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Grade 12 Mathematics (Social Science) national exam, 2015 E.C. (2023 G.C.)
43 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.Which one of the following defines a prime number?
- A. It is a natural number where one of its factors is . 3
- B. It is a natural number that has exactly two distinct factors.
- C. It is an odd natural number that has only two factors.
- D. It is a natural number that has an odd number of factors.
Answer: B. A prime number is a natural number with exactly two distinct factors: $1$ and itself. Option B states this definition. A only describes numbers with a factor of $3$; C wrongly requires the number to be odd; D describes a different factor count. Textbook: Grade 9, unit 2 · section 2.1. - 2.Which one of the following is a vector quantity?
- A. Temperature of a room.
- B. Floor area of a classroom.
- C. Speed of a motorbike.
- D. Force to push a car.
Answer: D. The key idea is that a vector has both size and direction. Force has size and a direction in which it pushes, so D is a vector. A is wrong because temperature has no direction. B is wrong because area has no direction. C is wrong because speed has no direction. Textbook: Grade 11, unit 5 · section 5.1. - 3.Which one of the following is true for the values of the trigonometric functions for the indicated angles? 5π) −1 tan(−3π) = −1
- B. sin = ( 2 = ( 2 3π) 0 tan(7π) = −1
- D. cos −
Answer: C. The key idea is to use the repeating values of sine, cosine, and tangent at special angles. Since $\cos\left(-\frac{3\pi}{2}\right)=0$, option C is true. A. $\sin\left(\frac{5\pi}{2}\right)=1$, not $-1$. B. $\tan(-3\pi)=0$, not $-1$. D. $\tan(7\pi)=0$, not $-1$. Textbook: Grade 10, unit 4 · section 4.3. - 4.Which one of the following is true about the function $f(x)=\cos x$?
- A. Domain of $f$ is $[-1,1]$.
- B. It is periodic with period $2\pi$.
- C. Range of $f$ is $\{y\mid y\in\mathbb{R}\}$.
- D. The difference between its maximum and minimum values is $2\pi$.
Answer: B. The key idea is that cosine repeats every $2\pi$. Thus $f(x)=\cos x$ has period $2\pi$, making B true. A is wrong because $[-1,1]$ is the range, not the domain. C is wrong because the range is $[-1,1]$. D is wrong because the maximum minus the minimum is $1-(-1)=2$, not $2\pi$. Textbook: Grade 10, unit 4 · section 4.2. - 5.If the polynomial $f(x)=x^4-9x^2+22$ is divided by $x-2$, which one of the following are the quotient and remainder respectively?
- A. $x^3-2x^2-5x+10$ and $2$
- B. $x^3-2x^2-5x+10$ and $-2$
- C. $x^3+2x^2-5x-10$ and $-2$
- D. $x^3+2x^2-5x-10$ and $2$
Answer: D. The key idea is that polynomial division gives a quotient and a remainder. Dividing $x^4-9x^2+22$ by $x-2$ gives quotient $x^3+2x^2-5x-10$ and remainder 2. A, B, and C have an incorrect quotient or remainder. Textbook: Grade 10, unit 2 · section 2.2. - 6.Consider the graph of which is shown below. Which one of the following is NOT true about the function ?
- A. The domain of is the set of all real numbers greater than . f −3 f −2
- B. The x-intercept of the graph of is .
- C. The y-axis is a vertical asymptote of the graph of . f
- D. The function increases as the value of increases. x
Answer: C. For a logarithmic function, its input must be positive, and its graph has a vertical asymptote where the input is zero. Option C is not true because $x+3=0$ gives the asymptote $x=-3$, not the $y$-axis. A. The domain is $x>-3$, so this is true. B. Setting $\log_5(x+3)=0$ gives $x+3=1$, so the $x$-intercept is $-2$. D. The function increases because its base, $5$, is greater than $1$. Textbook: Grade 10, unit 3 · section 3.3. - 7.In shown below, the medians and of are concurrent at point . If cm, cm and , then which one of the following is NOT true?
- A. BO = 10 cm
- B. OE = 5 cm
- C. AO = 5 cm
- D. AE = 15 cm
Answer, explanation and textbook page: open in the app. - 8.Which one of the following interval(s) is the solution set of the quadratic inequality 4x −12 ?
- B. (1, 3) (3, ∞)
- D. (−∞, 1) (−∞, 1) ∪(3, ∞)
Answer, explanation and textbook page: open in the app. - 9.Which of the following is the solution set of the equation ? 1} − { 2 1} {1} {−1}
- B. { 2
Answer, explanation and textbook page: open in the app. - 10.Which one of the following expressions is equal to $\frac{x-3}{x+1}\div\frac{x+1}{x-1}$?
