Q.1) Find the real root of equation $f(x) = x^3 - x - 10$ by using Newton Raphson method correct up to 4 decimal places. (June-2025)

Q.4) Using Newton-Raphson method find the real root of the equation $3x = \cos x + 1$ correct to five decimal places. (Dec-2024)

Q.6) Show that Newton Raphson method is quadratic convergent. (Dec-2024)

Q.8) If the equation $f(x)$ is given as $x^3 - x^2 + 4x - 4 = 0$. Considering the initial approximation at $x = 2$ then find the value of next approximation correct upto 2 decimal places. (June-2024)

Q.9) What is the rate of convergence of Newton Raphson method? (June-2024)

Q.10) Find
i) $\Delta e^{ax}$
ii) $\Delta^2 e^x$
iii) $\Delta \log x$ (June-2024)

Q.11) Prove that $f(4) = f(3) + \Delta f(2) + \Delta^2 f(1) + \Delta^3 f(1)$ taking '1' as the interval of differencing. (June-2024)

Q.14) Find the real root of the equation $x \log_{10} x - 1.2 = 0$ correct to five places of decimal by Regula Falsi Method. (June-2024)

Q.15) Evaluate $\sqrt{12}$ to four decimal places by Newton Raphson Method. (Nov-2023)

Q.16) What is the rate of convergence of bisection method? (Nov-2023)

Q.17) Prove that
i) $\Delta = \frac{1}{2} \delta^2 + \delta \sqrt{1 + \left(\frac{\delta^2}{4}\right)}$
ii) $\Delta + \nabla = \frac{\Delta}{\nabla} - \frac{\nabla}{\Delta}$ (Nov-2023)

Q.23) Using Regula-Falsi method, find the real root of $x \log_{10} x = 1.2$, correct to four decimal places. (Nov-2023)

Q.24) Find the real root of the equation $x^3 - 3x - 5 = 0$ correct to four places of decimals by Newton Raphson method. (June-2023)

Q.25) Evaluate
$\Delta^6 (ax - 1)(bx^2 - 1)(cx^3 - 1); h=1$ (June-2023)

Q.27) Find the real root of the equation $xe^x - 3 = 0$ correct to three decimal places by Regula Falsi method. (June-2023)

Q.29) Using Newton-Raphson's method find the real root of $x^4 - x - 10 = 0$. (Nov-2022)

Q.30) Prove that
$$ \Delta^n 0^{n+1} = \frac{n(n+1)}{2} \Delta^n 0^n $$ (Nov-2022)

Q.32) Prove that
$$ \Delta^n \sin(ax + b) = \left( 2 \sin \frac{ah}{2} \right)^n \sin \left[ ax + b + n \left( \frac{ah + \pi}{2} \right) \right] $$ (Nov-2022)

Q.2) Solve the given set of equations by using Gauss elimination method:
$x + y + z = 4$
$x + 4y + 3z = 8$
$x + 6y + 2z = 6$ (June-2025)

Q.3) Apply the Simpson's $\frac{3}{8}$ rule to evaluate the following integral for six digit $\int_{1.0}^{1.30} \sqrt{x} \cdot dx$. (Dec-2024)

Q.4) Solve the following system of equation using Gauss-Seidel method.
$27x + 6y - z = 85$
$6x + 15y + 2z = 72$
$x + y + 54z = 110$ (Dec-2024, June-2023)

Q.6) Solve the system of equations
$3x + y - z = 3$
$2x - 8y + z = -5$
$x - 2y + 9z = 8$
Using Gauss elimination method. (Dec-2024)

Q.7) Using Simpson's 3/8 rule to solve the integral $\int_4^{5.2} \log_e x dx$ (June-2024)

Q.8) Apply Gauss Elimination method to solve the following equations:
$2x - y + 3z = 9$
$x + y + z = 6$
$x - y + z = 2$ (June-2024)

Q.10) Find $\int_0^6 \frac{e^x}{1+x} dx$ approximately using simpson's $\frac{3}{8}$th rule on integration. (Nov-2023)

Q.11) Use Simpson's 1/3 and 3/8 rule to evaluate the following.
$\int_0^1 \frac{dx}{1+x^2}$
Hence obtain the approximate value of $\pi$ in each case. (June-2023)

