FE MECHANICAL – Math_medium_difficulty

#1. Calculate the magnitude of the vector v = 4i – 3j + 12k.

11.00

15.93

19.00

13.00

#2. Find the dot product of vectors A = 2i + 5j – k and B = 3i – 2j + 4k.

0

4

12

-8

#3. Determine the unit vector in the direction of v = 6i + 8j.

i + j

0.5i + 0.5j

0.6i + 0.8j

0.8i + 0.6j

#4. Calculate the distance between points A(1, 3, -2) and B(4, -1, 2).

7.81

5.00

25.00

6.40

#5. Given vectors P = i + 2j + 3k and Q = 2i – j + k, find |P × Q|.

14.00

11.22

6.00

8.66

#6. Find the angle (in degrees) between vectors u = 3i + 4j and v = 4i – 3j.

90.0°

45.0°

120.0°

60.0°

#7. Calculate the scalar projection of A = 5i + 12j onto B = 3i + 4j.

13.00

15.00

8.60

11.40

#8. Determine the direction cosine of vector v = 2i + 3j + 6k with respect to the z-axis.

0.286

0.600

0.429

0.857

#9. Find the midpoint between P₁(-3, 5, 2) and P₂(7, -1, 6).

(2, 3, 4)

(4, 4, 8)

(5, 2, 8)

(2, 2, 4)

#10. If |A| = 5, |B| = 7, and A·B = 21, find the angle between A and B.

30°

0°

90°

53.1°

#11. Find f'(x) for f(x) = x⁴ – 3x² + 2x evaluated at x = 2.

22

26

18

30

#12. Evaluate the derivative of y = ln(x²) at x = e.

1/e

2

e

2/e

#13. Find dy/dx for y = e^(3x)cos(2x) at x = 0.

0

2

3

5

#14. Using the chain rule, find f'(x) for f(x) = sin(x³) at x = 1.

3cos(1)

cos(1)

3sin(1)

-3sin(1)

#15. Find the second derivative f”(x) for f(x) = x⁴e^x at x = 0.

8

0

4

12

#16. Determine the derivative of f(x) = arcsin(2x) at x = 0.

1/2

0

2

1

#17. For f(x) = (x² + 1)/(x – 1), find f'(2).

1

4

2

3

#18. Find dy/dx for the implicit function x³ + y³ = 6xy at point (3, 3).

-1

0

2

1

#19. Calculate f'(x) for f(x) = x^x at x = e (use ln differentiation).

2e^e

e^(e-1)

e^e

e^(e+1)

#20. Find the rate of change of the area of a circle with respect to its radius when r = 5.

25π

5π

10π

50π

#21. Evaluate ∫(3x² – 2x + 1)dx.

x³ – 2x² + x + C

x³ – x² + x + C

3x³ – 2x² + x + C

3x³ – x² + x + C

#22. Calculate ∫₀² (x² + 2x)dx.

8/3

8

12

20/3

#23. Evaluate ∫e^(2x)sin(x)dx (use integration by parts).

(1/3)e^(2x)sin(x) + C

(1/5)e^(2x)(sin(x) + 2cos(x)) + C

(1/5)e^(2x)(2sin(x) – cos(x)) + C

e^(2x)sin(x) + C

#24. Find ∫₀^(π/4) sec²(x)dx.

√2

1

π/4

0

#25. Evaluate ∫x·ln(x)dx.

x²ln(x) + C

xln(x) – x + C

(x²/2)ln(x) + C

(x²/2)ln(x) – x²/4 + C

#26. Calculate the area under y = 1/x from x = 1 to x = e.

1/e

0

1

e

#27. Evaluate ∫₋₁¹ x³dx.

-2

1/2

0

2

#28. Find ∫(1/(x² + 4))dx.

arctan(x/2) + C

(1/4)arctan(2x) + C

(1/2)arctan(x/2) + C

ln(x² + 4) + C

#29. Evaluate ∫₀¹ √(1 – x²)dx (area of quarter circle).