- A. $\frac{x-1}{x-3}$
- B. $\frac{x^2-4x+3}{x^2+2x+1}$
- C. $\frac{x^2+2x+1}{x^2-4x+3}$
- D. $\frac{x-3}{x-1}$
Answer, explanation and textbook page: open in the app. - 11.A frustum of a cone is formed from a right circular cone of height 15cm. If the base radii of the frustum are 2cm and 3cm, where the height from the smaller base to the vertex is 4cm, what is the volume of the frustum of the cone? πcm 3 209 3 πcm 3 285 3
- B. πcm 3 119 3 πcm 3 195 3
Answer, explanation and textbook page: open in the app. - 12.Which one of the following statements defines frustum of a pyramid?
- A. It is part of the pyramid included between the vertex and a cross-section made by a plane parallel to one of its lateral surfaces.
- B. It is part of the pyramid included between the vertex and the cross-section made by a plane parallel to the base.
- C. It is part of the pyramid included between the base and the cross-section made by a plane parallel to the base.
- D. It is part of the pyramid included between the base and a cross-section made by a plane parallel to one of its lateral surfaces.
Answer, explanation and textbook page: open in the app. - 13.If the foci of a hyperbola are and and the transverse axis is 8 cm long, which one of the following is the standard form of equation of the hyperbola? − 16 (x−3)2 = 9 (y−4)2 1 − 9 (y−4)2 = 16 (x−3)2 1
- B. − 25 (x−3)2 = 16 (y−4)2 1 − 16 (y−4)2 = 25 (x−3)2 1
Answer, explanation and textbook page: open in the app. - 14.Which one of the following is the equation of the tangent line to the circle $(x-2)^2+y^2=10$ at the point $(5,1)$?
- A. $x+3y-16=0$
- B. $3x+y-16=0$
- C. $-3x+y-14=0$
- D. $-x+3y+14=0$
Answer, explanation and textbook page: open in the app. - 15.Which one of the following is true about the determinant of a matrix?
- A. The determinant of any two-by-two matrix is non-negative.
- B. If and are any matrices, then . H = (a b d) T = (d c a) ∣H∣= −∣T∣ c b 2 1 (x + 1
- C. If is a matrix, then for all in . x −1) ∣P∣≥0 x R P = −2 Q = (m p) ∣Q∣= ∣q∣ t R t
- D. If is a matrix, then for all in and ≠ 0. n r
Answer, explanation and textbook page: open in the app. - 16.Let $M=\begin{pmatrix}-\frac{1}{2}&\frac{1}{2}\[2pt]\frac{1}{2}&-\frac{1}{2}\end{pmatrix}$ and $Q=\begin{pmatrix}1&-1\-1&1\end{pmatrix}$ be given matrices. Which one of the following is true?
- A. $2Q=M$
- B. $3M=\frac{3}{2}Q$
- C. $2M=-Q$
- D. $\frac{1}{2}Q=2M$
Answer, explanation and textbook page: open in the app. - 17.If we toss a coin three times, what is the number of possible events of getting at least two heads?
- A. 3
- B. 4
- C. 5
- D. 6
Answer, explanation and textbook page: open in the app. - 18.A committee of five members was formed from 6 females and 4 males. How many different committees can be formed that consists of 3 females and 2 males?
- A. 20
- B. 40
- C. 120
- D. 240
Answer, explanation and textbook page: open in the app. - 19.Let , and be propositions. If the truth value of the proposition is False. Then the converse of which proposition has truth value True?
- B. p ⟹¬q p ⟹r r ⟹q q ⟹¬p
Answer, explanation and textbook page: open in the app. - 20.Which one of the following is the converse of the conditional statement "If life in Ethiopia was simpler, then people in Ethiopia would not go to other nations."
- A. If life in Ethiopia was simpler, then people in Ethiopia would go to other nations.
- B. If life in Ethiopia was not simpler, then people in Ethiopia would go to other nations.
- C. If people of Ethiopia would go to other nations, then life in the Ethiopia was not simpler.
- D. If people of Ethiopia would not go to other nations, then life in Ethiopia was simpler.
Answer, explanation and textbook page: open in the app. - 21.Which one of the following is true about addition of complex numbers?
