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 4 - GENERAL MATHEMATICS -- Dr. Tariq A. AlFadhel Solution of the First Mid-Term Exam First semester H Q. Let A ( ) 4 and B 3 3 Compute (if possible) : AB and BA ( ) 4 AB 3 3 ( ) ( ) ( ) BA Q. Compute The determinant 3 3 Solution () : Using Sarrus Method (6+4+9) (++8) Solution () : Using The definition (6 ) (9 )+3(3 4) alfadhel@ksu.edu.sa

2 Q.3 Find all the elements of the conic section x y +x and sketch it. x x y x x+ y + ( (x ) y + ) The conic section is a parabola opens upwards. ( The vertex is V, ). 4a a 4. The focus is F (, + ) (,). The equation of the directrix is y y x y x 3 F 3 V y Q.4 Find the standard equation of the ellipse with vertex (,) and with foci (,) and (,), and then sketch it. The equation of the ellipse has the form (x h) a The center of the ellipse is the mid-point of the two foci. + (y k) b.

3 ( + The center is P, + ) (6,), hence h 6 and k. The two foci lie on a line parallel to the x-axis, hence a > b c is the distance between the center and one of the foci, hence c 4 a is the distance between the center (6,) and the vertex V (,), hence a 5. c a b 4 5 b b b 3. The equation of the ellipse is (x 6) 5 The other vertex is V (,). + (y ) 9. The end-points of the minor axis are W (6,5) and W (6, ) x 6 y W F P F V V 6 W Q.5 Solve by Cramer s Rule the following linear system ( ) ( ) ( ) A, A x, A 4 y 4 A 3 ( ) ( 3 ) ( 3) 5 A x (3 ) ( 3 4) 3 ( ) 5 A y 3 4 ( 4) (3 ) { x 3y 3 x + y 4 3

4 x A x 5 A 5 3 y A y A 5 5 The solution of the linear system is ( ) x y ( ) 3 4

5 M 4 - GENERAL MATHEMATICS -- Dr. Tariq A. AlFadhel Solution of the Second Mid-Term Exam First semester H Q. Solve by Gauss Elimination method the linear system : x + 3y + z x y 3z 3 3x 4y z 5 The augmented matrix is 3 3 R 3 3 +R R +R 3 3R +R 3 6z z y +z y + y 7 R x+3y +z x 3+4 x+ x x The solution of the linear system is y. z Q. Compute the integrals : (a) (b) (c) (d) (e) xe x dx x+ (x+)(x 3) dx 4x(x +3) 3 dx x lnx dx x (x ) dx alfadhel@ksu.edu.sa 5

6 (a) Using integration by parts : u x du dx dv e x dx v e x xe x dx [xe x ] e x dx [xe x ] [ex ] [( e ) ( e )] [ e e ] (e ) (e ) e e+ (b) Using the method of partial fractions : x+ (x+)(x 3) A x+ + A x 3 x+ A (x 3)+A (x+) Put x : + A ( 3) 5A A 5 Put x 3 : 3+ A (3+) 4 5A A 4 5 ( x+ 4 ) (x+)(x 3) dx 5 x+ + 5 dx x 3 5 x+ dx+ 4 5 x 3 dx 5 ln x ln x 3 +c (c) Using the formula [f(x)] n f (x) dx [f(x)]n+ +c, where n n+ 4x(x +3) 3 dx (x +3) 3 4x dx (x +3) 4 +c 4 (d) Using integration by parts : u lnx du x dx dv x dx v x x lnx dx x x lnx x lnx x dx x dx x lnx x +c (e) Using the method of partial fractions : x (x ) A (x ) + A (x ) 6

7 x A (x )+A A x A +A Comparing the coefficients of both sides A A +A +A A ( ) x (x ) dx x + (x ) dx x dx+ (x ) x dx+ (x ) dx ln x + (x ) +c Q.3 (a) Sketch the region R determined by the curves : y x, x, x and y. (b) Find the area of the region R. (a) y x is a parabola opens upwards with vertex (,). x is a straight line parallel to the y-axis and passes through (,). x is a straight line parallel to the y-axis and passes through (,). y is the x-axis. 4 3 y x 3 x x (b) Area [ x x 3 dx 3 ]

8 M 4 - GENERAL MATHEMATICS -- Dr. Tariq A. AlFadhel 3 Solution of the Final Exam First semester H 5 3 Q. (a) Compute AB and BA for A 3 3 and B (b) Compute the determinant 3 3. (c) Solve by Cramer s rule : 4x y x 3y 5 3 (a) AB BA (b) Solution (): Using Sarrus Method (3++6) (7+4+) 33 3 Solution (): R +R alfadhel@ksu.edu.sa 8