Q.12) Compute the value of the following integral by trapezoidal Rule.
$\int_{0.2}^{1.4} (\sin x - \log_e x + e^x) dx$ (June-2023)

Q.13) Use Simpson's rule to find approximate value of the following integral.
$\int_1^2 \frac{dx}{x}$ (June-2023)

Q.14) Solve the simultaneous linear equations using Crout's method.
$x_1 + x_2 + x_3 = 1$
$3x_1 + x_2 - 3x_3 = 5$
$x_1 - 2x_2 - 5x_3 = 10$ (Nov-2022)

Q.15) Evaluate $\int_0^1 \log x \cos x \, dx$ by (i) Trapezoidal rule (ii) Simpson 3/8 rule. (Nov-2022)

Q.1) Using modified Euler's method, find an approximate value of $Y$, when $x = 0.3$, Given that $\frac{dy}{dx} = x + y$ and $y = 1$, when $x = 0$. (June-2025)

Q.2) Consider an ordinary differential equation $\frac{dy}{dx} = x^2 + y^2, y(1) = 1.2$. Find $y(1.05)$ using the fourth order Runge-Kutta method. (June-2025)

Q.3) Using Milne's method find $y(4.4.)$ given $5xy + y^2 - 2 = 0$
Given $y(4) = 1, y(4.1) = 1.0049, y(4.2) = 1.0097$ and $y(4.3) = 1.0143$. (June-2025)

Q.4) Using Runge-Kutta method of fourth order solve
$\frac{dy}{dx} = \frac{y^2 - x^2}{y^2 + x^2}, y(0) = 1$ at $x = 0.4$ in step of $0.2$ (Dec-2024)

Q.5) Find $y(2.2)$ using Euler's method for $\frac{dy}{dx} = -xy^2$ where $y(2) = 1$. (Dec-2024)

Q.6) Use Runge-Kutta method to approximate $y$, when $x = 0.1$ and $x = 0.2$, given that $x = 0$ when $y = 1$ and $\frac{dy}{dx} = x + y$. (June-2024, Nov-2022)

Q.7) Use Milne's method to solve $\frac{dy}{dx} = x + y$ with initial condition $y(0) = 1$, from $x = 0.20$ to $x = 0.30$. (June-2024, Nov-2022)

Q.8) Solve the equation $\frac{dy}{dx} = 1 - y$ with initial condition $y(0) = 0$ using Euler's modified method and tabulate the solution at $x = 0.1, 0.2, 0.3$. (Nov-2023)

Q.9) Solve $\frac{dy}{dx} = x + y^2$ by using Runge-Kutta method of fourth order to find an approximate value of $y$ for $x = 0.2$, given that $y = 1$ when $x = 0$. (Take $h = 0.1$) (Nov-2023)

Q.10) Use Euler's method compute the solution of
$\frac{dy}{dx} = y^2 - x^2$ with $y(0) = 1$ at $x = 0.1$. (June-2023)

Q.11) Use Euler's Modified method to solve the equation
$\frac{dy}{dx} = x + y$, $y(0) = 1$, $h = 0.2$ compute $y(1)$ (June-2023)

Q.12) Find $y(0.1)$ for differential equation $\frac{dy}{dx} = x^2 y - 1, y(0) = 1$ using Taylor's series method. (Nov-2022)

Q.13) Solve $xy'' + y' + xy = 0$, where $y(0) = 1, y'(0) = 0$, for $x = 0$ to $x = 1.5$. (Nov-2022)

Q.1) Find the inverse Laplace transform of the following functions.
$$F(s) = \frac{1}{(s+1)(s^2+1)}$$ (June-2025)

Q.2) Suppose that the function $y(t)$ satisfies the DE $y'' - 2y' - y = 1$, with initial values, $y(0) = -1, y'(0) = 1$. Find the laplace transform of $y(t)$. (June-2025)

Q.3) Find the Laplace transform of the following function
i) $L(t^2 e^{-t} \cosh 2t)$
ii) $L(e^{3t} \sin^2 4t)$ (June-2025)

Q.4) Find the Inverse Laplace transform of $\frac{1}{p^3 (p^2 + a^2)}$. (Dec-2024)