1

π/2

π

π/4

#30. Calculate ∫x²e^x dx using integration by parts.

e^x(x² – 2x + 2) + C

(x² – 1)e^x + C

x²e^x + C

e^x(x² + 2x + 2) + C

#31. Solve dy/dx = 3y with initial condition y(0) = 2. Find y(1).

6e

2e³

2e

6

#32. Find the general solution of y” – 9y = 0.

y = C₁e^(3x) + C₂e^(-3x)

y = C₁e^(9x) + C₂e^(-9x)

y = C₁cos(3x) + C₂sin(3x)

y = (C₁ + C₂x)e^(3x)

#33. Determine the particular solution for y” + y = 2 (constant forcing term).

yₚ = 2e^x

yₚ = 2x

yₚ = 2cos(x)

yₚ = 2

#34. Solve dy/dx + y/x = x with initial condition y(1) = 2. Find y(2).

5

4

3

6

#35. Find the characteristic equation for y” + 6y’ + 9y = 0.

λ² + 6λ + 9 = 0

λ² + 6λ = 0

λ² – 6λ + 9 = 0

λ² + 9 = 0

#36. Determine the general solution of y” + 4y’ + 4y = 0.

y = C₁e^(-4x) + C₂xe^(-4x)

y = C₁e^(-2x) + C₂e^(2x)

y = (C₁ + C₂x)e^(-2x)

y = C₁cos(2x) + C₂sin(2x)

#37. Solve the separable equation dy/dx = xy with y(0) = 1. Find y at x = 2.

7.39

e⁴/2

e²

e

#38. Identify the integrating factor for dy/dx + 3y = 6x.

e^x

e^(6x)

3e^x

e^(3x)

#39. For y” – 2y’ + y = 0, the general solution is:

y = C₁cos(x) + C₂sin(x)

y = C₁e^(2x) + C₂e^(-x)

y = (C₁ + C₂x)e^x

y = C₁e^x + C₂e^(-x)

#40. Solve dy/dx = 2x + 1 with y(0) = 3. Find y(2).

7

9

5

11

#41. Find L{e^(-4t)}.

4/s

-4/(s + 4)

1/(s – 4)

1/(s + 4)

#42. Determine L{cos(4t)}.

s/(s² + 16)

s/(s² + 4)

4/(s² + 16)

1/(s² + 16)

#43. Evaluate L{t²}.

6/s³

2/s³

1/s³

2/s²

#44. Find L{sin(3t)}.

s/(s² + 9)

1/(s² + 9)

9/(s² + 9)

3/(s² + 9)

#45. Determine L⁻¹{1/(s – 5)}.

5e^t

te^(5t)

e^(-5t)

e^(5t)

#46. Find L{te^(2t)}.

1/(s – 2)

2/(s – 2)²

1/(s + 2)²

1/(s – 2)²

#47. Evaluate L{u(t – 1)} where u is the unit step function.

1/s

e^s/s

e^(-s)

e^(-s)/s

#48. Find L⁻¹{6/s⁴}.

t²

t⁴

t³

6t³

#49. Determine L{sinh(t)} (recall sinh(t) = (e^t – e^(-t))/2).

1/(s² + 1)

2/(s² – 1)

s/(s² – 1)

1/(s² – 1)

#50. If L{f(t)} = F(s), find L{e^(3t)f(t)}.

e³F(s)

F(s + 3)

3F(s)

F(s – 3)

#51. Calculate the determinant of matrix A = |3 1| |2 4|.

10

8

14

12

#52. Find the trace of matrix B = |5 2 1| |0 3 4| |1 0 2|.

12

10

8

15

#53. Given A = |2 3| |1 4| and B = |1 0| |2 1|, find element (2,1) of AB.

9

5

7

11

#54. Determine the rank of matrix C = |1 2| |2 4| |3 6|.

1

2

3

0

#55. Find the eigenvalues of matrix D = |6 -2| |-2 3|.

λ = 7, 2

λ = 8, 1

λ = 6, 3

λ = 5, 4

#56. Calculate A⁻¹ for A = |2 1| |3 2|. Find element (1,1).

