2015 E.C. Mathematics Social stream EUEE past paper

Q1 · Statistics - mean

1. A. 95
2. B. 89
3. C. 90
4. D. 93

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Q2 · Geometry - parallelogram area

1. A. $24\sqrt{3}$ cm²
2. B. 64 cm²
3. C. $4\sqrt{3}$ cm²
4. D. $12\sqrt{3}$ cm²

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Q3 · Geometry - symmetry

1. A. Eight
2. B. Seven
3. C. Six
4. D. Five

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Q4 · Functions - domain and range

1. A. Domain of $f$ is {$x \in \mathbb{R} | x \geq 7$}.
2. B. Range of $f$ is {$f(x) \in \mathbb{R} | f(x) \geq 1$}.
3. C. Range of $f$ is {$f(x) \in \mathbb{R} | f(x) > 0$}.
4. D. Domain of $f$ is {$x \in \mathbb{R} | x \geq \frac{7}{5}$}.

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Q5 · Sets - power set

1. A. $\left\{ \emptyset, \left\{ \left\{ \frac{1}{2} \right\} \right\}, \emptyset, \left\{ \left\{ \frac{1}{2} \right\} \right\} \right\}$
2. B. $\left\{ \emptyset, \left\{ \emptyset \right\}, \left\{ \frac{1}{2} \right\}, \left\{ \emptyset,\left\{ \frac{1}{2} \right\} \right\} \right\}$
3. C. $\left\{ \emptyset, \left\{ \left\{ \frac{1}{2} \right\} \right\}, \left\{\frac{1}{2}\right\} \right\}$
4. D. $\left\{ \emptyset,\left\{ \emptyset \right\}, \left\{ \left\{ \frac{1}{2} \right\} \right\}, \left\{ \emptyset,\left\{ \frac{1}{2}\right\} \right\} \right\}$

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Q6 · Exponents - equations

1. A. $\left\{\frac{5}{11}\right\}$
2. B. $\left\{\frac{7}{34}\right\}$
3. C. $\left\{\frac{7}{17}\right\}$
4. D. $\left\{\frac{14}{17}\right\}$

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Q7 · Absolute value equations

1. A. $\{\frac{1}{7}, \frac{1}{6}\}$
2. B. $\{0, \frac{1}{7}\}$
3. C. $\{0, \frac{1}{6}\}$
4. D. $\{-\frac{1}{7}, 0\}$

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Q8 · Real numbers - number line

1. A. For every real number there is exactly one point corresponding to it on the number line.
2. B. It is not possible to find a point on a number line corresponding to an irrational number.
3. C. If a point on a number line is represented by a rational number, then we can use the same point to represent an irrational number.
4. D. The set of integers has a one-to-one correspondence with the set of all points on a number line

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Q9 · Number theory - prime numbers

1. A. It is a natural number where one of its factors is $3$.
2. B. It is a natural number that has exactly two distinct factors.
3. C. It is an odd natural number that has only two factors.
4. D. It is a natural number that has an odd number of factors.

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Q10 · Vectors

1. A. Temperature of a room.
2. B. Floor area of a classroom.
3. C. Speed of a motorbike.
4. D. Force to push a car.

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Q13 · Trigonometry - function values

1. A. $\sin\left(\frac{5\pi}{2}\right)=-1$
2. B. $\tan(-3\pi)=-1$
3. C. $\cos\left(-\frac{3\pi}{2}\right)=0$
4. D. $\tan(7\pi)=-1$

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Q14 · Trigonometry - cosine function

1. A. Domain of $f$ is $[-1,1]$.
2. B. It is periodic with period $2\pi$.
3. C. Range of $f$ is $\{y|y\in\mathbb{R}\}$.
4. D. The difference between its maximum and minimum values is $2\pi$.

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Q15 · Polynomials - division

1. A. $x^3 - 2x^2 - 5x + 10$ and $2$
2. B. $x^3 - 2x^2 - 5x + 10$ and $-2$
3. C. $x^3 + 2x^2 - 5x - 10$ and $-2$
4. D. $x^3 + 2x^2 - 5x - 10$ and $2$

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Q16 · Logarithmic functions - graph

This item uses a figure in the original paper (not redrawn here).