Q.5) Using Laplace transform solve the differential equation
$\frac{d^3y}{dt^3} + 2 \frac{d^2y}{dt^2} - \frac{dy}{dt} - 2y = 0$
Where $y = 1, \frac{dy}{dt} = 2, \frac{d^2y}{dt^2} = 2$ at $t = 0$. (Dec-2024)

Q.6) Evaluate $\int_0^\infty \frac{e^{-t} \cdot \sin t}{t} dt$ (Dec-2024)

Q.7) Find Fourier sine transform of $\frac{1}{x}$. (Dec-2024)

Q.8) Find the inverse Laplace transform of $\frac{2s+1}{s(s-1)}$. (June-2024)

Q.9) State convolution theorem and hence evaluate $L^{-1} \left[ \frac{s}{(s^2+1)(s^2+4)} \right]$. (June-2024)

Q.10) Evaluate
$L^{-1} \left[ \frac{s+7}{s^2 + 4s + 8} \right]$ (Nov-2023)

Q.11) Find the inverse Laplace transform of $\frac{s+4}{s(s-1)(s^2+4)}$. (Nov-2023)

Q.12) Prove that if $L\{f(t)\} = F(s)$ then $L \left\{ \frac{1}{t} f(t) \right\} = \int_s^\infty F(s) ds$ hence, evaluate $\frac{e^{at} - \cos bt}{t}$. (Nov-2023)

Q.13) Use Laplace transform to solve the Initial Value Problem.
$y' + 2y = 26 \sin 3t, y(0) = 3$ (June-2023)

Q.14) Evaluate
$L(t \cos at)$ (June-2023)

Q.15) Evaluate
$L^{-1} \left[ \frac{1}{(1+s)^3} \right]$ (June-2023)

Q.16) Evaluate
$L \{ t^2 e^{-3t} \}$ (June-2023)

Q.17) Find Laplace transform of
$$f(t) = \begin{cases} \sin t & 0 < t < 2\pi \\ 0 & 2\pi < t \end{cases}$$ (Nov-2022)

Q.18) Find the Fourier series for periodic extension of
$$f(t) = \begin{cases} \sin t, & 0 \le t \le \pi \\ 0, & \pi \le t \le 2\pi \end{cases}$$ (Nov-2022)

Q.1) A device is used to measure the speed of cars on a highway. The speed is normally distributed with a 90 km/hr mean and a standard deviation of 10. What is the probability that the car picked was travelling at more than 100 km/hr? (June-2025)

Q.2) The mean and the standard deviation of a binomial distribution are 100 and 5. Find the binomial distribution. (June-2025)

Q.3) In a normal distribution, 10.03% of the items are under 25 kg. weight and 89.97% of the items are under 70 kg. weight. What are the mean and standard deviation of the distribution? (June-2025)

Q.4) Find mean of Poisson distribution. (June-2025, June-2023)

Q.5) Write short note on Exponential distribution. (June-2025, June-2023)

Q.7) Calculate the mean and standard deviation of Binomial Distribution. (Dec-2024, Nov-2023)

Q.8) A binomial variable X satisfies the relation $9P(X = 4) = P(X = 2)$ when $n = 6$. Find the value of the parameter $p$ and $P(X = 1)$. (June-2024)

Q.9) A large number of measurement is normally distributed with a mean 65.5" and S.D. of 6.2". Find the percentage of measurements that fall between 54.8" and 68.8". (June-2024)

Q.10) If 'm' balls are distributed among 'a' men and 'b' women show that the probability that the number of balls received by men is odd, shall be
$$ \frac{1}{2} \left[ \frac{(b+a)^m - (b-a)^m}{(b+a)^m} \right] $$ (Nov-2022)

Q.11) Two independent random variable X and Y are both normally distributed with means 1 and 2 and standard deviations 3 and 4 respectively. If Z = X - Y, write the probability density function of Z. Also state the median, s.d. and mean of the distribution of Z. Find P[Z + 1 $\le$ 0]. (Nov-2022)

Q.12) Prove that for normal distribution, the Quartile Deviation (QD), Mean Deviation (MD) and Standard Deviation (SD) follows QD : MD : SD :: 10 : 12 : 15. (Nov-2022)