1

-1

-3

2

#57. For matrix E = |1 0 0| |0 2 0| |0 0 3|, what is det(E)?

1

3

0

6

#58. If A is a 3×4 matrix with rank 3, what is the dimension of its null space?

1

3

0

2

#59. Find the transpose of matrix F = |1 2 3| |4 5 6|. What is element (3,2)?

4

3

5

6

#60. Calculate |A|² if A = |1 2| |3 4| (where |A| denotes determinant).

4

1

16

9

#61. For f(x,y) = x²y + xy², find ∂f/∂x at (2,3).

24

18

27

21

#62. Calculate the gradient ∇f at (1,1) for f(x,y) = x³ + y³.

i + j

3i + j

i + 3j

3i + 3j

#63. Evaluate ∫₀² ∫₀³ (2x + y) dy dx.

42

36

24

30

#64. For f(x,y) = xy², find the directional derivative at (1,2) in direction of i + j.

8

4

4√2

2√2

#65. Find the critical points of f(x,y) = x² + y² – 2x – 4y.

(2, 1)

(1, 2)

(0, 0)

(2, 4)

#66. Calculate div F for F = x²i + y²j + z²k at point (1,1,1).

3

6

2

4

#67. For f(x,y,z) = xyz, find ∂²f/∂x∂y.

yz

0

z

xy

#68. Evaluate ∫₀¹ ∫₀¹ e^(x+y) dx dy.

e² – 2e + 1

2e – 2

(e – 1)²

e² – 1

#69. Find the maximum value of f(x,y) = 4xy subject to x + y = 6.

48

36

24

72

#70. Calculate curl F for F = yi – xj + 0k.

-2k

-k

0

2k

#71. Find the center of the circle x² + y² – 4x + 6y – 12 = 0.

(-4, 6)

(-2, 3)

(4, -6)

(2, -3)

#72. Determine the eccentricity of ellipse x²/36 + y²/20 = 1.

0.800

0.600

0.667

0.745

#73. Find the equation of the line perpendicular to 2x + 3y = 6 passing through (0,0).

y = (-2/3)x

y = (-3/2)x

y = (2/3)x

y = (3/2)x

#74. Calculate the distance from point (1,2) to line 3x + 4y – 10 = 0.

1.0

1.4

0.2

0.6

#75. Find the vertex of parabola y = 2x² – 8x + 10.

(1, 4)

(0, 10)

(4, 10)

(2, 2)

#76. Determine the slope of line passing through (3,-2) and (7,6).

4

2

1

3

#77. For hyperbola x²/9 – y²/16 = 1, find the length of transverse axis.

6

8

4

3

#78. Find the equation of plane through (1,0,0), (0,2,0), and (0,0,3).

3x + y + z = 3

x + y + z = 1

2x + y + z = 2

6x + 3y + 2z = 6

#79. Calculate the radius of sphere (x-2)² + (y+1)² + (z-3)² = 49.

√49

7

14

49

#80. Find focus of parabola x² = 16y.

(2, 0)

(0, 4)

(0, 2)

(4, 0)

#81. Using Newton’s method with x₀ = 1, find x₁ for f(x) = x² – 3.

1.75

1.50

2.00

2.50

#82. Apply trapezoidal rule with n=2 to approximate ∫₀² x³ dx.

3

4

6

5

#83. Using Euler’s method with h=0.5, solve dy/dx = 2x with y(0)=1. Find y(0.5).

1.5

2.0

1.25

1.0

#84. Bisection method for f(x)=x²-4 on [1,3]: what is midpoint after first iteration?

1.5

1.0

2.5

2.0

#85. Using Simpson’s 1/3 rule, approximate ∫₀² x² dx with n=2.

4

10/3

8/3

3

#86. For secant method with x₀=1, x₁=2 to solve x²-5=0, calculate x₂.