1. A. The domain of $f$ is the set of all real numbers greater than $-3$.
2. B. The x-intercept of the graph of $f$ is $-2$.
3. C. The y-axis is a vertical asymptote of the graph of $f$.
4. D. The function increases as the value of $x$ increases.

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Q17 · Geometry - medians and centroid

This item uses a figure in the original paper (not redrawn here).

1. A. BO = 10 cm
2. B. OE = 5 cm
3. C. AO = 5 cm
4. D. AE = 15 cm

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Q18 · Quadratic inequalities

1. A. $(1, 3)$
2. B. $(3, \infty)$
3. C. $(-\infty, 1)$
4. D. $(-\infty, 1) \cup (3, \infty)$

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Q20 · Rational equations

1. A. ${\left\{\frac{1}{2}\right\}}$
2. B. ${\left\{-\frac{1}{2}\right\}}$
3. C. ${\{1\}}$
4. D. ${\{-1\}}$

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Q21 · Rational expressions

1. A. $$\frac{x - 1}{x - 3}$$
2. B. $$\frac{x^2 - 4x + 3}{x^2 + 2x + 1}$$
3. C. $$\frac{x^2 + 2x + 1}{x^2 - 4x + 3}$$
4. D. $$\frac{x - 3}{x - 1}$$

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Q23 · Solid geometry - definitions

1. 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.
2. B. It is part of the pyramid included between the vertex and the cross-section made by a plane parallel to the base.
3. C. It is part of the pyramid included between the base and the cross-section made by a plane parallel to the base.
4. 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.

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Q24 · Functions - composition

1. A. The domain of the function $f\circ g$ is $\{x\in\mathbb{R}|x > \sqrt{7}\}$.
2. B. The range of the function $f\circ g$ is $\{y\in\mathbb{R}|y = (f\circ g)(x) > 7\}$.
3. C. The domain of the function $h\circ m$ is $\{x\in\mathbb{R}|x > 0\}$.
4. D. The range of $h\circ m$ is $\{y\in\mathbb{R}|y = (h\circ m)(x) > 0\}$.

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Q25 · Conics - hyperbola

1. A. $\frac{(x - 3)^2}{16} - \frac{(y - 4)^2}{9} = 1$
2. B. $\frac{(y - 4)^2}{9} - \frac{(x - 3)^2}{16} = 1$
3. C. $\frac{(x - 3)^2}{25} - \frac{(y - 4)^2}{16} = 1$
4. D. $\frac{(y - 4)^2}{16} - \frac{(x - 3)^2}{25} = 1$

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Q26 · Functions - inverses

1. A. $f(x) = 5^x$ and $g(x) = \log_5 x$
2. B. $f(x) = \frac{2x - 1}{x + 3}, x$ ≠ $-3$ and $g(x) = \frac{3x + 1}{2 - x}, x$ ≠ $2$
3. C. $f(x) = ^5\sqrt{x + 11}$ and $g(x) = x^5 + 11$
4. D. $f(x) = (x - 13)^2$ and $g(x) = x + \sqrt{13}$

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Q27 · Coordinate geometry - tangent to circle

1. A. $x + 3y - 16 = 0$
2. B. $3x + y - 16 = 0$
3. C. $-3x + y - 14 = 0$
4. D. $-x + 3y + 14 = 0$

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Q29 · Matrices - determinants

1. A. The determinant of any two-by-two matrix is non-negative.
2. B. If $H = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$ and $T = \begin{pmatrix} d & c \\ b & a \end{pmatrix}$ are any matrices, then $|H| = -|T|$.
3. C. If $P = \begin{pmatrix} x+1 & \frac{1}{2} \\ -2 & x-1 \end{pmatrix}$ is a matrix, then $|P| \geq 0$ for all $x$ in $\mathbb{R}$.
4. D. If $Q = \begin{pmatrix} m & n \\ r & p \end{pmatrix}$ is a matrix, then $|Q| = |q|$ for all $t$ in $\mathbb{R}$ and $t$ ≠ 0.

