EUEE 2004, 2005, 2006, 2007 Past Papers PDF

ETHIO NATIONAL SCHOOL - MATH FOR GRADE 11 AND 12 Maths EUEE 2004 E.C Grade 11 Unit One x | x | 1. If x1, then which one of the following is the inverse off?  x 1  e 2...

ETHIO NATIONAL SCHOOL - MATH FOR GRADE 11 AND 12 Maths EUEE 2004 E.C Grade 11 Unit One x | x | 1. If x1, then which one of the following is the inverse off?  x 1  e 2 x ex  2 ex x A. G(x) = x B. G(x) = x C. G(x) = x 2 D. G(x) = x 1 e 3 e 1 e 1 e 2 Maths EUEE 2005 E.C Grade 11 Unit One 1. What is the value of |x| + 2x if x A. 6 B. 18 C. 27 D.  4. A business association gets a net profit of Birr 3,000 at the end of month. Just after the fifth year, its amount was Birr 230,000. What the starting amount of the business? A. Birr 47,000 B. Birr 50,000 C. Birr 53,000 D. Bir 56,000320-1 Maths EUEE 2005 E.C Grade 12 Unit One 1. What is the 50th term of the sequence 3,10,17,24,31…? A. 310 B. 346 C. 510 D. 531 35 2. If {an} is a sequence such that a1 = 2, and an+1 = an + 4 for all n> 1, then a n 1 n is equal to: A. 2460 B. 2458 C. 2450 D. 2442 2 n 1 2   3. What is the sum of the series ? n 1 5 n 1 A. 40 B. 20 C. 10 D. 8 4. The population of certain country is currently 80 million with growth rate 0f 2% per year. Given: (0,02)9 = 5.12 x 10-16, (1.02) 9= 1.19 (0.02)10 = 1.024 x 10-17, (1.02) 10 = 1.22 Which one of the following is the best approximation of the population after 10 years? A. 81.9 million B. 86.8 million C. 95.2 million D. 97.6 million Maths EUEE 2006 E.C Grade 12 Unit One 1. Which one of the following represents a geometric sequence? 1 1 1 A. 3,1,, , ,... C. 1,3,6,10,15,… 3 9 27 1 1 1 1 1 B. , , , , ,... D. -3,6,-9,12,-15,… 2 3 4 5 6  2 n  5n  n1  10n  2. What is the actual value of the sum  ?   5 37 A. 0.325 B. 1 C. D. 4 9  3. What is the sum of the series  (1) n 1 n 32 n 1 1 A. B. -0.13 C. -0.1 D. 8 8 Maths EUEE 2007 E.C Grade 12 Page 21  ETHIO NATIONAL SCHOOL - MATH FOR GRADE 11 AND 12 Unit One 1. Which one of the following is an arithmetic sequence? A. 3, 5, 7, 9, 11… C. -3, 6,-9, 12,-15… B. 3, 6, 12, 24, 48… D. 1, 3, 6, 10, 15, 21… 2. Which one of the following sequence is a convergent sequence?  1 1 1  109 1  A. 1, ,1, ,1, ,... C. 10  n 2 3 4  100  n 1  B.  1  n  n 1 D.   1  sin    n  n 1 3. A certain meeting hall has 20 rows of seats. There are 20 seats in the first row, 22 seats in the second row, 24 seats in the third row, and so on. How many seats are there on the last (20th) row of the hall? A. 46 B. 58 C. 760 D. 5240 4. A ball is thrown vertically from ground up to a height of 16m. Each time it drops h meters, it rebounds 0.80h m. Nothing that the ball travels every height of h twice, what is the total vertical distance travelled by the ball before it comes to rest? A. 40m B. 80m C. 160m D. 320m 5. What is the sum of all multiples of 3 between 20 and 200? A. 7,227 B. 6,570 C. 6,150 D. 5,166 Maths EUEE 2008 E.C Grade 12 Unit One If An n 1 is an arithmetic sequence such that its 1st term A1 = -5 and its 5th term A5 =15, then its 11th term A11 is  1. equal to: A. 40 B. 50 C. 45 D. 55 