GRADE 12 MATHEMATICS N.scMODEL EXAMINATIONS 2015

Welcome to your GRADE 12 MATHEMATICS N.scMODEL EXAMINATIONS 2015

ADDIS ABABA CITY ADMINISTRATION EDUCATION BUREAU GRADE 12 MATHEMATICS MODEL EXAMINATIONS FOR NATURAL SCIENCE 2015 / 2022 NUMBER OF QUESTIONS: 65 TIME ALLOWED:

1.

Which of the following is true about the relation R given by the graph below?

(1,0) is belongs to R⁻¹.

The relation R= {(x, y): y − x ≥ 1and y + x > 1, x, y ∈ ℜ}.

The range of R −¹ = (0, ∞) .

Domain of R = (1, ∞).

2.

Which one of the following functions is a power function?

f (x) = x −¹

f (x) = x ² + 1

f (x) = e ˣ

f (x) = x ˣ

3.

Which one of the following functions is a one-to-one correspondence?

f : [1, ∞) → [0, ∞), defined by f (x) = (x − 1)2 + 1 .

f : [1, ∞) → (0, ∞), defined by f (x) = x − 1 .

f : (0, ∞) → (0, ∞), defined by f (x) = log₃ x .

f : [0, ∞) → [0, ∞), defined by f (x) = x ⁴ .

4.

If f (x) = (x − 2) / (x − 1) and h(x) = f (1/x), then which of the following is not true?

Domain of h is ℜ \ {0,1}.

The line x = 0 is a vertical asymptote to the graph of h .

h(1/x) = f (x) , for x ≠ 0 and x ≠ 1.

h(x) = (1 − 2x) / (1 − x), for x ≠ 0 and x ≠ 1.

5.

Which of the following is the universal set of the expression (x ⁴ − 8x) / (3x ³ − 2x ² − 8x)?

ℜ / {− 4/3 }

ℜ / {− 4/3 ,0,2}

ℜ /{−2, 4/3 ,0}

ℜ /{0,2}

6.

Suppose that ℓ₁ and ℓ ₂ represent two perpendicular lines in a plane. If ℓ₁ passes through the point (0, 4) and the equation of ℓ ₂ is 2x − y − 3 = 0 , then what is the equation ofℓ₁ ?

2y + x = 8

y − 4x + 3 = 0

x − 2 y − 4 = 0

2x + y − 8 = 0

7.

What is the equation of the circle whose center is at (2,-3) and which passes through the intersection of the lines 3x + 2y =11 and 2x + 3y = 4 ?

x² + y² − 4x + 6 y + 3 = 0

x² + y² + 4x + 6 y + 3 = 0

x² + y² − 4x + 6 y − 3 = 0

x² + y² + 4x + 6 y − 3 = 0

8.

What is the focus of the parabola y² − 8x + 6 y +1 = 0 ?

(8,0)

(-8,6)

(2,0)

(1,-3)

9.

If base of a triangle is the major axis of the ellipse x²/16 + y²/9 =1 and third vertex moves on the ellipse, then what is the maximum area of the triangle?

24

6

72

12

10.

Suppose a proposition p ⇒ ¬q is false (F), then which of the following is true?

¬q ∧ (p ⇒ q)

(p ∨ q)⇔ ¬p

¬p ∨ (q ⇒ ¬p)

(¬q ∨ p)⇔ q

11.

Which of the following is a valid argument?

¬p ⇒ ¬q, q − ¬p

p ⇒ q, ¬q ⇒ r − p

p ⇒ ¬q, p, r ⇒ q − ¬r

¬p ∨ q, r ⇒ p, r − ¬q

12.

What is the negation of the statement “If it is raining, then you take your umbrella.”?

It is not raining and you do not take your umbrella.

It is not raining and you take your umbrella.

If it is raining, then you do not take your umbrella.

It is raining and you do not take your umbrella.

13.

The following table gives the age distribution in a certain class. Which of the following is not true about the data ?( to the nearest whole number)

The mode is 21

The fifth deciles (D₅) is 20

The standard deviation is 15

The mean is 19

14.

