List of practice Questions

The values of a function \(f(x)\) at three different values of \(x\) are given below:
\(f(0.90) = 7.75\), \(f(1.00) = 9.00\), \(f(1.10) = 10.35\).
Using the central difference approximation method, which is the CORRECT estimate of the first differential \(f'(1.00)\) and second differential \(f''(1.00)\) respectively?

A differentiable function \(f(x)\) is such that \(f(-0.1) = 2\), \(f(0) = 1\), and \(f(0.1) = 2\).

Then, \(\dfrac{d^2f}{dx^2}\) at \(x = 0\) is ______ (answer in integer).

Secant formula is applicable for

The exact solution of \( \displaystyle\int_0^4 \frac{dx}{1+x} \) is represented as \( n \).
If \( m \) represents the numerically evaluated value of the above integral using the Trapezoidal rule by considering four equal subintervals in the range of \( x \), then \( (m-n) \) is

Newton-Raphson method for solving algebraic equations is based on

Secant formula is applicable for

The coefficient of \(\dfrac{1}{z}\) in the Laurent's series expansion of the function \[ f(z)=\frac{1}{z^2(1-z)} \] about \(z=0\), is ____.

The value of the integral \[ \oint_{|z|=2}\frac{\cos\left(\dfrac{z}{2}\right)}{z^3}\,dz \] is ____.

Suppose the probability that an individual suffers an adverse reaction from a particular drug is known to be \(0.001\). The probability that out of \(2000\) individuals, exactly three will suffer an adverse reaction is ____.

If \(X\) is a normal random variable with mean \(\mu=100\) and variance \(\sigma^2=25\), then \[ P(90<X<120) \] is same as ____.

The values of \(x\) and \(f(x)\) of a curve are as follows: \[ \begin{array}{|c|c|c|c|c|c|} \hline x & 2 & 2.5 & 3 & 3.5 & 4 \hline f(x) & 3 & 6 & 8 & 10 & 12 \hline \end{array} \] By Simpson's one-third rule, the area under the curve \(f(x)\) is ____.

The integral \[ \int_{0}^{\pi/2}\frac{\sin x}{x^p}\,dx \] converges if ____.

Suppose \(C\) is the closed curve defined as the circle \(x^2+y^2=1\) with \(C\) oriented anti-clockwise. The value of the line integral \[ \oint_C (x\,dy-y\,dx) \] is equal to ____.

If \(y=A(x)\cdot e^{-x^2\) is a solution of the differential equation} \[ \frac{dy}{dx}+2xy=2e^{-x^2} \] with \(y(0)=-3\), then the value of \(A(1)\) is ____.

If \(F(s)\) is the Laplace transform of the function \[ f(t)=e^{-3t}(2\cos5t-3\sin5t), \] then \(F(0)=\_\_\_\_.\)

If the matrix \[ A=\begin{pmatrix} -8 & 5 2 & 4 \end{pmatrix} \] satisfies the equation \(x^2+4x-k=0\), then the value of \(k\) is ____.

If the initial guess of the solution of \[ 1+x-x^3=0 \] is \(x_0=1\), then the \(2^{\text{nd}}\) approximate root of \[ 1+x-x^3=0 \] using Newton-Raphson method is

A fifth-degree polynomial in \(x\) is defined for \(x > 0\). All coefficients of the polynomial are positive. The first derivative of the polynomial is obtained numerically at a point by using the first-order forward as well as the first-order backward difference methods. Identical step lengths are used for both the methods.
Following statements are made.
(I) Forward difference method underestimates the true derivative.
(II) Backward difference method overestimates the true derivative.
Which one of the following options is CORRECT?

Using single step trapezoidal rule, the value of
\[ \int_{0}^{0.5} (1+16x-20x^2)\,dx \]
is ______ (rounded off to two decimal places).

\[ f(x) = e^{-x} - x \]

Values of \(y\) for different values of \(x\) are tabulated below.

If a second-degree interpolating polynomial \(P_2(x)\) is used to represent \(y\), the value of \(P_2(0)\) is (rounded off to the nearest integer).

From the following plot of \(\dfrac{dy}{dx}\) versus \(x\), and if \(y(2) = 5\), the value of \(y(3)\) is . (rounded off to one decimal place)

Consider the equation \[ \frac{dy}{dx} = \frac{1}{x} \] The value of the integral \(\int_{1}^{2} dy\) using trapezoidal method and interval \(h = 0.25\) is ________. (rounded off to two decimal places)

While solving the equation \(e^{-x} - x = 0\) for \(x\), using the Newton-Raphson method with initial guess \(x_0 = 2\), the solution after the first iteration (\(x_1\)) is ________ (Rounded off to two decimal places)

Consider the simultaneous equations
\[ 2x + 4y = -7; \quad 3x + 5y = 1 \] While solving using Gauss elimination without pivoting, after the forward elimination, the diagonal elements of the coefficient matrix are