Q. In a factory which manufactures bolts, machines A, B and C manufacture $30\%,50\%$ and $20\%$ of the bolts respectively. Of their output $3\%,4\%$ and $1\%$ respectively are defective bolts. A bolt is drawn at random from the product and is found to be defective. Find the probability that this is not manufactured by machine B.

Q. Evaluate: $\int (3-2x).\sqrt{2+x-x^2}dx.$

Q. The equation of a line is $5x-3=15y+7=3-10z$. Write the direction cosines of the line.

Q. Find $\lambda$, if the vectors $\overrightarrow{a}=\hat{i}+3\hat{j}+\hat{k},\overrightarrow{b}=2\hat{i}-\hat{j}-\hat{k}$ and $\overrightarrow{c}=\lambda \hat{j}+3\hat{k}$ are coplanar.

Q. If $y=e^{msi{n}^{-1}x}$, then show that $(1-x^2)\frac{d^{2}y}{d{x}^2}-x\frac{dy}{dx}-m^{2}y=0$.

Q. Find $\left(\begin{array}{c}{\int }_0^{\pi /4}\frac{dx}{{\cos }^{3}x\sqrt{2\sin 2x}}\end{array}\right)$.

Q. Using integration, prove that the curves $y^2=4x$ and $x^2=4y$ divide the area of the square bounded by $x=0,x=4,y=0$ and $y=4$ into three equal parts.

Q. Solve for $x$:
${\tan }^{-1}(x+1)+{\tan }^{-1}(x-1)={\tan }^{-1}\frac{8}{31}$

Q. If $\overrightarrow{a}=\hat{i}+2\hat{j}+\hat{k},\overrightarrow{b}=2\hat{i}+\hat{j}$ and $\overrightarrow{c}=3\hat{i}-4\hat{j}-5\hat{k}$, then find a unit vector perpendicular to both of the vectors $(\overrightarrow{a}-\overrightarrow{b})$ and $(\overrightarrow{c}-\overrightarrow{b})$.

Q. To promote the making of toilets for women, an organization tried to generate awarencess through:

Q. Write the element $a_{23}$ of a $3\times 3$ matrix $A\left(a_{ij}\right)$ whose elements are represented by $a_{ij}=\frac{|i-j|}{2}$.

Q. Find the adjoint of the matrix $A=\left(\begin{array}{ccc}-1& -2& -2\\ 2& 1& -2\\ 2& -2& 1\end{array}\right)$ and
hence show that

Q. Find the equation of a line passing through the point $(1,2,-4)$ and perpendicular to two lines $\overrightarrow{r}=(8\hat{i}-19\hat{j}+10\hat{k})+\lambda (3\hat{i}-16\hat{j}+7\hat{k})$ and $\overrightarrow{r}=(15\hat{i}+29\hat{j}+5\hat{k})+\mu (3\hat{i}+8\hat{j}-5\hat{k})$.

Q. Show that the function $f\left(x\right)=|x-1|+|x+1|$, for all $x\in R$, is not differentiable at the points $x=-1$ and $x=1$.

Q. Three cards are drawn successively with replacement from a well shuffled pack of $52$ cards. Find the probability distribution of the number of spades. Hence find the mean of the distribution.

Q. Using properties of determinants, prove the following:
$\left|\begin{array}{ccc}{a}^2& bc& ac+c^2\\ a^2+ab& b^2& ac\\ ab& b^2+bc& c^2\end{array}\right|=4a^2{b}^2{c}^2$.

Q. A company manufactures three kinds of calculators : A, B and C in its two factories I and II, The company has got an order for manufacturing at least $6400$ calculators of kind A, $4000$ of kind B and $4800$ of kind C. The daily output of factory I is of $50$ calculators of kind A, $50$ calculators of kind B, and $30$ calculators of kind C. The daily output of factory II is of $40$ calculators of kind A, $20$ of kind B and $40$ of kind C. The cost per day to run factory I is Rs. $12,000$ and of factory II is Rs. $15,000$.
How many days do the two factories have to be in operation to produce the order with the minimum cost? Formulate this problem as an LPP and solve it graphically.

Q. Write the sum of the order and degree of the following differential equation: $\frac{d}{dx}\left\{{\left(\frac{dy}{dx}\right)}^3\right\}=0$.

Q. If $f\left(x\right)=\sqrt{x^2+1},g\left(x\right)=\frac{x+1}{x^2+1}$ and $h\left(x\right)=2x-3,$ then find $f^{\prime }\left[h^{\prime }|g^{\prime }\right(x\left)|\right]$.

Q. Write a unit vector perpendicular to both the vectors $\overrightarrow{a}=\hat{i}+\hat{j}+\hat{k}$ and $\overrightarrow{b}=\hat{i}+\hat{j}.$

Q. Find $\int \left(\begin{array}{c}\frac{\log x}{(x+1{)}^2}dx\end{array}\right).$

Q. Write the value of $△=\left|\begin{array}{ccc}x+y& y+z& z+x\\ z& x& y\\ -3& -3& -3\end{array}\right|$.

Q. Find the distance of the point P$(3,4,4)$ from the point, where the line joining the point A$(3,-4,-5)$ and B$(2,-3,1)$ intersects the plane $2x+y+z=7$.

Q. Show that the differential equation $\frac{dy}{dx}=\frac{y^2}{xy-x^2}$ is homogeneous and also solve it.

Q. If $\overrightarrow{A}=2\hat{i}+7\hat{j}+3\hat{k}$ and $\overrightarrow{B}=3\hat{i}+2\hat{j}+5\hat{k}$, then find the projection of $\overrightarrow{a}$ on $\overrightarrow{b}$.

Q. Find the value of $p$ for which the curves $x^2=9p(9-y)$ and $x^2=p(y+1)$ cut each other at right angles.

Q. Write the integrating factor of the following differential equation:
$(1+y^2)+(2xy-\cot y)\frac{dy}{dx}=0$.

Q. If $\hat{a},\hat{b}$ and $\hat{c}$ are mutually perpendicular unit vectors, then find the value of $|2\hat{a}+\hat{b}+\hat{c}|$.

Q. Consider $f:R^{+}\to [-9,\infty ]$ given by $f\left(x\right)=5x^2+6x-9$. Prove that $f$ is invertible with $f^{\prime }\left(y\right)=\left(\frac{\sqrt{54+5y}-3}{5}\right)$.

Q. If a line makes angles ${90}^o,{60}^0$ and $\theta$ with $x,y$ and $z-$ axis respectively, where $\theta$ is acute, then find $\theta$.