Maths 2010 – SMART ETHIO E-Learning

Welcome to your Maths 2010

1.

Which one of the following is true about signum, absolute value and greatest integer functions?

∣ X ∣ = Xsgn(x), for all x ∈ ℝ.

sgn(x) ≤ ∣ x ∣, for all x ≤ 0.

sgn(x) = ± ∣ x ∣, for all x ∈ ℝ.

sgn(x) ≤ ∣ x ∣, for all x ≥ 0

2.

What is the equation of the line that passes through (1, 1) and is parallel to the line 3y - x = 1?

3x - y = 2

3y - x + 2 = 0

x + 3y = 4

x - 3y + 2 = 0

3.

Which one of the following is an equation of the circle whose end points of a diameter are (0, -2) and (2, 2)?

x2 + y2 = 4

(x - 1)2 + y2 = 4

x2 + y2 - 2x - 4 = 0

x2 + y2 - 2y - 4 = 0

4.

What are the greatest lower bound and least upper bound of the sequence

, respectively?

-2 and 3/2

-2 and -3/2

-2 and 2

-3/2 and 2

5.

What is the maximum value of the function f(x)= x4 - 2x2 on [-2, 1] ?

8

24

12

40

6.

If the truth value of (p ∧ ¬p) ⇔ [(q ∨ ¬q) ⇒ r] is True, then which one of the following must be True?

p

¬r

¬q

q

7.

Let A and B be two events. Suppose that the probability that neither event occurs is 3/8. What is the probability that at least one of the event occur?

1/8

5/8

3/4

1/4

8.

What is the value of

9.

Which one of the following is a convergent sequence?

10.

If a function f is differentiable at a, then what is the value of

0

f '(a) - f(a)

f(a)

f '(a)

11.

. If

and (1)=5, what is f(x) ?

(x)=e^(x-1)-x³+lnx+5?

f(x)=e^(x-1)+x³3-lnx+3?

12.

If A is a square matrix of order 3 and det(A)=5, then what is the value of det(A.adj(A)) ?

125

5

25

3

13.

A salesman sold items x1, x2 and x3, with different rates of commissions as shown in the table below.

Months	Sales of Unit			Total Commission (in Birr)
Months	X₁	X₂	X₃	Total Commission (in Birr)
February	90	100	20	800
March	130	50	40	900
April	60	100	30	850

What are the rates of commission on items x₁, x₂ and x₃, respectively.

4, 11 and 2

11, 2 and 4

2, 4 and 11

4, 2 and 11

14.

If z = -3 + 4i and w= 1 + 2i, then what is the value of

3 + 5i

3 - 2i

2 + 3i

3 + 2i

15.

The population of a certain city is increasing at a rate of 3% per year. If the population was 100,000 in 2010 E.C., then what will be the population in 2020 E.C?

(Given: (1.03)⁹ =1.30, (1.03)¹⁰= 1.34, (1.3)⁹ = 10.60, (1.3)¹⁰=13 .78)

1,378,000

130,000

134,000

1,060,000

16.

What is the slope of the tangent line to the graph of f(x)= 3ex + sinx + 2 at the point (0, 5)?

3

4

2

5

17.

What is the value of

ln4 - ln3

ln2

ln3

2

18.

What is the area of the region enclosed by the graph of y2= x + 1 and y2 = -x + 1?

4/3 sq. units

3/4 sq. units

3/8 sq. units

8/3 sq.units

19.

What is the partial fraction decomposition of

20.

What is the area of the triangle (in sq. units) formed by the lines joining the vertex of the parabola x2 = -36y to the end points of the latus rectum?

216

162

126

261

21.

The time needed to type a sample of 8 business letters in an office is 7, 8, 6, 8, 9, 7, 5, 6 minutes. What are the mean

and the standard deviation(s) of the data in minute?

22.

Let A be a 3 invertible matrix and B any be any3 matrix. If |A|= a and, |B|= b, then which one of the following is NOT true?