2. What is the sum of all multiples of 4 that are between 30 and 301? A. 12,882 B. 11,288 C. 6,288 D. 6,882 3. If f : A B and g : B C are functions, then which one of the following is true about the composition function? A. Domain of gof   Domain of f B. Range of gof   Range of g C. Domain of gof   Domain of f D. Range of gof   Range of f 4. The nth term of the sequence: 1, -4, 9, -16,... is: A. an = (-2)n C. an = (-1)2n n2 B. an = (-1)n n2 D. an = (-1)n-1 n2  n 2 5. The sum of  5  is. n 0  3  10 A. 0 B. 15 C. D. 5 3 Maths EUEE 2004 E.C Grade 12 Unit Two Page 22  ETHIO NATIONAL SCHOOL - MATH FOR GRADE 11 AND 12 3 n  6n  5 1. To which of the following is lim ? n 4n  1 3 3 5 A. B. 0 C. D. 2 4 4 x4 1 2. To which of the following is lim equals to? n 1 / x  1/ A. 4 B. 0 C. - D. -4  a  , ifx  1  x2  3 3. Let f(s) =   x  1 , ifx  1   x 1 What is the value of a if f is continuous at x = 1? A. 0 B. 2 C. 4 D. 8 2 x4  1 4. lim1   is equal to: n   x A. e B. e2 C. e4 D.  Maths EUEE 2005 E.C Grade 12 Unit Two f ( 2  h )  f ( 2) 1. If f ( x)  x 2  2nx , then what is lim ? h 0 h A. 5 B. 4 C. 2 D. 0 x  1 2 2. What is the value of lim1   ? h   x 1 A. B. e C. e-2 D.  e 1  n  3n 2 3. Which one of the following is equal to lim ? h  6n 2  1 1 1 1 A. B. C. D. -  6 2 6 3 x  k , ifx  0  4. Let f ( x)   sin(2 x) 3 , ifx  0  x If f is continuous at x = 0, then what is the value of k? A. 6 B. 5 C. 2 D. 0 x 1 5. Which one of the following is equal to lim ? h1 x 12 1 1 A.  B. C. D. 0 2 4 Page 23  ETHIO NATIONAL SCHOOL - MATH FOR GRADE 11 AND 12 1 x 6. Which one of the following is equal to lim ? h 1 1 1 x 1 A. 1 B. 0 C. D. Doesn‟t exist 2 Maths EUEE 2006 E.C Grade 12 Unit Two 1. Which of the following expression is a polynomial expression? 2 A. x2 – 3x + sin x C. 1  2 4 x 3  12x 2  x 2 5 B. D. 2  3x 3  7x 2  3x 1 x 2  n  1)(2n  1  2. The sequence   converges to:  1 n  n 1 2 A. - B. -2 C. 0 D. 1  ( f ( x)  g ( x))(g ( x)  2 f ( x))  3. Given that lim x3   ?  g ( x) 2  f ( x) 2  66 1 A.  B.  C. 0 D. does not exist 96 16  sin x a , ifx  0 4. Let f ( x)   x  | x | e  x  cos x, ifx  0  If f continuous at x=0, then what is the value of a? 1 A. 4 B. 2 C. D. -4 2 3 x  x  5. Which one of the following is equal to lim  ? x  x  2   3 A. e6 B. e 3 C. e 2 D. e 6  n  3 n 6. If an    , then the limit of the sequence a n n1 is equal to:  n 1  Maths EUEE 2007 E.C Grade 12 Unit Two  sin 2 x a , ifx  0 1. Let f ( x)   x e 2 x  2, ifx  0  If f is continuous at x = 0, then what is the value of a? Page 24  ETHIO NATIONAL SCHOOL - MATH FOR GRADE 11 AND 12 1 1 A. B. 2 C. D. -1 2 2 3 x  3x  2. Which one of the following is equal to lim  ? x  3 x  2   A. e2 B. e-3 C. e-2 D. e3 Maths EUEE 2008 E.C Grade 12 Unit Two xe x  x 1. The left hand side limit, lim is equal to. x 0 x A. 0 B. 2 C. 1 D. Does not exist 2 x  tan x 2. Which one of the following is equal to lim ? x 0 x sec x A. 2 B. 0 C. 1 D. 3    1  n 3. In which interval the sequence   is bounded?  3 n  n1  1 1   1 1   1 1  1 1  A.  9 , 12  B.  3 , 6  C.  6 , 3  D.  12 , 9  4. Which one of the following is true about the function  x2 , x  0 f x    x ? 