M GENERAL MATHEMATICS -2- Dr. Tariq A. AlFadhel 1 Solution of the First Mid-Term Exam First semester H

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1 M - GENERAL MATHEMATICS -- Dr. Tariq A. AlFadhel Solution of the First Mid-Term Exam First semester H 3 Q. Let A =, B = and C = 3 Compute (if possible) : A+B and BC A+B is impossible. ( ) BC = 3 + +( ) = + + = Q. Compute The determinant Solution () : Using Sarrus Method ( ) = (++) (8++) = 8 = Solution () : By the definition (using second row) 3 = 3 = (3 ) = alfadhel@ksu.edu.sa

2 Q.3 Solve by Gauss the linear system : x y + z = 5 y + 3z = 5 x + 3y z = 6 The augmented matrix is R+R R +R 3 3z = 6 = z = y +3z = 5 = y +6 = 5 = y = x y +z = 5 = x ( )+ = 5 = x = x The solution is y = z Q. Find the standard equation of the parabola with focus F(5,) and vertex V(6,), then sketch it. From the position of the focus and the vertex the parabola opens to the left. The equation of the parabola has the form (y k) = a(x h) The vertex is V(6,), hence h = 6, k =. a is the distance between V and F, hence a =. The standard equation of the parabola is (y ) = (x 6) The equation of the directrix is x = 7.

3 (y-) =-(x-6) F V x=7 Q.5 Find all the elements of the conic section y x + y + 8x + 7 = and sketch it. y x +y +8x+7 = y +y x +8x = 7 y +y (x x) = 7 By completing the square. (y +y +5) (x x+) = 7+5 (y +5) (x ) = (y +5) (y +5) (x ) (x ) = = The conic section is Hyperbola. The center is P = (, 5) a = = a = b = = b = c = a +b = + = 5 = c = 5 The vertices are V = (, 3) and V = (, 7) The foci are F = (, 5+ 5 ) and F = (, 5 5 ) The equations of the asymptotes are L : (y +5) = (x ) and L : (y +5) = (x ) 3

4 L L F V -5 P y -x +y+8x+7= V F

5 M - GENERAL MATHEMATICS -- Dr. Tariq A. AlFadhel Solution of the Second Mid-Term Exam First semester H Q. Compute the integrals : (a) x(x +) 7 dx (b) x cos ( x 3) dx (c) x lnx dx (d) (x+)e x dx (e) x +6x+ dx x+ (f) (x )(x ) dx (a) x(x +) 7 dx = (x +) 7 (x)dx = (x +) 8 +c 8 Using the formula [f(x)] n f (x) dx = [f(x)]n+ +c, where n n+ (b) x cos ( x 3) dx = cos ( x 3) (3x ) dx = 3 3 sin( x 3) +c Using the formula cos(f(x))f (x) dx = sin(f(x))+c (c) x lnx dx Using integration by parts u = lnx du = x dx dv = x dx v = x3 3 x lnx dx = x3 3 lnx x = x3 3 lnx 3 x 3 3 dx x dx = x3 3 lnx x c 5

6 (d) (x+)e x dx Using integration by parts : u = x+ dv = e x dx du = dx v = e x (x+)e x dx = (x+)e x e x dx = (x+)e x e x +c = xe x +c (e) x +6x+ dx = (x +6x+9)+ dx = (x+3) + dx = tan (x+3)+c Using the formula f (x) a +[f(x)] dx = a tan ( f(x) a ) +c, where a > (f) x+ (x )(x ) dx Using the method of partial fractions x+ (x )(x ) = A x + A x x+ = A (x )+A (x ) Put x = : + = A ( ) = = A = A = Put x = : + = A ( ) = 6 = A = A = 3 ( x+ (x )(x ) dx = x + 3 ) dx x = x dx+3 dx = ln x +3ln x +c x Q. (a) Sketch the region R determined by the curves y = x, y =, x = (b) Find the area of the region R described in part (a). (a) y = x is a parabola opens upwards with vertex (, ) y = is a straight line parallel to the x-axis and passes through (, ) x = is a straight line parallel to the y-axis and passes through (,) 6

7 y=- - x= (b) Area = [ x 3 = 3 ] [ (x ) ( ) ] dx = = 3 = 3 x dx Q.3 (a) Sketch the region R determined by the curves y = x, x =, y = (b) Find the volume of the solid generated by rotating the region R in part (a) about the y-axis. (a) y = is the x-axis y = x is a parabola opens upwards with vertex (,) x = is a straight line parallel to the y-axis and passes through (,) x= 7

8 (b) Using Cylindrical Shells method : Volume = π [ x = π ] x ( x ) dx = π [ ] = π = π [ 6 x 3 dx Another solution : Using Washer Method y = x = x = y [ Volume = π () ( y) ] dy = π = π [y y ] = π ] = 8π ) ] [( ( ) ( y) dy = π(6 8) = 8π 8

