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Grade 12 Mathematics (Natural Science) national exam, 2015 E.C. (2023 G.C.)
52 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.What is the value of x that satisfies the equation $(5 - \frac{25}{8})^{\frac{1}{2}x^{2} - \frac{2}{3}x} = 2$
- A. 1
- B. -3/2
- C. 3/2
- D. 4
- 2.The solution set of the inequality $(2x - 3)(x + 5) \leq 0$ is
- A. $\left(5,-\frac{3}{2}\right)$
- B. $(-\infty, -5] \cup \left[\frac{3}{2}, \infty\right)$
- C. $[-\infty, -\frac{3}{2}] \cup [5, \infty)$
- D. $\left(-5,\frac{3}{2}\right)$
Answer: D. Textbook: Grade 12, unit 1 · section 1.4. - 3.The histogram below shows the recorded temperature (in °C) of a certain town in Ethiopia for the first 15 days of March 2023. Then which one of the following statement is true about the data represented by the histogram?
- A. The temperature was 24°C for two days.
- B. The mode of the data is 22°C.
- C. The highest frequency is 3-.
- D. For 45% of the days, the temperature was more than 19°C.
- 4.Which of the following statement is true about ?
- A. for all x in the domain of f. f(x) ≥2 x = 1
- B. The line is the axis of symmetry of the graph of f.
- C. The graph of f opens downward.
- D. The vertex of the graph of f is (3, 1).
- 5.What is the area of a regular hexagon inscribed in a circle of radius 6cm? 48 cm 3 2
- B. 54 cm 3 2
- C. 48cm 2
- D. 36 cm 3 2
Answer: B. The key idea is that each side of a regular hexagon inscribed in a circle equals the circle's radius. Divide the hexagon into $6$ equilateral triangles: each has area $\frac{\sqrt{3}}{4}(6\,\mathrm{cm})^2=9\sqrt{3}\,\mathrm{cm}^2$, so the total area is $6(9\sqrt{3})=54\sqrt{3}\,\mathrm{cm}^2$. A, $48\sqrt{3}\,\mathrm{cm}^2$, is too small. C, $48\,\mathrm{cm}^2$, has the wrong area value. D, $36\sqrt{3}\,\mathrm{cm}^2$, is too small. Textbook: Grade 9, unit 6 · section 6.4. - 6.Which of the following defines a prime number?
- A. It is a number that has more than two positive factors.
- B. It is a number whose only factor is 1.
- C. It is a number that has only two positive factors.
- D. It is a number whose only factor is itself.
Answer: C. A prime number has exactly two positive factors. Option C gives this definition. A has more than two factors; B and D each name only one factor, not both factors. Textbook: Grade 9, unit 2 · section 2.1. - 7.The solution set of the equation $5\log_{9}x - 2\log_{4}(x^{2}) - \log_{4} 8 = -1$ is
- A. {4}
- B. {3}
- C. {9}
- D. {3/2}
- 8.The solution set of the equation $3|2x-4|=6-2|1-5x|$ is
- A. $\left\{\frac{1}{2},\frac{5}{4}\right\}$
- B. $\left\{-1,\frac{1}{2},\frac{5}{4}\right\}$
- C. $\{\}$
- D. $\left\{-1,\frac{1}{2}\right\}$
Answer: C. Textbook: Grade 10, unit 2 · section 2.2. - 9.The domain of the function $f(x)=2x^{\frac{2}{3}}$ is
- A. $\mathbb{R}\setminus\{0\}$
- B. $[0,\infty)$
- C. $\mathbb{R}$
- D. $(0,2)$
Answer: C. The key idea is that a function's domain is every input for which its rule gives a real value. Since the cube root is defined for negative and positive real numbers, $x^{2/3}=(\sqrt[3]{x})^2$ is defined for every real $x$. A excludes $0$, even though the function is defined there; B wrongly excludes negative inputs; D includes only a small interval, not all valid inputs. Textbook: Grade 10, unit 3 · section 3.1. - 10.The slope of the line that makes an angle of 135° with the positive x-axis is equal to
- A. 2
- B. -1
- C. 1
- D. 3
Answer, explanation and textbook page: open in the app. - 11.If tan31° = 0.81 then what is the value of tan509°
- A. 0.19
- B. -0.81
- C. 0.81
- D. -0.91
Answer, explanation and textbook page: open in the app. - 12.What is the value of x in the equation $\sqrt{125x^{2} 5^{x-1}} = 25^{3x + 1}$?