0 , x  0 A. f is continuous except at x = 0 B. f has an infinite discontinuity at x = 0 C. f is continuous every where D. f has x = 0 as a vertical asymptote 1 5. lim x sin is equal to: x  x A. 0 B. 1 C.  D. -1 Maths EUEE 2004 E.C Grade 12 Unit Three x csc 3 x 1. lim is equal to: n0 x 1 1 A. B. 2 C. 1 D. 0 3 2. If f(x) =  2 , then f‟(1) is equal to A. 2 B. 2 C.1 D.  ax 3. Let f(x) =. For what value of a is f‟(1) = 1? xa Page 25  ETHIO NATIONAL SCHOOL - MATH FOR GRADE 11 AND 12 1 2 3 A. B. C. D. 3 3 3 2 4. Which one of the following is necessarily true about a function f(x)? A. If f is continuous at x  a, then f is differentiable at x = 1. B. If f is not differentiable at x = a, then lim f ( x)  lim f ( x). x a  x a C. If f is differentiable at x  a, then lim f (a)  lim f ( x) x a x a D. If the derivative f‟(a) = 0, then f attains its maximum value at = a. Maths EUEE 2005 E.C Grade 12 Unit Three 1. If f ( x)  e 2 x  x  3 cos(x), then what is f”(X)? A. e 2 x  x  3 cos(x) C. 4e 2 x  3 cos(x) B. e 2 x  x  3 cos(x) D. 4e 2 x  3 cos(x) 2. If g(X) = xf(x) - f (x) and f(2) = f‟(2) = 4, then which of the following is equal to g‟(2)? A. 11 B. 8 C. 2 D. 0 3. Which one of the following is true about the derivative of f(x) = x|x|? A. f is not differentiable ft x = 0 B. f‟(x) = 2|x|, for every x  (  ,  ) C. f‟(x) = 2x, for every x  (  ,  ) D. f‟(x) = |x|+x, for every x  (  ,  ) 1 4. Which one of the following is the equation of the line tangent to the graph of f(x) =  cos x at (0,f(0))? x 1 A. x+y = 1 B. x-y = -2 C. x+y=2 D. x+4y = 2 Maths EUEE 2006 E.C Grade 12 Unit Three x2 1. If f ( x)  , g (2)  1 and g’(x)=10, then which one of the following is equal to f’(2)? 1  xg ( x) 8 4 8 A. -8 B. C. D. 9 3 9 1  xinx 2. The simplified form of the derivative of f(x)= is cos x 1 A. Sec x + tan x C. 1  tan x 1  sin x cos x B. 2 D. cos x sin 2 x 3. If f(x)=e2x sinx, then f”(x) is equal to A. 3e2x sin – 4e2x, cos x C. e2x( 3 sin x + 4 cos x) 2x 2x B. 3e sin x + 2e cos x D. e2x ( 4sinx – 3 sin x) Page 26  ETHIO NATIONAL SCHOOL - MATH FOR GRADE 11 AND 12 d2y 4. If y=sin(3x2), then the simplified form of is: dx 2 A. -6 sin (3x2) C. 6 cos (3x2)-36x2 sin (3x2) B. Cos (6x) – 6 sin(3x2) D. x2 cos (3x2)+6sin (3x2) 5. If f ( x)  x 2 2x  12 , what is the slope of the tangent line to the graph of f at x=2? A. -4 B. 2 C. 18 D. 17 6. If F ( x)  f (2 x  2).g (1  x ), with f(2)=-3, f’(2)=4, g(1)=-5, and g’(1)=1, then what is the actual value of F’(0)? 2 A. -40 B. -20 C. 0 D. 19 Maths EUEE 2007 E.C Grade 12 Unit Three If f ( x)  n( x  1) , which of the following is equal to f‟(x)? 2 1. x x 2x 2x A. B. C. D. x 32 x 1 2 x 12 x 1 2 2. If f(x) = 2x (x2 + 1)4, then which of the following is an anti derivative of f(x)? A. 2x 2 5 5  x 1  c  C. x 2 5 5 x 1 1  B. 2 2 5 5 x 1  c  D. 1 2 5 5 x 1 1  3. The Ozone level (in ppb – parts per billion) on a sunny day in a metropolitan area is given by the formula: p(t) = 80 + 12t – t2, where t is time in hours and t = 0 corresponds to 9 A.M. what is the rate of increase of the ozone level after 3 hours (i.e. at 12 A.M.)