Maths 2008 – SMART ETHIO E-Learning

Welcome to your Maths 2008

1.

Which one of the following is a one to one correspondence function from A=[0, 1] to B=[1, 2]?

f(x) = x

f(x) = 1/3x3 + 1

f(x) = x2 + 1

f(x) = 2x + 1

2.

What is the solution set of

?

{1/3}

{-1/3}

{-1, 1/3}

{3, -1/3}

3.

Suppose

, where Q(x) is a quadratic function. Which one of the following is necessarily true about the graph of f.

x = 0, x = 1 and x = -1 are vertical asymptotes of graph of f.

the vertical asymptotes of graph of f are x = 1and x = -1 if Q(x) = 2x2.

the vertical asymptotes of the graph of f is only x = -1

the graph of f doesn't intersect with its horizontal asymptote.

4.

Which one of the following is equation of a circle whose center is on y-axis and radius is 3?

(x-2)2 + y2 = 9

x2 + y2 + 6y = 0

x2 - 2x + y2 = 8

x2 + (y-2)2 = 9

5.

The planet Mercury's orbit around the sun is an ellipse with eccentricity 0.206, length of the major axis 1.16 x 108 km and the sun at one focus. What is the max distance from Mercury to the sun?

6.99 x 107 km

9.66 x 108 km

9.66 x 107 km

6.99 x 108 km

6.

The graph of a hyperbola and the lines of its asymptotes are given as shown in the following figure. Which one of the following is an equation of the hyperbola?

y2 - 2y - x2 = 0

y2 - 3y -x2 = 0

x2 - (y-1)2 = 1

(x-1)2 - y2 = 1

7.

If the truth value of a proposition p is False, then which one of the following compound proposition has a truth value True?

p ⇒ ¬p

¬p ∧ p

¬p ⇒ p

¬(¬p ∨ p)

8.

What is the contrapositive of "If x ∈ N, then x is integer and x > 0."?

if x is not integer or x ≤ 0, then x ∉ N

if x is integer or x > 0, then x ∈ N

if x ∉ N, then x is not integer and x ≤ 0

if x is not integer or x < 0, then x ∈ N

9.

Which one of the following compound propositions is a tautology?

p ∨ (q ∧ ¬q)

p ⇒ (q ∧ ¬q)

p ⇒ (p ∨ ¬q)

(q ∨ ¬q) ⇒ p

10.

The following is sample frequency distribution of data with variable x.

X	3	5	6	7
Frequency	2	5	2	1

11.

A box contains 10 items of which 3 are defective. If 2 items are randomly taken out of the box, what is the probability that both items are not defective?

49/100

4/7

7/10

7/15

12.

Items produced by accompany are subjected to two kinds of defects D1 and D2. Out of the total product, 5% have the defect D1, 10% have defect D2, and 2% have both defects. What is the probability that a randomly selected item has neither defect D1 nor defect D2?

0.87

0.98

0.5

0.13

13.

14.

Let

. If det(A)=3, then what is the solution set of the system AX = b?

{(6, -2, -8)T }

{(-3, 1, 4)^T }

∅

15.

In set of complex numbers, what is the solution set of x2 + 4x + 5 = 0?

{∅}

{2 - i, 2 + i}

{-2 - i, -2 + i}

{1 - 2i, 1 + 2i}

16.

If z = (1 + i)10, then which one of the following is equal to z?

1 + 10i

1 + 32i

32i

10i

17.

If

is an arithmetic sequence such that its 1^st term A₁ = -5 and its 5^th term A₅ = 15, then its 11^th term A₁₁ is equal to:

55

45

50

40

18.

What is the sum of all multiples of 4 that are between 30 and 301?

11,288

6,288

6,882

12,882

19.

The left hand side limit

is equal to:

2

0

1

does not exist

20.

Which one of the following is equal to

?

1

0

2

3

21.

Which one of the following is true about the horizontal asymptote(s) of the graph of

?

y = 1 & y = -1 are the horizontal asymptote of the graph.

y = 1 is the only horizontal asymptote of the graph.

y = 2 & y = -2 are the horizontal asymptote of the graph.

y = 2 is the only horizontal asymptote of the graph.

22.