9 M - GENERAL MATHEMATICS -- Dr. Tariq A. AlFadhel Solution of the Final Exam First semester H 3 Q. (a) Compute (if possible) AB for A = 3 6 and B = (b) Compute the determinant 3 5. x y + z = (c) Solve by Gauss Method : x 5y + z = 3 3x 6y z = 8 3 (a) AB = = = (b) Solution (): Using Sarrus Method = (++) (3++) = 3 = 5 Solution () : Using the definition (using the first row) : 3 5 = = ( )+( 5) = ( 8) 3 = 8 3 = (c) Using Gauss Method : R+R

10 3R +R z = 6 = z = 6 3 = R+R y z = 5 = 3y = 5 = 3y = 3 = y = 3 3 = x y +z = = x + = = x+ = = x = x The solution is y = z Q. (a) Find the standard equation of the ellipse with foci (,3) and (,3), and with vertex (5,3). (b) Find the elements of the conic section 9x y +8x y = 63. (a) The two foci and the vertex are located on a line parallel to the x-axis. The standard equation of the ellipse is (x h) + a > b. ( + P(h,k) = a (y k) b, 3+3 ) = (,3), hence h = and k = a is the distance between the vertex (5,3) and P, hence a = c is the distance between one of the foci and P, hence c = 3 c = a b = 9 = 6 b = b = 6 9 = 7 = b = 7 =, where The standard equation of the ellipse is (x ) 6 The other vertex is ( 3,3) (y 3) 7 = The end-points of the minor axis are (,3 7 ) and (,3+ 7 ). (b) 9x y +8x y = 63 9x +8x y y = 63 9(x +x) (y +6y) = 63 By completing the square 9(x +x+) (y +6y +9) = (x+) (y +3) = 36

11 9(x+) 36 (x+) (y +3) 36 (y +3) 9 = = The conic section is a hyperbola. The center is P(, 3). a = = a =. b = 9 = b = 3. c = a +b = +9 = 3 = c = 3. The vertices are V ( 3, 3) and V (, 3) The foci are F ( 3, 3 ) and F ( + 3, 3 ). The equations of the asymptotes are : L : y +3 = 3 (x+) and L : y +3 = 3 (x+) Q.3 (a) Compute the integrals : x+ (i) dx (ii) xlnx dx (iii) xsin ( x ) dx (x )(x 3) (b) Find the area of the region bounded by the curves : y = x and y = x. (c) The region R between the curves y =, x = and y = x is rotated about the x-axis to form a solid of revolution S. Find the volume of S. x+ (a) (i) (x )(x 3) dx Using the method of partial fractions x+ (x )(x 3) = A x + A x 3 x+ = A (x 3)+A (x ) Put x = then + = A ( 3) = 3 = A = A = 3 Put x = 3 then 3+ = A (3 ) = A = ( x+ 3 (x )(x 3) dx = x + ) dx x 3 = 3 x dx+ dx = 3ln x +ln x 3 +c x 3

12 (ii) xlnx dx Using integration by parts u = lnx du = x dx dv = x dx v = x xlnx dx = x x lnx = x lnx (iii) xsin ( x ) dx xsin ( x ) dx = x x dx x dx = x lnx sin ( x ) (x) dx x +c = ( ( cos x )) +c = cos( x ) +c Using the formula sin(f(x)) f (x) dx = cos(f(x))+c (b) y = x is a parabola opens upwards with vertex (,). y = x is the upper-half of the parabola x = y which opens to the right with vertex (, ), y=x y= x Points of intersection of y = x and y = x : x = x = x = x = x x = = x(x 3 ) = = x =, x 3 = = x =, x = Area = ( x x ) dx = ( x x ) dx = [ x 3 3 ] [ x3 x 3 ] = 3 3 x3 3

13 ( () 3 ) ( () 3 ) ( = 3 ()3 3 3 ()3 = 3 3 ) ( ) = 3 3 (c) y = is the x-axis. x = is a straight line parallel to the y-axis and passes through (,). y = x is the upper-half of the parabola x = y which opens to the right with vertex (, ), y= x x= Using Disk Method : ( ) [ x Volume = π x dx = π x dx = π [ ] ( ) () = π () = π = π ] Q. (a) Find f x, f y and f z for the function f(x,y,z) = xy 3 z +ln ( xz ). (b) Solve the differential equation dy dx 3x y =. (a) f x = y 3 z()+ z () xz = y 3 z + x f y = xz ( 3y ) + = 3xy z f z = xy 3 ()+ x( z 3) xz = xy 3 + z 3

14 (b) dy dx 3x y = dy dx = 3x y y dy = 3x dx It is a separable differential equation. y dy = 3x dx y dy = 3x dx y = x3 +c y = x3 +c y = x3 c y = x 3 c = x 3 +c