- A. $(a + bi) + (b + bi)i = (a + b) + (a + b)i$
- B. $(x + yi) + x(i)^5 - y(i)^{10} = (x + y) + (x + y)i$
- C. $(3 + 12i) + (-7 - \sqrt{169})i = -4 + i$
- D. $(5i + 9i)^2 = 14i$
Answer, explanation and textbook page: open in the app. - 22.Which one of the following is true about the system of linear equations ⎩ ⎨ ⎧3x + y −z = 1 2x −y + 2z = 2 = −1 −x + y + z 3 2 −1 1 −1 1 2 −1
- B. 1 2 −1 1 −1 3 2 −1 1 x = 3 2 −1 1 −1 −2 1 2 1 z = 3 2 −1 1 −1 −2 1 2 1 1
- D. 3 2 −1 1 −1 −1 1 2 −1 1 −1 −1 2 1 x = 2 1 1 1 y = 3 2 −1 1 −1 −2 3 2 −1 1 −1 −1 2 1 2 1 1 1
Answer, explanation and textbook page: open in the app. - 24.Which one of the following is the minimum value of the objective function $Z=2x-5y$ subject to the constraints:
$3x+2y\leq 6$
$2x+5y\leq 10$
$x\geq 0$
$y\geq 0$
- A. $-\frac{150}{11}$
- B. $4$
- C. $-10$
- D. $0$
Answer, explanation and textbook page: open in the app. - 25.Which one of the following geometric series is convergent?
- A. $\sum_{n=1}^{\infty}(-3)^n$
- B. $\sum_{n=1}^{\infty}\left(\frac{1}{2}\right)^n$
- C. $\sum_{n=1}^{\infty}\left(\frac{1}{10}\right)^{-n}$
- D. $\sum_{n=1}^{\infty}(\sqrt{3})^n$
Answer, explanation and textbook page: open in the app. - 26.A woman wants to buy a new car that costs Birr 1,100,000. She paid Birr 870,000 and agreed with the seller to pay Birr 10,000 per month to cover the remaining amount at 0% interest rate. How many months will it take the woman to pay all the remaining money?
- A. Twenty Three months.
- B. Seventeen months.
- C. Thirteen months.
- D. Seven months.
Answer, explanation and textbook page: open in the app. - 27.If , , and are functions defined by , , and respectively, which one of the following is true?
- A. The function is continuous on . f + g [−1, 1] h −g [−1, 1]
- B. The function is continuous on . 3g g
- C. The function is continuous on the interval where is continuous.
- D. The function is continuous on the interval . hg (−∞, 1]
Answer, explanation and textbook page: open in the app. - 28.Which one of the following limit values is correct for the limit of the given function?
- A. $\lim_{x\to 2}\frac{x^3-8}{2-x}=12$
- B. $\lim_{x\to 1}\frac{1-x^2}{x-1}=-2$
- C. $\lim_{x\to 10}2^x=100$
- D. $\lim_{x\to \sqrt{5}}\frac{x^2-5}{x^4-25}=\frac{1}{5}$
Answer, explanation and textbook page: open in the app. - 29.If , , , and are functions defined by , , , and x+7 g(x) = f g h m f(x) = x +9x+14 2 x sin x cos x , which one of the following is the correct result? lim m(x) = x→0 0 lim f(x) = x→−7 − 5 1
- B. lim h(x) = x→3+ 15 lim g(x) = x→0+ 1
Answer, explanation and textbook page: open in the app. - 30.Which one of the following is the correct definition of continuity of a function on the given interval?
- A. If a function is continuous on the open interval , then . h (a, b) lim h(x) = x→k h(k), ∀k ∈[a, b] g [a, b] g (a, b) b
- B. If a function is continuous on the interval , then is continuous on and continuous from the right of .
- C. m [a, ∞) m (a, ∞) a If a function is continuous on the interval , then is continuous on and continuous from the right of .
- D. If a function is continuous on the interval , then is continuous for every point in the interval . f [a, b] f c [a, b]
Answer, explanation and textbook page: open in the app. - 31.Which one of the following is NOT true about the derivative of a function?
- A. The derivative of a function at a point is equal to the area of the region enclosed by the graph of the function and the coordinate axes.
- B. The derivative of a function at a point is the instantaneous rate of change of at . f x 0 f(x) (x , f(x )) 0 0 g t g x t x −t x t f x (x, f(x))
- C. The derivative of a function at a point is equal to the limit of the ratio of the difference of functional values of at and at to the difference as approaches .
- D. The derivative of a function at a point is the slope of a line tangent to the graph of the function at the point .
Answer, explanation and textbook page: open in the app. - 32.If $f$ and $g$ are functions defined by $f(x)=x^7-10x^5+x^3$ and $g(x)=e^{-2x}$ respectively, then which one of the following is true?
- A. $f^{(n)}(x)=0,\ \forall n\geq 6$
- B. $g^{(n)}(x)=(-1)^n2^ne^{-2x},\ \forall n\in\mathbb{N}$
- C. $(\frac{1}{2}f)^{(n)}(x)=0,\ \forall n\geq 6$
- D. $(-g)^{(n)}(x)=(-1)^n2^{n+1}e^{-2x},\ \forall n\in\mathbb{N}$
Answer, explanation and textbook page: open in the app. - 33.If a rectangular region of length and width has an area of , what are the values of and such that its perimeter is minimum?
