MTH 174

Syllabus

Unit 1 : Matrix Algebra

Unit 2 and Unit 3 

Unit 4 : Fourier Series

Unit 5 : Multivariate Calculus

Unit 6 : Integral Calculus

Practice Paper

Youtube Tutorial

What is the rank of the matrix <pre>| 1  2  3 |<br>| 0  1  2 |<br>| 0  0  1 |</pre>?

If A is a 3×3 matrix with eigenvalues 1, 2, 3, then the eigenvalues of A² are:

The general solution of the differential equation y'' + 4y = 0 is:

Which method would be most appropriate to solve y'' + y = sec x?

The Euler's formula for Fourier coefficients is valid when the function f(x) is:

For an odd function f(x), the Fourier series contains:

If u = x² + y², then ∂u/∂x at (1,2) is:

Euler's theorem for homogeneous functions states that if u = f(x,y) is homogeneous of degree n, then:

The Jacobian ∂(u,v)/∂(x,y) for u = x + y, v = x - y is:

The value of the double integral ∫∫<sub>R</sub> (x + y) dxdy where R is the rectangle [0,1]×[0,1] is:

For what value of k does the system <pre>x + 2y = 3<br>2x + ky = 6</pre> have infinitely many solutions?

The particular solution of y'' + y = x² using undetermined coefficients method is of the form:

The Fourier series of f(x) = x in (-π, π) contains:

The critical points of f(x,y) = x² + y² - 2x - 4y are:

The value of ∫<sub>0</sub>¹∫<sub>0</sub>ˣ y dydx is:

If A is a 3×3 matrix with det(A) = 5, then det(2A) is:

The general solution of the Euler-Cauchy equation x²y'' + 5xy' + 4y = 0 is:

The Fourier coefficient a₀ for f(x) = x² in (-π, π) is:

If f(x,y) = x³ + y³ is a homogeneous function, its degree is:

Changing the order of integration in ∫<sub>0</sub>¹∫<sub>x²</sub>ˣ f(x,y) dydx gives:

The number of linearly independent eigenvectors for the matrix <pre>| 2 1 |<br>| 0 2 |</pre> is:

The method of undetermined coefficients can be used to solve:

The half-range sine series expansion is used for:

Using Lagrange multipliers to maximize f(x,y) = xy subject to x + y = 10 gives the maximum value as:

The volume bounded by z = 1 - x² - y² and z = 0 is:

The solution to the system <pre>x + y + z = 6<br>2x + y + 3z = 14<br>x + 2y + z = 8</pre> is:

The particular integral of (D² + 1)y = sinx is:

<div class='p-4 rounded-lg mb-2'><h3 class=' text-lg text-blue-800'>Find the inverse of the matrix A = </h3><pre class='bg-white p-2 rounded mt-2'>| 1  2  3 |
| 0  1  2 |
| 0  0  1 |</pre><p class='mt-2'>using elementary row operations. Show all steps.</p></div>

<div class=' p-4 rounded-lg mb-2'><h3 class=' text-lg text-blue-800'>Solve the following system of linear equations using matrix method:</h3><pre class='bg-white p-2 rounded mt-2'>x + y + z = 6
2x - y + 3z = 14
3x + y - z = 2</pre></div>

<div class=' p-4 rounded-lg mb-2'><h3 class=' text-lg text-blue-800'>Find the general solution of the differential equation</h3><pre class='bg-white p-2 rounded mt-2'>y'' - 4y' + 4y = e<sup>2x</sup></pre><p class='mt-2'>using the method of variation of parameters.</p></div>

<div class=' p-4 rounded-lg mb-2'><h3 class=' text-lg text-blue-800'>Find the Fourier series expansion of f(x) = x in the interval (-π, π).</h3><p class='mt-2'>Hence show that 1 - 1/3 + 1/5 - 1/7 + ... = π/4.</p></div>

<div class=' p-4 rounded-lg mb-2'><h3 class=' text-lg text-blue-800'>Verify Euler's theorem for the function</h3><pre class='bg-white p-2 rounded mt-2'>f(x,y) = x<sup>3</sup> + y<sup>3</sup> + 3xy<sup>2</sup></pre><p class='mt-2'>and find its degree of homogeneity.</p></div>

<div class=' p-4 rounded-lg mb-2'><h3 class=' text-lg text-blue-800'>Find the extreme values of the function</h3><pre class='bg-white p-2 rounded mt-2'>f(x,y) = x<sup>3</sup> + y<sup>3</sup> - 3xy</pre><p class='mt-2'>using the second derivative test.</p></div>

<div class=' p-4 rounded-lg mb-2'><h3 class=' text-lg text-blue-800'>Evaluate ∬<sub>R</sub> (x² + y²) dxdy where R is the region bounded by y = x² and y = x in the first quadrant.</h3></div>

<div class=' p-4 rounded-lg mb-2'><h3 class='text-lg text-blue-800'>Solve the differential equation</h3><pre class='bg-white p-2 rounded mt-2'>(x²D² - 3xD + 4)y = x²lnx</pre><p class='mt-2'>using the method for Euler-Cauchy equations.</p></div>

<div class=' p-4 rounded-lg mb-2'><h3 class=' text-lg text-blue-800'>Find the volume bounded by the cylinder x² + y² = 4 and the planes z = 0 and z = 3 using triple integration.</h3></div>

<div class=' p-4 rounded-lg mb-2'><h3 class=' text-lg text-blue-800'>Using Lagrange multipliers, find the maximum and minimum values of</h3><pre class='bg-white p-2 rounded mt-2'>f(x,y) = x² + y²</pre><p class='mt-2'>subject to the constraint x + y = 1.</p></div>