15 M - GENERAL MATHEMATICS -- Dr. Tariq A. AlFadhel Solution of the First Mid-Term Exam Second semester H Q. Let A =, B = 3 and C = Compute (if possible) : AB and B+C 3 3 AB = = = B+C = = = Q. Compute The determinant 3 Using Sarrus Method 3 3 = (++6) (+3+) = 9 = 3 Q.3 Solve by Gauss the linear system : x y + z = x 3y z = x + y z = The augmented matrix is 3 R+R 5

16 R +R z = 8 = z = 8 5 = 8 5 ( ) 8 y z = = y 5 = y = x y +z = = x = x = = 5 = 3 x The solution is y = z 6R+R3 3+6 = y = = = y 6 5 = = 5 = y = 5 ( ) = = x = Q. Find all the elements of the conic section x y +x+8y 7 = and sketch it. x y +x+8y 7 = x +x y +8y = 7 x +x (y y) = 7 By completing the square. (x +x+) (y y +) = 7+ (x+) (y ) = (x+) (x+) (y ) (y ) = = The conic section is Hyperbola. The center is P(,) a = = a = b = = b = c = a +b = + = 5 = c = 5 6

17 The vertices are V ( 3,) and V (,) The foci are F ( 5, ) and F ( + 5, ) The equations of the asymptotes are L : (y ) = (x+) and L : (y ) = (x+) L x -y +x+8y-7= L P F V V F -3 - Q.5 Find the standard equation of the ellipse with foci F (5,) and F (5,7) with vertex V(5,8), then sketch it. The two foci and the vertex are located on a line parallel to the y-axis. The standard equation of the ellipse is (x h) + b > a. P(h,k) = a (y k) b ( 5+5, +7 ) = (5,), hence h = 5 and k = c is the distance between one of the foci and P, hence c = 3 b is the distance between the vertex (5,8) and P, hence b = c = b a = 9 = 6 a = a = 6 9 = 7 = a = 7 =, where The standard equation of the ellipse is (x 5) 7 The other vertex is (5,) + (y ) 6 = The end-points of the minor axis are ( 5 7, ) and ( 5+ 7, ). 7

18 8 V (x-5) 7 + (y-) = 6 7 F P W W F V

19 M - GENERAL MATHEMATICS -- Dr. Tariq A. AlFadhel Solution of the Second Mid-Term Exam Second semester H Q. Compute the integrals : x (a) x + dx (b) xcosx dx (c) xlnx dx (d) xe x dx x (e) (x ) dx (f) (x )(x ) dx x (a) x + dx = ln(x +)+c Using the formula f (x) f(x) dx = ln f(x) +c (b) xcosx dx Using integration by parts u = x dv = cosx dx du = dx v = sinx x cosx dx = x sinx sinx dx = x sinx ( cosx)+c = x sinx+cosx+c (c) xlnx dx Using integration by parts u = lnx du = x dx dv = x dx v = x 9

20 x lnx dx = x lnx x = x lnx x dx x dx = x lnx x +c (d) xe x dx = e x (x) dx = ex +c Using the formula e f(x) f (x) dx = e f(x) +c (e) x (x ) dx Using the method of partial fractions x (x ) = A x + A (x ) = A (x )+A (x ) x = A (x )+A = A x A +A By comparing the coefficients of the two polynomials in each side : A = A +A = = +A = = A = ( ) x (x ) dx = x + (x ) dx = x dx+ (x ) dx = ln x + (x ) +c = ln x x +c (f) (x )(x ) dx Using the method of partial fractions (x )(x ) = A x + A x = A (x )+A (x ) (x )(x ) = A (x )+A (x ) Put x = : = A ( ) = = A = A = Put x = : = A ( ) = A =

21 (x )(x ) dx = = x dx+ ( x + ) x x dx dx = ln x +ln x +c Q. (a) Sketch the region R determined by the curves y = x, y =, x = (b) Find the area of the region R described in part (a). (a) y = x is a parabola opens upwards with vertex (,) y = is the x-axis x = is a straight line parallel to the y-axis and passes through (,) y=x x= (b) Area = [ x x 3 dx = 3 ] = = 8 3

22 Q.3 (a) Sketch the region R determined by the curves y = x, y = x+ (b) Find the volume of the solid generated by rotating the region R in part (a) about the x-axis. (a) y = x is a parabola opens upwards with vertex (,) y = x+ is a straight line passes through (,) with slope equals y=-x+ y=x - (b) Points of intersection of y = x and y = x+ : x = x+ = x +x = = (x+)(x ) = = x =, x = Using Washer method : [ Volume = π ( x+) ( x ) ] dx = π = π [ x +x x+ ] dx = π [ x5 [ (x x+) x ] dx 5 + x3 3 x +x ) )] = π [( ( )+() ( ( )5 + ( )3 (( ) )+( ) 5 3 = π [ ( )] 8 8 = π ( 335 ) ( = π 33 ) = 7π 5 5 ]