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UNIVERSITY OF THEPUNJAB$\newline$B.S. $4$ Years Program / First Semester-Fall $2022$$\newline$Roll N$\newline$Roll$\newline$Time: $3$$\newline$Calculus - $1$$\newline$Course Code: MATH$-1001$$\newline$THE ANSWERS MUST BE ATTEMPTED ON THE ANSWER SHEET PROVIDEI$\newline$Q.$1$. Solve the following:$\newline$$$(6 \times 5=30)$$$\newline$\begin{tabular}{|l|l|}$\newline$\hline (i) & \begin{tabular}{l} $\newline$Replace the polar equation by equivalent Cartesian equations, and identify the \\$\newline$graphs. $\mathrm{r}=\frac{4}{2 \cos \theta-\sin \theta}$$\newline$\end{tabular} \\$\newline$\hline (ii) & Evaluate the integral $\int \frac{r^{2}}{\sqrt{1-r^{2}}} d r=?$ \\$\newline$\hline (iii) & Find the values of $\mathrm{x}$ for which $\mathrm{f}(\mathrm{x})$ is continuous: $f(x)=\frac{x^{4}+20}{5 x(x-2)}$. \\$\newline$\hline (iv) & Find derivative of $\mathrm{f}(\mathrm{x})$ if $: f(x)=\left(\frac{\sin x}{1+\cos x}\right)^{2}$. \\$\newline$\hline (v) & Solve: $\lim _{x \rightarrow 0}\left(\frac{1}{\sin x}-\frac{1}{x}\right)$. \\$\newline$\hline (vi) & Find $\frac{d y}{d x}$ if $y=\int_{1}^{x^{2}} \cos t d t$. \\$\newline$\hline$\newline$\end{tabular}$\newline$Solve the following:$\newline$$\int \frac{r^{2}}{\sqrt{1-r^{2}}} d r=?$$0$$\newline$\begin{tabular}{|l|l|}$\newline$\hline Q. $2$ & Evaluate the integral $\int \frac{r^{2}}{\sqrt{1-r^{2}}} d r=?$$1$. \\$\newline$\hline Q. $3$ & Find the intervals on which the function $\int \frac{r^{2}}{\sqrt{1-r^{2}}} d r=?$$2$ is increasing and decreasing. \\$\newline$\hline Q. $4$ & Find the area enclosed by parabola $\int \frac{r^{2}}{\sqrt{1-r^{2}}} d r=?$$3$ and the line $\int \frac{r^{2}}{\sqrt{1-r^{2}}} d r=?$$4$. \\$\newline$\hline Q. $5$ & Find $\int \frac{r^{2}}{\sqrt{1-r^{2}}} d r=?$$5$ of $\int \frac{r^{2}}{\sqrt{1-r^{2}}} d r=?$$6$. \\$\newline$\hline Q. $6$ & Solve the integral $\int \frac{r^{2}}{\sqrt{1-r^{2}}} d r=?$$7$. \\$\newline$\hline$\newline$\end{tabular}