If Qi, Di and Pi are respectively the iᵗʰ quartile, decile and percentile of data arranged in an increasing order, then which one of the following is necessarily true?

P25>Q1

Q2=(Q1 + Q3)/2

D3>P25

Q2=mean of the data

15.

16.

A bag contains 3 red, 4 blue and 3 white balls. Three balls are drawn one after the other. What is the probability of getting a red ball on the first draw, a blue ball on the second draw and a white ball on the third draw if the balls are drawn without replacement?

4/5

9/250

1/20

3/80

17.

What is the constant term in the expansion of (2/x + 3x)^6?

540

4320

216

160

18.

A student needs to select 3 books from 3 different mathematics, 3 different physics and 1 history book. What is the probability that one of them is mathematics and the other two are either physics or history books?

3/35

15/35

18/35

9/25

19.

What is the value of x such that the matrix A = [[x, -3, x], [-1, 0, 2], [2, 1, -5]] is singular (no inverse)?

-2

2

1

-1

20.

What should be the values of a and b so that the system {x + y + z = 0, 3x - ay - z = 4, x + 5y + 5z = b} will have no solution?

a ≠ 1, b = 4

a ≠ -1, b = -1

a = 1, b ≠ -4

a = -1, b ∈ R

21.

Suppose A and B be 3x3 matrices such that A = [[2, 0, 0], [-4, 6, 0], [0, -1, 2/3]] and |B| = 1/4, then which of the following is equal to |3A^T B|?

27/32

54

18

6

22.

Let A = [[2, 3, 5], [4, 5, 7], [8, -3, 3]] and B = [[-4, 3, -5], [2, 1, 7], [6, 5, 0]]. If C = AB + A^T, then which of the following is the entry C23?

28

33

12

15

23.

Which one of the following is the conjugate of |1 - 4i| - (2+i)(3-i)/(3+4i)?

-4 - i

4 - i

-1 - i

4 + i

24.

In the set of complex numbers, what is the solution set of x^2 + 2x + 5 = 0?

∅

{-2 - 4i, -2 + 4i}

{-1 - 2i, -1 + 2i}

{1 - 2i, 1 + 2i}

25.

If z = 2 + 2√3i, then which one of the following is the polar representation of z?

4(cos(π/6) + i sin(π/6))

4(cos(π/3) + i sin(π/3))

16(cos(π/6) + i sin(π/6))

16(cos(π/3) + i sin(π/3))

26.

A contest offers 10 prizes with a total value of Birr 13,250. If the difference in value between consecutive prizes is Birr 250, what is the value of the first prize?

Birr 2,450

Birr 2,400

Birr 200

Birr 2,700

27.

Suppose f₁ = 3, and for n = 2,3,4,..., fₙ = 1 + fₙ₋₁. Which of the following is true?

For any natural number n, f(fₙ) = n + 6.

Σᵢ₌₁ⁿ fᵢ = n(n+5)/2

If fₙ is an odd composite number, then n is even.

f₃₇ is a prime number.

28.

Which of the following is the sum of the series 1/2 - 1/3 + 2/9 - 4/27 + ...?

5/6

3/10

∞

3

29.

Which one of the following is not true about the arithmetic progression whose sixth term and tenth terms are 18 and 30 respectively?

The arithmetic mean of the first 10 terms is 30.

The sum of the first one hundred terms is 15,150.

The common difference is 3.

The first term is 3.

30.

Which of the following is true about the sequence {2 - (-1)ⁿ/(n+1)}ₙ₌₁∞?

8/5 is its greatest lower bound.

The sequence converges to the least upper bound.

The sequence converges to 2.

The sequence is divergent.

31.

Which of the following is equal to limₙ→∞ (3-4n)²/(2-√n-2n²)?

2

-8

3/2

8

32.

Which of the following is equal to limₓ→₄ (√x - 2)/(x - 4)?

1/4

4

does not exist

0

33.

Which one of the following is true about the function

f(x) = { (x²-9)/|x-3|, if x ≠ 3,

{ 6, if x = 3

f is continuous everywhere.

f is continuous from the right at x=3.

f is continuous from the left at x=3.

f has x=3 as a vertical asymptote.

34.

