BMATS201 Mathematics II for CSE Set 2 Solved Model Question Paper

BMATS201 Set – 2 Solved Model Question Paper 2nd Semester P Cycle for Computer Science and Engineering (CSE) Stream 22 Scheme

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MODULE – 1

1.A] Evaluate \iiint_{-a}^{a} (x^2 + y^2 + z^2) , dx,dy,dz

1.B] Evaluate \int_{0}^{1} \int_{y}^{1} xy \sqrt{x} , dx , dy by changing the order of integration.

1.C] Show that \Gamma \left( \frac{1}{2} \right) = \sqrt{\pi}

OR

2.A] Evaluate \int_{0}^{1} \int_{0}^{\sqrt{1-y}} (x^2 + y^2) , dx , dy by converting to polar coordinates.

2.B] Find by double integration, the area between the parabolas y^2 = 4ax and x^2 = 4ay .

2.C] Write the code to find the area of an ellipse by double integration: A = 4 \int_{0}^{a} \int_{0}^{b \sqrt{1 - \frac{x^2}{a^2}}} dy,dx

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MODULE – 2

3.A] Find the directional derivative of \phi = x^2yz + 4xz^2 at (1, -2, -1) in the direction of vector 2\hat{i} - \hat{j} - 2\hat{k} .

3.B] If \vec{F} = \nabla(xy^3z^2) , find \nabla \cdot \vec{F} and \nabla \times \vec{F} at (1, -1, 1).

3.C] Prove that the spherical coordinate system is orthogonal.

OR

4.A] Find the angle between the surfaces x^2 + y^2 - z^2 = 4 and z = x^2 + y^2 - 13 at (2, 1, 2).

4.B] Show that \vec{F} = \frac{x\hat{i} + y\hat{j}}{x^2 + y^2} is both solenoidal and irrotational.

4.C] Write the code to find the curl of \vec{F} = xy^2\hat{i} + 2x^2yz \hat{j} - 3yz^2 \hat{k}

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MODULE – 3

5.A] Let V = \mathbb{R}^3 . Is the set W = { (a, a², b) } a subspace of V?

5.B] Find basis and dimension of the subspace spanned by
{ (2, 4, 2), (1, −1, 0), (1, 2, 1), (0, 3, 1) }

5.C] Find the kernel and range of the linear transformation T(x, y, z) = (x + y, z)

OR

6.A] Show that h(x) = 4x^2 + 3x - 7 lies in the span of f(x) = 2x^2 - 5 and g(x) = x + 1 .

6.B] Prove that the transformation T(x, y) = (3x, x + y) is linear. Find T(1, 3) and T(-1, 2) .

6.C] Show that f(x) = 3x - 2 and g(x) = x are orthogonal in P_n with
\langle f, g \rangle = \int_0^1 f(x)g(x),dx .

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MODULE – 4

7.A] Find the root of x e^x = 3 using Regula-Falsi method (3 iterations).

7.B] Using Newton’s divided difference formula, find f(8) for
x: 4 5 7 10 11 13
f(x): 48 100 294 900 1210 2028

7.C] Evaluate \int_{0}^{1} \frac{1}{1 + x^2} , dx using Trapezoidal Rule (6 divisions)

OR

8.A] Solve \cos x = x e^x using Newton-Raphson method near x = 0.5 (3 decimal accuracy)

8.B] Using Newton’s forward interpolation, find \sin 48^\circ
Given:
\sin 45^\circ = 0.7071 , \sin 50^\circ = 0.7660 , \sin 55^\circ = 0.8192 , \sin 60^\circ = 0.8660

8.C] Evaluate \int_{0}^{30} \frac{1}{4x + 5} , dx using Simpson’s 1/3 Rule (7 ordinates)

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MODULE – 5

9.A] Solve \frac{dy}{dx} = x^2 y - 1, \quad y(0) = 1 using Taylor’s series at x = 0.1 and 0.2 (5 decimals)

9.B] Solve \frac{dy}{dx} = 3e^x + 2y, \quad y(0) = 0 at x = 0.1 using RK-4, h = 0.1

9.C] Given \frac{dy}{dx} = x - y^2 , with
y(0) = 0, y(0.2) = 0.02, y(0.4) = 0.0795, y(0.6) = 0.1762,
find y(0.8) using Milne’s method

OR

10.A] Solve \frac{dy}{dx} = x^2 + y, \quad y(0) = 1 using Modified Euler’s method (h = 0.05, 2 iterations)

10.B] Solve \frac{dy}{dx} = \frac{y - x}{y + x}, \quad y(0.1) = 1.0912 at x = 0.2 using RK-4, h = 0.1

10.C] Write code to solve \frac{dy}{dx} = 1 + \frac{y}{x}, \quad y(1) = 2 at x = 2, using RK-4 method with h = 0.2