Board Paper of Class 12Commerce 2014 Maths (SET 1)  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
If $\mathrm{R}=\left[\left(x,y\right):x+2y=8\right]$ is a relation on N, write the range of R. VIEW SOLUTION
 Question 2
If ${\mathrm{tan}}^{1}\mathrm{x}+{\mathrm{tan}}^{1}\mathrm{y}=\frac{\pi}{4},\mathrm{xy}1,$ then write the value of $\mathrm{x}+\mathrm{y}+\mathrm{xy}$. VIEW SOLUTION
 Question 3
If A is a square matrix, such that ${\mathrm{A}}^{2}=\mathrm{A}$, then write the value of $7\mathrm{A}{\left(\mathrm{I}+\mathrm{A}\right)}^{3}$, where I is an identity matrix. VIEW SOLUTION
 Question 4
If $\left[\begin{array}{cc}\mathrm{x}\mathrm{y}& \mathrm{z}\\ 2\mathrm{x}\mathrm{y}& \mathrm{w}\end{array}\right]=\left[\begin{array}{cc}1& 4\\ 0& 5\end{array}\right]$, find the value of $\mathrm{x}+\mathrm{y}$. VIEW SOLUTION
 Question 5
If $\left[\begin{array}{cc}3\mathrm{x}& 7\\ 2& 4\end{array}\right]=\left[\begin{array}{cc}8& 7\\ 6& 4\end{array}\right]$, find the value of x. VIEW SOLUTION
 Question 6
If $\mathrm{f}\left(\mathrm{x}\right)={\int}_{0}^{x}\mathrm{t}\mathrm{sin}\mathrm{t}\mathrm{dt}$, then write the value of f ' (x). VIEW SOLUTION
 Question 7
$\underset{2}{\overset{4}{\int}}\frac{\mathrm{x}}{{\mathrm{x}}^{2}+1}\mathrm{dx}$ VIEW SOLUTION
 Question 8
Find the value of 'p' for which the vectors $3\hat{\mathrm{i}}+2\hat{\mathrm{j}}+9\hat{\mathrm{k}}$ and $\hat{\mathrm{i}}2\mathrm{p}\hat{\mathrm{j}}+3\hat{\mathrm{k}}$ are parallel. VIEW SOLUTION
 Question 9
Find $\overrightarrow{\mathrm{a}}\xb7(\overrightarrow{\mathrm{b}}\times \overrightarrow{\mathrm{c}}),\mathrm{if}\overrightarrow{\mathrm{a}}=2\hat{\mathrm{i}}+\hat{\mathrm{j}}+3\hat{\mathrm{k}},\overrightarrow{\mathrm{b}}=\hat{\mathrm{i}}+2\hat{\mathrm{j}}+\hat{\mathrm{k}}\mathrm{and}\overrightarrow{\mathrm{c}}=3\hat{\mathrm{i}}+\hat{\mathrm{j}}+2\hat{\mathrm{k}}.$ VIEW SOLUTION
 Question 10
If the Cartesian equations of a line are $\frac{3x}{5}=\frac{y+4}{7}=\frac{2z6}{4}$, write the vector equation for the line. VIEW SOLUTION
 Question 11
If the function f : R → R be given by f[x] = x^{2} + 2 and g : R → R be given by $g\left(x\right)=\frac{x}{x1},x\ne 1$, find fog and gof and hence find fog (2) and gof (−3). VIEW SOLUTION
 Question 12
Prove that
${\mathrm{tan}}^{1}\left[\frac{\sqrt{1+x}\sqrt{1x}}{\sqrt{1+x}+\sqrt{1x}}\right]=\frac{\mathrm{\pi}}{4}\frac{1}{2}{\mathrm{cos}}^{1}x,\frac{1}{\sqrt{2}}\le x\le 1$
OR
If ${\mathrm{tan}}^{1}\left(\frac{x2}{x4}\right)+{\mathrm{tan}}^{1}\left(\frac{x+2}{x+4}\right)=\frac{\mathrm{\pi}}{4}$, find the value of x. VIEW SOLUTION
 Question 13
Using properties of determinants, prove that
$\left\begin{array}{ccc}x+y& x& x\\ 5x+4y& 4x& 2x\\ 10x+8y& 8x& 3x\end{array}\right={x}^{3}$ VIEW SOLUTION
 Question 14
Find the value of $\frac{\mathrm{dy}}{\mathrm{dx}}$ at $\mathrm{\theta}=\frac{\mathrm{\pi}}{4}$ if x = ae^{θ} (sin θ − cos θ) and y = ae^{θ} (sin θ + cos θ). VIEW SOLUTION
 Question 15
If y = P e^{ax} + Q e^{bx}, show that
$\frac{{\mathrm{d}}^{2}\mathrm{y}}{{\mathrm{dx}}^{2}}(\mathrm{a}+\mathrm{b})\frac{\mathrm{dy}}{\mathrm{dx}}+\mathrm{aby}=0.$ VIEW SOLUTION
 Question 16
Find the value(s) of x for which y = [x(x − 2)]^{2} is an increasing function.Find the equations of the tangent and normal to the curve $\frac{{\mathrm{x}}^{2}}{{\mathrm{a}}^{2}}\frac{{\mathrm{y}}^{2}}{{\mathrm{b}}^{2}}=1$ at the point $\left(\sqrt{2}\mathrm{a},\mathrm{b}\right)$. VIEW SOLUTION
OR
 Question 17
Evaluate :
$\underset{0}{\overset{\mathrm{\pi}}{\int}}\frac{4\mathrm{x}\mathrm{sin}\mathrm{x}}{1+{\mathrm{cos}}^{2}\mathrm{x}}\mathrm{dx}$
OR
Evaluate :
$\int \frac{\mathrm{x}+2}{\sqrt{{\mathrm{x}}^{2}+5\mathrm{x}+6}}\mathrm{dx}$ VIEW SOLUTION
 Question 18
Find the particular solution of the differential equation $\frac{\mathrm{dy}}{\mathrm{dx}}$ = 1 + x + y + xy, given that y = 0 when x = 1. VIEW SOLUTION
 Question 19
Solve the differential equation (1 + x^{2}) $\frac{\mathrm{dy}}{\mathrm{dx}}+\mathrm{y}={{\mathrm{e}}^{\mathrm{tan}}}^{1}\mathrm{x}.$ VIEW SOLUTION
 Question 20
Show that the four points A, B, C and D with position vectors $4\hat{\mathrm{i}}+5\hat{\mathrm{j}}+\hat{\mathrm{k}},\hat{\mathrm{j}}\hat{\mathrm{k}},3\hat{\mathrm{i}}+9\hat{\mathrm{j}}+4\hat{\mathrm{k}}\mathrm{and}4\left(\hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}}\right)$, respectively, are coplanar.