KA|=K3 |A|, for any K ∈ ℝ

|AT A|= a2

If b = 0, then B is not invertible.

|A-1 B|= ab

23.

Let .

Then what is the value of z ?

1 - i

i

-i

-1

24.

-9/8

∞

-7/8

9/8

25.

what is the value of

2

4

1/2

1/4

26.

Which one of the following is NOT true about the function f(x)= 3x4 - 4x3?

f is concave upward on (0, 2/3) and concave downward on (-∞, 0) and on (2/3, ∞)

0 and 1 are critical numbers of f.

(0, 0) is point of inflection of f.

f is decreasing on (-∞, 1) and increasing on (1,∞).

27.

What is the value of

28.

What is the value of

2(e³ - e)

e³ - e

29.

A man running a race-course noted that the sum of the distance from the two flag posts from him is always 10 meters and the distance between the flag post is 8 meters. What is the equation of the path traced by the man?

30.

Suppose the following are the premises of an argument.

He is healthy and he is not angry.
He is angry or he is plan fails.
His plan does not fail if he does not travel abroad.
Given that the premises are true, which one of the following can be a conclusion that makes the argument valid?

He travels abroad.

His plan fails and he is angry.

His plan does not fail.

His plans fails and he does not travel abroad.

31.

The age distribution of students in a certain class is given below:

Age	10 - 14	15 - 19	20 - 24	25 - 29
No. of Students	2	10	6	7

What is the modal value of the distribution?

17.38

18.73

17.83

18.37

32.

Let A={1, 2, 3, 4, 5, 6, 7}, B={7, 8, 9} and C={8, 9, 10}. If one of the number is deleted randomly from each of these sets, what is the probability that all the three deleted numbers are even or are multiples of 3?

8/63

1/9

2/21

4/5

33.

Let A be a 33 matrix and|A|= -2. Then what is the value of |adj(A)|?

4

1/2

-2

-8

34.

If the sum of the first three consecutive terms of an arithmetic progression {An}, with An > 0 for all n, is 9 and the sum of their squares is 35, then what is the sum sn of the first n terms?

n2 - 1

n2 + 1

2n2 + 1

n2

35.

Let {an} be a sequence with a1 = a1, a2 = f(a1) = f(a), a3 = f(a2 ) = f(f(a)), an+1 = f(an), where f is a continuous function.

5n

5n-1

5

1

36.

If f(x)=1/3 x3 + cx2 + ax + 5 has a local minimum value at x = 1, then which one of the following is true about the possible value of a and c?

a = -2c - 1, c any real number.

a = -2c - 1, c < -1

a = -2c - 1, c > -1

a = 3, c = -2

37.

What is the value of

38.

Let

. Then what is the range of f(x)?

ℝ

ℝ\{-1l3}

ℝ{3}

ℝ\{2}

39.

Which one of the following is true about the graph of

?

The graph has at least one x-intercept.

The graph has y-intercept at (0, 3).

The vertical asymptote of the graph is only x = -1 and its oblique asymptote is y = 2x.

The vertical asymptotes of the graph are at x = 0 and x = -1 but it has no horizontal asymptote.

40.

A private college has 1000 students. 60% of these students are males. 45% of these students pay their payment by credit card including 175 females. What is the probability that the student is a male or a credit card user?

0.325

0.675

0.775

0.225

41.

If z = (1 + √3i)(1 + i), then which one of the following is the polar representation of z?

z = 4(cos(1050) + isin(1050))

z= 2√2 (cos(1050) + i sin(1050))

z = 4(cos(750) + i sin(750))

z = 2√2 (cos(150) + isin(150))

42.

What is the sum of the series

∞

3/6

4

3

43.

Which one of the following is equal to

44.

Let f(x) = ln(x√x). Then what is f '(x) equal to?

3/2x

2x/3

2/x√x

√x/2

45.

What is the maximum possible area of a rectangle in square units with diagonal of length 16 units?

128

256

64

48

46.

Let U = ℕ (the set of natural numbers) be the universe. Which one of the following proposition is true?

47.

