal this surface area. Textbook: Grade 10, unit 6 · section 6.2.
  4. 4.Which of the following is the measure of an exterior angle of a 10 sided regular polygon?
    • A. 72°
    • B. 144°
    • C. 180°
    • D. 36°
    Answer: D. The exterior angles of a regular polygon add up to $360^\circ$. Divide by the $10$ equal angles: $360^\circ\div10=36^\circ$. $72^\circ$ is too large; $144^\circ$ is an interior angle; $180^\circ$ is not the measure of one exterior angle. Textbook: Grade 9, unit 6 · section 6.4.
  5. 5.If the volume of a right circular cone is 64π cm 3 and the diameter of its base is 8 cm, what is the height of the cone?
    • A. 4cm
    • B. 12cm
    • C. 8cm
    • D. 3cm
    Answer: B. The key idea is to use the cone volume formula and the base radius. The diameter is $8\,\mathrm{cm}$, so $r=4\,\mathrm{cm}$; then $64\pi=\frac{1}{3}\pi(4)^2h$, giving $h=12\,\mathrm{cm}$. Option A is too short; option C does not satisfy the volume formula; option D is also too short. Textbook: Grade 10, unit 6 · section 6.3.
  6. 6.What are the quotient and remainder respectively, when 75 is divided by 20?
    • A. 3 and 15
    • B. 4 and 5
    • C. 15 and 3
    • D. 2 and 19
    Answer, explanation and textbook page: open in the app.
  7. 7.In rolling a fair die, what is the probability of obtaining 3 or 5?
    • A. $1/3$
    • B. $1/4$
    • C. $1/6$
    • D. $1/2$
    Answer, explanation and textbook page: open in the app.
  8. 8.Which one of the following is true?
    • A. The range of arctangent function is [-π, π]
    • B. The range of arcsine function is (-∞, ∞)
    • C. The domain of arccosine function is [-1, 1]
    • D. The domain of arcsine function is [0, π]
    Answer, explanation and textbook page: open in the app.
  9. 9.The angle of depression of the top of a flag pole from the top of a building that is 40.3 m away from the flag pole is 30°. If the height of the flag pole is 10m, what is the height of the building?
    • A. 20.3 m
    • B. 10(4/√3 + 1) m
    • C. 50 m
    • D. 40 m
    Answer, explanation and textbook page: open in the app.
  10. 10.A −B Given matrices and . Which one of the following is ?
    • B. 1 −3 3 −4 5 2 2 1 −3 −3 −4 −5 2 2 9 1 9 1
    • D. 1 −3 −9 3 −4 5 −2 1 −3 −9 −3 −4 −1 5 2 −2 1 2
    Answer, explanation and textbook page: open in the app.
  11. 11.Let R = {(x, y) | x² + y² + 1 and y ≤ 5} be a relation. Then which one of the following defines the inverse of R?
    • A. {(x, y) | x² + y² + 1, x ≤ 5}
    • B. {(x, y) | x² + y² + 1, x ≤ 5}
    • C. {(x, y) | x² + y² - 1, x ≤ 5}
    • D. {(x, y) | x² - y² + 1, x ≤ 5}
    Answer, explanation and textbook page: open in the app.
  12. 12.Given matrices \( A = \begin{pmatrix} 2 & 0 & 5 \\ 3 & 1 & 4 \\ 0 & 6 & -2 \end{pmatrix} \) and \( B = \begin{pmatrix} 1 & 3 & 0 \\ 6 & 5 & 2 \\ 9 & 7 & 0 \end{pmatrix} \). Which one of the following is \( A - B \)?
    • A. $$\begin{pmatrix} 1 & 3 & 5 \\ -3 & -4 & 2 \\ 9 & 1 & 2 \end{pmatrix}$$
    • B. $$\begin{pmatrix} 1 & -3 & -5 \\ -3 & -4 & 2 \\ 9 & 1 & 2 \end{pmatrix}$$
    • C. $$\begin{pmatrix} 1 & 3 & 5 \\ -3 & -4 & -2 \\ -9 & 1 & 2 \end{pmatrix}$$
    • D. $$\begin{pmatrix} 1 & -3 & 5 \\ -3 & -4 & 2 \\ -9 & -1 & -2 \end{pmatrix}$$
    Answer, explanation and textbook page: open in the app.
