Engineering Mathematics Practice Problems

On this page, you will find Engineering Mathematics practice problems.

Study Materials:

nth Derivative	Coplanar Vectors
Indeterminate Forms and L’Hospital Rule	Characteristic Equation of a Matrix
Rolle’s Theorem	Cayley-Hamilton Theorem
Taylor Series Expansion	Volume generated by revolving a curve
Eulers Theorem for Homogeneous Functions	Area generated by revolving a curve
Vector Algebra	Change of Order of Integration

Previous Year Question Papers

• December 2023
• December 2024
• January 2023

Differential Calculus and Vector Algebra

Q1: Find the limits:

1. $\lim\limits_{x \to 0} \dfrac{1-\cos2x}{4x^2}$
2. $\lim\limits_{x \to 0} \dfrac{x-\sin x}{x^3}$

Q2: Find the value of $a$ for which the limit $\displaystyle \text{Lt}_{x \to 0} \dfrac{a\sin x -\sin 2x}{x^2}$ is finite. Find the limit. [Indeterminate Forms and L’Hospital Rule]

Q3: Find the n^th derivatives of the following functions: (any three) [nth Derivative]

1. cos2x
2. ln x
3. $\dfrac{1}{2x}$
4. 2^2x

Q4: State Leibnitz’s theorem. If $y=e^{\sin^{-1}x}$, then show that

$(1-x^2)y_{n+2} -(2n+1)xy_{n+1}$ $- (n^2+1)y_n =0.$

Q5: State Rolle’s thorem and Lagrange’s mean value theorem. Verify these theorems for f(x) = x(x-1)(x-2) on [0, $\frac{1}{2}$]. [Rolle’s Theorem]

Q6: Find the maximum and minimum values of the functions: $y=x^3-3x^2-9x+12$.

Q7: Write down the Taylor series expansion of log(1+x) about x=1 with Lagrange’s form of remainder after 3 terms. [Taylor Series Expansion]

Q8: For $u=x^3y+y^3z+z^3x$, find the value of $\dfrac{\partial{u}}{\partial{x}} +\dfrac{\partial{u}}{\partial{y}}+\dfrac{\partial{u}}{\partial{z}}$.

Q9: Find $\dfrac{\partial{u}}{\partial{x}}$ and $\dfrac{\partial{u}}{\partial{y}}$ for $u=\ln\sqrt{x^2+y^2}$.

Q10: Define a homogeneous function. State Euler’s theorem. [Eulers Theorem for Homogeneous Functions]

Let $u =\dfrac{x^2y^2}{x+y}$. Apply Euler’s theorem to find $x \dfrac{\partial{u}}{\partial{x}} + y \dfrac{\partial{u}}{\partial{y}}$. Hence, show $x^2 \dfrac{\partial^2{u}}{\partial{x^2}} + 2xy\dfrac{\partial^2{u}}{\partial{x}\partial{y}}$ $+y^2\dfrac{\partial^2{u}}{\partial{y^2}}=6u$.

More Practice Problems

Q11: Compute $\hat{i} \cdot \hat{i}$ and $\hat{i} \times \hat{i}$.

Q12: The position vector of a moving particle is given by .

$\vec{r}=x\hat{i}+y\hat{j}$

where

$x=4t-\dfrac{t^2}{2}$ and $y=3+6t-\dfrac{t^3}{6}$.

Find the velocity and the acceleration of the particle at time t=2.

Q13: Find a unit vector parallel to the resultant of the vectors $\vec{r_1}=(2,4,-5)$ and $\vec{r_2}=(1,2,3)$.

Q14: If $\vec{A}=2x^2 \hat{i}-3yz \hat{j} +xz^2 \hat{k}$ and f= 2z-x³y, then find $\vec{A} \cdot \vec{\nabla}f$ and $\vec{A} \times \vec{\nabla}f$.

Q15: Find the value of $p$ for which the vectors $\vec{a}=2\hat{i}+\hat{j}+3\hat{k}$, $\vec{b} = 4\hat{i} + p\hat{j}+\hat{k}$ and $\vec{c}=2\hat{i}-\hat{j}+2\hat{k}$ are coplanar. [Coplanar Vectors]

Q16: Definde divergence and curl of vector. Find div $\vec{F}$ and curl $\vec{F}$ at (1, -1, 1), where [Vector Algebra]

$\vec{F}=(3x^2 – 3yz)\hat{i} + (3y^2 – 3xz)\hat{j} + (3z^2 – 3xy)\hat{k}$.

More Practice Problems

Linear Algebra and Integral Calculus

Q1: Find the rank of the following matrix:

$A = \begin{pmatrix} 1 & 1 & -2 \\ -1 & 2 & 1 \\ 0 & 1 & -1 \end{pmatrix}$

Q2: Find the eigenvalue and eigenvectors of the matrix

\begin{equation} A = \begin{pmatrix} 2 & 1 & 1 \\ 1 & 2 & 1 \\ 0 & 0 & 1 \end{pmatrix}. \end{equation}

Q3: Find the inverse of the following matrix using elementary row operations (Gauss-Jordon method): $A=\begin{pmatrix} 1&-1&-2 \\ -1&2&4 \\ 0&0&3 \end{pmatrix}$.

Q4: Find the of the surface generated by revolving the parabola $y^2=2ax, \, 0 \leq x \leq 3a$ about the $x$-axis.

Q5: Derive the reduction formula for

1. $I_n=\displaystyle \int \sec^n x \,dx$
2. $I_n=\displaystyle \int \tan^n x \,dx$

Using this reduction formula, evaluate $\displaystyle \int_0^{\pi/4}\tan^4 x \ dx$

Q6: Find the value of $\Gamma(6)$ and B(4,2).

Q7: Applying the change of order of integration, compute the following integrals:

1. $\displaystyle \int_{0}^1 \displaystyle \int_{y}^{1} x^2 e^{xy} dx dy \quad$ [Solution]
2. $\displaystyle \int_0^\infty \displaystyle \int_0^x e^{-xy} dy \,dx$