FE MECHANICAL – Math_high_difficulty

#1. Find the volume of parallelepiped formed by vectors A = 2i + 3j – k, B = i – j + 4k, and C = 3i + j + 2k using scalar triple product.

31

35

43

39

#2. Given vectors P = ai + 2j + 3k and Q = i + bj – k are perpendicular, and |P| = 7, find a if b = 1.

2.83

4.12

3.74

3.00

#3. Find the shortest distance from point (2,3,1) to the line r = (1,1,0) + t(2,1,3).

2.45

2.83

2.14

1.83

#4. Calculate the angle between diagonal of cube and any of its edges (exact answer).

60.00°

45.00°

54.74°

35.26°

#5. Find the equation of plane containing line r = i + t(j + k) and parallel to vector 2i – j.

2x – y + z = 2

x + 2y – z = 1

x – y + 2z = 1

2x + y – z = 2

#6. Vectors A, B, C form right-handed orthogonal system. If A = 3i + 4j and B·k = 0, |B| = 5, find C.

0

Cannot determine

±k

±(4i – 3j)/5

#7. Find work done by force F = yzi + xzj + xyk along curve x = t², y = 2t, z = t³ from t=0 to t=1.

4.0

3.5

3.0

2.5

#8. A parallelepiped has edges A = i + j, B = j + k, C = k + i. Find angle between diagonals d₁ = A + B + C and d₂ = A – B + C.

60.0°

109.5°

70.5°

90.0°

#9. Find the area of triangle with vertices A(1,2,3), B(3,0,1), C(2,1,2) using cross product method.

2.65

3.00

1.87

2.24

#10. Vector r makes angles α, β, γ with x, y, z axes respectively. If α = 60° and β = 45°, find γ.

60°

45°

30°

Not uniquely determined

#11. Evaluate lim(x→0) [sin(3x) – 3x + (9x³/2)]/(x⁵) using Taylor series.

27/40

-27/40

-9/40

9/40

#12. Find d³y/dx³ for y = e^x·sin(x) at x = π.

2e^π

4e^π

-4e^π

-2e^π

#13. Evaluate ∫₀^(π/2) ln(sin(x))dx (known integral).

-π·ln(2)/2

π/2

-π/4

0

#14. Find the radius of curvature for y = x³ at x = 1.

5√10/9

√10/9

10/9

5/9

#15. Evaluate ∫₀^∞ x²e^(-x²)dx using substitution.

√π/2

1/2

√π/4

1

#16. Find the length of astroid x = a·cos³(t), y = a·sin³(t) for 0 ≤ t ≤ 2π.

4a

6a

8a

2πa

#17. Determine d/dx[∫ₓ^(x²) sin(t²)dt] using Leibniz rule.

x·sin(x⁴) – sin(x²)

sin(x⁴) – sin(x²)

2x·sin(x⁴) – sin(x²)

2x·sin(x⁴)

#18. Find area between curves y = x³ – x and y = x – x³.

2

1/2

1

0

#19. Evaluate lim(x→∞) x[√(x² + 1) – x].

1

∞

1/2

0

#20. Find coefficient of x⁵ in Taylor expansion of e^x·cos(x) about x = 0.

1/30

1/15

-1/30

-1/15

#21. Evaluate ∫(x² + 1)/((x² – 1)(x² + 4))dx using partial fractions.

ln|x⁴ + 3x² – 4| + C

(1/5)ln|x-1| – (1/10)ln|x+1| + (1/10)arctan(x/2) + C

Complex expression

(1/4)ln|x²-1| + (1/8)arctan(x/2) + C

#22. Calculate ∫₀^(π/4) x²·sec²(x)dx using integration by parts twice.

π²/32 – ln(2)/8 + 1/4

1/2

π/8

π²/16 + 1/2

#23. Evaluate improper integral ∫₁^∞ (ln(x))/x² dx.

0

∞ (diverges)

1

2

#24. Find ∫√(x² + 4)dx using trigonometric substitution.

(x/2)√(x²+4) + 2ln|x + √(x²+4)| + C

x√(x²+4) + C

2ln|x + √(x²+4)| + C

√(x²+4) + C

#25. Evaluate ∫₀^1 x³√(1-x²)dx using substitution u = 1-x².

2/15

4/15

1/5

1/15

#26. Calculate surface area of revolution for y = √x from x = 0 to x = 4 about x-axis.