OR
The scalar product of the vector $\overrightarrow{\mathrm{a}}=\hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}}$ with a unit vector along the sum of vectors $\overrightarrow{\mathrm{b}}=2\hat{\mathrm{i}}+4\hat{\mathrm{j}}5\hat{\mathrm{k}}$ and $\overrightarrow{\mathrm{c}}=\mathrm{\lambda}\hat{\mathrm{i}}+2\hat{\mathrm{j}}+3\hat{\mathrm{k}}$ is equal to one. Find the value of λ and hence, find the unit vector along $\overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{c}}.$ VIEW SOLUTION
 Question 21
A line passes through (2, −1, 3) and is perpendicular to the lines $\overrightarrow{r}=\left(\hat{i}+\hat{j}\hat{k}\right)+\mathrm{\lambda}\left(2\hat{i}2\hat{j}+\hat{k}\right)\mathrm{and}\overrightarrow{r}=\left(2\hat{i}\hat{j}3\hat{k}\right)+\mathrm{\mu}\left(\hat{i}+2\hat{j}+2\hat{k}\right)$. Obtain its equation in vector and Cartesian from. VIEW SOLUTION
 Question 22
An experiment succeeds thrice as often as it fails. Find the probability that in the next five trials, there will be at least 3 successes. VIEW SOLUTION
 Question 23
Two schools A and B want to award their selected students on the values of sincerity, truthfulness and helpfulness. School A wants to award Rs x each, Rs y each and Rs z each for the three respective values to 3, 2 and 1 students, respectively with a total award money of Rs 1,600. School B wants to spend Rs 2,300 to award 4, 1 and 3 students on the respective values (by giving the same award money to the three values as before). If the total amount of award for one prize on each value is Rs 900, using matrices, find the award money for each value. Apart from these three values, suggest one more value which should be considered for an award. VIEW SOLUTION
 Question 24
Show that the altitude of a right circular cone of maximum volume that can be inscribed in a sphere of radius r is $\frac{4\mathrm{r}}{3}$. Also, show that the maximum volume of the cone is $\frac{8}{27}$ of the volume of the sphere. VIEW SOLUTION
 Question 25
Evaluate:
$\int \frac{1}{{\mathrm{cos}}^{4}\mathrm{x}+{\mathrm{sin}}^{4}\mathrm{x}}\mathrm{dx}$ VIEW SOLUTION
 Question 26
Using integration, find the area of the region bounded by the triangle whose vertices are (−1, 2), (1, 5) and (3, 4). VIEW SOLUTION
 Question 27
Find the equation of the plane through the line of intersection of the planes x + y + z = 1 and 2x + 3y + 4z = 5 which is perpendicular to the plane x − y + z = 0. Also find the distance of the plane, obtained above, from the origin.
OR
Find the distance of the point (2, 12, 5) from the point of intersection of the line $\overrightarrow{\mathrm{r}}=2\hat{\mathrm{i}}4\hat{\mathrm{j}}+2\hat{\mathrm{k}}+\mathrm{\lambda}(3\hat{\mathrm{i}}+4\hat{\mathrm{j}}+2\hat{\mathrm{k}})$ and the plane $\overrightarrow{\mathrm{r}}.(\hat{\mathrm{i}}2\hat{\mathrm{j}}+\hat{\mathrm{k}})=0.$ VIEW SOLUTION
 Question 28
A manufacturing company makes two types of teaching aids A and B of Mathematics for class XII. Each type of A requires 9 labour hours for fabricating and 1 labour hour for finishing. Each type of B requires 12 labour hours for fabricating and 3 labour hours for finishing. For fabricating and finishing, the maximum labour hours available per week are 180 and 30, respectively. The company makes a profit of Rs 80 on each piece of type A and Rs 120 on each piece of type B. How many pieces of type A and type B should be manufactured per week to get maximum profit? Make it as an LPP and solve graphically. What is the maximum profit per week? VIEW SOLUTION
 Question 29
There are three coins. One is a twoheaded coin (having head on both faces), another is a biased coin that comes up heads 75% of the times and the third is also a biased coin that comes up tails 40% of the time. One of the three coins is chosen at random and tossed and it shows heads. What is the probability that it was the twoheaded coin?
OR
Two numbers are selected at random (without replacement) from the first six positive integers. Let X denote the larger of the two numbers obtained. Find the probability distribution of the random variable X and hence, find the mean of the distribution. VIEW SOLUTION
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