9 3R +R R +R 3 3 Solution (3) : Using the definition of the determinant (3 4) ( 6)+3 ( 9) + (c) Using Cramer s rule : ( ) ( ) ( ) 4 4 A, A 3 x, A 3 y A 4 3 (4 3) ( ) ( ) + A x 3 ( 3) ( ) 6 4 A y 4 (4 ) ( ) 8 x A x A y A y A The solution of the linear system is ( ) x y ( ) Q. (a) Find the standard equation of the ellipse with foci (,3) and (6,3) and the length of its major axis is, and then sketch it. (b) Find The elements of the conic section 9x 4y 6y 8x 43 (a) The standard equation of the ellipse is (x h) a The center P (h,k) is located between the two foci. ( +6 The center is P (h,k), 3+3 ) (,3) + (y k) b The major axis (where the two foci are located) is parallel to the x-axis, hence a > b. The length of the major axis equals means a a 5. 9

10 c is the distance between the two foci, hence c 4 c a b 6 5 b b b 3 The standard equation of the ellipse is (x ) 5 The vertices are V ( 3,3) and V (7,3) + (y 3) 9 The end-points of the minor axis are W (,6) and W (,) x y W F P F V 3 V W (b) 9x 4y 6y 8x 43 9x 8x 4y 6y 43 9(x x) 4(y +4y) 43 By completing the square 9(x x+) 4(y +4y +4) (x ) 4(y +) 36 9(x ) 36 (x ) 4 4(y +) 36 (y +) 9 The conic section is a hyperbola with transverse axis parallel to the x-axis. The ceneter is P (, ). a 4 a and b 9 b 3. c a +b c 3. The vertices are V (, ) and V (3, ).

11 ( The foci are F ) ( 3, and F + ) 3,. The equations of the asymptotes are L : y + 3 (x ) and L : y + 3 (x ) L L P F V V F 5 9x 4y 6y 8x 43 Q.3 (a) Compute the integrals : (i) x dx (ii) x lnx dx (iii) +6x+ (b) Find the area of the surface delimited by the curves : y, x, x 4 and y x+5. x (x+)(x ) dx (c) The region R between the curves y, y x, and y x + is rotated about the y-axis to form a solid of revolution S. Find the volume of S. (a) (i) x +6x+ dx f (x) Using the formula [f(x)] +a dx ( ) f(x) a tan +c a x dx +6x+ (x +6x+9)+ dx

12 (x+3) + dx tan (x+3)+c (ii) x lnx dx Using integration by parts u lnx dv x dx du x dx v x3 3 x lnx dx x3 3 lnx x 3 x 3 dx x3 3 lnx x dx x3 3 3 lnx 3 (iii) x (x+)(x ) dx Using the method of partial fractions (x+)(x ) A x+ + A x x 3 3 +c (x )(x+) dx (x+)(x ) A (x ) (x+)(x ) + A (x+) (x+)(x ) A (x )+A (x+) Put x then A A Put x then A A ( ) (x+)(x ) dx x+ + dx x x+ dx+ x dx ln x+ + ln x +c (b) y is the x-axis. x is the y-axis. x 4 is a straight line parallel to the y-axis and passes through (4,). y x+5 (y 5) x is the upper-half of the parabola with vertex (,5) and opens to the right

13 7 y x 5 5 x 4 4 Area 4 ( 3 (4) ( x+5 ) dx [x 3 ) 3 +5x ] 4 ( 3 ()3 +5 ) 3 [ ] 4 3 x3 +5x (8) (c) y x y x y is the x-axis. y x+ is a straight line passes through (,) and its slope is -. y x is a parabola opens upwards with vertex (,). Points of intersection of y x and y x+ : 3

14 x x+ x +x (x+)(x ) x, x Points of intersection are (,4) and (,). Using Washer Method : In this case y x+ x y + and y x x y Volume π ( ( y +) ( y) ) dy π [ ] y π (y 3 5y +4) dy π 3 5y +4y [( π 3 5 ) ] ( ) ( +) π 6 (4 4y +y y) dy 6 π Q.4 (a) Find f x and f y for the function f(x,y) x y 3 +ye x + x x+y (b) Solve the differential equation x dy dx y x e x (a) f x f x xy3 +ye x + (x+y) x(+) (x+y) xy 3 +ye x y + (x+y) f y 3x y +e x + (x+y) x(+) (x+y) 3x y +e x x (x+y) (b) x dy dx y x e x xy y x e x y x y xex It is a first-order linear differential equation. P(x) x and Q(x) xex The integrating factor is: u(x) P(x) dx e x dx e lnx e lnx x x The general solution of the differential equation is : y u(x) Q(x) dx u(x) x x xe x dx x x(e x +c) xe x +cx e x dx 4