Which of the following is equal to limₓ→₀ sin(2x³)/sin³(4x)?

1/2

1/8

2

1/32

35.

Which of the following is equal to limₓ→∞ (2x/(2x+4))⁻⁵ˣ?

e⁻⁵/²

e¹⁰

e²⁰

e⁻¹⁰

36.

If f(x) = ln x, then what is the value of f⁽ⁿ⁾(x)?

f⁽ⁿ⁾(x) = (-1)ⁿ⁺¹ (n+1)!/xⁿ⁺¹

f⁽ⁿ⁾(x) = (-1)ⁿ⁺¹ (n-1)!/xⁿ

f⁽ⁿ⁾(x) = (-1)ⁿ n!/xⁿ⁺¹

f⁽ⁿ⁾(x) = (-1)ⁿ⁺¹ n!/xⁿ

37.

If f(x) = 2x³ - 6x + 2 ln x, then which of the following is equal to limₓ→₁ (f(x) - f(1))/(x-1)?

2

1

-4

0

38.

Which one of the following is not true about the derivative of f(x) = x|x|?

f is differentiable at x = 0.

f'(-1) = -2

f'(x) = 2, at x = 1.

f'(x) = 2|x|, for every x ∈ (-∞, ∞).

39.

If F(x) = f(x² + 1) + f(1-x)/cos(ln x), f'(2) = 10, f'(0) = f(2) = -4 and f(0) = 6, then F'(1) is equal to

-12

24

16

-4

40.

Which one of the following is the equation of the tangent line to the graph of

f(x) = { x - 2, if x ≤ 0
{ -2 sin x, if x > 0
at the point (π, f(π))?

y = 2x - 3π

y = -2x - 3π

y = x - 2π

y = 2x - π

41.

Which one of the following is not true about the function f(x) = 5x⁴ - x⁵?

f is concave downward on (3, ∞).

f is decreasing on (-∞, 0] and [4, ∞).

f has a local maximum value at x = 4 and local minimum value at x = 0.

(0, 0) and (3, 162) are inflection points of f.

42.

A box is to have a square base, an open top and volume of 32m³. What is the dimension (base (x) and height (h)) of the box that uses the least amount of material?

x = 2 m and h = 4 m

x = 2∛4 m and h = 8∛2 m

x = 8√2 m and h = 2∛4 m

x = 4 m and h = 2 m

43.

The height of rectangular box is 10 m. Its length increases at the rate of 2 m/sec, its width decreases at the rate of 4 m/sec. When the length is 8 m and the width is 6 m, then at what rate the volume of the box is changing?

-200 m³/sec

200 m³/sec

-20 m³/sec

80 m³/sec

44.

For what value of c the conclusion of mean value theorem is satisfied for the function f(x) = x³ + x - 1 on [0, 2]?

±2√3/3

-2√3/3

2√3/3

2√3

45.

Which one of the following is the set of all critical numbers of f(x) = 2√x(6-x)?

{-2, 0}

{0}

{0, 2}

{2}

46.

Which of the following is equal to ∫₀¹ e⁻²ˣ dx?

1/2(e-1)

2(e-1)

1

-1/2(e-1)

47.

Which of the following is equal to ∫ sin(2x)/(2 sin x) dx?

1/2 cos x + c

sin x + c

1/2 sin x + c

-sin x + c

48.

Which of the following is equal to ∫ x³/((x+1)(x+2)) dx?

ln|x+2| - 8 ln|x+1| + c

8 ln|x+2| - ln|x+1| + c

x²/2 - 3x + 8 ln|x+2| - ln|x+1| + c

x²/2 + 3x + ln|x+2| - 8 ln|x+1| + c

49.

If F(x) is an anti-derivative of f(x) = 2x + √x e^√x and F(0) = 6, then F(4) is equal to

-12e⁴ + 32e² + 2

-20e² + 18

20e² + 18

4e² + 18

50.

What is the area of the region bounded by the lines y = e, y = e², x = -1 and the curve f(x) = 1/x?

e² - e + 1

1

-e² + e + 1

e² - e - 1/e + 1

51.

What is the volume of the solid of revolution about the x-axis generated by revolving the region enclosed by the curve y = x² and the line y = 4?