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Q30 · Matrices - scalar multiplication

1. A. $2Q = M$
2. B. $3M = \frac{3}{2}Q$
3. C. $2M = -Q$
4. D. $\frac{1}{2}Q = 2M$

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Q32 · Combinations

1. A. 20
2. B. 40
3. C. 120
4. D. 240

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Q35 · Logic - propositions

1. A. $p \implies \neg q$
2. B. $p \implies r$
3. C. $r \implies q$
4. D. $q \implies \neg p$

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Q36 · Logic - converse

1. A. If life in Ethiopia was simpler, then people in Ethiopia would go to other nations.
2. B. If life in Ethiopia was not simpler, then people in Ethiopia would go to other nations.
3. C. If people of Ethiopia would go to other nations, then life in the Ethiopia was not simpler.
4. D. If people of Ethiopia would not go to other nations, then life in Ethiopia was simpler.

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Q37 · Complex numbers - addition

1. A. $(a + bi) + (b + bi)i = (a + b) + (a + b)i$
2. B. $(x + yi) + x(i)^5 - y(i)^{10} = (x + y) + (x + y)i$
3. C. $(3+12i) + (-7 - \sqrt{169})i = -4 + i$
4. D. $(5i + 9i)^2 = 14i$

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Q39 · Linear systems - Cramer's rule

1. A. $x=\frac{\left|\begin{array}{ccc}3&1&1\\2&-1&2\\-1&1&-1\end{array}\right|}{\left|\begin{array}{ccc}3&1&-2\\2&-1&2\\-1&1&1\end{array}\right|}$
2. B. $$z=\frac{\left|\begin{array}{ccc}1&1&3\\2&-1&2\\-1&1&-1\end{array}\right|}{\left|\begin{array}{ccc}3&1&-2\\2&-1&2\\-1&1&1\end{array}\right|}$$
3. C. $$y=\frac{\left|\begin{array}{ccc}3&1&-1\\2&-1&2\\-1&1&1\end{array}\right|}{\left|\begin{array}{ccc}3&1&-2\\2&-1&2\\-1&1&1\end{array}\right|}$$
4. D. $$x=\frac{\left|\begin{array}{ccc}1&1&-1\\2&-1&2\\-1&1&1\end{array}\right|}{\left|\begin{array}{ccc}3&1&-1\\2&-1&2\\-1&1&1\end{array}\right|}$$

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Q40 · Financial math - compound interest

1. A. Birr 6554
2. B. Birr 1554
3. C. Birr 5554
4. D. Birr 1654

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Q42 · Sequences and series - sigma notation

1. A. $1+3+5+7+...=\sum_{k=1}^{\infty}(2k+1)$, where $k \in \mathbb{N}$
2. B. $({x_1 + x_2 + ... + x_n})+({y_1 + y_2 + ... + y_n})=\sum_{j=1}^{n}(x_j + y_j)$, where $n \in \mathbb{N}$
3. C. $2y_1 + 2y_2 + ... + 2y_k=2\sum_{i=1}^{k}y_i$, where $k \in \mathbb{N}$
4. D. $1 + 16 + 81 + 256 + ... = \sum_{n=0}^{\infty}(1 + n)^4$

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Q43 · Linear programming

1. A. $-\frac{150}{11}$
2. B. $4$
3. C. $-10$
4. D. $0$

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Q44 · Series - convergence

1. A. $\sum_{n=1}^{\infty}(-3)^{n}$
2. B. $\sum_{n=1}^{\infty}(\frac{1}{2})^{n}$
3. C. $\sum_{n=1}^{\infty}(\frac{1}{10})^{-n}$
4. D. $\sum_{n=1}^{\infty}(\sqrt{3})^{n}$

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Q45 · Financial math - installments

1. A. Twenty Three months.
2. B. Seventeen months.
3. C. Thirteen months.
4. D. Seven months.

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Q46 · Functions - continuity

1. A. The function $f + g$ is continuous on $[-1, 1]$.
2. B. The function $h - g$ is continuous on $[-1, 1]$.
3. C. The function $3g$ is continuous on the interval where $g$ is continuous.
4. D. The function $hg$ is continuous on the interval $(-\infty, 1]$.