- A. -5
- B. -5/7
- C. 5
- D. 5/7
Answer, explanation and textbook page: open in the app. - 13.The solution set of the equation $3|2x - 4| = 6 - 2|1 - 5x|$ is
- A. $\left\{\frac{1}{2}, \frac{5}{4}\right\}$
- B. $\left\{-1,\frac{1}{2}, \frac{5}{4}\right\}$
- C. $\left\{\right\}$
- D. $\left\{-1,\frac{1}{2}\right\}$
Answer, explanation and textbook page: open in the app. - 14.Given $A = \{x \in \mathbb{N} : x < 3\}$ and B is the set of all possible factors of 13. Then which one of the following is equal to B x A?
- A. {(1, 1), (2, 1), (1, 13), (2, 13)}
- B. {(1, 1), (1,2), (13, 1), (13, 2), (1, 13), (13, 2)}
- C. {(1, 1), (13, 2)}
- D. {(1, 2), (13, 1), (13, 2)}
Answer, explanation and textbook page: open in the app. - 15.The object shown in the figure below is made of wood. Its top cover is a solid hemisphere and its bottom is a cylindrical part of base diameter 6cm and height of 12cm. From the bottom of the cylindrical part of height 8cm is cut off. Thus, what is the volume of the solid in cubic centimeter?
- A. 102π
- B. 72π
- C. 96π
- D. 126π
Answer, explanation and textbook page: open in the app. - 16.The domain of the function $f(x) = 2x^{\frac{2}{3}}$ is
- A. $\mathbb{R}\setminus\{0\}$
- B. [0,∞)
- C. $\mathbb{R}$
- D. (0, 2)
Answer, explanation and textbook page: open in the app. - 17.On the set of Natural numbers $\mathbb{N}$, which one of the following propositions has a truth value True ?
- A. $(\exists x)(\exists y)(x-y=y-x)$
- B. $(\forall x)(\forall y)(x\ne y\Longrightarrow x>y)$
- C. $(\forall x)(\exists y)(x-y=5)$
- D. $(\forall x)(\exists y)(xy<x)$
Answer, explanation and textbook page: open in the app. - 18.In a certain month of the year, the temperature of a city was recorded and presented as follows:
Temperature (in °C) | 10-14 | 15-19 | 20-24 | 25-29 | 30-34
Frequency (number of days) | 2 | 9 | 11 | 5 | 3
Which one of the following is the cumulative frequency of the fourth class?
- A. 22
- B. 27
- C. 18
- D. 20
Answer, explanation and textbook page: open in the app. - 19.A fair die is going to be rolled once, and let: E1 be the event that an even number occurs; E2 be the event that a prime number occurs; E3 be the event that a multiple of 3 occurs; E4 be the event that a multiple of 5 occurs; Then which of the following pairs of events are mutually exclusive?
- A. E1 and E3
- B. E3 and E4
- C. E1 and E2
- D. E2 and E4
Answer, explanation and textbook page: open in the app. - 20.The following is the distribution of the weight of 40 students in a certain class:
Weight (in Kg) | 40-48 | 49-57 | 58-66 | 67-75 | 76-84
f | 8 | 18 | 7 | 5 | 2
Then for data given above, what is the median weight in kg?
- A. 54.5
- B. 55
- C. 53
- D. 53.8
Answer, explanation and textbook page: open in the app. - 21.A group of students, 5 from grade 10, 8 from grade 11 and 2 from grade 12 is to be formed. If a student is chosen randomly from this group, what is the probability that the chosen student is either from grade 10 or grade 12?
- A. 7/15
- B. 5/15
- C. 12/15
- D. 1/7
Answer, explanation and textbook page: open in the app. - 22.If the truth value of $p$ is T, then which of the following compound propositions has a truth value T for any proposition $q$?
- A. $\neg p \Leftrightarrow (p \lor \neg q)$
- B. $(\neg p \land q) \Rightarrow q$
- C. $(p \Rightarrow \neg p) \land q$
- D. $(p \Rightarrow q) \land \neg p$
Answer, explanation and textbook page: open in the app. - 23.Which of the following functions is one to one?
- A. $f=\{(1,5),(2,3),(5,4),(6,5)\}$
- B. $f:(0,\infty)\to\mathbb{R},\ f(x)=|x|-1$
- C. $f:\mathbb{R}\to\mathbb{R},\ f(x)=x^2-1$
- D. $f=\{(x,y)\mid y\text{ is the mother of }x\}$
Answer, explanation and textbook page: open in the app. - 24.If $\frac{x^2+14}{(x+2)(x-1)^2}=\frac{A}{x+2}+\frac{B}{x-1}+\frac{C}{(x-1)^2}$ then what are the values of A, B and C respectively?
- A. 1, -1, 3
- B. 3, -2, 6
- C. 2, -2, -5
- D. 2, -1, 5
Answer, explanation and textbook page: open in the app. - 25.Which of the following line is true about the intercept and symmetry of the graph of $f(x)=\frac{x^3-3x}{x^2-6}$?
