List of top Engineering Mathematics Questions

Let \( a_0 = 0 \) and define \( a_n = \frac{1}{2} (1 + a_{n-1}) \) for all positive integers \( n \geq 1 \). The least value of \( n \) for which \( |1 - a_n|<\frac{1}{2^{10}} \) is ______.
Consider a nonlinear algebraic equation, \( e^x - 2 = 0 \). Using the Newton-Raphson method, with the initial guess of \( x_0 = 1 \), the approximated value of the root of the equation after one iteration is ________.
The value of \( k \), for which the linear equations \( 2x + 3y = 6 \) and \( 4x + 6y = 3k \) have at least one solution, is ________. (Answer in integer)
An approximate solution of the equation \( x^3 - 17 = 0 \) is to be obtained using the Newton-Raphson method. If the initial guess is \( x_0 = 2 \), the value at the end of the first iteration is \( x_1 = \) __________ (rounded off to two decimal places).
Find the limit: \[ \lim_{x \to 0} \frac{1 - \cos(2x)}{x^2} \]
The equation of a closed curve in two-dimensional polar coordinates is given by \( r = \frac{2}{\sqrt{\pi}} (1 - \sin \theta) \). The area enclosed by the curve is __________ (answer in integer).
\( \hat{i} \) and \( \hat{j} \) denote unit vectors in the \( x \) and \( y \) directions, respectively. The outward flux of the two-dimensional vector field \( \vec{v} = x \hat{i} + y \hat{j} \) over the unit circle centered at the origin is ___________ (rounded off to two decimal places).
The maximum value of the function \( f(x) = (x - 1)(x - 2)(x - 3) \) in the domain [0, 3] occurs at \( x = \) ___________ (rounded off to two decimal places).
The directional derivative of a function \( f(x, y, z) = 4x^2 + 8y^2 + 9z^2 \) at the point \( P(3, 4, 5) \) in the direction vector \( \vec{b} = 2\hat{i} - 3\hat{j} + 4\hat{k} \) is _______ (rounded off to 1 decimal place).
The series \[ \sum_{n=0}^{r} q^n = 1 + q + q^2 + \cdots + q^r \] has the sum:
Consider the infinite series
\[ (P): \sum_{n=2}^{\infty} \frac{1}{(n \log n)^{1/n}} {and} (Q): \sum_{n=1}^{\infty} \frac{n^n}{(2n)!}. \] Then which one of the following statements is correct?
What is the respective ratio between the numbers of female blood donors from the Anand Lok and Green Park?
The ratio of the present age of A and B is 4:7, and 6 years back the ratio was 2:5. What will be the ratio of their ages after 9 years?
A bag contains \( 4 \frac{2}{3} \) dozens of candies, out of which \( a \) are lemon candies. If one candy is drawn at random from the bag, there is some probability that it will be a lemon candy. Now, \( 4 \frac{2}{3} \) dozen more lemon candies are put in the bag, and the probability of drawing a lemon candy will be \( 2 \frac{1}{2} \) times higher than in the first case. Find \( a \).
A trader gives a discount of 25\% and makes a profit of 20\%. He wants to earn more and he reduces the discount to 15\%. Find the new profit percentage.
Consider a nonlinear algebraic equation, \( e^x - 2 = 0 \). Using the Newton-Raphson method, with the initial guess of \( x_0 = 1 \), the approximated value of the root of the equation after one iteration is ________.
Consider the differential equation \(\frac{dy}{dx} + \frac{y}{x} = 0\). Choose the CORRECT option(s) for the solution \(y\).
Let the difference equation \( y[n] = \alpha y[n-1] + x[n] \), where \( \alpha>1 \) and \( \alpha \) is real, represent a causal discrete-time linear time invariant system. The system is initially at rest. If \( x[n] = -\delta[n-p] \), where \( p>10 \), the value of \( y[p+1] \) is:
If \[ P = \begin{pmatrix} \cos\alpha & \sin\alpha \\ -\sin\alpha & \cos\alpha \end{pmatrix} \] and \(P + P^T = I\), the value of \(\alpha \; (0 \leq \alpha \leq \pi/2)\) is
The length of the edge of a variable cube is increasing at the rate of 25 cm s\(^{-1}\). If the initial length of the edge of the cube is 10 cm, the rate of increase of the surface area of the cube is .............. cm\(^2\) s\(^{-1}\). (answer in integer)
The area bounded by the curve \( y = \sin x \) and the x-axis between \( x = 0 \) and \( x = \frac{3\pi}{2} \) is ............. sq. units. (answer in integer)
Let \(X\) and \(Y\) be two random variables with mean \(0\), variance \(1\), and correlation coefficient \(\tfrac{1}{3}\). Then the value of \(\mathrm{Var}(X+3Y)\) is equal to:
Let $f(z)$ be an analytic function such that \[ Re(f'(z)) = 3x^2 - 4y - 3y^2, f(i)=0, f'(0)=0, \] where $i = \sqrt{-1}$. Then the value of $f(1)$ is:
For $a,b \in \mathbb{R}$, consider the system of linear equations: \[ x + y + az = 2, \] \[ 2y + 2z = 1, \] \[ ax + 2z = b \] If the system has infinitely many solutions, then which of the following statements is/are correct?
Consider the function \[ f(x,y) = x^2y + 2xy^2 - 2x^2y^2. \] Then which one of the following statements is correct?