If f(x) = 2x² - 3x, then

is equal to:

-1

7

∞

1

23.

If

, then f '(0) is equal to:

3/2

3 - π/2

5/2

7/4

24.

If

then which of the following is equal to f'(x)?

25.

Let f be twice differentiable function on R. which one of the following is necessarily true?

if f(x) is increasing, then f"(x) = 0 for all x ∈ ℜ

if f'(x) is increasing, then graph of y= f(x) is concave upward.

if f'(c) =0, at some c ∈ ℜ, then f has relative extreme at x = c.

if f'(x) = 0 for all x ∈ ℜ, then f(x) =0 for all x ∈ ℜ

26.

Suppose f(x) is differentiable on (-∞, ∞) and the graph of its derivative, y= f(x) is as shown in the following figure.

f(x) has a local extreme value at x = 2

f(x) is increasing on (-∞, 0) ∪ (2, ∞)

f(x) has a local maximum value at x = 0

f(x) has a local minimum value at x = -2

27.

A closed cylindrical can is to be made to hold 1000 cm3 of oil. What are the dimensions (radius r and height h) that will minimize the total surface area of the can?

28.

If

, then which one of the following is anti-derivative of f(x)?

29.

Which one of the following is equal to

?

1/x2 lnx + x sinx + cosx + c

1/3 (lnx)3 + x sinx + cosx + c

1/x2 lnx + x sinx - cosx +c

1/3 (lnx)3 + x sinx - cosx + c

30.

The volume of the solid generated when the region bounded between graph of

and x-axis is rotated about the x-axis:

32π/5

64π/5

112π/5

112π/3

31.

If f: A ⟶ B and g: B ⟶ C are functions, then which one of the following is true about the composition function?

domain of g ∘ f ⊈ domain of f

domain of g ∘ f ⊆ domain of f

range of g ∘ f range of f

range of g ∘ f ⊈ range of g

32.

If the point (3, -2) is on the graph of y = f(x), then which point is on the graph of y = f -1(x)?

(1/3,-2)

(-2, 3)

(3, -1/2)

33.

The equation of the line that passes through (2,-1) and perpendicular to the line 3x + 4y = 6 is:

-4x + 3y = -11

4x + 3y = 11

-4x + 3y = 5

4x - 3y = 5

34.

2√3

√10

2√2

3√2

35.

The nth term of the sequence: 1, -4, 9, -16,... is:

an = (-1)nn2

an = (-1)n-1n2

an = (-1)2nn2

an = (-2)n

36.

Which one of the following is true about the horizontal asymptote(s) of the graph of

y = 1 is the only horizontal asymptote of the graph.

y = 2 is the only horizontal asymptote of the graph.

y = 1 and y = -1 are the horizontal asymptotes of the graph.

y = 2 and y = -2 are the horizontal asymptote of the graph.

37.

For any n x n square matrix A, which of the following is true? Not Answered

If k is a scalar, then det (kA) = kn det (A)

Det (A) = - det (AT), where AT is the transpose of A

If B is a matrix obtained from A by interchanging of two rows of A, then det (B) = det (A)

If A is invertible, then det (A) = det (A(-1))

38.

The solution of the system of linear equation of

is:

x = -1, y = -3, z = 2

x = 1, y = 3, z =-2

x = -1, y = -3, z = -2

x = 1, y = -3, z = 2

39.

there are three children in a room, ages three, four, and five. If a four- year- old child enters the room then which one of the following is true?

Mean age will stay the same but the standard deviation will decrease.

Mean age ad standard deviation will stay the same.

Mean age will stay the same but the standard deviation will increase.

Mean age and standard deviation will increase.

40.

In how many more ways can 4 people be arranged in a row than if they were arranged in a circle?

22

6

18

1

41.

Two machines A and B work independently. The probability that both machines A and B work is 0.4. If the conditional probability that machine B works given that machine A works is 0.5, then the conditional probability that machine A works given that machine B works is

0.8

0.3

0.7

0.5

42.

Which of the following is true about the function f defined by f(x)=x2 + e2x ?

f has a relative maximum at x = 0

f is decreasing for x = 0

f has a relative minimum x = 0

f is increasing for x = 0

43.