- A. and
- B. and x = 16 y = 4 x = 8 y = 8 x = 64 y = 1 x = 32 y = 2
- C. and
- D. and
Answer, explanation and textbook page: open in the app. - 34.Which one of the following is true about the zeros of a function?
- A. The zero of a function is the functional value of the function at zero. f h h(x) = x−3 x −27 3 h
- B. If is a function defined by , then 3 is the zero of . g h g(x) = + 1 −x2 1 −x2 h(x) = 1
- C. The functions and given by and have identical zeros. 1−x2
- D. The zeros of a given function and for are identical, if it exist(s). f c/f c =0
Answer, explanation and textbook page: open in the app. - 35.Which one of the following is true about the derivative of the given functions? d ( x4 x−ln x) , x = x6 −4x +x−1+4x ln x 2 0 (x sin x) = dx d cos x −x sin x
- B. = dx cos x e 27 82 27 f(x) = 1+sin x [f(x)]∣ = dx d x=π 0 h(x) =3 [h(x)]∣ = dx d x=27
- C. If , then
- D. If , then e x x
Answer, explanation and textbook page: open in the app. - 36.Which one of the following is an equation of the line tangent to the graph of the function x + 1 (1, 0) at ?
- B. y = 2x + 2 y = −2x −2
- D. y = −2x + 2 y = 2x −2
Answer, explanation and textbook page: open in the app. - 37.Which one of the following is true about integration and differentiation?
- A. Integral of a function is an anti-derivative of its derivative.
- B. The process of finding from is called integration. f(x) f(x)dx ∫
- C. Integration is the reverse process of differentiation.
- D. Any continuous function has only one anti-derivative.
Answer, explanation and textbook page: open in the app. - 38.Which one of the following statements is true?
- A. If the price of an egg reduced from Birr 13.00 to Birr 11.25, then the percent decrease is 0.142%.
- B. If a woman buys 20 kg of sugar by Birr 2000.00, and she sells one kilogram of it by Birr 85.00, then she sells by 30% price decrease.
- C. If a shop owner offered 75 T-shirts by a price of Birr 325.00 for each and he bought all by Birr 25,000.00, then the percent decrease he had made for one T-shirt is 2.5%.
- D. A person earns Birr 175.00 commission per service he provides. If he provided 13 services and got Birr 2000.00, then the percent decrease of his income is approximately 10%.
Answer, explanation and textbook page: open in the app. - 39.Which one of the following integrals of trigonometric functions is the true? (sin 2x)dx = ∫ cos 2x + 2 1 C 10(cos 10x)dx = ∫ −sin 10x + C
- B. 5(cos 7x + ∫ 3)dx = − sin 7x + 7 5 C cos(−5x)dx = ∫ sin(5x) + 5 1 C
Answer, explanation and textbook page: open in the app. - 40.The following table shows the mean score and the standard deviation of scores of the students in each class P, Q, R and S who took a similar test which is marked out of 20. Which of the following statements is true about these data? Class Mean Standard deviation P 16 3.05 Q 16 2.62 R 16 3.05 S 16 3.02
- A. The most consistent test result is class P students' result.
- B. Test result of class Q students is highly variable.
- C. Test result of class P students is more consistent than class R students.
- D. Test result of class S students is more consistent than class R students.
Answer, explanation and textbook page: open in the app. - 41.What is the area (in square units) of the regions bounded by the graph of the equation ?
- A. 41
- B. 20.5
- C. 40
- D. 21.5
Answer, explanation and textbook page: open in the app. - 42.What is the area (in square units) of the region enclosed by the curve , the x-axis, the vertical lines and ?
- A. 8
- B. 12
- C. 5
- D. 17
Answer, explanation and textbook page: open in the app. - 43.The following table shows the first quartile and the third quartile of two data sets X and Y. Which of the Q 1 Q 3 Q 1 Q 3 following is true about the given data? X Y 5 3 9 6
- A. Data X possesses higher variability than data Y.
- B. Data Y possesses higher variability than data X.
- C. The inter quartile range of data X is 7.
- D. The inter quartile range of data Y is 4.5
Answer, explanation and textbook page: open in the app. - 44.Suppose the following data represents the weight in kg of 10 students in a classroom: 61, 49, 52, 68, 61, 55, 64, 63, 67, 60 Which one of the following statements is NOT true about the data?
- A. Mean deviation about the mean of the data is 4.8
- B. Mean deviation about the median of the data is 4.6
- C. Mean deviation about the mode of the data is 4.6
- D. Mean deviation about the mean of the data is 4.7
Answer, explanation and textbook page: open in the app.