16π/3

56π/9

98π/9

8π

#27. Evaluate ∫e^x·cos(x)dx using integration by parts.

e^x·sin(x) + C

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

e^x·cos(x) + C

(e^x/2)(cos(x) – sin(x)) + C

#28. Find ∫₀^(π/2) sin⁴(x)dx using reduction formula.

3π/8

π/8

π/4

3π/16

#29. Evaluate ∫(2x + 3)/(x³ + x² + x + 1)dx after factoring denominator.

2ln|x²+1| + C

Complex expression

ln|x+1| – arctan(x) + C

ln|(x+1)| + arctan(x) + C

#30. Calculate ∫₀^1 ∫₀^√(1-x²) √(x² + y²) dy dx (polar coordinates recommended).

π/6

π/12

(2-√2)/3

1/3

#31. Solve y” + 4y’ + 4y = e^(-2x) using method of undetermined coefficients. Find yₚ.

yₚ = xe^(-2x)

yₚ = x²e^(-2x)/2

yₚ = x²e^(-2x)

yₚ = e^(-2x)

#32. For system dx/dt = y, dy/dt = -4x – 5y, determine stability of equilibrium at origin.

Center

Stable spiral

Unstable spiral

Saddle point

#33. Solve Bernoulli equation dy/dx + y = xy³ using substitution v = y^(-2).

y² = e^x/(1 + Ce^x)

1/y² = 1 + Ce^(-x)

y = 1/√(1 + Ce^(-x))

Complex solution

#34. Find particular integral for y” – 2y’ + y = x²e^x using undetermined coefficients.

yₚ = x²(Ax² + Bx + C)e^x

yₚ = (x²/2)e^x

yₚ = (x⁴/12 + x³/3)e^x

yₚ = (x⁴/12)e^x

#35. Solve exact equation (2xy + y²)dx + (x² + 2xy – 1)dy = 0.

x²y + xy²/2 – y = C

xy² + x²y = C

x²y + xy² – y = C

2x²y + xy² = C

#36. For Cauchy-Euler equation x²y” + 4xy’ + 2y = 0, find general solution.

y = C₁x + C₂x²

y = C₁/x + C₂/x²

y = (C₁ + C₂ln(x))/x

y = C₁ + C₂ln(x)

#37. Solve using Laplace: y” + 9y = cos(2t), y(0) = 0, y'(0) = 1. Find Y(s).

Y(s) = 1/(s²+9)

Y(s) = s/((s²+4)(s²+9))

Y(s) = (s²+s+4)/((s²+4)(s²+9))

Y(s) = 1/(s²+9) + s/((s²+4)(s²+9))

#38. Find Green’s function for y” + y = 0 with y(0) = y(π) = 0.

Cannot determine

G(x,t) = sin(x)sin(π-t) for x<t

G(x,t) = sin(x-t)

G(x,t) = 0

#39. Solve variation of parameters for y” + y = sec(x). Form of yₚ?

yₚ = xsec(x)

yₚ = Asec(x)

yₚ = u₁cos(x) + u₂sin(x) where u₁’, u₂’ determined

yₚ = ln|sec(x) + tan(x)|

#40. For y” – 2y’ + 5y = 0, classify the type of solution.

Oscillatory with exponential growth

Pure oscillation

Exponential decay without oscillation

Oscillatory with exponential decay

#41. Find L{t·cos(ωt)} using differentiation property.

2s/(s²+ω²)²

d/ds[s/(s²+ω²)] = (s²-ω²)/(s²+ω²)²

s/(s²+ω²)²

(s²-ω²)/(s²+ω²)²

#42. Evaluate L⁻¹{s/((s²+1)(s²+4))}.

(1/3)sin(t)sin(2t)

cos(t)cos(2t)

sin(t) + sin(2t)

(1/3)(cos(t) – cos(2t))

#43. Using convolution theorem, find L⁻¹{1/((s²+1)²)}.

sin(t) + t·cos(t)

t·sin(t)

(1/2)t·sin(t)

(1/2)(sin(t) – t·cos(t))

#44. Solve integral equation y(t) = t + ∫₀^t y(τ)dτ using Laplace transforms.

y(t) = e^t

y(t) = 2e^t – t – 1

y(t) = te^t

y(t) = e^t – 1 + t

#45. Find L{∫₀^t τsin(τ)dτ} without evaluating the integral.