64/5 π

384/5 π

256/5 π

640

52.

Suppose a and b are vectors with |a| = 4, |b| = 3 and the angle between a and b is π/3, then what is the cosine of the angle between a - b and a?

5/√13

5/(2√13)

10

√13

53.

What is the image of the line given by (x, y) = (-1, 0) + t(3, 6) under the translation that takes (1, 0) to (0, 1) followed by the reflection about the line y = 2x?

y = 2x + 3

y = 2x + 6

y = 2x - 3

y = 2x - 5

54.

What is the image of the circle (x-3)² + (y+5)² = 1 when it is rotated through 50π/3 about the point (4,-3)?

(x - 7/2 - √3)² + (y + 4 + √3/2)² = 1

(x + 7/2 + √3)² + (y - 4 - √3/2)² = 1

(x + 7/2 - √3)² + (y - 4 + √3/2)² = 1

(x - 7/2 + √3)² + (y + 4 - √3/2)² = 1

55.

Which of the following is the vector equation of the line passing through the points (3,5) and (-2,3)?

(x,y) = (3,-2) + t(-5,-2)

(x,y) = (2,-3) + t(-5,2)

(x,y) = (3,5) + t(-5,-2)

(x,y) = (3,5) + t(-5,2)

56.

Which of the following is true about the graph f(x) = -3 sin(1/2 x - 1/3) + 1?

The amplitude of f is -3.

The period of f is 4π.

The graph completes one cycle on [2π/3, 13π/3].

The maximum value of f is 6.

57.

Which of the following is true about the function f(x) = csc x?

The range of f is (-∞,-1] ∪ [1,∞).

The domain is the set of real numbers.

The domain is {x ∈ ℝ: x ≠ (2k+1)π/2, k ∈ ℤ}.

The period is π.

58.

As shown in the figure below, the angle of elevation of the top a cliff when seen from the foot and the top of a building are 45° and 30° respectively. If the height of the building is 10 m, then how high is the top of the cliff in meters?

5(3 + √3)

15 + √3

10/(√3 - 1)

10√3/(√3 - 2)

59.

Which of the following is equal to the value of sin⁻¹(sin(5π/4))?

-π/4

5π/4

π/4

-√2/2

60.

Suppose A = 2i - j - 3k, B = i - 2j + ak and D = ai - 4k be vectors in space and i, j and k are standard unit vectors. Then for what value of a is the vector from A to B perpendicular to the vector from A to D?

5

-1

3

1

61.

Suppose that the equation x² + y² + z² + 2x - 6y + 8z = 7 represents a sphere. Where is the point (3,-1,-4) located relative to the sphere?

On the sphere.

Inside the sphere.

At the center of the sphere.

Outside the sphere.

62.

If one of the end point of the line segment is (-3,2,4) and the midpoint is (2,-5,-3), then the coordinates of the other end point in space is

(-8,-9,11)

(-7,12,10)

(7,-12,-10)

(-1,-3,1)

63.

Consider the following assertion of a person and his proof.

"If n is odd, then n² is odd"

Proof: Suppose that n is odd. Then n = 2k + 1 for some integer k. Then it follows that n² = (2k + 1)² = 4k² + 4k + 1 = 2(2k² + 2k) + 1 = 2m + 1. (where m = 2k² + 2k which is an integer). Therefore, n² is odd. This can be proved or disproved by which one of the following mathematical proofs?

Proof by exhaustion

Direct proof

Proof by contradiction

Indirect proof

64.

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

n < 2ⁿ, for every integer n ≤ 100.

2ⁿ⁻¹ ≤ n!, for every integer n ≥ 1.

n² - 11n + 121 is prime, for every natural number n ≥ 1.

6ʳ - 1 is a multiple of 5, for every real number r ≥ 1.

65.

Let p(x) = x² + x is positive. Which one of the following is equivalent to ¬(∃x)p(x)?

(∃x)(x² + x > 0)

(∀x)(x² + x < 0)

(∃x)(x² + x ≤ 0)

(∀x)(x² + x ≤ 0)

Please Share This Share this content

Leave a Reply Cancel reply

Share this content