15 M 4 - GENERAL MATHEMATICS -- Dr. Tariq A. AlFadhel 4 Solution of the First Mid-Term Exam Second semester H Q. Let A ( ) 4 and B 3 4 Compute (if possible) : AB and BA ( ) 4 AB 3 4 ( ) ( ) ( ) BA Q. Compute The determinant Solution () : Using Sarrus Method (6+8+) (3++8) Solution () : Using The definition (6 ) (9 4)+4(3 8) alfadhel@ksu.edu.sa 5

16 Q.3 Find all the elements of the conic section y 4y 4x+4 and sketch it. By completing the square y 4y 4x+4 y 4y +4 4x+4+4 y 4y +4 4x+8 (y ) 4(x+) The conic section is a parabola opens to the right. The vertex is V(,). 4a 4 a 4 4. The focus is F( +,) (,). The equation of the directrix is x 3 y 4y 4x 4 V F x 3 3 Q.4 Find the standard equation of the ellipse with vertex (,4) and with foci (,4) and (,4), and then sketch it. The equation of the ellipse has the form (x h) a + (y k) b. The center of the ellipse is the mid-point of the two foci. ( + The center is P, 4+4 ) (6,4), hence h 6 and k. 6

17 The two foci lie on a line parallel to the x-axis, hence a > b c is the distance between the center and one of the foci, hence c 4 a is the distance between the center (6,4) and the vertex V (,4), hence a 5. c a b 4 5 b b b 3. The equation of the ellipse is (x 6) 5 The other vertex is V (,4). + (y 4) 9. The end-points of the minor axis are W (6,7) and W (6,) 7 W x 6 5 y P V F F V W 6 Q.5 Solve by Cramer s Rule the following linear system { 4x 3y x + y 3 ( ) ( ) ( ) A, A x, A 3 y 3 A 4 3 (4 ) ( 3) 4 ( 6) 4+6 A x 3 3 ( ) (3 3) ( 9) +9 A y 4 3 (4 3) ( ) x A x A 7

18 y A y A The solution of the linear system is ( ) x y ( ) 8

19 M 4 - GENERAL MATHEMATICS -- Dr. Tariq A. AlFadhel 5 Solution of the Second Mid-Term Exam Second semester H Q. Find the area of the surface bounded by the curves : y x + and y 3. y x + y x is a parabola opens upwards with vertex (,). y 3 is a straight line parallel to the x-axis and passes through (,3). Points of intersection of y x + and y 3 : x + 3 x x ±. y x 3 y 3 Area ] [x x3 3 [ 3 (x +) ] dx ( ) ( 3 3 ( x ) dx ) 3 ( ) Q. Compute the integrals : x+5 (a) (x+)(x+) dx x+3 (b) x +3x+6 dx 3 (c) x +x+5 dx 5 alfadhel@ksu.edu.sa 9

20 (d) (e) (f) 3x sin(x 3 +3) dx xcosx dx x(x +) 5 dx (a) x+5 (x+)(x+) dx Using the method of partial fractions: x+5 (x+)(x+) A x+ + A x+ x+5 A (x+)+a (x+) Put x : ( )+5 A ( +) A A Put x : ( )+5 A ( +) A +5 3 ( x+5 (x+)(x+) dx x+ + 3 ) dx x+ x+ dx+3 dx ln x+ +3ln x+ +c x+ (b) (c) x+3 x +3x+6 dx Using the formula f (x) f(x) dx ln f(x) +c x+3 x +3x+6 dx ln x +3x+6 +c 3 x +x+5 dx Using the formula 3 x +x+5 dx 3 3 tan ( x+ f (x) a +[f(x)] dx ( f(x) a tan a (x +x+)+4 dx 3 ) +c 3 tan ( x+ ) +c ) +c, where a > (x+) + dx (d) 3x sin(x 3 +3) dx

21 Using the formula sin(f(x)) f (x) dx cos(f(x))+c 3x sin(x 3 +3) dx sin(x 3 +3) (3x ) dx cos(x 3 +3)+c (e) xcosx dx Using integration by parts : u x dv cosx dx du dx v sinx xcosx dx xsinx sinx dx xsinx ( cosx)+c xsinx+cosx+c (f) x(x +) 5 dx Using the formula [f(x)] n f (x) dx [f(x)]n+ +c, where n n+ x(x +) 5 dx (x +) 5 (x) dx (x +) 6 +c 6 Q.3 Find the area of the srface bounded by the curves : y x, y and y x+ y x y x is the upper half of a parabola opens to the right with vertex (,). y x+ is a straight line passes through (,) with slope -. y is the x-axis. y x y x and y x+ x y + Points of intersection of x y and x y + : y y + y +y (y +)(y ) y, y