- A. The graph is symmetrical with respect to the y-axis.
- B. Its y-intercept is 3.
- C. Its x-intercepts are ±√6.
- D. The graph is symmetrical with respect to the origin.
Answer, explanation and textbook page: open in the app. - 26.Consider the lines and . Then the angle between and is
- A. π/4
- B. 3π/2
- C. π/2
- D. π/6
Answer, explanation and textbook page: open in the app. - 27.The equation of the parabola with vertex $V(2, -1)$ and focus $F(-1, -1)$ is
- A. $(x+1)^2=-12(y+1)$
- B. $(y+1)^2=-12(x-2)$
- C. $(y+1)^2=12(x-2)$
- D. $(x-2)^2=-12(y+1)$
Answer, explanation and textbook page: open in the app. - 28.What is the value of $\lim_{x\to1}\frac{x^2-1}{x^2+1}$?
- A. 3
- B. 1
- C. 2
- D. 0
Answer, explanation and textbook page: open in the app. - 29.Which one of the following is the value of $a$ so that the function given by $f(x)=\begin{cases}ax+1 & \text{for } x\geq 1\x^2-4 & \text{for } x<1\end{cases}$ is continuous at 1?
- A. 4
- B. -4
- C. -1
- D. 1
Answer, explanation and textbook page: open in the app. - 30.Let $f(x)=-\frac{1}{3}\cos\left(2^{x^3}-\frac{\pi}{3}\right)+1$. Then what is the period and Phase shift of the graph of $f$ respectively?
- A. $4\pi/9$ and $2\pi/9$
- B. $2\pi/3$ and $2\pi/9$
- C. $3\pi/2$ and $4\pi/9$
- D. $\pi/9$ and $4\pi/9$
Answer, explanation and textbook page: open in the app. - 31.Which of the following functions is one to one?
- A. $f = \{ (1,5), (2,3), (5,4), (6,5) \}$
- B. $f : (0, \infty) \to \mathbb{R}, \, f(x) = |x| - 1$
- C. $f: \mathbb{R} \to \mathbb{R}, \, f(x) = x^{2} - 1$
- D. $f = \{ (x,y) \mid y \text{ is the mother of } x \}$
Answer, explanation and textbook page: open in the app. - 32.Which one of the following numbers is the upper bounds of the sequence ?
- A. 3/2
- B. 0
- C. 1
- D. 4
Answer, explanation and textbook page: open in the app. - 33.Let the sequence {a n } converges to 3 while {b n } converges to -1. Then the sequence {a n -b n } converges to
- A. 1
- B. 4
- C. 3
- D. 1/3
Answer, explanation and textbook page: open in the app. - 34.Which of the following argument is valid?
- A. $p\lor q,\ p\to q\vdash\neg q$
- B. $p\to q,\ q,\ r\to p\vdash r$
- C. $p,\ p\to q,\ r\lor\neg q\vdash p\land r$
- D. $p\to q,\ \neg r\land q\vdash p\lor r$
Answer, explanation and textbook page: open in the app. - 35.Which of the following assertions is true applying the Principle of Mathematical Induction?
- A. $1+2+2^2+\ldots+2^n=2^{n+1}-1$ for $n\geq 0$
- B. $n!\geq 2^n$ for $n\geq 0$
- C. $2+3+4+\ldots+n=\frac{n(n+1)}{2}+1$ for $n\geq 2$
- D. $3+5+7+\ldots+(2n+1)=n^2$ for $n\geq 1$
Answer, explanation and textbook page: open in the app. - 36.Which of the following is equal to $\int 3\cos 3x\,dx$?
- A. $\frac{1}{3}\sin 3x+c$
- B. $\sin 3x+c$
- C. $\sin x+c$
- D. $-\sin x+c$
Answer, explanation and textbook page: open in the app. - 37.What is the area of the region bounded by the graph of and the x-axis in square units?
- A. 16
- B. 32
- C. 16/3
- D. 32/3
Answer, explanation and textbook page: open in the app. - 38.Which of the following set of equations define the xz-plane?
- A. {(x,y,z)|x,y,z ∈ ℝ and x==9}
- B. {(x,y,z)|x,y,z ∈ ℝ and y=0}
- C. {(x,y,z)|x,y,z ∈ ℝ and x=0}
- D. {(x,y,z)|x,y,z ∈ ℝ and z=0}
Answer, explanation and textbook page: open in the app. - 39.Which of the following functions has a graph that completes one full cycle on $\left[-\pi,\frac{11\pi}{4}\right]$?