? A. 6 ppb B. 12 ppb C. 107 ppb D. 113 ppb 4. Suppose that a function f has the property that f(x + y) = f(x) f(y) for all value of x and y and that f (0) = 2, f‟(0) = 1. Then which one of the following represents the formula for the derivative f‟(x)? A. f‟(x) = 2f‟(x) + 1 C. f‟(x) = f(x) + 2 B. f‟(x) = f(x) + 2f‟(x) D. f‟(x) = 2f(x) – 1 5. if f(2)=-3, f‟(2) = 4, g(1) = -5, g‟(1) and f(x) = f(2x + 2). g (1-x2), then what is the value of F‟ (0)? A. 19 B. 0 C. -20 D. -40 6. For what values of a and b is the function f ( x)  1  3x 2 , for x  1 ax + b, for x > 1 differentiable at x = 1? A. a = 6, b = 0 C. a = 0, b = -2 B. a = -3, b = 1 D. a = -6, b = Maths EUEE 2008 E.C Grade 12 Unit Three f  x   f 1 If f x   2 x  3x, then 5 1. lim is equal to. x 1 x 1 A. 1 B. -1 C. 7 D.  Page 27  ETHIO NATIONAL SCHOOL - MATH FOR GRADE 11 AND 12 x  2. If f  x   e 3 x cos x  , then f 0 is equal to. x2  2  3 7 5 A. 3 B. C. D. 2 2 4 2 3. If f x   In   x 2  5 , which one of the following is equal to f x  ? x x 2x x A. B. C. D. x 5 2 x2  5 x2  5 x2  5 4. d dx   Ine 2 x is equal to: 1 2 A. 2x B. 2 x C. 2x D. 2 e e 5. If f x   2  x  3 for all x, then the value of the derivative f x  at x= 3 is. A. -1 B. does not exist C. 1 D. 2 Maths EUEE 2004 E.C Grade 12 Unit Four 1. If f(x) = xe3x-cos(2x), then f‟(0) is equal to A. 0 B. 2 C. 6 D. 10 1 1 2. At what value (s) of x does f(x) = x3  x 2  2 x  1 have a local maximum? 3 2 A. x=2 B. x=-1 C. x=5 D. x=2,x=-1 3. Look at the following graph of f‟(x). Which of the following is true about f? A. F is increasing on [c,d] B. F if decreasing of [b,c] C. F has a local minimum at d. D. F has a local extreme value at c. 4. A rectangular field of length 1 and width w meters f w P25 C. P25> Q1 D. Q2 = mean of the data 2 Page 33  ETHIO NATIONAL SCHOOL - MATH FOR GRADE 11 AND 12 x  n( x  1) 6. Which one of the following is equal to  ( x  1) 2 dx ? x 1 A. n( x  1)  c C. ( x  1) 2  c x 1 x 1 1 xn( x  1) B. ( x  1) 2  c D. c x 1 x 1 7. What is the area of the region between the graph of f(x) =-x2+4x-3 and the x-axis from x=0 to x=3? 2 2 4 8 A.  B. C. D. 3 3 3 3 Maths EUEE 2007 E.C Grade 12 Unit Five 1. What is the area of the region between the graphs of y = sin x and x – axis where 0 < x < 2  ? A. 4 B. 4  C. 2 D. 2  nx  x e 2 x 2. Which one of the following is equal to  x dx ? 1 2 1 A. n x  e x ( x 2  1)  c C. 2 nx  e x ( x  1)  c 2 x 1 2 1 B. n x  e x ( x  1)  c D. 2 nx  e x ( x  1)  c 2 x x 1 3. The derivative of the function F(x) =  dt is: x 1  t 2 1 1 x A. B. In |1 + x| C. D. In 1 x2 1 x 1 x Maths EUEE 2008 E.C Grade 12 Unit Five If f ( x )  3 x x 3  1 , then which of the following is an anti-derivative of f(x)? 2 1. 3x 3 3 3 A. ( x  1) 3 / 2  c C. ( x  1) 3 / 2 2 2 2 3 3 3 B. ( x  1) 3 / 2 D. ( x  1) 2/3 c 3 2 Inx2  x 2 cos x dx ? 