2/(s²+1)²

1/s · 2s/(s²+1)² = 2/(s²+1)²

s/(s²+1)²

1/(s(s²+1))

#46. Determine L{e^(at)sinh(bt)}.

1/((s-a)²-b²)

b/((s-a)²-b²)

(s-a)/((s-a)²-b²)

ab/((s-a)²-b²)

#47. Using initial value theorem, find lim(t→0⁺) f(t) if F(s) = (2s²+3s+1)/(s³+s²).

0

3

Does not exist (∞)

2

#48. Solve y” + 2y’ + 2y = δ(t-π) with y(0)=0, y'(0)=0 where δ is Dirac delta.

y = u(t-π)sin(t-π)

y = e^(-t)sin(t)

y = u(t-π)e^(-(t-π))sin(t-π)

y = δ(t-π)

#49. Find L{f(t)} where f(t) is periodic with period 2π and f(t) = sin(t) for 0 ≤ t < 2π.

e^(-πs)/(s²+1)

Cannot determine

1/((s²+1)(1-e^(-2πs)))

1/(s²+1)

#50. Evaluate L⁻¹{ln((s+1)/s)} using differentiation property of inverse transforms.

e^(-t)/t

1/t

(1-e^(-t))/t

ln(1+t)

#51. Matrix A = |2 1 0| |1 2 1| |0 1 2| is tridiagonal. Find its eigenvalues.

1, 2, 3

2, 2±√2

2, 2, 2

0, 2, 4

#52. For matrix A with eigenvalues 2, 3, 5, what is det(A²)?

900

300

100

30

#53. Find the Jordan canonical form of A = |4 1| |0 4|.

J = |4 1| |0 4|

J = |4 0| |0 4|

J = |4 0| |1 4|

J = |1 4| |0 1|

#54. Determine whether vectors v₁=(1,2,1), v₂=(2,1,3), v₃=(1,-1,2) span R³.

Yes, but not uniquely

Yes, determinant ≠ 0

Cannot determine

No, linearly dependent

#55. Find the spectral radius of A = |3 1| |-1 1| (largest absolute eigenvalue).

2.414

2.236

2.000

3.000

#56. Using Cayley-Hamilton theorem, if A = |1 2| |3 4| satisfies A² – 5A + kI = 0, find k.

0

4

-2

2

#57. For orthogonal matrix Q, which statement is always true?

det(Q) = 1

Q^T = Q^(-1)

Q² = I

All eigenvalues = 1

#58. Find the minimal polynomial of A = |2 0 0| |0 2 1| |0 0 2|.

(λ-2)

(λ-2)²

(λ-2)³

λ²-4λ+4

#59. Matrix A is 5×3 with rank 3. What is the dimension of column space?

3

5

0

2

#60. For positive definite matrix A, all eigenvalues must be:

Complex conjugates

Zero

Real and negative

Real and positive

#61. For f(x,y) = x³ – 3xy², classify critical point at (0,0) using Hessian.

Saddle point

Local maximum

Degenerate

Local minimum

#62. Calculate ∫∫∫_E (x² + y² + z²)dV where E is sphere x² + y² + z² ≤ a².

4πa⁴/3

πa⁵

πa⁵/5

4πa⁵/5

#63. Verify Stokes’ theorem for F = -yi + xj over hemisphere x² + y² + z² = 1, z ≥ 0.

∫∫(∇×F)·dS = ∮F·dr = 2π

Cannot verify without calculation

Both equal π

Both equal 0

#64. Find the maximum of f(x,y,z) = xyz subject to x + y + z = 9 and x,y,z > 0.

27

54

243

81

#65. Evaluate ∫₀^∞ ∫₀^∞ e^(-(x²+y²))dx dy using polar coordinates.

π/4

2π

π

π/2

#66. For transformation u = x + y, v = x – y, find Jacobian ∂(u,v)/∂(x,y).

-2

2

0

1

#67. Calculate flux of F = xi + yj + zk through closed surface of cube [0,1]³.

3

2

6

1

#68. Find equation of tangent plane to x² + 2y² + 3z² = 21 at (1,2,2).

2x + 8y + 12z = 42

x + 4y + 6z = 21

2x + 4y + 6z = 21

x + 2y + 3z = 11

#69. Evaluate ∫_C (y²dx + 2xydy) where C is ellipse x²/4 + y² = 1, counterclockwise.