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Q47 · Limits

1. A. $\lim_{x\to 2}\frac{x^3 - 8}{2 - x} = 12$
2. B. $\lim_{x\to 1}\frac{1-x^2}{x-1} = -2$
3. C. $\lim_{x\to 10}2^x = 100$
4. D. $\lim_{x\to \sqrt{5}}\frac{x^2 - 5}{x^4 - 25} = \frac{1}{5}$

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Q48 · Limits

1. A. $\lim_{x\to 0}m(x) = 0$
2. B. $\lim_{x\to -7}f(x) = -\frac{1}{5}$
3. C. $\lim_{x\to 3^+}h(x) = \sqrt{15}$
4. D. $\lim_{x\to 0^+}g(x) = 1$

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Q50 · Derivatives - concept

1. 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.
2. B. The derivative of a function $f$ at a point $x_0$ is the instantaneous rate of change of $f(x)$ at $(x_0, f(x_0))$.
3. C. The derivative of a function $g$ at a point $t$ is equal to the limit of the ratio of the difference of functional values of $g$ at $x$ and at $t$ to the difference $x - t$ as $x$ approaches $t$.
4. D. The derivative of a function $f$ at a point $x$ is the slope of a line tangent to the graph of the function at the point $(x, f(x))$.

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Q52 · Higher-order derivatives

1. A. $f^{(n)}(x)=0, \forall n\geq 6$
2. B. $g^{(n)}(x)=(-1)^n 2^n e^{-2x}, \forall n\in\mathbb{N}$
3. C. $\left(\frac{1}{2}f\right)^{(n)}(x)=0, \forall n\geq 6$
4. D. $(-g)^{(n)}(x)=(-1)^{n}2^{n+1}e^{-2x}, \forall n\in\mathbb{N}$

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Q53 · Optimization

1. A. $x=16$ and $y=4$
2. B. $x=8$ and $y=8$
3. C. $x=64$ and $y=1$
4. D. $x=32$ and $y=2$

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Q55 · Derivatives - computation

1. A. $\frac{d}{dx}(x\sin x)=\cos x-x\sin x$
2. B. $\frac{d}{dx}\left(\frac{x-\ln x}{x^4}\right)=\frac{-4x^2+x-1+4x\ln x}{x^6}, x$≠$0$
3. C. If $f(x)=\frac{\cos x}{1+\sin x}$, then $\frac{d}{dx}[f(x)]|_{x=\pi}=0$
4. D. If $h(x)=^3\sqrt{x}e^x$, then $\frac{d}{dx}[h(x)]|_{x=27}=\frac{82}{27}e^{27}$

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Q56 · Tangent lines

1. A. $y=2x+2$
2. B. $y=-2x-2$
3. C. $y=-2x+2$
4. D. $y=2x-2$

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Q58 · Integration - concept

1. A. Integral of a function is an anti-derivative of its derivative.
2. B. The process of finding $f(x)$ from $\int f(x)dx$ is called integration.
3. C. Integration is the reverse process of differentiation.
4. D. Any continuous function has only one anti-derivative.

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Q59 · Percentages

1. A. If the price of an egg reduced from Birr 13.00 to Birr 11.25, then the percent decrease is 0.142%.
2. 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.
3. 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%.
4. 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%.

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Q60 · Integration - trigonometric

1. A. $\int(\sin2x)dx = \frac{1}{2}\cos2x+C$
2. B. $\int10(\cos10x)dx = -\sin10x+C$
3. C. $\int5(\cos7x+3)dx = -\frac{5}{7}\sin7x+C$
4. D. $\int\cos(-5x)dx = \frac{1}{5}\sin(5x)+C$

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Q61 · Statistics - consistency

1. A. The most consistent test result is class P students' result.
2. B. Test result of class Q students is highly variable.
3. C. Test result of class P students is more consistent than class R students.
4. D. Test result of class S students is more consistent than class R students.

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Q62 · Definite integrals - area

1. A. 41
2. B. 20.5
3. C. 40
4. D. 21.5

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Q63 · Definite integrals - area

1. A. 8
2. B. 12
3. C. 5
4. D. 17

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Q64 · Statistics - quartiles

1. A. Data X possesses higher variability than data Y.
2. B. Data Y possesses higher variability than data X.
3. C. The inter quartile range of data X is 7.
4. D. The inter quartile range of data Y is 4.5

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Q65 · Statistics - mean deviation

1. A. Mean deviation about the mean of the data is 4.8
2. B. Mean deviation about the median of the data is 4.6
3. C. Mean deviation about the mode of the data is 4.6
4. D. Mean deviation about the mean of the data is 4.7

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