  13. 13.Which one of the following is the derivative of f(x) = tanx + 3 x ?
    • A. sec²x + 3 x lnx
    • B. -csc²x + 3 x lnx
    • C. -tanx + 3 x lnx
    • D. sec²x + 3 x
    Answer, explanation and textbook page: open in the app.
  14. 14.Which one of the following points is, inside the sphere with equation $x^2+y^2+z^2=5$?
    • A. $(1,-2,2)$
    • B. $(1,1,1)$
    • C. $(1,2,3)$
    • D. $(0,-2,3)$
    Answer, explanation and textbook page: open in the app.
  15. 15.Which one of the following gives the polar form of the complex number $z=2-2\sqrt{3}i$?
    • A. $4\left(\cos\frac{5}{3}\pi+i\sin\frac{5}{3}\pi\right)$
    • B. $4\left(\cos\frac{5}{3}\pi+i\sin\frac{5}{3}\pi\right)$
    • C. $4\left(\cos\frac{3}{4}\pi+i\sin\frac{3}{4}\pi\right)$
    • D. $4\left(\cos\frac{1}{3}\pi+i\sin\frac{1}{3}\pi\right)$
    Answer, explanation and textbook page: open in the app.
  16. 16.What is the product of the two complex number z = 2 - 3i and w = 5 + 2i?
    • A. 16 + 11i
    • B. 4 - 11i
    • C. 16 - 11i
    • D. 29
    Answer, explanation and textbook page: open in the app.
  17. 17.Which one of the following gives the polar form of the complex number \( z = 2 - 2\sqrt{3}i \)?
    • A. \( 4 \left( \cos \frac{5}{3} \pi + i \sin \frac{5}{3} \pi \right) \)
    • B. \( 4 \left( \cos \frac{5}{3} \pi + i \sin \frac{5}{3} \pi \right) \)
    • C. \( 4 \left( \cos \frac{3}{4} \pi + i \sin \frac{3}{4} \pi \right) \)
    • D. \( 4 \left( \cos \frac{1}{3} \pi + i \sin \frac{1}{3} \pi \right) \)
    Answer, explanation and textbook page: open in the app.
  18. 18.If $\vec{a} = (4, 3, 2)$ and $\vec{b} = (1, 2, -3)$ are two vectors in space, then which one of the following is NOT correct?
    • A. \( \vec{a} \to \vec{b} = (5,5-1) \)
    • B. \( \vec{a} + \vec{2b} = (-6,7,-4) \)
    • C. \( \vec{a} - \vec{b} = (-3,-1,-5) \)
    • D. \( \vec{a} - \vec{2b} = (2,1,8) \)
    Answer, explanation and textbook page: open in the app.
  19. 19.The integral x 1 −3x + 3 e−3x − 3 e−3x
    • B. + x 5 C 2 + x 5 C + x + x 2 2 − x − 3 e−3x + x 5 C 2 − x + 3 e−3x
    • D. + x 5 C
    Answer, explanation and textbook page: open in the app.
  20. 20.Let f be a continuous function on [a, b]. Then which of the following is NOT true?
    • A. b a f(x)dx = ∫ − f(x)dx ∫ a b
    • B. [\int_{a}^{b} k f(x) \, dx = k \int_{a}^{b} f(x) \, dx \quad \text{for any constant } k]
    • C. [\int_{a}^{b} f(x) \, dx = \int_{a}^{c} f(x) \, dx - \int_{c}^{b} f(x) \, dx \quad \text{for any } c \text{ in } (a,b).]
    • D. c f(x)dx = ∫ 0 for any c in (a, b). c
    Answer, explanation and textbook page: open in the app.
  21. 21.The simplified form of the expression $\frac{6\sqrt{20}-3\sqrt{45}}{3\sqrt{75}}$ is:
    • A. $\sqrt{3}$
    • B. $\frac{\sqrt{15}}{15}$
    • C. 1/5
    • D. 5
    Answer, explanation and textbook page: open in the app.