4π

2π

8π

0

#70. For f(x,y) = e^(-x²-y²), find the maximum directional derivative at origin.

√2

1

2

0

#71. Find the equation of sphere passing through points (0,0,0), (1,0,0), (0,1,0), (0,0,1).

(x-1)² + (y-1)² + (z-1)² = 1

(x-1/2)² + (y-1/2)² + (z-1/2)² = 3/4

x² + y² + z² – x – y – z = 0

x² + y² + z² = 1

#72. Determine angle between planes 2x – y + z = 3 and x + y + z = 1.

60.0°

70.5°

90.0°

73.4°

#73. Find parametric equations of line intersection of planes x + y + z = 6 and x – y + 2z = 3.

x = 3+3t, y = 3-t, z = t

Cannot determine uniquely

x = 3t, y = 3-t, z = t

x = 3-t, y = 3+t, z = t

#74. Calculate the volume of ellipsoid x²/a² + y²/b² + z²/c² = 1.

4πabc

πabc

(2/3)πabc

(4/3)πabc

#75. For rotated conic 3x² + 2xy + 3y² = 4, find the rotation angle to eliminate xy term.

22.5°

30.0°

60.0°

45.0°

#76. Find the distance between parallel planes 2x + 2y + z = 4 and 2x + 2y + z = 10.

1

6

2

3

#77. Determine the center of mass of region bounded by y = x² and y = 4 (uniform density).

(0, 12/5)

(0, 3)

(0, 8/5)

(0, 2)

#78. For cone z² = x² + y², 0 ≤ z ≤ h, calculate surface area.

πh√2

πh²√2

πh√(2h²)

2πh²

#79. Find locus of points equidistant from point (0,0,1) and plane z = -1.

Plane z = 0

Sphere x² + y² + z² = 1

Paraboloid x² + y² = 4z

Paraboloid x² + y² = 2z

#80. Calculate moment of inertia of circle x² + y² ≤ a² about x-axis (uniform density ρ).

πρa⁴

πρa⁴/4

2πρa⁴

πρa⁴/2

#81. Using Modified Euler method (Heun’s) with h=0.1 for dy/dx = x+y, y(0)=1, find y(0.1).

1.105

1.120

1.115

1.110

#82. Apply 4th-order Runge-Kutta with h=0.5 for dy/dx = -y, y(0)=1. Find y(0.5).

0.609

0.606

0.612

0.603

#83. Using Richardson extrapolation, improve trapezoidal approximation of ∫₀¹ e^x dx with h=1 and h=0.5.

1.359

2.000

1.500

1.718

#84. For stiff ODE y’ = -1000y + 3000 – 2000e^(-t), which method is most appropriate?

Runge-Kutta 4

Implicit (backward) Euler

Explicit Euler

Predictor-corrector

#85. Gauss-Seidel iteration for 10x + y = 11, x + 10y = 11 with (0,0): find (x₁,y₁).

(1.1, 1.0)

(1.1, 0.99)

(1.21, 1.1)

(1.0, 1.0)

#86. Using cubic spline interpolation, how many conditions are needed for n+1 data points?

n+1

4n

2n

4n-2

#87. Newton’s method for finding √N uses iteration xₙ₊₁ = (xₙ + N/xₙ)/2. Order of convergence?

Quadratic

Super-linear

Linear

Cubic

#88. For function f with f(0)=1, f(1)=2, f(2)=5, estimate f'(1) using central difference.

2.0

1.5

3.0

2.5

#89. LU decomposition of |4 3| |6 3| gives U₁₁ = 4. Find L₂₁.

2

1.5

0.67

0.75

#90. Adaptive quadrature subdivides interval when estimated error exceeds tolerance. Initial method used?

Trapezoidal rule

Gauss quadrature

Simpson’s rule

Romberg integration

#91. Find Fourier series coefficient a₁ for f(x) = x on [-π,π].

π

0

2/π

1

#92. Determine Laurent series of f(z) = 1/(z(z-1)) about z = 0 for 0 < |z| < 1.

-1/z – 1 – z – z² – …

1/z + 1 + z + z² + …

1/z – 1 + z – z² + …

Cannot expand

#93. Does series Σ(sin(n)/n²) converge absolutely, conditionally, or diverge?

Diverges

Converges absolutely

Oscillates

Converges conditionally

#94. Find radius of convergence for Σ(n!x^n)/n^n.