The value of

is:

-

e

e³ -1

3

44.

14/3

21/2

16/3

7

45.

-1/2

https://www.smartethio.com/wp-content/uploads/2023/10/B-21.bmp

https://www.smartethio.com/wp-content/uploads/2023/10/A-39.bmp

1/2

46.

The sum of

15

10/3

0

5

47.

In which interval the sequence

is bounded?

48.

Which one of the following is true about the function given by the formula

f is continuous everywhere

f is continuous except at x = 0

f has an infinite discontinuity at x = 0

f has x = 0 as a vertical asymptote

49.

is equal to 😕

-1

∞

0

1

50.

is equal to 😕

2

https://www.smartethio.com/wp-content/uploads/2023/10/B-23.bmp

https://www.smartethio.com/wp-content/uploads/2023/10/A-41.bmp

2x

51.

If f(x)= 2 + ∣ x - 3 ∣ for all x, then the value of the derivative f ' (x) at x = 3

1

-1

2

does not exist

52.

The graph of y = 5x4 - x5 has a point of inflection at:

(0, 0) only

(4, 256) only

(3, 162) only

(0, 0) and (3, 162)

53.

4/3

5/4

3/4

4/5

54.

Consider the following assertion of a person and his proof "If x and y are equal positive integers, then x + y = y."

Proof: the following steps and reasons are used to proof the assertion.

Step	Reason
1. x = y	Given hypothesis
2. x² = xy	Multiply both sides of (1) by x
3. x²- y² = xy - y²	Subtract y² from both sides of (2)
4. (x - y) (x + y)=(x - y)y	Factor both sides of (3)
5. x + y = y	Divide both sides of (4) by x - y

Step 5 completes the proof.
Which one of the following is true about this proof?

The proof is invalid because Step 4 does not follow from Step 3.

The proof is invalid because Step 4 does not lead to Step 5.

It is a correct direct proof of the assertion.

If follows the technique of a proof by contradiction because the steps lead to a contradiction.

55.

Which one of the following is a valid assertion that can be proved by the principle of mathematical induction?

2n > 10n for every integer n such that n ≥ 6.

r2 > 0 for every real number r such that r ≥ 1.

2n > 8n for every integer n such that n ≥ 3.

n2 + 10n > 2n2 for every natural number n ≥ 1.

56.

The image of a figure with vertices A(1, 2), B(3, 6), C(-1, 2), and D(-2, -2) after reflection across the x-axis is:

A'(-1, 2), B'(-3, 6), C'(1, -2), and D'(2, -2)

A'(1, -2), B'(-3, -6), C'(1, -2), and D'(2, 2)

A'(1, -2), B'(3, 6), C'(-1, 2), and D'(-2, -2)

A'(1, 2), B'(3, -6), C'(-1, -2), and D' (-2, 2)

57.

If

, where A and B are distinct points in the coordinate plane, then which one of the following is equal to

?

58.

2

1/2

1/4

4

59.

60.

If cot(θ) = 2, then which of the following is equal to csc(θ)?

1/√5

2/√5

1/2

√5

61.

What is the amplitude and period, respectively, of the graph of f(x)= 4sin (x/3) cos(x/3)?

2, 2π/3

4, π/3

4, 3π

2, 3π

62.

A boat on a sea sailed from its station toward North with constant speed of 80 km/h. Another boat from the same station sailed 600 NE (North East) with constant speed of 100km/h. If the two boats started sailing at the same time, what is the straight distance between them after they have sailed for just 30 minutes?

10 √41 km

90km

10 √21km

10 √42km

63.

What is the value k, for which the two vectors

are perpendicular?

4

3

-4

-3

64.

If one of the end point of the line segment is (3, 2, -4) and the mid-point is (4, 1, -2), then the coordinate of the other end point is:

(5, 0, 0)

(3, 1, 0)

(5, 1, 2)

(2, 0, 5)

65.

What is the value of arcsin

-π/4

-π/2

π/4

π/2

66.

In set of complex numbers, what is the solution set of x2 + 4x + 5 = 0?

{1 - 2i, 1 + 2i}

{2 - i, 2 + i}

{-2 - i, -2 + i}

{∅}

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