ETHIOPIAN UNIVERSITY PRE - ENTRANCE

MATHEMATICS EXAM.

SUBJECT CODE: 03
TIME ALLOWED: 3 HOURS

GENERAL DIRECTIONS

THERE ARE A TOTAL OF 65 MULTIPLE CHOICE QUESTIONS. CAREFULLY SELECT THE

BEST ANSWER AND CHOOSE ONLY THE LETTER OF YOUR CHOICE ON YOUR DEVICE.

YOU CAN PASS A QUESTION THAT YOU ARE UNCERTAIN ABOUT AND BACK LATER IF
YOU HAVE TIME TO CHECK AGAIN. BUT IF TIME IS UP, QUESTIONS THAT HAVE BEEN
ANSWERED ON TIME WILL GET SUBMITTED AND THE BOT WILL MARK ONLY IT.

YOU ARE ALLOWED TO WORK ON THE EXAM FOR 3 HOURS.

TARGETED GROUP: NATURAL SCIENCE STUDENTS

PREPARED BY FETENA NET

- @fetena_net
1) Let f(x)=3x+1 and g(x)=2x², then which of the following is TRUE about fog(x)?

A/ Domain of fog(x) ç Domain of f(x)

B/ Range of fog(x) ç Range of f(x)
C/ Range of fog(x) ç Domain of f(x)
D/ Domain of fog(x) ç Range of f(x)

2) The inverse of the function y=x²+10x+13 is:

A/ ⎷(x+10) - 13
B/ ⎷(x+12) - 5
C/ ⎷(x+5) - 13
D/ √( x+12) + 5

3) Which one of the following is TRUE about the graph of f(x)=(x⁴)-1/(x²+1)(x-1) ?

A/ y=0 is its Horizontal Asymptote

B/ x=1 is its Vertical Asymptote
C/ x+1 is its Oblique Asymptote
D/ The graph has hole at x=1

4) The general term of a sequence is aₙ={½}¹⁻ⁿ. Then which of the following is TRUE about the
sequence?

A/ Its terms are even numbers

B/ 0 is one of its term
C/ Its terms are natural numbers
D/ Its terms are non-positive.

5) What is the sum of the infinite series ∑ 7[(3ⁿ+2ⁿ)/6ⁿ]?

A/ 5/6
B/ 35/6
C/ 49/2
D/ 28/3

6) What is the slope of the tangent line to the graph of f(x)=5eˣ + sin(x) + 7x at (0,5)?

A/ 13
B/ 3
C/ 11
D/ 5
7) What is the period and range of f(x)=3sin(¹/₅x + 3)+8 respectively ?

A/ 5π , [5,11]
B/ 10π , [5,11]
C/ 5π , [11,5]
D/ 10π , [5,5]

8) Let f(x)=⁵ˣ⁺³/₂ₓ₊₁ ,then what is the range of f(x)?

A/ ℝ\0
B/ ℝ\-½
C/ ℝ
D/ ℝ\(⁵/₂).

9) Which one of the following statement on the set of real numbers is TRUE?

A/ ∀y∃x(x²+y²=1)
B/ ∀x(x²+1=0)
C/ ∃x∀y(x=y+7)
D/ ∀x∃y(y=x+3)

10) If x = arc tan(3), then the value of cos2x________

A/ ²/√₁₀
B/ ¹/√₅
C/ ³/√₁₀
D/ ²/√ ₅.

11) Which of the following represents equation of line tangent to circle x²+y²+2x+4y+1=0 at point (0,-
4)?

A/ 2y+4-x=0
B/ 2x-y-4=0
C/ 2y+8-x=0
D/ y-½x+8=0

12) Which one of the following represents equation of a circle whose center is on x-axis and radius 2 ?

A/ x²+y²-4y+4=6
B/ x²+y²-4x=4
C/ x²+y²-4y=8
D/ x²+y²-4x-8=0
13) Which of the following is NOT TRUE about the function f(x)=4x/(x²+2 )?

A/ (-√2,√2) are critical numbers

B/ f is increasing between [√-2,√2)
C/ f is decreasing between [-√2, √2]
D/ √2 is local maxima .

14) If f(x)=²/ₓ₋₁ and fog(x)=²ˣ/₁₋ₓ , then g(x)=__________

A/ ¹/ₓ₋₁
B/ ¹/ₓ
C/ ²/ₓ
D/ ²/x-1.

15) Let p,q and r are propositions such that p⇒(r∨¬q) has truth value false. Then which of the
following propositions has truth value True?

A/ q⇔r
B/ r⇒q
C/ ¬pvr
D/ ¬p∧q.

16) If {an} is a sequence such that a₁=3 and aₙ₊₁=an+2 for all n≥1 then ∑10anⁿ⁼¹ is equal to:________

A/ 24
B/ 12
C/ 144
D/ 21.

17) which one of the following is equal to lim(5x/5x+4)⁻⁵ˣ ?