∞

1/e

1

e

#95. Evaluate Σ(n=1 to ∞) n/2^n using differentiation of geometric series.

1

4

3

2

#96. For power series Σaₙx^n with aₙ = 1/n!, find interval of convergence.

(-∞,∞)

(-1,1)

[0,∞)

[-1,1]

#97. Using Parseval’s identity, if f(x) = x on [-π,π], evaluate Σ(1/n²).

π²/4

π²/12

π²/8

π²/6

#98. Find sum of series Σ((-1)^(n+1))/(2n-1) from n=1 to ∞.

ln(2)

1

π/2

π/4

#99. Determine Maclaurin series for f(x) = x/(1-x²): coefficient of x⁵?

0

5

1

-1

#100. For Fourier series of f(x) = |x| on [-π,π], what type of terms appear?

Only even harmonics (cosines)

Only odd harmonics (cosines)

Only sines

Both sines and cosines

Math – High DifficultyARE YOU READY?

LET’S PRACTICE!

Results

#1. Find the volume of parallelepiped formed by vectors A = 2i + 3j – k, B = i – j + 4k, and C = 3i + j + 2k using scalar triple product.

#2. Given vectors P = ai + 2j + 3k and Q = i + bj – k are perpendicular, and |P| = 7, find a if b = 1.

#3. Find the shortest distance from point (2,3,1) to the line r = (1,1,0) + t(2,1,3).

#4. Calculate the angle between diagonal of cube and any of its edges (exact answer).

#5. Find the equation of plane containing line r = i + t(j + k) and parallel to vector 2i – j.

#6. Vectors A, B, C form right-handed orthogonal system. If A = 3i + 4j and B·k = 0, |B| = 5, find C.

#7. Find work done by force F = yzi + xzj + xyk along curve x = t², y = 2t, z = t³ from t=0 to t=1.

#8. A parallelepiped has edges A = i + j, B = j + k, C = k + i. Find angle between diagonals d₁ = A + B + C and d₂ = A – B + C.

#9. Find the area of triangle with vertices A(1,2,3), B(3,0,1), C(2,1,2) using cross product method.

#10. Vector r makes angles α, β, γ with x, y, z axes respectively. If α = 60° and β = 45°, find γ.

#11. Evaluate lim(x→0) [sin(3x) – 3x + (9x³/2)]/(x⁵) using Taylor series.

#12. Find d³y/dx³ for y = e^x·sin(x) at x = π.

#13. Evaluate ∫₀^(π/2) ln(sin(x))dx (known integral).

#14. Find the radius of curvature for y = x³ at x = 1.

#15. Evaluate ∫₀^∞ x²e^(-x²)dx using substitution.

#16. Find the length of astroid x = a·cos³(t), y = a·sin³(t) for 0 ≤ t ≤ 2π.

#17. Determine d/dx[∫ₓ^(x²) sin(t²)dt] using Leibniz rule.

#18. Find area between curves y = x³ – x and y = x – x³.

#19. Evaluate lim(x→∞) x[√(x² + 1) – x].

#20. Find coefficient of x⁵ in Taylor expansion of e^x·cos(x) about x = 0.

#21. Evaluate ∫(x² + 1)/((x² – 1)(x² + 4))dx using partial fractions.

#22. Calculate ∫₀^(π/4) x²·sec²(x)dx using integration by parts twice.

#23. Evaluate improper integral ∫₁^∞ (ln(x))/x² dx.

#24. Find ∫√(x² + 4)dx using trigonometric substitution.

#25. Evaluate ∫₀^1 x³√(1-x²)dx using substitution u = 1-x².

#26. Calculate surface area of revolution for y = √x from x = 0 to x = 4 about x-axis.

#27. Evaluate ∫e^x·cos(x)dx using integration by parts.

#28. Find ∫₀^(π/2) sin⁴(x)dx using reduction formula.

#29. Evaluate ∫(2x + 3)/(x³ + x² + x + 1)dx after factoring denominator.

#30. Calculate ∫₀^1 ∫₀^√(1-x²) √(x² + y²) dy dx (polar coordinates recommended).

#31. Solve y” + 4y’ + 4y = e^(-2x) using method of undetermined coefficients. Find yₚ.

#32. For system dx/dt = y, dy/dt = -4x – 5y, determine stability of equilibrium at origin.

#33. Solve Bernoulli equation dy/dx + y = xy³ using substitution v = y^(-2).

