MATHEMATICS ENTRANCE 2015 – SMART ETHIO E-Learning

Welcome to your MATHEMATICS ENTRANCE 2015

Mathematics 2015

1.

Which of the following functions is one to one?

2.

If the truth value of p is T , then which of the following compound propositions has a truth value T for any proposition q ?

(p∧¬p)∧q

(¬p∧ q)⇒ q

(p⇒ q)∧¬p

¬p ⇔ (p∨ ¬q)

3.

The object shown in the figure below is made of wood. Its top cover is a solid hemisphere and its bottom is a cylinder of base diameter 6 cm and height of 12 cm . morn the bottom of the cylindrical part a right circular cone of height 8 cm is cut off.

Thus what is the the volume of the solid in cubic centimeters

102Ω

126Ω

96Ω

72Ω

4.

A regular pyramid has a hexagonal base whose side length is 12 cm . What is the lateral surface area of the pyramid in square centimeters if the length of each lateral edge is 10 cm ?

360

720

72

288

5.

If ABCD is a parallelogram with P a mid-point of AB, and Q a mid-point of A CD,. Then which of the following is NOT necessarily true?

ΔADQ ☰ ΔCBP

APCQ is rhombus

AQ☰PC

APCD is a trapezium

6.

If tan 31° =0.81 then what is the value of tan 509°

0.81

-0.81

-0.91

0.19

7.

If sin θ -3/5 and θ is a 4^th quadrant angle, then what is the value of sec θ ?

-5/4

5/4

-5/3

5/3

8.

The slope of the line that makes an angle of 135° with the positive x -axis is equal to

1

-1

3

2

9.

The domain of the function ƒ(x)=2x^⅔

[0,∞]

ℝ\{0}

ℝ

(0,2)

10.

What is the area of a regular hexagon inscribed in a circle of radius 6 cm ?

36√3cm²

48 cm

48√3 cm²

54√3 cm²

11.

Consider the lines e₁:y=2x+3 e₂:y=1-3x then the angle between e₁ and e₂

3π/2

π/4

π/2

π/6

12.

Which of the following defines a prime number?

It is a number whose only factor is itself.

It is a number that has more than two positive factors

It is a number whose only factor is 1 .

It is a number that has only two positive factors.

13.

Which of the following rational numbers is written in its standard form?

5 x 10⁵

10.1 x 10^-2

0.8 x 10²

0.09 x 10⁵

14.

Which of the following line is true about the intercept and symmetry of the A graph of

The graph is symmetrical with respect to the origin.

its y-intercept is 3

The graph is symmetrical with respect to the y-axis.

Its x-intercepts are ±√6.

15.

The equation of the parabola with vertex V (2,-1)and focus F(-1,-1)is

(x-1)²=-12(y+1)

(y+1)²=12(x-2)

(x-2)²=-12(y+1)

(y+1)²=-12(x-2)

16.

What is the value of x in the equation

5

-5/7

-5

5/7

17.

What is the equation of the hyperbola with foci F₁ =( -2, 1), F₂=(8,1) and the length of the conjugate axis is 8 units?

18.

When the polynomial p(x)= 3x⁶+ 5x⁴- 7x³ + 2kx³+3 is divided by x+1 the remainder is 4 . What is the value of k ?

3

6

-7

-5

19.

20) A 20 m high building casts a shadow of 4 m , at the same time a 35 m tree casts a shadow of x m. What is the value of x ?

7 m

15 m

11 m

8 m

20.

which of the following statement is true about (x)=x²-6x+10 ?

ƒ(x)≥2 for all x in the domain of ƒ

the vertex of graph of ƒ is (3,1)

the line x=1 is the axis of symmetry of graph of ƒ

the graph of ƒ opens downward

21.

Given A={x ∈ 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 ?

{(1,1),(1,2)(13,1),(13,2)}

{(1,2),(13,1)(13,2)}

{(1,1),(13,2)}

{(1,1),(2,1)(1,13),(2,13)}

22.

The solution set of the equation 3|2x-4|=6-2|1-5x| is

23.

The solution set of the inequality (2x-3)(x+5)≤0 is

24.

The solution set of the equation 5 log₉ x-2 log₉ (x²)-log₄ 8=- is

{9}

{4}

{3/2}

{3}

25.

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?

The highest frequency is 3 .

The mode of the data is 22 °C.

for 45% of the days, the temperature was more than 19 °C.

The temperature was 24 °C for two days.

26.

A car started moving 9 km to the East then move 14 km to the North and then 12 km towards West to reach its destination. How far the car it travelled from its starting point?

5 km

25 km

4 km

12 km

27.

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 is to be formed. If a student is chosen student is either from grade 10 or grade

7/15

5/15

12/15

1/7

28.

3,2

0,-1

-1,2

2,-2

29.

3

2

1

0

30.

A fair die is going to be rolled once, and let;
E₁ be the event that an even number occurs:
E₂be the event that a prime number occurs;
E₃ be the event that a multiple of 3 occurs;
E₄ he the event that a multiple of 5 occurs Then which of the following pairs of events are mutually exclusive ?

E₂and E₄

E₁and E₂

E₃and E₄

E₁and E₃

31.

The following is the distribution of the weight of 40 students in a certain class.

Then for data given above, what is the median weigh in kg ?

53.8

53

55

54.5

32.

In a certain month of the year, the temperature of a city was recorded and presented as follows.

Which one of the following is the cumulative frequency of the fourth class?

22

27

20

18

33.

34.

On the set of Natural numbers ℕ, which one of the following propositions D has a truth value True?

35.

-3/2

1

3/2

4

36.

4ℼ/3 and 2ℼ/9

3ℼ/2 and 2ℼ/9

ℼ/9 and 4ℼ/9

2ℼ/ and 2ℼ/9

37.