2. Which of the following is equal to  x 1 1 A. Inx  x sin x  cos x  c C. ( Inx) 3  x sin x  cos x  c x2 3 1 1 B. ( Inx) 3  x sin x  cos x  c D. 2 Inx  x sin x  cos x  c 3 x Page 34  ETHIO NATIONAL SCHOOL - MATH FOR GRADE 11 AND 12  x 2 ,0  x  2 3. The volume of the solid generated when the region bounded between the graph of y   and x-axis is 4, 2  x  3 rotated about the x-axis is: 32 112 112 64 A. B. C. D. 5 5 3 5 3  x  1e ( x2 2 x) 4. The value of dx is: 0 e 3 e4  e e3  1 A. B. C. D. e3 – 1 2 2 2 3  x  1 1 5. 2 dx =. 0 21 14 16 A. B. C. 7 D. 2 3 3  x  1, forx  0 1 6. Given f x    , then  f x dx =. cosx, forx  0 1 1 A.  1 B. 1  1 C. 1 D.  1 2  2  2 2 Maths EUEE 2004 E.C Grade 12 Unit Six 1. If a sphere with center C (0,1,1) interests the z-axis at P(0,0,3), then the radius of the sphere is equal to A. 5 B. 3 C. 3 D. 5 2. Suppose A and B are the end points of a diameter of the sphere whose equation is x2 + y2 + (z+2)2 = 1. If A = (1,0,-2), then B is equal to A. (0,1,-2) C. (-41,0,-2) B. (0,0,-1) D. (0,-1,-2) 3. Let 1 and 2 be two lines in space intersecting at the origin, (0, 0, 0). If 1 and 2 pass through point A(1,1,0) and V(0,1,1) respectively, there the angle between 1 and 2 is equal to A. 30° B. 45° C. 60° D. 90° 4. Let a  i  3k andb  i  j be vectors in the space. Which one of the following is the cosine of the angle between a and a  b ? A. 9 C. 3 10 10 3 9 B. D.  3 10 5. Let V  3i  4k , when i and k are the standard unit vectors in the direction of positive x-axis and positive z-axis, respectively, and AB be a vector from the point A(0,1,2) to a point B in space iv AB is parallel to V and AB = 10, then point B is at A. (-6,-1,10) C. (6,1,-6) B. (6,-1,-10) D. (-6,-1,6) Page 35  ETHIO NATIONAL SCHOOL - MATH FOR GRADE 11 AND 12 6. Consider the following statement: x2  2  0 for every real number x x To show this, a person constructed the following proof. “Proof: Take x = 1 Then, x 2  2  12  2  3  0 x 1 In the same way, if we take x = n for any real number n we get x2  2 n2  2  0 x n It follows that x  2  0 , for every number x,” 2 x Which one of the following is true about this proof? A. The proof is correct by the principle of induction B. The proof is correct by the method of exhaustion C. The proof is correct and it uses the method of direct proof D. It is not a valid proof since its argument cannot lead to the conclusion Maths EUEE 2005 E.C Grade 12 Unit Six         1. Suppose A  2 j  k and B  5i  15k , where i , j and k are the standard unit vectors in the directions of positive s, y  1  and z – axis, respectively. Which one of the following is unit vector in the direction of A  B ? 5 3  4  4  3 A. i k C. i  k 5 5 5 5 1 2  2  2 2  2  B. i  k k D. i  k  k 3 3 3 3 3 3     2. Suppose p (1,2,1) and Q(1,0,2) are points in space and A  PQ. If B is parallel to PQ and A.B = -10, then which one of the following is true?     1 A A. A and B has the same direction C. B 10    A  A B. B  10 D. B2 3. Which one of the following points is closer to the sphere x2+y2+z2-2x+6z+9=0? A. (1,0,0) B. (0,0,0) C. (0,-1,0) D. (0,0,-1)           4. Let a  2i  ( x  1) j  kandc  i  j  yk be vectors. If a.c  0 and a = 3, which one of the following is a possible value of y? A. -4 B. -1 C. 3 D. 4 Maths EUEE 2006 E.C Grade 12 Unit Six 1. (x)( p( x)  Q( x)) ? Which