Applied Mathematics I Tutorial Questions PDF

## MANJARA CHARITABLE TRUST ### RAJIV GANDHI INSTITUTE OF TECHNOLOGY ### DEPARTMENT OF APPLIED SCIENCES & HUMANITIES ### APPLIED MATHEMATICS - I (BSE 101) ### TUTORIAL NO. 01 **Q.1 Solve the equations** i) xº - x5 + x4 -1 = 0 ii)x7 + 64x4 + x3 + 64 = 0 iii) x4 - x3 + x2 - x + 1 = 0 **Q.2 Show t...

## MANJARA CHARITABLE TRUST ### RAJIV GANDHI INSTITUTE OF TECHNOLOGY ### DEPARTMENT OF APPLIED SCIENCES & HUMANITIES ### APPLIED MATHEMATICS - I (BSE 101) ### TUTORIAL NO. 01 **Q.1 Solve the equations** i) xº - x5 + x4 -1 = 0 ii)x7 + 64x4 + x3 + 64 = 0 iii) x4 - x3 + x2 - x + 1 = 0 **Q.2 Show that all the roots of (x + 1)⁶ + (x - 1)⁶ = 0 are given by - i Cot ((2k+1)π/12) where k = 0, 1, 2, 3, 4, 5** **Q.3 Find the cube root of unity. If w is complex cube root of unity, then prove that (1 - w⁶) = -27** **Q.4 Express Sin70 and Cos70 in terms of sine & cosθ.** **Q.5 Express cos8θ in a series of cosines of multiples of θ.** **Q.6 Prove that cos5θsin3θ = (1/27)(Sin80 + 2sin60 - 2sin40 - 6sin20)** **Q.7 Prove that cos6θ - sin6θ = (1/sin2θ)(Cos60 + 15Cos20)** **Q.8 Prove that sin6θ/sin2θ = 16cos⁴θ - 16cos²θ + 3** **Q.9 Find the continued product of the roots of √3/2 + (i√3)/2** **Q.10 Show that sin5θ = (sin5θ - 5sin30 + 10sine)/16** **Q.11 If Sin6θ = acos5θsine + bcos3θ + ccosθsin5θ Find the value of a, b, c** **Q.12 If sinθcos3θ = acosθ + bcos3θ + ccos5θ + dcos7θ thn find a, b, c, d.** --- ## MANJARA CHARITABLE TRUST ### RAJIV GANDHI INSTITUTE OF TECHNOLOGY ### DEPARTMENT OF APPLIED SCIENCES & HUMANITIES ### APPLIED MATHEMATICS - I (BSE 101) ### TUTORIAL NO.-2 **Q.1 If tanhx = 2/3, find the value of x and then cosh2x** **Q.2 Solve the equation for real values of x, 17 coshx + 18 sinhx = 1** **Q.3 If coshβ = √(x²/y²) and cosasinhβ = 4xy/(x²+y²), show that** i) cosec (α - iβ) + cosec (α + iβ) = (4x)/(x²+y²) ii) cosec (α - iβ) - cosec (α + iβ) = (4iy)/(x²+y²) **Q.4 If coshx = secθ, prove that** i) x = log (secθ + tanθ) ii) tanh(x/2) = tan(θ/2) **Q.5 If u + iv = (π/7 + i√(x² -1), prove that (u² + v²)² = 2(u² - v²)** **Q.6 Prove that cosh⁻¹(√1 + x²) = tanh⁻¹(x²)** **Q.7 Separate into real and imaginary parts tan⁻¹(x + iy)** **Q.8 Separate into real and imaginary parts of tan⁻¹(eiθ)** **Q.9 Prove that coth⁻¹x = (1/2)log((x+1)/(x-1))** **Q.10 Prove that sin[ilog(a+b)/(a²+b²)] = a + iβ, find a and β** **Q.11 If (a+ib)x+iy = a + iβ, find a and β ** **Q.12 Show that for real value of a and b e^(2aicot⁻¹(b)/(bi⁻¹ - 1)⁻¹a)[(bi+1)/(bi-1)] = 1** --- ## MANJARA CHARITABLE TRUST ### RAJIV GANDHI INSTITUTE OF TECHNOLOGY ### DEPARTMENT OF APPLIED SCIENCES & HUMANITIES ### APPLIED MATHEMATICS - I (BSE 101) ### TUTORIAL NO. 6 1. Find the roots of the equation x³ + 2x - 5 = 0 by Regula falsi method (Take 3 iterations) 2. Find the roots of the equation 2x + 3sin(x) - 5 = 0 by Regula falsi method 3. Find the real positive root of the equations log₁₀ X - 1.2 = 0 lying between 2 and 3 correct upto four decimal places by Regula falsi method 4. Using Regula falsi method to find the root of equation 3x - cosx - 1 = 0 lying Between 0 and 1. 5. Find the real root of the equation correct to three decimal places by Newton-Raphson method. 1) eˣ - 4x = 0 2) x³ + x - 1 = 0 3) cos(x) - xex = 0 4) e⁻ˣ - sin(x) = 0 6. Solve the following equations by Jacobi's iteration method. 1) 4x + y + 3z = 17, x + 5y + z = 14, 2x -y + 8z = 12 2) 20x + y - 2z = 17, 3x + 20y - z = -18, 2x - 3y + 20z = 25 3) 5x - y + z = 10, 2x + 4y = 12, x + 5y + 5z = -1 7. Solve the following equations by Gauss-Seidel method. 1) 27x + 6 y - z = 85, 6x + 15y + 2z = 72, x + y + 54z = 110. up to 3 iterations. 2) 10x + y + z = 12 2x + 10y + z = 13, 2x + 2 y + 10z = 14 up to 3 - iterations. 3) 3x₁ - 0.1x₂ - 0.2x₃ = 7.85, 0.1x₁ + 7x₂ - 0.3x₃ = -19.3, 0.3x₁ - 0.2x₂ + 10x₃ = 71.4 up to 3 - iterations