#34. Find particular integral for y” – 2y’ + y = x²e^x using undetermined coefficients.

#35. Solve exact equation (2xy + y²)dx + (x² + 2xy – 1)dy = 0.

#36. For Cauchy-Euler equation x²y” + 4xy’ + 2y = 0, find general solution.

#37. Solve using Laplace: y” + 9y = cos(2t), y(0) = 0, y'(0) = 1. Find Y(s).

#38. Find Green’s function for y” + y = 0 with y(0) = y(π) = 0.

#39. Solve variation of parameters for y” + y = sec(x). Form of yₚ?

#40. For y” – 2y’ + 5y = 0, classify the type of solution.

#41. Find L{t·cos(ωt)} using differentiation property.

#42. Evaluate L⁻¹{s/((s²+1)(s²+4))}.

#43. Using convolution theorem, find L⁻¹{1/((s²+1)²)}.

#44. Solve integral equation y(t) = t + ∫₀^t y(τ)dτ using Laplace transforms.

#45. Find L{∫₀^t τsin(τ)dτ} without evaluating the integral.

#46. Determine L{e^(at)sinh(bt)}.

#47. Using initial value theorem, find lim(t→0⁺) f(t) if F(s) = (2s²+3s+1)/(s³+s²).

#48. Solve y” + 2y’ + 2y = δ(t-π) with y(0)=0, y'(0)=0 where δ is Dirac delta.

#49. Find L{f(t)} where f(t) is periodic with period 2π and f(t) = sin(t) for 0 ≤ t < 2π.

#50. Evaluate L⁻¹{ln((s+1)/s)} using differentiation property of inverse transforms.

#51. Matrix A = |2 1 0| |1 2 1| |0 1 2| is tridiagonal. Find its eigenvalues.

#52. For matrix A with eigenvalues 2, 3, 5, what is det(A²)?

#53. Find the Jordan canonical form of A = |4 1| |0 4|.

#54. Determine whether vectors v₁=(1,2,1), v₂=(2,1,3), v₃=(1,-1,2) span R³.

#55. Find the spectral radius of A = |3 1| |-1 1| (largest absolute eigenvalue).

#56. Using Cayley-Hamilton theorem, if A = |1 2| |3 4| satisfies A² – 5A + kI = 0, find k.

#57. For orthogonal matrix Q, which statement is always true?

#58. Find the minimal polynomial of A = |2 0 0| |0 2 1| |0 0 2|.

#59. Matrix A is 5×3 with rank 3. What is the dimension of column space?

#60. For positive definite matrix A, all eigenvalues must be:

#61. For f(x,y) = x³ – 3xy², classify critical point at (0,0) using Hessian.

#62. Calculate ∫∫∫_E (x² + y² + z²)dV where E is sphere x² + y² + z² ≤ a².

#63. Verify Stokes’ theorem for F = -yi + xj over hemisphere x² + y² + z² = 1, z ≥ 0.

#64. Find the maximum of f(x,y,z) = xyz subject to x + y + z = 9 and x,y,z > 0.

#65. Evaluate ∫₀^∞ ∫₀^∞ e^(-(x²+y²))dx dy using polar coordinates.

#66. For transformation u = x + y, v = x – y, find Jacobian ∂(u,v)/∂(x,y).

#67. Calculate flux of F = xi + yj + zk through closed surface of cube [0,1]³.

#68. Find equation of tangent plane to x² + 2y² + 3z² = 21 at (1,2,2).

#69. Evaluate ∫_C (y²dx + 2xydy) where C is ellipse x²/4 + y² = 1, counterclockwise.

#70. For f(x,y) = e^(-x²-y²), find the maximum directional derivative at origin.

#71. Find the equation of sphere passing through points (0,0,0), (1,0,0), (0,1,0), (0,0,1).

#72. Determine angle between planes 2x – y + z = 3 and x + y + z = 1.

#73. Find parametric equations of line intersection of planes x + y + z = 6 and x – y + 2z = 3.

#74. Calculate the volume of ellipsoid x²/a² + y²/b² + z²/c² = 1.

#75. For rotated conic 3x² + 2xy + 3y² = 4, find the rotation angle to eliminate xy term.

#76. Find the distance between parallel planes 2x + 2y + z = 4 and 2x + 2y + z = 10.

#77. Determine the center of mass of region bounded by y = x² and y = 4 (uniform density).

Math –
High Difficulty

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