one of the following is equivalent to A. (x)(P ( x)  Q ( x)) C. (x)(P( x)  Q( x)) Page 36  ETHIO NATIONAL SCHOOL - MATH FOR GRADE 11 AND 12 B. (x)(P ( x)  Q ( x)) D. (x) P( x)  Q( x)) 2. Suppose P and Q are points in space such that the midpoint of PQ is on the negative z-axis and the distance between P and Q is 6. If P=(2,-1,0), then what is the coordinate of Q? A. (-2,1,4) B. (2,-1,6) C. (2,-1,-6) D. (-2,1,-4) Maths EUEE 2007 E.C Grade 12 Unit Six 1. If P= (3,  - 1,  + 2) and Q = (2  + 1, 3, 3  ) are points in space, what should be the value (s) of  so that the distance between the two points is 6? A.  = -2 or  = 5 C.  = - 1 or  = 3 B.  = 0 or  = 5 D.  = - 3 or  = 2 2. If (-1, 2, 2) and (1, 0, -2) are endpoints of a diameter of a sphere, then which one of the following is true about the sphere? A. (0,1,0) is a point on the sphere B. The equation of the sphere is x2 + (y-1)2 + z2 = 6 C. The equation of the sphere is x2 + (y – 2)2 + z2 = 6 D. The radius of the sphere is 6. 3. Suppose  is the line through the center of the sphere x2 + y2 = (z – 2)2 = 9 and the intersects sphere at (1,2,4). What is the cosine of the angle between  and positive z – axis? 2 1 3 4 A. B. C. D. 3 3 5 5 4. Suppose that the equation x2 + y2 + z2 + 2x + 8z = 6(y + 1). Represents a sphere. Where is the point (1, -1, 4) located relative to the sphere? A. Inside the sphere, B. On the sphere C. At the center of the sphere D. Outside the sphere Maths EUEE 2008 E.C Grade 12 Unit Six 1. Let the angle between V  2i  j  2k and PQ be 600, where P and Q are points in space. If V  PQ  2 , then what is that distance between P and Q? 3 4 4 5 A. B. C. D. 4 5 3 4 2. 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: A. (5, 0, 0) B. (2, 0, 5) C. (5, 1, 2) D. (3, 1, 0) Maths EUEE 2004 E.C Grade 12 Unit Seven 1. Let an = n2 – n, when n  N (set of natural numbers). Which one of the following is true, when k is an arbitrarily chosen natural number and m is an integers? A. an is not a multiple of 2 for some n  N because a1 = 0. B. an is a multiple of 2 for all n  N because a1 = 0 and if ak = 2m, then ak+1 = 2(m+1) Page 37  ETHIO NATIONAL SCHOOL - MATH FOR GRADE 11 AND 12 C. an is a multiple of 2 for all n  N because a1 = 0 and if ak = 2m, then ak+1 = 2(m+k) D. an is a multiple of 3 for all n  N because a1 = 0 and if ak = 3m, then ak+1 = 3(m+k) Maths EUEE 2005 E.C Grade 12 Unit Seven 1. Which one of the following describes the principle of Mathematical induction on a set of natural numbers? A. If an assertion is true for a natural number n, then it is true for n+1 B. If an assertion is true for 1 and it is true for n +1, then it is true for some n. C. If an assertion holds for n = 20 and for any n> 20, then it is true for n implies true for n + 1 D. If an assertion is true for n = 1, and is true for n = k, whenever is true for n = k + 1. Maths EUEE 2006 E.C Grade 12 Unit Seven 1. The following is an assertion of a person and his proof. “For any natural numbers n, n! < 10n. Proof: Step 1. Let N = 1. Since 1! = 1 and 101 = 10, it is true that 1! < 