BMATS201 Set – 1 Solved Model Question Paper 2nd Semester P Cycle for Computer Science and Engineering (CSE) Stream 22 Scheme
MODULE – 1
1.A] Evaluate \iiint e^{x+y+z} , dz,dy,dx with the limits: 0 \le x \le a,\quad 0 \le y \le x,\quad 0 \le z \le x+y
1.B] Evaluate \iint_0^\infty \iint_0^\infty e^{-(x^2 + y^2)} , dx,dy by changing into polar coordinates.
1.C] Show that beta(m, n)= \frac{\Gamma(m)\Gamma(n)}{\Gamma(m+n)}
OR
2.A] Evaluate \iint_{0}^{a} \int_{y}^{a} \frac{x}{x^2 + y^2} , dx,dy by changing the order of integration.
2.B] Find the area of a plate shaped like a quadrant of the ellipse \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 using double integration.
2.C] Write the code to find the volume of the tetrahedron bounded by x = 0,\quad y = 0,\quad z = 0,\quad ax + by + cz = 1 using Mathematical tools.
MODULE – 2
3.A] Find \nabla\phi , where \phi = x^3 + y^3 + z^3 - 3xyz at point (1, −1, 2)
3.B] If \vec{F} = \nabla(x^3 + y^3 + z^3 - 3xyz) , find \nabla \cdot \vec{F} and \nabla \times \vec{F}
3.C] Express the vector \vec{A} = z \hat{i} - 2x \hat{j} + y \hat{k} in cylindrical coordinates.
OR
4.A] Find the directional derivative of \phi = x^2yz + 4xz^2 at (1, -2, -1) in the direction of 2\hat{i} - \hat{j} - 2\hat{k}
4.B] Show that the spherical coordinate system is orthogonal.
4.C] Write code to compute \nabla \phi for \phi = x^2 y z
MODULE – 3
5.A] Prove that the set W = {(x, y, z) \in \mathbb{R}^3 \mid x - 3y + 4z = 0} is a subspace of \mathbb{R}^3 .
5.B] Determine whether the matrix A = \begin{bmatrix} -1 & 7 \ 8 & -1 \end{bmatrix} is a linear combination of the matrices B_1 = \begin{bmatrix} 1 & 0 \ 2 & 1 \end{bmatrix},\quad B_2 = \begin{bmatrix} 2 & -3 \ 0 & 2 \end{bmatrix},\quad B_3 = \begin{bmatrix} 0 & 1 \ 2 & 0 \end{bmatrix}
5.C] Find the matrix of the linear transformation T: \mathbb{R}^2 \to \mathbb{R}^3 such that T(-1, 1) = (-1, 0, 2),\quad T(2, 1) = (1, 2, 1)
OR
6.A] Show that the set S = {(1, 2, 4), (1, 0, 0), (0, 1, 0), (0, 0, 1)} is linearly dependent.
6.B] Show that the transformation T: P_2 \to P_1 defined by T(ax^2 + bx + c) = (a + b)x + c is linear.
6.C] Verify the Rank-Nullity Theorem for the transformation T: \mathbb{R}^3 \to \mathbb{R}^2 defined by T(x, y, z) = (y - x, y - z)
MODULE – 4
7.A] Find the real root of the equation 3x = \cos x + 1 using Newton-Raphson method (correct to 3 decimal places)
7.B] Use Lagrange’s interpolation to find y(5) Given:
x: 1, 3, 4, 6
y: -3, 9, 30, 132
7.C] Evaluate \int_{0}^{10} \frac{1}{1 + x} , dx using Simpson’s 3/8 rule with 7 ordinates.
OR
8.A] Find an approximate root of x^3 - 3x + 4 = 0 between -3 and -2 using the method of false position (3 iterations)
8.B] Use Newton’s interpolation to find values of y at x = 8 and x = 22
Given:
x: 0, 5, 10, 15, 20, 25
y: 7, 11, 14, 18, 24, 32
8.C] Evaluate \int_{0}^{\pi/2} \sqrt{\cos x} , dx using Trapezoidal rule with 7 ordinates.
MODULE – 5
9.A] Solve \frac{dy}{dx} = 3x + y^2, \quad y(0) = 1 Find y(0.2) using Modified Euler’s method, h = 0.2
9.B] Apply RK-4 method to solve \frac{dy}{dx} = x + y, \quad y(0) = 1 Find y(0.2)
9.C] Given \frac{dy}{dx} = xy + y^2,\quad y(0) = 1,\quad y(0.1) = 1.1169,\quad y(0.2) = 1.2773,\quad y(0.3) = 1.5049 Find y(0.4) using Milne’s method
OR
10.A] Use Taylor series method to approximate y at x = 0.2 for \frac{dy}{dx} = 2y + 3e^x, \quad y(0) = 0
10.B] Use RK-4 method to solve \frac{dy}{dx} = \frac{y^2 - x^2}{y^2 + x^2},\quad y(0) = 1 Find y(0.2)
10.C] Write code to solve \frac{dy}{dx} - 2y = 3e^x,\quad y(0) = 0 Using Taylor series at x = 0.1, 0.2, 0.3