A/ e⁻⁴
B/ e²
C/ e⁴
D/ e⁵.

18) f(x) =e²ˣ+2x² - 2cos(x) then what is f''(x)?

A/ 4+2cosx
B/ e²ˣ+4-2cosx
C/ 4e²ˣ+4-2cosx
D/ 4e²ˣ+4+2cosx
19) The simplest form of (2-3i/3+2i) + conjugate 1-4i is______

A/ 1+3i
B/ 1-3i
C/ 3-2i
D/ 3+2i

20) If U is a unit vector in the direction of A and |A| =16, then A.U is equal to __

A/ 4
B/ 1/16
C/ 16
D/ 1/4

21) If a point p(3,6) is reflected under a line to the point (1,-2) , what is the line of reflection ?

A/ y-4x=6
B/ 4y-x =6
C/ y-4x=-6
D/ y+4x=6

22) What is the equation of the tangent line to the graph of f(x) =x³+2x-5 at (1,2) ?

A/ 5y-x=5
B/ y-5x=3
C/ y-5x=-3
D/ 3y-2x=6

23) d/dx ln(√x²-4) ______

A/ ⁻ˣ/√x²⁻4
B/ ˣ/x²-4
C/ ²ˣ/x²-5
D/ ˣ/√x-5

24) How many possible ways a committee of 2 members can be formed from 6 members ?

A/ 21
B/ 15
C/ 12
D/ 34
 25) cos[sin⁻¹(x)] is equal to :

A/ √(1-x²)
B/ 1/1-x²
C/ √(x²+1)
D/ √(1-x²).

26) The proposition ¬p⇒q is equivalent with __

A/ ¬(¬p∨¬q)
B/ ¬(p∧q)
C/ p∨q
D/ p∧q

27) Which of the following propositions are NOT equivalent to each other?

A/ p⇒q≡q⇒p
B/ ¬(p∨q)≡¬p∧¬q
C/ ¬p⇒¬q≡p⇒q
D/ p∧q≡q∧p.

28) If there are 5 red and 2 black balls in a box and if 2 balls are drawn one after another without
replacement, the probability of getting both balls are black is:

A/ 2/7
B/ 1/42
C/ 12/7
D/ 2/42.

29) What are the greatest lower bound and least upper bound of the sequence (-1ⁿ)(1+¹/ₙ) respectively?

A/ -2 and 2
B/ -2 and -3/2
C/ - 3/2 and 2
D/ -2 and 3/2.

⃗A Let⃗
30)AB and B⃗⃗⃗ be unit vectors A=(2i,3j) and B=(-i,2j), then 3⃗ ⃗- will be:

A/ 5i+11j
B/ 7i-11j
C/ 5i+7j
D/ 7i+7j.
31) For what value of k that the system [kx-2y=5] - [-3x+3y=2] has a unique solution?

A/ k≠-2
B/ k≠2
C/ k=2
D/ k=-2.

32) If f(x)=²/ₓ₋₁ and fog(x)=²ˣ/₁₋ₓ , then g(x)= ___

A/ ¹/ₓ₋₁
B/ ¹/ₓ
C/ ²/ₓ
D/ ˣ⁻¹/x .

33) On which of the following intervals is the graph of f(x)=ln(x²+1)?

A/ (-∞, -1]∪[0, ∞)
B/ [0,∞)
C/ (-∞,0]
D/ [-1,1] .

34) The graph of f(x)=x³-2x²+1 has an inflection point at:

A/ 4/3
B/ 2/3
C/ 23/27
D/ 8/27 .

35) ∫e²ˣ-4x/xe²ˣ is equal to:

A/ ln|x|+2e⁻²ˣ+c
B/ ln|x|-4e²ˣ+c
C/ ln|x|+4e⁻²ˣ+c
D/ 1/x² +2e⁻²ˣ+c.

36) LetA be 2x2 matrix and |A|=8.Then what is the value of |adj(A)|?

A/ 16
B/ 64
C/ 1/8
D/ 8 .
37) The variance of 8 observations is 4. If each observation is multiplied by 3 and 5 is added on old
variance, what is the new variance?
A/ 36
B/ 12
C/ 4
D/ 16

38) Which of the following series is divergent?

A/ ∑(3/2ⁿ)
B/ ∑3ⁿ⁻²
C/ ∑3(¹/₂)ⁿ
D/ ∑(¹/ₙ)

39) Which of the following is the domain f(x)=csc(x) for k∈Z?

A/ (x∈R X≠kπ)
B/ (x∈R x=(2k+1)π/2)
C/ X∈R
D/ X∈R X=(k+1)π

40) The equation of the directrix of parabola y-1=x²+2x is:

A/ x=-1/4
B/ y=-1/4
C/ x=1/4
D/ y=1/4 .