  22. 22.The integral \( \int \left( \frac{1}{\sqrt{x}} - e^{-3x} + \frac{5}{x^2} \right) \, dx \text{ is equal to:} \)
    • A. \( 2\sqrt{x} + \frac{e^{-3x}}{3} + \frac{5}{x} + C \)
    • B. \( 2\sqrt{x} + \frac{e^{-3x}}{3} - \frac{5}{x} + C \)
    • C. \( 2\sqrt{x} - \frac{e^{-3x}}{3} - \frac{5}{x} + C \)
    • D. \( 2\sqrt{x} - \frac{e^{-3x}}{3} + \frac{5}{x} + C \)
    Answer, explanation and textbook page: open in the app.
  23. 23.The graph of a certain relation R is represented by the shaded region shown on the figure below. Which one of the following pairs of sets respectively gives the domain and range of this relation?
    • A. {x/x ≤ 2} and {y/y ≤ 6}
    • B. {x/x ≤ 2} and R
    • C. R and {y/y ≤ 2}
    • D. R and R
    Answer, explanation and textbook page: open in the app.
  24. 24.Which of the following pairs of events are dependent?
    • A. Tossing two coins simultaneously and obtaining head from one and tail from the other.
    • B. Drawing two cards one after the other from a well shuffled pack of cards without replacement.
    • C. Drawing two balls successively from a box containing balls having the same size and shape with replacement.
    • D. Any events obtaining from tossing a coin and rolling a die at the same time.
    Answer, explanation and textbook page: open in the app.
  25. 25.What is the value of $$\int_a^b x e^x dx$$
    • A. e
    • B. -e
    • C. -1
    • D. 1
    Answer, explanation and textbook page: open in the app.
  26. 26.Let f and g be two functions. Which one of the following statements is true about f or g?
    • A. The line y = 0 is the a tangent line to the graph of f(x) = |x| at (0, 0).
    • B. If f' differentiable at a, then it is continuous at a.
    • C. If f'(a) = 0, then the tangent line to the graph of f at (a, f(a)) is y = a.
    • D. If f and g differentiable at a, then f/g is differentiable at a.
    Answer, explanation and textbook page: open in the app.
  27. 27.What is the value of $a$ for which the function defined by, $f(x)=\begin{cases}ax+3,&\text{for }x\ge 1\x^2-2x+3,&\text{for }x<1\end{cases}$ is continuous at 1?
    • A. 1
    • B. -1
    • C. 3
    • D. 4
    Answer, explanation and textbook page: open in the app.
  28. 28.What is the derivative of the function $f(x)=\frac{xe^x}{\cos x}$ at $x=0$?
    • A. 2
    • B. -1
    • C. 1
    • D. 0
    Answer, explanation and textbook page: open in the app.
  29. 29.Which of the following is true about bounded sequence?
    • A. Every bounded sequence is convergent.
    • B. Every increasing bounded sequence converges to the greatest lower bound of the sequence.
    • C. Every decreasing bounded sequence converges to the least upper bound of the sequence.
    • D. Every monotone bounded sequence converges.
    Answer, explanation and textbook page: open in the app.
  30. 30.If f and g are continuous functions at a, then which of the following functions may NOT be continuous at a?
    • A. 2f + 3g
    • B. fg
    • C. f/g
    • D. f - 2g
    Answer, explanation and textbook page: open in the app.
  31. 31.Let $f$ be a continuous function on R and $c$ be a number in the domain of $f$. Which one of the following must be true about the function?
    • A. An absolute maximum value of $f$ which is attained at a critical number is also a local maximum value of $f$.
    • B. If $f'(c) = 0$, then $(c, f(c))$ is an inflection point of $f$.
    • C. If $f(c)$ is a local minimum value of $f$, then $f'(c) = 0$.
    • D. If $c$ is a critical number of $f$, then $f(c)$ is a local extreme value of $f$.
    Answer, explanation and textbook page: open in the app.
  32. 32.If the perimeter of a rectangle is 120 m, what are the length l and width w of the rectangle that maximizes its area?