Which of the following functions has a graph that completes one full cycle on

38.

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?

724

846

3565

2635

39.

what is the fifth term of the sequence {2ⁿ + 4n - 5}^∞_{n=0 ?}

47

27

15

17

40.

Let the sequence {a_n} converges to 3 while {b_n} converges to -1 . Then the sequence {a_n-b_n} converges to

4

3

1

1/3

41.

Let A be a square matrix of order n and let B be a matrix obtained by multiplying one row of A by 2 and C be a matrix found by adding one row of A to another of A. If let (A)=10 what are let (B) and (let (c) respectively?

10 and 20

10 and 10

20 and 10

20 and 0

42.

3/2

1

4

0

43.

Let z₁,z₂ and z₃ be complex numbers. Then which of the following is NOT necessarily true?

44.

Which of the following is true about the graph of y=3sin (4x)

The graph rises up on [0, π/4]

The graph completes one cycle the interval [0,π]

The graph completes one cycle on the interval [0,π/4].

The graph lines below the x-axis on(π/4, π/2)

45.

which of the following complex numbers corresponds to the point (-3,-1) in the Argand plane?

-3-i

-3+i

-1-3i

3+i

46.

the solution set of the system
{
  2x - y + 3z = 1
  x + y + 3z = is
  x - y + z = -1
}

{ (k + 1, 2k, k - 1) : k ∈ ℝ }

{ (k, k + 1, k - 1) : k ∈ ℝ }

{ (2 - 2k, 3 - k, k) : k ∈ ℝ }

{ ( -4 - 2k, k, 2 - k ) : k ∈ ℝ }

47.

If a translation T takes a point (-1, 1) to a point (3, 2), then the image of the circle x² + y² - 2x + 4y + 1 = 0 under T is

(x - 2)² + (y + 3)² = 4

(x - 5)² + (y + 1)² = 4

(x + 3)² + (y + 3)² = 4

(x - 3)² + (y - 3)² = 4

48.

Let u = (-1,6) and v = (2,4) be position vectors. Then the vector 4u - 3v is

(-15, 18)

(-10,11)

(-10, 12)

(-4, 24)

49.

Let F be the anti-derivative of f. Then which of the following is NOT true about F and f?

F'(x) = f(x)

∫ (d/dx)f(x) dx = f(x) + c

∫ F(x) dx = f(x) + c

d/dx ∫ F(x) dx = F(x)

50.

Which of the following assertions is true applying the Principle of Mathematical Induction?

3 + 5 + 7 + ... + (2n+1) = n² for n ≥ 1

n! ≥ 2ⁿ for n ≥ 0

1 + 2 + 2² + ... + 2ⁿ = 2^(n+1) - 1 for n ≥ 0

2 + 3 + 4 + ... + n = n(n+1)/2 + 1 for n ≥ 2

51.

What is the area of the region bounded by the graph of f(x) = x² - 4 and the x-axis in square units?

32

16

32/3

16/3

52.

Which of the following set of equations define the xz-plane?

{(x,y,z) | x,y,z ∈ ℝ and x = z = 0}

{(x,y,z) | x,y,z ∈ ℝ and z = 0}

{(x,y,z) | x,y,z ∈ ℝ and x = 0}

{(x,y,z) | x,y,z ∈ ℝ and y = 0}

53.

Let f be a continuous function on [0, 1] and let F be an anti-derivative of f. D with F(1) = -11 and F(0) = 11, then ∫₀¹ f(x) dx is equal to

0

22

11

-22

54.

Which of the following is equal to ∫ 3 cos 3x dx ?

sin x + c

sin 3x + c

(1/3) sin 3x + c

-sin x + c

55.

hat is the coordinates of the midpoint of the line segment joining (-1, -3, 3) and (-1, 5, -7) in space?

(1, 4, 3)

(-1, 1, -2)

(-1, 3, -9/2)

(-1, 3, 5)

56.

What is the value of a number in the interval (0,1) that satisfies the Mean Value Theorem for the function f(x) = 3x² - 5x + 1 ?

1/2

5/6

3/5

1

57.

Which of the following arguments is valid?

p ∨ q, p → q ⊢ ¬q

P → q, ¬q ∧ q ⊢ p ∨ r

P, P → q, r ∨ ¬q ⊢ p ∧ ¬r

P → q, q ⊢ r → p ⊢ r

58.

Which of the following is true about the zeros of a function f(x)?

If f(a) = 0, then (x - a)² is a factor of f(x).

If f(a) = 0, then a is not necessarily in the domain.

If f(a) = 0, then x - a is a factor for f(x).

If f(a) = 0, then y = a is the y-intercept.

59.

Water is running into a conical tank which is pointed down at a constant rate of 6π cm³/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?

6 cm/sec

6/π cm/sec

4 cm/sec

4/π cm/sec

60.

Let f(x) = 5x³ + 2x - sin x + eˣ. What is the value of f'(0) ?

4

-1

2

1

61.

Which one of the following is the value of α so that the function given by f(x) = { αx + 1 for x ≥ 1; x² - 4 for x < 1} is continuous at 1?

4

-1

-4

1

62.

Let f(x) = 1/√x and g(x) = 2 - (1/6)x³ be two functions. For what values of x do the graphs of f and g have tangent lines of the same slope?

1

1/4

4

√3/2

63.

Which of the following describes the mechanical meaning of derivative?

Area under the curve

Instantaneous rate of change

Slope of tangent line

Gradient of graphs of functions

64.

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)?

6

4

1

2

65.

If x²+14 / (x+2)(x-1)² = A/(x+2) + B/(x-1) + C/(x-1)², then what are the values of A, B and C respectively?

2, -2, -5

2, -1, 5

3, -2, 6

1, -1, 3

Mathematics 2015

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