Board Paper of Class 12Commerce 2011 Maths Abroad(SET 2)  Solutions
General Instructions:
i. All questions are compulsory.
ii. The question paper consists of 29 questions divided into three sections A, B and C. Section A comprises of 10 questions of one mark each, Section B comprises of 12 questions of four marks each, and Section C comprises of 7 questions of six marks each.
iii. All questions in section A are to be answered in one word, one sentence or as per the exact requirements of the question.
iv. There is no overall choice. However, internal choice has been provided in 4 questions of four marks each and 2 questions of six marks each. You have to attempt only one of the alternatives in all such questions.
v. Use of calculators is not permitted.
i. All questions are compulsory.
ii. The question paper consists of 29 questions divided into three sections A, B and C. Section A comprises of 10 questions of one mark each, Section B comprises of 12 questions of four marks each, and Section C comprises of 7 questions of six marks each.
iii. All questions in section A are to be answered in one word, one sentence or as per the exact requirements of the question.
iv. There is no overall choice. However, internal choice has been provided in 4 questions of four marks each and 2 questions of six marks each. You have to attempt only one of the alternatives in all such questions.
v. Use of calculators is not permitted.
 Question 1
What are the direction cosines of a line that makes equal angles with the coordinate axes? VIEW SOLUTION
 Question 2
If $\overrightarrow{\mathrm{a}}\xb7\overrightarrow{\mathrm{a}}=0\mathrm{and}\overrightarrow{\mathrm{a}}\xb7\overrightarrow{\mathrm{b}}=0$, then what can be concluded about the vector $\overrightarrow{\mathrm{b}}$? VIEW SOLUTION
 Question 3
Write the position vector of the midpoint of the vector joining the points P(2, 3, 4) and Q(4, 1, −2). VIEW SOLUTION
 Question 4
Evaluate :
$\underset{1}{\overset{\sqrt{3}}{\int}}\frac{\mathrm{dx}}{1+{\mathrm{x}}^{2}}$ VIEW SOLUTION
 Question 5
If $\left\begin{array}{cc}\mathrm{x}& \mathrm{x}\\ 1& \mathrm{x}\end{array}\right=\left\begin{array}{cc}3& 4\\ 1& 2\end{array}\right$, write the positive value of x. VIEW SOLUTION
 Question 6
Write the order of the product matrix:
$\left[\begin{array}{c}1\\ 2\\ 3\end{array}\right]\left[\begin{array}{ccc}2& 3& 4\end{array}\right]$ VIEW SOLUTION
 Question 7
Write the values of x − y + z from the following equation:
$\left[\begin{array}{ccccc}x& +& y& +& z\\ & x& +& z& \\ & y& +& z& \end{array}\right]=\left[\begin{array}{c}9\\ 5\\ 7\end{array}\right]$ VIEW SOLUTION
 Question 8
Write the principal value of tan^{−1}(−1). VIEW SOLUTION
 Question 9
Write fog if f : R → R and g : R → R are given by
f(x) = x and g(x) = 5x − 2. VIEW SOLUTION
 Question 10
 Question 11
Find the mean number of heads in three tosses of a fair coin. VIEW SOLUTION
 Question 12
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
 Question 13
Find the particular solution of the differential equation:
(1 + e^{2x}) dy + (1 + y^{2}) e^{x} dx = 0, given that y = 1 when x = 0. VIEW SOLUTION
 Question 14
 Question 15
Prove that :
$\frac{\mathrm{d}}{\mathrm{dx}}\left[\frac{\mathrm{x}}{2}\sqrt{{\mathrm{a}}^{2}{\mathrm{x}}^{2}}+\frac{{a}^{2}}{2}\mathrm{sin}{}^{1}\left(\frac{\mathrm{x}}{\mathrm{a}}\right)\right]=\sqrt{{\mathrm{a}}^{2}{\mathrm{x}}^{2}}\phantom{\rule{0ex}{0ex}}$OR
$\mathrm{If}\mathrm{y}=\mathrm{log}\left[\mathrm{x}+\sqrt{{\mathrm{x}}^{2}+1}\right],\mathrm{prove}\mathrm{that}({\mathrm{x}}^{2}+1)\frac{{\mathrm{d}}^{2}\mathrm{y}}{{\mathrm{dx}}^{2}}+\mathrm{x}\frac{\mathrm{dy}}{\mathrm{dx}}=0.$ VIEW SOLUTION
 Question 16
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.OR
Find the points on the curve y = x^{3} at which the slope of the tangent is equal to the ycoordinate of the point. VIEW SOLUTION
 Question 17
Prove the following :
$\frac{9\pi}{8}\frac{9}{4}{\mathrm{sin}}^{1}\left(\frac{1}{3}\right)=\frac{9}{4}\mathrm{sin}{}^{1}\left(\frac{2\sqrt{2}}{3}\right)$OR
Solve the following equation for x :
${\mathrm{tan}}^{1}\left(\frac{1\mathrm{x}}{1+\mathrm{x}}\right)=\frac{1}{2}{\mathrm{tan}}^{1}\left(\mathrm{x}\right),\mathrm{x}0$ VIEW SOLUTION
 Question 18
Consider $\mathrm{f}:{\mathrm{R}}_{+}\to \left[4,\infty \right]$ given by f(x) = x^{2} + 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 nonnegative real numbers. VIEW SOLUTION
 Question 19
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
 Question 20
Find the value of k, so that the function f defined by
$\mathrm{f}\left(\mathrm{x}\right)=\left\{\begin{array}{ll}\mathrm{kx}+1,& \mathrm{if}\mathrm{x}\le \mathrm{\pi}\\ \mathrm{cos}\mathrm{x},& \mathrm{if}\mathrm{x}\mathrm{\pi}\end{array}\right.$
is continuous at x = π. VIEW SOLUTION
 Question 21
Solve the following differential equation:
$\frac{\mathrm{dy}}{\mathrm{dx}}+2\mathrm{y}\mathrm{tan}\mathrm{x}=\mathrm{sin}\mathrm{x},$ given that y = 0 when $x=\frac{\mathrm{\pi}}{3}$. VIEW SOLUTION
 Question 22
Find the shortest distance between the given lines:
$\overrightarrow{\mathrm{r}}=\left(\hat{\mathrm{i}}+2\hat{\mathrm{j}}+3\hat{\mathrm{k}}\right)+\mathrm{\lambda}\left(\hat{\mathrm{i}}3\hat{\mathrm{j}}+2\hat{\mathrm{k}}\right)\mathrm{and}\phantom{\rule{0ex}{0ex}}\overrightarrow{\mathrm{r}}=\left(4\hat{\mathrm{i}}+5\hat{\mathrm{j}}+6\hat{\mathrm{k}}\right)+\mathrm{\mu}\left(2\hat{\mathrm{i}}+3\hat{\mathrm{j}}+\hat{\mathrm{k}}\right).$ VIEW SOLUTION
 Question 23
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
 Question 24
Evaluate $\underset{1}{\overset{4}{\int}}\left({\mathrm{x}}^{2}\mathrm{x}\right)\mathrm{dx}$ as a limit of sums.OR
Evaluate :
$\underset{0}{\overset{\frac{\pi}{4}}{\int}}\frac{\mathrm{sin}\mathrm{x}+\mathrm{cos}\mathrm{x}}{9+16\mathrm{sin}2\mathrm{x}}\mathrm{dx}$ VIEW SOLUTION
 Question 25
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
 Question 26
A window is in the form of a rectangle surmounted by a semicircular 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
 Question 27
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.
OR
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
 Question 28
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
 Question 29
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
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