Mathematics II old question papers

A function $f(x)$ is defined as $f(x)=2x+3$ for $x<1$ $f(x)=4$ for $x=1$ $f(x)=6x-1$ for $x>1$ Is the function $f(x)$ is continuous at $x=1$? If not, how can you make it continuous at $x=1$?

Evaluate the limit, ${\lim }_{x\to \infty }\left(\frac{x\cos \theta -\theta \cos x}{x-\theta }\right)$

Find the maximum and minimum values of the function $f(x)=4x^3-15x^2+12x-1$. Also, find the point of inflection.

Find $\frac{dy}{dx}$ if (a) $x=t^2-1,y=t^4-1$ (b) $x^2+y^2=\sin xy$

Evaluate ${\int }_0^2\sqrt{1+x^3}dx$ by using Simpson's $\frac{1}{3}$ rule by taking $n=4$.

Find the derivative of the function $f(x)=\frac{1}{\sqrt{x}}$ by using first principle.

Solve the differential equations $2\frac{dy}{dx}=2\frac{y}{x}+\frac{y^2}{x^2}$

State Rolle's theorem and Lagrange's mean value theorem with their geometrical interpretation. Verify Rolle's theorem for the function $f(x)=\sin x,x\in [0,\pi ]$. Also find a point on the curve represented by given function where the target is parallel to x-axis.

Using the simplex method, find the optimal solution of the following linear programming problem. Maximize, $z=15x+12y$ Subject to, $2x+3y\le 21$ $3x+2y\le 24$ $x,y\ge 0$

(a) State L' Hospital Rule. Use it to evaluate ${\lim }_{x\to 0}\left(\frac{x{e}^x-\ln (1+x)}{x^2}\right)$ (b) Solve the following system of equations by Gauss-Seidel method. $x-2y+3z=1$ $3x-y+4z=3$ $2x+y-2z=-1$