NMCV IMP QUESTIONS PDF

## NMCV (UNIT-I) ### 1. Expand *f(x) = sin x* in a fourier series in 0 ≤ *x* ≤ π. ### 2. Find the fourier series of *f(x) = x + x²*, -π ≤ *x* ≤ π and hence deduce the series $1/1² + 1/2² + 1/3² +.... = π²/12$. ### 3. Find the fourier series to represent the function *f(x)* given by *f(x)* = {...

## NMCV (UNIT-I) ### 1. Expand *f(x) = sin x* in a fourier series in 0 ≤ *x* ≤ π. ### 2. Find the fourier series of *f(x) = x + x²*, -π ≤ *x* ≤ π and hence deduce the series $1/1² + 1/2² + 1/3² +.... = π²/12$. ### 3. Find the fourier series to represent the function *f(x)* given by *f(x)* = { (2π - *x*, 0 ≤ *x* ≤ π) (0, π ≤ *x* ≤ 2π) }. Deduce that $1/1² + 1/3² + 1/5² +.... = π²/8$. ### 4. Obtain the fourier cosine series for *f(x) = x sin x*, 0 ≤ *x* ≤ π and show that $1/1.3 + 1/3.5 + 1/5.7 + 1/7.9 +.... = (π - 2) /4$. ### 5. Find the half-range sine series expansion of *f(x) = x²* in 0, π. ### 6. Using fourier integral, show that: $e^{ax} e^{-bx} - 2(b² - a²) / π $∫₀^∞ sin *λx* d*λ* / (λ² + a²) (*λ² + b²)* = 1 (a,b > 0) ### 7. Using fourier integral, show that: ∫₀^∞ (1 - cos πλ) / λ sin *λx* d*λ* = { (π/2, if 0 ≤ *x* ≤ π) (0, if *x* > π) }. ### 8. Using fourier integral, show that: $e^{an} - 2a / π $∫₀^∞ cos *λx* d*λ* / λ² + a² = 1 ### 9. Find the fourier transform of *f(x)* = { (1, for |*x*| ≤ 1) (0, for |*x*| > 1) }. Hence, evaluate ∫₀^∞ sin *λx* / *x* d*x*. ### 10. Find the fourier series transform of *f(x)* defined by *f(x)* = { (1 - *x²*, if |*x*| ≤ 1) (0, if |*x*| > 1) }. ### 11. Find the fourier sine transform of *e^2x*. ## Unit-II ### 1. Solve the equation *x² - 2x - 1 = 0* by bisection method. ### 2. Find a real root of the equation x log *x* = 1.2 by Regula-Falsi method correct to four decimal places. ### 3. Using Regula-Falsi method, solve *x² + 2x - 4 = 0* for a negative root. ### 4. Find the real root of the equation 2^x - 10⁹ *x* = 7 which lies between 3.5 and 4 by using Regula-Falsi method. ### 5. Find the positive real root of the equation *x³ -5x - 7 = 0* by the Newton Raphson method. ### 6. Find the positive root of *x⁴ - 2x = 10* correct to three decimal places using Newton Raphson method. ### 7. Apply Jacobi Iteration method to solve the equations 20*x* + *y* - 2*z* = 17, 3*x* + 120*y* - *z* = -18, and 2*x* - 3*y* + 202*z* = 25. ### 8. Apply Gauss-Seidal iteration method to solve the equations: 20*x* + *y* - 2*z* = 17 3*x* + 120*y* - *z* = -18 ### 9. Use Newton's backward difference formula to find **y(9)** *x* | 2 | 5 | 8 | 11 | ---|---:|---:|---:|---:| *y* | 94.8 | 87.9 | 81.03 | 75.01 ### 10. Using Newton's forward formula, find the value of *f(1.06)* from the following data *x* | 1.04 | 1.08 | 2.2 | ---|---:|---:|---:| *f(x)* | 3.49 | 4.82 | 5.96 | 6.05 ### 11. Using Newton's backward difference formula to the data given below *x* | 1 | 2 | 3 | 4 | 5 | ---|---:|---:|---:|---:|---:| *y* | 1 | -1 | 1 | -1 | 1 ### 12. From the data find number of students who obtained marks between 30 and 45 | Marks | No. of Students | |---|---| | 30-40 | 31 | | 40-50 | 42 | | 50-60 | 51 | | 60-70 | 35 | | 70-80 | 31 | ## Unit-III ### 13. Find *y(43)* if *y(20) = 0.929, y(25) = 0.906, y(32) = 0.848, y(49) = 0.56* using lagrange's interpolation formula. ### 14. Find the polynomial function *f(x)* by using lagrange's formula and hence find *f(3)* for *x* | 0 | 1 | 2 | 5 | ---|---:|---:|---:|---:| *y* | 2 | 3 | 12 | 147 | ### 1. Evaluate ∫₀² *e^-x²* d*x* by using (i) Trapezoidal rule (ii) Simpson's 1/3 rule taking *h* = 0.25 ### 2. Evaluate ∫₀⁶ d*x* / (1 + *x*²) by using (i) Trapezoidal rule (ii) Simpson's 1/8 rule ### 3. Using Simpson's 1/8 rule to find ∫_0.6¹ *e^-x²* d*x* by taking 8 ordinates. ### 4. Use Simpson's 3/8 rule to evaluate ∫_0.2^1.4 (*sin x* - log *x* *e^x*) d*x* ### 5. Given that: *dy* / *dx* = 1 + 2*xy* at *y* = 1 at *x* = 0; find an approximate value of *y* at *x* = 0.1, 0.2 by taylor's method. ### 6. Find *y*(0,1), and *y*(0,2)* using taylor series given that *dy* / *dx* = 2*x* - *y*, *y*(0) = 1. ### 7. Find the value of *y* for *x* = 0.02, 0.04 by picard's method, given that *dy* / *dx* = *x²* + *y²*, *y*(0) = 0. ### 8. Find the second successive approximation of *dy* / *dx* = *x* - *y* such that *y* = 1 when *x* = 0 using picard's method. ## Unit-IV ### 9. Apply R-K fourth order method find *y*(0.1), y(0.2)* when *dy* / *dx* = *x²* - *y*, *y*(0) = 1. ### 10. Apply Runge-Kutta method to find approximate value of *y* for *x* = 0.2 in steps of 0.1, if *dy* / *dx* = *x* + *y²* given that *y* = 1 at *x* = 0. ### 11. Find the values of *y*(1.1) and *y*(1.2)* using R-K fourth order method for the equation *dy* / *dx* = *y²* + *x* *y*, *y*(1) = 1. ### 12. Apply R-K method of 4th order find the approximate value of *y* when *x* = 0.02 given that *dy* / *dx* = *y* + *x²* and *y*(0) = 1. ### 1. Determine the analytic function whose real part is *e^x* cos *y*. ### 2. Find *f’* such that the function *f(z) = log(x² + y²) + i tan⁻¹(y/x)*. ### 3. Determine the analytic function *w = u + iv*, if *u = e^2x(x cos 2y - y sin 2y)*. ### 4. If *f(z)* is an analytic function, then show that (d²/dx² + d²/dy²) |f(z)|² = 2 |f’(z)|². ### 5. Find an analytic function *f(z)* whose real part is *u = e^x(x sin y - y cos y)* ### 6. Find the values of *a* and *b* in the function *f(z) = x + ay - i(b + y)*, if it is analytic. ### 7. Find *b* such that *u = e^bx* cos 3*y* is harmonic. ### 8. If *f(z) = u + iv* is an analytic function of *z* and if *u - v = e^x(cos y - sin y)*, find *f(z)* in terms of *z*. ### 9. If *f(z)* is an analytic function, show that (d²/dx² + d²/dy²) |f(z)|² = 4 |f’(z)|². ### 10. Using Milne-Thompson method, construct the analytic function whose real part is *y/x² + y²*. ### 11. Find all values of *k* such that *f(z) = e^x(cos y + i k sin y)* is analytic. ### 12. Find the bilinear transformation which maps the points *z = 2, 1, -2* into the points *w = 1, i, -1*. ### 13. Find the bilinear transformation which maps the points *z = -1, 1, i* to *w = 0, 1, i*. ## Unit-V ### 1. Evaluate ∫_C (*y² - xy - 3x²*) d*z* where *C* is the straight line from *z* = 0 to *z* = 1 + *i*. ### 2. Evaluate ∫_C d*z* / *z³(2 + 4y)* where *C* is |*z*| = 2. ### 3. Expand 1 / *z²(z - 3)²* as laurent series in the region (i) |*z*| < 1 (ii) 1 < |*z*| < 3 (iii) |*z*| > 3 ### 4. Find the value of ∫_C *z²* d*z* along the line *n* = *y*. ### 5. Evaluate ∫_C (*z² + 1*) / (*z⁴*) d*z* where *C* is the circle |*z*| = 3 using Cauchy's integral formula. ### 6. Find the laurent series expansion of *f(z) = 1 / (z - 1)(z - 3)* in the following regions: (i) |*z*| < 1 (ii) 1 < |*z*| < 3 (iii) |*z*| > 3 ### 7. Find the laurent series of *f(z) = 1 / (z + 1)(z + 2)* for (i) |*z*| < 1, (ii) 1 < |*z*| < 2, (iii) |*z*| > 2. ### 8. Evaluate ∫_C *e^zz* d*z* where *C* is the circle |*z*| = 3 using Cauchy's residue theorem. ### 9. Evaluate ∫_C d*z* / *e^zt(z - 1)²* where *C* is the circle |*z*| = 2 using Cauchy's integral theorem. ### 10. Expand *f(z) = 1 / (z² - 4z + 3)* as laurent series in the region (i) |*z*| < 1 (ii) 1 < |*z*| < 3 ### 11. Find the laurent series for *f(z) = z²e^z* about *z = 0*. ### 12. Using Cauchy integral formula, find ∫_C *e^zz* / (*z + 1)³* where *C* is the curve |*z*| = 2. ### 13. Apply the Cauchy’s residue theorem to evaluate ∫_C (cos π*z* + sinπ*z*) / (*z - 1)(*z - 2)* d*z* where |*z*| = 2. ### 14. Evaluate ∫_C (cos π*z*) / (*z - 1)(*z - 2)* d*z* where *C* is |*z*| = 4 using Cauchy's integral formula. ### 15. Evaluate ∫_C (4 - 3*z*) / *z²(z - 1)(z - 2)* d*z* where *C* is the circle |*z*| = 3/2 using residue theorem.