41) The distance between point (2,-1) and line x-2y-3 is:

A/ √5/5
B/ √5
C/ 2/√5
D/ √5/3

42) The simplified form of (¹/ₓ₊₁)-(¹/ₓ²₋₁) __________

A/ (X+1)²
B/ (1/x) - (²/ₓ₋₁)
C/ 1/(x+1)²
D/ ¹/x²
43) In the set of complex numbers,what is the solution set of x²+4x+5=0?

A/ {1-2i,1+2i}
B/ {2-i,2+i }
C/ {-2-1,-2+i}
D/ ∅

44) The simplified form of i(1+√3i )⁶ - 65i is equal to ____________

A/ 1
B/ i
C/ -i
D/ -1

45) Which of the following relation holds the sequence: -10,-3,4,11, ...

A/ an=aₙ₋₁ + 8
B/ an=aₙ₋₁ - 7
C/ an=aₙ₋₁ - 8
D/ an=aₙ₋₁ + 7

46) The derivative of f(x)=2π²+3π+7 is equal with:

A/ 4π+3
B/ π³+3
C/ 2
D/ 0

47) Which of the following is the distance from the center to one of foci of hyperbola (y- 3)²/16+
(x+1)²/9 =1?

A/ 10
B/ 5
C/ 6
D/ 8 .

48) The angle between vectors U=i-j and V=2j is:

A/ π/4
B/ π/2
C/ 3/4π
D/ 4/3π
49) Let line PQ be a vector with initial point P=(1,5) and terminal point Q=(4,0). If V=x⃗i+2⃗j is
parallel to line PQ, then what is th
e value of x?

A/ 3
B/ -6/5
C/ -3
D/ -5/6

50) The asymptote of the hyperbola: y²-x²+4x-5=0 is ___________

A/ y=-x+2
B/ y=x+2
C/ y=-x-2
D/ y=x-4.

51) A business association gets net profit of birr 3,000 at the end of each month. Just after the fifth
year,its amount was birr 230,000.What was the starting amount of the business association?

A/ 56,000
B/ 53,000
C/ 47,000
D/ 50,000.

52) ∫²(x+1)¹/³ dx?

A/ 39/2
B/ 81/2
C/ 9/2
D/ 78/3

53) What are the values of U and V that satisfy the equation u+3i/4-2i=2+vi/20?

A/ U=2 ,V=3
B/ U=6 , V=10
C/ U=2 , V=16
D/ U=-4 , V=6
54) If the truth value of the proposition P is false, then which one of the following compound
proposition has a truth value True?

A/ If X is not integer or X≤0 ,then X∉N

B/ If x is ∉N,then X is not an integer & X ≤0
C/ If X is an integer or X>0,then X∉N
D/ If X is not integer or X≤0
55) If angle θ is an acute angle of a right triangle, what is the length of the side adjacent to ,given the
hypotenuse has 6 units length and sec θ=10/3?

A/ 1.8 units
B/ 2 units
C/ 18 units
D/ 20 units

56) If F(x)=f(2x+2), g(1-x²) with f´(2)=4, f(2)=-3, g(1)=-5 and g'(1)=1, then what is the actual value of
F'(0)?

A/ 40
B/ -20
C/ 0
D/ 19.

57) What is the vector equation of the line passing through (-1,4) and parallel to the line y- 3x-10=0?

A/ (x,y)=(0,7) + t(1,3)
B/ (x,y)=(1,10) + t(1,3)
C/ (x,y)=(0,7) + t(1,-3 )
D/ (x,y)=(-1,4) + t(1,3)

58) What two positive numbers whose product is 98 and sum is minimum?

A/ 40&58
B/ 38&60
C/ 49&49
D/ 28&70

59) What is the maximum value of function x⁴-2x² on [-2,1]?

A/ 16
B/ 8
C/ 24
D/ 64

60) Air is being pumped in to a spherical balloon so that its volume increase at a rate of 100cm cube/s.
How fast is the radius of the balloon increases when the diameter is 10cm?

A/ 1/π
B/ 2π
C/ π
D/ 3/4π
61) lim (3ˣ⁺²)/4ˣ⁺²) as x⇒∞ is:

A/ 9/8
B/ 3/4
C/ ⁹ /₁₆(3ⁿ/4ⁿ)
D/ 0

62) lim(x-π/₂)/cos(x) as x⇒π/2?

A/ 0
B/ -1
C/ -1
D/ π/2

63) If A= 2x -4 × 2 , then det(2A)=2x² what is the value of x?

A/ 16
B/ 8
C/ 4
D/ 32

64) Which one of the following is equal to lim [√(x)-1]/x²-1 as n⇒1?

A/ -1/4
B/ 1/2
C/ 1/4
D/ 0

65) f(x)=cos²x and h(x)=(x³+5)/(x+2), then

1
∫(cos²x+x³+5/ x+2) is ________
0

A/ sin2+24
B/ sin(2)/4 + 23/6 - ln2/ln3
C/ sin(2)/4 +23/6 + ln2/ln3
D/ 25.