Board Paper of Class 12-Commerce 2011 Maths Abroad(SET 2) - Solutions

Find the mean number of heads in three tosses of a fair coin. VIEW SOLUTION

If vectors $\overrightarrow{a}=2\hat{i}+2\hat{j}+3\hat{k}, \overrightarrow{b}=-\hat{i}+2\hat{j}+\hat{k} \mathrm{and} \overrightarrow{c}=3\hat{i}+\hat{j}$ are such that $\overrightarrow{a}+\mathrm{\lambda }\overrightarrow{b}$ is perpendicular to $\overrightarrow{c}$, find the value of λ. VIEW SOLUTION

Find the particular solution of the differential equation:
(1 + e^2x) dy + (1 + y²) e^x dx = 0, given that y = 1 when x = 0. VIEW SOLUTION

Evaluate:
∫ e^2x sin x dx
  ​
Evaluate:
$\int \frac{3x+5}{\sqrt{x^2-8x+7}}dx$ VIEW SOLUTION

Prove that :
$\frac{\mathrm{d}}{\mathrm{dx}}\left[\frac{\mathrm{x}}{2}\sqrt{{\mathrm{a}}^2 -{\mathrm{x}}^2} + \frac{a^2}{2} \sin { }^{-1} \left(\frac{\mathrm{x}}{\mathrm{a}}\right)\right] = \sqrt{{\mathrm{a}}^2 - {\mathrm{x}}^2}$
$\mathrm{If} \mathrm{y} = \log \left[\mathrm{x} +\sqrt{{\mathrm{x}}^2 +1}\right], \mathrm{prove} \mathrm{that} ({\mathrm{x}}^2 + 1) \frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2} + \mathrm{x}\frac{\mathrm{dy}}{\mathrm{dx}} = 0.$ VIEW SOLUTION

Find the intervals in which the function f given by
      f(x) = sin x + cos x, 0 ≤ x ≤ 2π
is strictly increasing or strictly decreasing.
Find the points on the curve y = x³ at which the slope of the tangent is equal to the y-coordinate of the point. VIEW SOLUTION

Prove the following :

$\frac{9\pi }{8} -\frac{9}{4}{\sin }^{-1} \left(\frac{1}{3}\right) = \frac{9}{4} \sin { }^{-1}\left(\frac{2\sqrt{2}}{3}\right)$
Solve the following equation for x :

${\tan }^{-1} \left(\frac{1-\mathrm{x}}{1+\mathrm{x}}\right) = \frac{1}{2} {\tan }^{-1} \left(\mathrm{x}\right), \mathrm{x}>0$ VIEW SOLUTION

Consider  $\mathrm{f} : {\mathrm{R}}_{+} \to \left[4, \infty \right]$ given by f(x) = x² + 4. Show that f is invertible with the inverse (f⁻¹) of f

given by  ${\mathrm{f}}^{-1} \left(\mathrm{y}\right) = \sqrt{\mathrm{y} -4,}$ where R₊ is the set of all non-negative real numbers. VIEW SOLUTION

Prove using properties of determinants :

$\left|\begin{array}{ccc}\mathrm{a}-\mathrm{b}-\mathrm{c}& 2\mathrm{a}& 2\mathrm{a}\\ 2\mathrm{b}& \mathrm{b}-\mathrm{c}-\mathrm{a}& 2\mathrm{b}\\ 2\mathrm{c}& 2\mathrm{c}& \mathrm{c}-\mathrm{a}-\mathrm{b}\end{array}\right| = {\left(\mathrm{a} + \mathrm{b} + \mathrm{c}\right)}^3$ VIEW SOLUTION

Find the value of k, so that the function f defined by

$\mathrm{f}\left(\mathrm{x}\right)=\left\{\begin{array}{ll}\mathrm{kx}+1,& \mathrm{if} \mathrm{x}\le \mathrm{\pi }\\ \cos \mathrm{x},& \mathrm{if} \mathrm{x}>\mathrm{\pi }\end{array}\right.$

is continuous at x = π. VIEW SOLUTION

Solve the following differential equation:

$\frac{\mathrm{dy}}{\mathrm{dx}}+2\mathrm{y} \tan \mathrm{x}= \sin \mathrm{x},$ given that y = 0 when $x=\frac{\mathrm{\pi }}{3}$. VIEW SOLUTION

Find the shortest distance between the given lines:

$$\begin{gathered} \overrightarrow{\mathrm{r}}=\left(\hat{\mathrm{i}}+2\hat{\mathrm{j}}+3\hat{\mathrm{k}}\right)+\mathrm{\lambda }\left(\hat{\mathrm{i}}-3\hat{\mathrm{j}}+2\hat{\mathrm{k}}\right) \mathrm{and} \\ \overrightarrow{\mathrm{r}}=\left(4\hat{\mathrm{i}}+5\hat{\mathrm{j}}+6\hat{\mathrm{k}}\right)+\mathrm{\mu }\left(2\hat{\mathrm{i}}+3\hat{\mathrm{j}}+\hat{\mathrm{k}}\right). \end{gathered}$$ VIEW SOLUTION

A cottage industry manufactures pedestal lamps and wooden shades, each requiring the use of grinding/cutting machine and a sprayer. It takes 2 hours on the grinding/cutting machine and 3 hours on the sprayer to manufacture a pedestal lamp. It takes one hour on the grinding/cutting machine and 2 hours on the sprayer to manufacture a shade. On any day, the sprayer is available for at most 20 hours and the grinding/cutting machine for at most 12 hours. The profit from the sale of a lamp is Rs 5 and that from a shade is Rs 3. Assuming that the manufacturer can sell all the lamps and shades that he produces, how should he schedule his daily production in order to maximise his profit? Make an L.P.P. and solve it graphically. VIEW SOLUTION

Evaluate $\underset{1}{\overset{4}{\int }} \left({\mathrm{x}}^2-\mathrm{x}\right) \mathrm{dx}$ as a limit of sums.
Evaluate :

$\underset{0}{\overset{\frac{\pi }{4}}{\int }} \frac{\sin \mathrm{x}+\cos \mathrm{x}}{9+16 \sin 2\mathrm{x}}\mathrm{dx}$ VIEW SOLUTION

Using the method of integration, find the area of the region bounded by the following lines :
2x + y = 4
3x − 2y = 6
x − 3y + 5 = 0 VIEW SOLUTION

A window is in the form of a rectangle surmounted by a semi-circular opening. The total perimeter of the window is 10 metres. Find the dimensions of the rectangle so as to admit maximum light through the whole opening. VIEW SOLUTION

Use product $\left[\begin{array}{ccc}1& -1& 2\\ 0& 2& -3\\ 3& -2& 4\end{array}\right]\left[\begin{array}{ccc}-2& 0& 1\\ 9& 2& -3\\ 6& 1& -2\end{array}\right]$ to solve the system of equation:
x − y + 2z = 1
2y − 3z = 1
3x − 2y + 4z = 2.
 
Using elementary transformations, find the inverse of the matrix :
$\left(\begin{array}{ccc}2& 0& -1\\ 5& 1& 0\\ 0& 1& 3\end{array}\right)$ VIEW SOLUTION

Find the vector equation of the plane passing through the points A(2, 2, −1), B (3, 4, 2) and C (7, 0, 6) Also, find the Cartesian equation of the plane. VIEW SOLUTION

Bag I contains 3 red and 4 black balls and Bag II contains 4 red and 5 black balls. One ball is transferred from bag I to bag II and then a ball is drawn from bag II at random. The ball so drawn is found to be red in colour. Find the probability that the transferred ball is black. VIEW SOLUTION