2.33

2.00

2.25

2.50

#87. Using forward difference with h=0.1, approximate f'(1) for f(x)=x². (f'(x) = [f(x+h)-f(x)]/h)

2.0

2.1

2.5

1.9

#88. Fixed-point iteration: g(x)=√x, x₀=4. Find x₁.

4

1

3

2

#89. Apply Runge-Kutta 2nd order (midpoint) with h=1 for dy/dx=x, y(0)=0. Find y(1).

0.25

0.75

1.0

0.5

#90. Using 3-point central difference, the approximation of f”(x) requires function values at:

x, x+h, x+2h

x-h, x+h

x-h, x, x+h

x-2h, x-h, x

#91. Find the Taylor series of e^x about x=0 up to x² term, then evaluate at x=1.

1.5

2.5

3.0

2.0

#92. Determine radius of convergence for Σ(x^n)/n² from n=1 to ∞.

∞

0

1

e

#93. Does the series Σ(1/n²) from n=1 to ∞ converge or diverge?

Conditionally converges

Converges by p-test (p=2>1)

Diverges

Converges by ratio test

#94. Maclaurin series for sin(x): coefficient of x³ is:

1/6

1/3!

-1/3!

-1/6

#95. Find the sum of geometric series Σ(1/2)^n from n=0 to ∞.

∞

1

2

3

#96. Using ratio test, does Σ(n!/n^n) converge?

Oscillates

Inconclusive

Converges

Diverges

#97. Taylor expansion of ln(1+x) at x=0: coefficient of x²?

1

-1/2

-1

1/2

#98. Find limit: lim(n→∞) (n²+1)/(2n²-3).

1/2

1

∞

0

#99. Does alternating series Σ(-1)^n/(2n+1) from n=0 to ∞ converge?

Converges conditionally

Converges absolutely

Oscillates

Diverges

#100. Evaluate Σ(1/n(n+1)) from n=1 to ∞ using partial fractions.

1

∞

2

0

Math – Medium DifficultyARE YOU READY?

LET’S PRACTICE!

Results

#1. Calculate the magnitude of the vector v = 4i – 3j + 12k.

#2. Find the dot product of vectors A = 2i + 5j – k and B = 3i – 2j + 4k.

#3. Determine the unit vector in the direction of v = 6i + 8j.

#4. Calculate the distance between points A(1, 3, -2) and B(4, -1, 2).

#5. Given vectors P = i + 2j + 3k and Q = 2i – j + k, find |P × Q|.

#6. Find the angle (in degrees) between vectors u = 3i + 4j and v = 4i – 3j.

#7. Calculate the scalar projection of A = 5i + 12j onto B = 3i + 4j.

#8. Determine the direction cosine of vector v = 2i + 3j + 6k with respect to the z-axis.

#9. Find the midpoint between P₁(-3, 5, 2) and P₂(7, -1, 6).

#10. If |A| = 5, |B| = 7, and A·B = 21, find the angle between A and B.

#11. Find f'(x) for f(x) = x⁴ – 3x² + 2x evaluated at x = 2.

#12. Evaluate the derivative of y = ln(x²) at x = e.

#13. Find dy/dx for y = e^(3x)cos(2x) at x = 0.

#14. Using the chain rule, find f'(x) for f(x) = sin(x³) at x = 1.

#15. Find the second derivative f”(x) for f(x) = x⁴e^x at x = 0.

#16. Determine the derivative of f(x) = arcsin(2x) at x = 0.

#17. For f(x) = (x² + 1)/(x – 1), find f'(2).

#18. Find dy/dx for the implicit function x³ + y³ = 6xy at point (3, 3).

#19. Calculate f'(x) for f(x) = x^x at x = e (use ln differentiation).

#20. Find the rate of change of the area of a circle with respect to its radius when r = 5.

#21. Evaluate ∫(3x² – 2x + 1)dx.

#22. Calculate ∫₀² (x² + 2x)dx.

#23. Evaluate ∫e^(2x)sin(x)dx (use integration by parts).

#24. Find ∫₀^(π/4) sec²(x)dx.

#25. Evaluate ∫x·ln(x)dx.