What are the values of a and b so that the function

is continuous everywhere?

a = 4, b = 2

a = -4, b = -2

a = -4, b = 2

a = 4, b = -2

48.

A cylindrical tank whose inner diameter is 2m contains 4π m3 oil. If the oil discharged from the tank at the rate of (2π/3) m3/min, then how long (in min) does it take for the tank to be empty?

6

4

4/3

12

49.

Let f(x) = x - x² and g(x) = 1/x . Then what is

equals to

50.

The variance of 20 observation is 5. If each observation is multiplied by 2, then what is the variance of the resulting observations?

20

10

40

5

51.

If f(x) = klnx + esinx and f ''(π) = -1, then what is the value of k?

π

2π2

2π

π2

52.

If θ is the fourth quadrant angel and sec θ = √2 then what is cscθ equals to?

https://www.smartethio.com/wp-content/uploads/2023/10/b-24.bmp

-√2

https://www.smartethio.com/wp-content/uploads/2023/10/a-32.bmp

√2

53.

Consider the following assertion:

P+2ⁿ is an odd number for any prime p and any n ∈ N.
Which one of the following is correct about a prove or disprove of the assertion?

The assertion can be proved by the indirect method: because if n ∉ N, then 2n ∉ N and hence p + 2n is not odd

The assertion can be proved by direct method; because p is odd and 2n = 2(2n-1) is even imply that p + 2n is odd since the sum of odd and even is odd.

There is a counter example that disproves the assertion.

The assertion can be disproved by the method of contradiction.

54.

What is the standard equation of the line passing through the point (2,3) and parallel to the line given by

55.

If, in ∆ABC, AB = 3, BC = 4 and m(∠B)= 60°, then what are the length of AC and the cosine of ∠A, respectively?

56.

Let P(n) be an open proposition on the set of natural numbers (N). Which one of the following is a correct application of the principle of mathematical induction?

If p(1) is true; and p(n) ⇒ p(n+1) is true for any n ∈ N, then p(n) is true for all n?N.

If P(10) is true; and if p(n) is true implies that p(n+1) is true, then p(n) is true for all n ∈ N.

If p(10) is true; and assuming p(n) is true for any n > 10 if it follows that p(n+1) is true, then p(n) is true for all n ≥ 10

If P(1) is true for n = 1; and if both P(n) and P(n+1) are true for certain n ∈ N, then P(n) is true for all n ∈ N.

57.

If the image of the line 2x - 3y = 7 under a translation is 2x - 3y = 0, which one of the following is a translation vector of the translation line?

U = (-2, 1)

U = (1, -2)

U = (-1, 2)

U = (2, -1)

58.

Let P = (1, α, α) and Q=(α - 1, 1, 1) be two points in space and the distance between P and Q is 3. Then what is the value(s) of α?

α = 3, α = -1/3

α = -1, α =9

α = -3, α = 1?3

&alpha = 1, α = -9

59.

What is the image of the circle x2 + y2 - 4x - 6y + 12 = 0 when it is reflected with respect to the line y = -x?

(x + 2)2 + (y + 3)2 = 1

(x + 3)2 + (y + 2)2 = 1

(x - 2)2 + (y - 3)2 = 1

(x - 3)2 + (y - 2)2 = 1

60.

If the dot products of a vector

with the vectors I - j + k, 2i + j - 3k and I + j + k are 4, 0 and 2, respectively, what is

?

= (2, 1, 1)

= (-2, -1, 1)

= (2, -1, 1)

= (-2, 1, -1)

61.

In order to measure the height of a tower, suppose a surveyor takes two sightings from a transit 1 meter high which are positioned d meter s apart on the same ground level as in the figure below. If the first measured angle of elevation is α and the second is β and d ?

62.

Let l be the line given by the vector equation (x, y)=(-2, 1)+λ(1,1), λ∈R, which one of the following is the equation of the image of l after being translated by the vector u=(2, -1) followed by a rotation through 45° about the origin?

y = √2 x

x = 0

y = -2√2 x

y = √2

63.

what is the period (p) and the range of