101. Step 2. Let N = 2. Since 2! = 2 and 102 = 100, it is true that 2! < 102. Step 3. Let N = 3. Since 3! = 6 and 103 = 1000, it is true that 2! < 103. Step 4. Continuing in this manner, we can see that whenever K! < 10k is true, then (K + 1)! < 10k+1 is also true. Therefore, by induction, n! < 10n for all natural numbers.” Which one of the following is true about the proof? A. The proof is correct by the principle of mathematical induction, though step 2 and step 3 can be omitted. B. The proof is correct by the principle of mathematical induction, and step 2 and step 3 are necessary since they provide additional information. C. The proof is invalid because step 4 did not justify the desired induction step. D. The proof follows the technique of a proof by exhaustion. Maths EUEE 2007 E.C Grade 12 Unit Seven 1. If each of the compound propositions P  Q, P  R and  R is True, then which one of the following is True? A. P B. Q C. Q  P D. P   R 2. Consider the formula for a natural number n  N: 2 + 4 + 8 - … + 2n = 2n+1 + 1 To proof this formula a person has used the following argument. “Assume the formula is true for n = k, for some k  N. Then the person has shown that the formula is also true for n = k + 1. And then, the person has concluded that, by Page 38  ETHIO NATIONAL SCHOOL - MATH FOR GRADE 11 AND 12 the principle of mathematical induction, the formula is true for all natural numbers n  N.” Which one of the following statements is true about the above arguments? A. The formula holds true though it does not work for n = 1. B. Since the left – hand – side is an even number and the right – hand – side an odd number, the principle of Mathematical induction if False. C. This is one example where the principle of Mathematical induction fails to work. D. The above formula does not work for all natural numbers n  N Maths EUEE 2007 E.C Grade 12 Unit Seven 1. What is the contra positive of “If x  N, then x is integer and x  0 ”? A. If x is not integer or x  0 , then x  N. B. If x is integer and x  0 , then x  N C. If x is not integer or x  0 , then x  N D. If x  N, then x is not integer and x  0 2. 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. x 2  xy Multiply both sides of (1) by x 3. x  y  xy  y 2 2 2 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 A. It is a correct direct proof of the assertion. B. It follows the technique of a proof by contradiction because the steps lead to a contradiction. C. The proof is invalid because Step 4 does not lead to Step 5. D. The proof is invalid because Step 4 does not follow from step 3. 3. Which one of the following is a valid assertion that can be proved y the principle of mathematical induction? A. 2 n  10n for every integer n such that n  6. B. r  0 for every real number r such that r  1 2 C. n  10n  2n for every natural number n  1 2 2 D. 2 n  8n For every integer n such that n  3 4. The valid conclusion from the premises: P V Q, Q  R, P  M , M is. A. P  R  R  B. P  P  R  C. Q  P  R  D. R  P  Q Page 39  ETHIO NATIONAL SCHOOL - MATH FOR GRADE 11 AND 12 Page 40