22 y x y x Area [( y +) y ] dy ( y y +) dy ( 3 ) + ( +) ] [ y3 3 y +y Another solution : Area [ 3 ] x dx+ ( x+) dx [ + ( +4) ( + )] [ x 3 3 ] ] + [ x +x

23 M 4 - GENERAL MATHEMATICS -- Dr. Tariq A. AlFadhel 6 Solution of the Final Exam Second semester H Q. (a) Compute BA+AB for A and B 3 3 (b) Compute the determinant 3 4. x + y + z 3 (c) Solve by Gauss Elimination Method : x y z x + y z 3 (a) BA AB BA+AB (b) Using Sarrus Method (++) ( 6 9+) 4 ( 5) alfadhel@ksu.edu.sa 3

24 (c) Using Gauss Elimination Method 3 R+R 3 R +R 3 R +R z z y +z y + y 3 3 R x+y +z 3 x++ 3 x x The solution of the linear system is y z Q. Find all the elements of the two following conic sections and sketch their graphs : (a) 9y +4x +8y 8x 3 (b) 9x 4y 6y 8x 43 (a) 9y +4x +8y 8x 3 4x 8x+9y +8y 3 4(x x)+9(y +y) 3 By completing the square 4(x x+)+9(y +y +) (x ) +9(y +) 36 4(x ) + 36 (x ) + 9 9(y +) 36 (y +) 4 The conic sectionis an ellipse with center P (, ) a 9 a 3 and b 4 b c a b c 5 4

25 The vertices are V (, ) and V (4, ) The Foci are F ( 5, ) and F ( + 5, ) The end-points of the minor axis are W (,) and W (, 3) W V F P F V 3 W x y 9 4 (b) 9x 4y 6y 8x 43 9x 8x 4y 6y 43 9(x x) 4(y +4y) 43 By completing the square 9(x x+) 4(y +4y +4) (x ) 4(y +) 36 9(x ) 36 (x ) 4 4(y +) 36 (y +) 9 The conic section is a hyperbola with transverse axis parallel to the x-axis. The ceneter is P (, ). a 4 a and b 9 b 3. c a +b c 3. The vertices are V (, ) and V (3, ). ( The foci are F ) ( 3, and F + ) 3,. The equations of the asymptotes are L : y + 3 (x ) and 5

26 L : y + 3 (x ) L L F P V V F 5 9x 4y 6y 8x 43 Q.3 (a) Compute the integrals : (i) dx (ii) xsinx dx (iii) (x )(x+) (b) Find the area of the region delimited by the curves : y and y x +. x 4 x 4x+5 dx (c) The region R in the first quadrant lying between the curves y and y x is rotated about the x-axis to form a solid of revolution S. Find the volume of S. (a) (i) (x )(x+) dx Using the method of partial fractions (x )(x+) A x + A x+ A (x+)+a (x ) Put x : A A Put x : A A (x )(x+) x + x+ 6

27 (x )(x+) dx x dx (ii) xsinx dx x+ Using integration by parts ( x + ) x+ dx dx ln x ln x+ +c u x dv sinx dx du dx v cosx xsinx dx xcosx cosx dx xcosx+ cosx dx xcosx+sinx+c x 4 (iii) x 4x+5 dx Using the formula f (x) f(x) dx ln f(x) +c x 4 x 4x+5 dx ln x 4x+5 +c (b) y is a straight line parallel to the x-axis and passes through (,) y x + y x is a parabola with vertex (,) and opens upwards Points of intersection of y and y x + : x + x (x )(x+) x, x y x y Area ( 3 [ (x +) ] dx ) ( 3 ( x ) dx ) 3 ( ) ] [x x

28 (c) y is the x-axis. y x (y ) x is a parabola opens downwards with vertex (,). Points of intersection of y x and y : x x (x )(x+) x, x y x y Using Disk Method : Volume π [( π ( x ) dx π ) ] ( +) ( x +x 4 ) dx 5 +3 π π ] [x x 3 + x5 5 Q.4 (a) Find f x, f y and f z for the function f(x,y,z) x y 3 +z 5 ye y (b) Solve the differential equation xdy +y dx, y() (a) f x xy 3 + xy 3 f y x (3y )+z 5 ( e y +ye y ) 3x y +z 5 (e y +ye y ) f z +5z 4 ye y 5z 4 ye y (b) xdy +y dx x dy y dx y dy x dx y dy x dx It is a separable differential equation 8

29 y dy x dx y ln x +c y ln x +c The general solution of the differential equation is : y ln x +c y() ln()+c +c c c The particular solution of the differential equation is : y ln x + 9