- A. $f(x)=3\sin\left(3x+\frac{\pi}{6}\right)$
- B. $f(x)=3\sin\left(\frac{4}{3}x+\frac{\pi}{2}\right)$
- C. $f(x)=2\sin\left(\frac{2}{3}x-\frac{\pi}{6}\right)$
- D. $f(x)=2\sin\left(2x-\frac{\pi}{6}\right)$
Answer, explanation and textbook page: open in the app. - 40.If the 6 th and the 11 th term of an arithmetic sequence are 35 and 75 respectively, then what is the sum of the first 31 terms of the sequence?
- A. 2145
- B. 724
- C. 2635
- D. 846
Answer, explanation and textbook page: open in the app. - 41.The solution set of the system $\begin{cases}2x-y+3z=1\x+y+3z=5\x-y+z=-1\end{cases}$ is
- A. $\{(-4-2k,k,2-k):k\in\mathbb{R}\}$
- B. $\{(k+1,2k,k-1):k\in\mathbb{R}\}$
- C. $\{(2-2k,3-k,k):k\in\mathbb{R}\}$
- D. $\{(k,k+1,k-1):k\in\mathbb{R}\}$
Answer, explanation and textbook page: open in the app. - 42.Let $z1, z2, and z3$ be complex numbers. Then which of the following is NOT necessarily true? z1 −(z2 + z3) = (z1 −z2) −z3 z1 −(z2 −z3) = (z1 −z2) −z3
- D. (z1z2)z3 = (z2z1)z3 (z1z2)z3 = z1(z2z3)
Answer, explanation and textbook page: open in the app. - 43.Which of the following argument is valid?
- A. $p \lor q, \, p \Rightarrow q \vdash \neg q$
- B. $p \Rightarrow q, \, q, \, r \Rightarrow p \vdash r$
- C. $p, \, p \Rightarrow q, \, r \lor \neg q \vdash p \land r$
- D. $p \Rightarrow q, \, \neg r \land q \vdash p \lor r$
Answer, explanation and textbook page: open in the app. - 44.Let u = (-1,6) and v = (2, 4) be position vectors. Then the vector 4u - 3v is
- A. (-10,12)
- B. (-10,11)
- C. (-4,24)
- D. (-15,18)
Answer, explanation and textbook page: open in the app. - 45.Water is running into a canonical tank, which is pointed down at a constant rate of $6\pi\,\text{cm}^3/\text{sec}$. The cone has a height of 10 cm and a base radius of 5 cm, then how fast is the water level rising when the water reaches 2 cm deep?
- A. $\frac{4}{\pi}$ cm/sec
- B. 6 cm/sec
- C. 4 cm/sec
- D. $\frac{6}{\pi}$ cm/sec
Answer, explanation and textbook page: open in the app. - 46.Let F be the anti-derivative of f. Then which of the following is NOT true about F and f?
- A. $\frac{d}{dx}\int F(x)\,dx=f(x)$
- B. $F'(x)=f(x)$
- C. $\int F(x)\,dx=f(x)+c$
- D. $\int \frac{d}{dx}f(x)\,dx=f(x)+c$
Answer, explanation and textbook page: open in the app. - 47.Let $f(x)=\frac{1}{\sqrt{x}}$ and $g(x)=2-\frac{1}{6}x^3$ be two functions. For what values of x do the graphs of f and g have tangent lines of the same slope?
- A. $\frac{\sqrt{3}}{2}$
- B. 1
- C. 4
- D. 1/4
Answer, explanation and textbook page: open in the app. - 48.What is the value of a number in the interval (0, 1) that satisfies the Mean Value Theorem for the function f(x) = 3x − 2 5x + 1 ?
- A. 1/2
- B. 3/5
- C. 5/6
- D. 1
Answer, explanation and textbook page: open in the app. - 49.Let $f(x)=3x+2x-\sin x+e^x$. What is the value of $f'(0)$?
- A. 2
- B. 1
- C. -1
- D. 4
Answer, explanation and textbook page: open in the app. - 50.Let $f$ and $g$ be differentiable functions, and $h(x)=g(f(3x+1))$. If $f'(1)=f(1)=2$ and $g'(2)=1$, then what is the value of $h'(0)$?
- A. 2
- B. 1
- C. 6
- D. 4
Answer, explanation and textbook page: open in the app. - 51.Which of the following describes the mechanical meaning of derivative?
- A. Gradient of graphs of functions
- B. Area under the curve
- C. Instantaneous rate of change
- D. Slope of tangent line
Answer, explanation and textbook page: open in the app. - 52.Which of the following is true about the zero(s) of a function f(x)?
- A. If f(a)=0, then x-a is a factor for f(x).
- B. If f(a)=0, then (x-a)² is a factor of f(x).
- C. If f(a)=0, then a is not necessarily in the domain.
- D. If f(a)=0, then y=a is they-intercept.
Answer, explanation and textbook page: open in the app.