#26. Calculate the area under y = 1/x from x = 1 to x = e.

#27. Evaluate ∫₋₁¹ x³dx.

#28. Find ∫(1/(x² + 4))dx.

#29. Evaluate ∫₀¹ √(1 – x²)dx (area of quarter circle).

#30. Calculate ∫x²e^x dx using integration by parts.

#31. Solve dy/dx = 3y with initial condition y(0) = 2. Find y(1).

#32. Find the general solution of y” – 9y = 0.

#33. Determine the particular solution for y” + y = 2 (constant forcing term).

#34. Solve dy/dx + y/x = x with initial condition y(1) = 2. Find y(2).

#35. Find the characteristic equation for y” + 6y’ + 9y = 0.

#36. Determine the general solution of y” + 4y’ + 4y = 0.

#37. Solve the separable equation dy/dx = xy with y(0) = 1. Find y at x = 2.

#38. Identify the integrating factor for dy/dx + 3y = 6x.

#39. For y” – 2y’ + y = 0, the general solution is:

#40. Solve dy/dx = 2x + 1 with y(0) = 3. Find y(2).

#41. Find L{e^(-4t)}.

#42. Determine L{cos(4t)}.

#43. Evaluate L{t²}.

#44. Find L{sin(3t)}.

#45. Determine L⁻¹{1/(s – 5)}.

#46. Find L{te^(2t)}.

#47. Evaluate L{u(t – 1)} where u is the unit step function.

#48. Find L⁻¹{6/s⁴}.

#49. Determine L{sinh(t)} (recall sinh(t) = (e^t – e^(-t))/2).

#50. If L{f(t)} = F(s), find L{e^(3t)f(t)}.

#51. Calculate the determinant of matrix A = |3 1| |2 4|.

#52. Find the trace of matrix B = |5 2 1| |0 3 4| |1 0 2|.

#53. Given A = |2 3| |1 4| and B = |1 0| |2 1|, find element (2,1) of AB.

#54. Determine the rank of matrix C = |1 2| |2 4| |3 6|.

#55. Find the eigenvalues of matrix D = |6 -2| |-2 3|.

#56. Calculate A⁻¹ for A = |2 1| |3 2|. Find element (1,1).

#57. For matrix E = |1 0 0| |0 2 0| |0 0 3|, what is det(E)?

#58. If A is a 3×4 matrix with rank 3, what is the dimension of its null space?

#59. Find the transpose of matrix F = |1 2 3| |4 5 6|. What is element (3,2)?

#60. Calculate |A|² if A = |1 2| |3 4| (where |A| denotes determinant).

#61. For f(x,y) = x²y + xy², find ∂f/∂x at (2,3).

#62. Calculate the gradient ∇f at (1,1) for f(x,y) = x³ + y³.

#63. Evaluate ∫₀² ∫₀³ (2x + y) dy dx.

#64. For f(x,y) = xy², find the directional derivative at (1,2) in direction of i + j.

#65. Find the critical points of f(x,y) = x² + y² – 2x – 4y.

#66. Calculate div F for F = x²i + y²j + z²k at point (1,1,1).

#67. For f(x,y,z) = xyz, find ∂²f/∂x∂y.

#68. Evaluate ∫₀¹ ∫₀¹ e^(x+y) dx dy.

#69. Find the maximum value of f(x,y) = 4xy subject to x + y = 6.

#70. Calculate curl F for F = yi – xj + 0k.

#71. Find the center of the circle x² + y² – 4x + 6y – 12 = 0.

#72. Determine the eccentricity of ellipse x²/36 + y²/20 = 1.

#73. Find the equation of the line perpendicular to 2x + 3y = 6 passing through (0,0).

#74. Calculate the distance from point (1,2) to line 3x + 4y – 10 = 0.

#75. Find the vertex of parabola y = 2x² – 8x + 10.

#76. Determine the slope of line passing through (3,-2) and (7,6).

#77. For hyperbola x²/9 – y²/16 = 1, find the length of transverse axis.

Math –
Medium Difficulty

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