    • A. l = 50 m and w = 10 m
    • B. l = 30 m and w = 30 m
    • C. l = 35 m and w = 25 m
    • D. l = 40 m and w = 20 m
    Answer, explanation and textbook page: open in the app.
  33. 33.The sum of the first 8 terms of a geometric sequence with first term $0.3$ and common ratio $1/10$ is equal to:
    • A. $\frac{3}{10}\left(\frac{10^8-1}{10^8}\right)$
    • B. $\frac{1}{3}\left(\frac{10^8-1}{10^8}\right)$
    • C. $\frac{1}{3}\left(\frac{10^8-1}{10^7}\right)$
    • D. $\frac{3}{10}\left(\frac{10^8-1}{10^9}\right)$
    Answer, explanation and textbook page: open in the app.
  34. 34.Suppose f is a continuous function on an interval [a, b] and differentiable on (a, b) with the property f(a) = f(b). Which one of the following must be true?
    • A. The function has a zero in (a, b).
    • B. The function has minimum value in (a, b).
    • C. The function has maximum value in (a, b).
    • D. The graph of f has horizontal tangent line at (c, f(c)) for some c ∈ (a, b).
    Answer, explanation and textbook page: open in the app.
  35. 35.Suppose f is a twice continuously differentiable function and f(x) = f(2x + 3). Then the second derivative of g at 0 is equal to:
    • A. 40
    • B. 20
    • C. 5
    • D. 10
    Answer, explanation and textbook page: open in the app.
  36. 36.A line segment chord in an ellipse which passes through the center and perpendicular to the major axis is called:
    • A. Semi-major axis
    • B. Minor axis
    • C. Eccentricity
    • D. Latus rectum
    Answer, explanation and textbook page: open in the app.
  37. 37.If A is a 3 × 3 matrix with det(A) = 5, then what is the determinant of 2A?
    • A. 10
    • B. 2/5
    • C. 40
    • D. 30
    Answer, explanation and textbook page: open in the app.
  38. 38.What is the seventh term of an arithmetic sequence whose third term is -4 and common difference is 3?
    • A. 15
    • B. -15
    • C. 16
    • D. -16
    Answer, explanation and textbook page: open in the app.
  39. 39.Which one of the following compound statements has a converse with truth value False?
    • A. If 6 is a prime integer, then 4 is an even number.
    • B. If man is immortal, then the earth does not rotate around the sun.
    • C. If athlete Derartu Tulu was born on October 21, 1971, then 6 is prime.
    • D. If man is mortal, then the earth does not rotate around the sun.
    Answer, explanation and textbook page: open in the app.
  40. 40.The solution set of the equation given by $\log(x^2-3)=2\log(x-1)$ is:
    • A. {1}
    • B. {2}
    • C. {4}
    • D. ∅
    Answer, explanation and textbook page: open in the app.
  41. 41.Which of the following is true about the trigonometric values of the given pairs of angles?
    • A. tan (75°) = tan (105°)
    • B. sin (75°) = cos (105°)
    • C. cos (120°) = cos (60°)
    • D. sin (120°) = sin (60°)
    Answer, explanation and textbook page: open in the app.
  42. 42.Which one of the following intervals is the solution set of the inequality x 2 - 5x + 6 ≤ 0?
    • A. [0, 2]
    • B. [2, 3]
    • C. (3, ∞)
    • D. (-∞, 2]
    Answer, explanation and textbook page: open in the app.
  43. 43.What are the coordinates of a point that divides the line segment with endpoints P(0, 1) and Q(5, 6) in the ratio 2:3 from P?
    • A. (2, 3)
    • B. (0, 2)
    • C. (3, 2)
    • D. (0, 3)
    Answer, explanation and textbook page: open in the app.
  44. 44.Which one of the following trigonometric values is correct?
    • A. sin (-120°) = 0.5
    • B. cos (-120°) = -0.5
    • C. tan (-120°) = -1
    • D. cos (-60°) = √3/2
    Answer, explanation and textbook page: open in the app.
  45. 45.Which one of the following is the augmented matrix associated to the system of equations: $$\left\{ \begin{aligned} 2x + y + 3z & = 4 \\ x - z & = 1 \\ -4x + y & = 3 \end{aligned} \right.$$
    • A. $$\begin{pmatrix} 3 & 2 & 1 & 1 \\ -1 & 1 & -1 & 3 \\ -1 & -4 & 1 & 4 \end{pmatrix}$$
    • B. $\begin{pmatrix} 3 & 1 & 2 & 4 \\ -1 & 0 & 1 & 1 \\ -4 & 1 & -3 & 0 \end{pmatrix}$
    • C. $\begin{pmatrix} 1 & 2 & 3 & 4 \\ -1 & 1 & -1 & 1 \\ 1 & -4 & -1 & 3 \end{pmatrix}$
    • D. $\begin{pmatrix} 2 & 1 & 3 & 4 \\ 1 & 0 & -1 & 1 \\ -4 & 1 & 0 & 3 \end{pmatrix}$
    Answer, explanation and textbook page: open in the app.
  46. 46.Which of the following is NOT true about argument and validity?
    • A. An argument which is not valid is false.
    • B. The validity of an argument can be checked by using a truth table.
    • C. In a valid argument, the conclusion may be true or false whenever all the premises are true.
    • D. In a valid argument if all the premises are true then the conclusion must also be true.
    Answer, explanation and textbook page: open in the app.
  47. 47.If Btu 20,000 is deposited in a Bank at a rate of 12% interest compounded monthly, how long will it take to double the amount?
    • A. 7.59 years
    • B. 6 years
    • C. 5.81 years
    • D. 5 years
    Answer, explanation and textbook page: open in the app.
  48. 48.Which one of the following is true about the matrix $ \begin{pmatrix} 1 & 3 & 5 \\ 6 & 4 & 2 \\ 9 & 7 & 0 \end{pmatrix} $?
    • A. The minor of the entry 3 is 18.
    • B. The cofactor of the entry 5 is -6.
    • C. The minor of the entry 0 is 14.
    • D. The cofactor of the entry 1 is -14.
    Answer, explanation and textbook page: open in the app.
  49. 49.What is the image of the circle with equation (x + 2) 2 + (y - 2) 2 = 1, when it is rotated through π/6 about the origin?
    • A. (x + (√3 - 1)) 2 + (y - (√3 - 1)) 2 = 1
    • B. (x - (√3 + 1)) 2 + (y + (√3 - 1)) 2 = 1
    • C. (x + (√3 + 1)) 2 + (y - (√3 - 1)) 2 = 1
    • D. (x - (√3 - 1)) 2 + (y + (√3 + 1)) 2 = 1
    Answer, explanation and textbook page: open in the app.
  50. 50.What is the dot (scalar) product of the vectors u = i + 2j and v = -2i + 4j?
    • A. -8
    • B. 5
    • C. 6
    • D. 7
    Answer, explanation and textbook page: open in the app.
  51. 51.The following is the distribution of the marks of students obtained in Mathematics test out of 50: Range of marks Frequency 20 - 25 15 26 - 31 25 32 - 37 25 38 - 43 20 44 - 50 10
    • A. The first quartile of the distribution is 28.
    • B. The first quartile of the distribution is 26.
    • C. The first quartile of the distribution is 26.5.
    • D. The first quartile of the distribution is 27.5
    Answer, explanation and textbook page: open in the app.
  52. 57.Consider the points A(-2, 1), B(0, 0), C(1, -1) and D(1, 1) in xy-plane. If a = AB and b = CD, then which one of the following is equal to a/b?
    • A. (3, 1)
    • B. (-7, 1)
    • C. (1, 3)
    • D. (-7, -1)
    Answer, explanation and textbook page: open in the app.
  53. 58.What is the point of intersection of the medians of ΔABC whose vertices are given by A(0, 0), B(6, 0) and C(0, 1)?
    • A. (2, 4/3)
    • B. (1, 2/3)
    • C. (9, 6/5)
    • D. (6, 4/5)
    Answer, explanation and textbook page: open in the app.
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