Board Paper of Class 12Commerce 2014 Maths (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
Find the projection of the vector $\hat{\mathrm{i}}+3\hat{\mathrm{j}}+7\hat{\mathrm{k}}$ on the vector $2\hat{\mathrm{i}}3\hat{\mathrm{j}}+6\hat{\mathrm{k}}$. VIEW SOLUTION
 Question 2
Write the vector equation of the plane, passing through the point (a, b, c) and parallel to the plane $\overrightarrow{\mathrm{r}}\xb7\left(\hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}}\right)=2$. VIEW SOLUTION
 Question 3
Write the antiderivative of $\left(3\sqrt{x}+\frac{1}{\sqrt{x}}\right).$ VIEW SOLUTION
 Question 4
If $2\left[\begin{array}{cc}3& 4\\ 5& x\end{array}\right]+\left\begin{array}{cc}1& y\\ 0& 1\end{array}\right=\left[\begin{array}{cc}7& 0\\ 10& 5\end{array}\right],\mathrm{find}(xy).$ VIEW SOLUTION
 Question 5
Solve the following matrix equation for x: $\left[x1\right]\left[\begin{array}{cc}1& 0\\ 2& 0\end{array}\right]=\mathrm{O}.$ VIEW SOLUTION
 Question 6
If $\left\begin{array}{cc}2x& 5\\ 8& x\end{array}\right=\left\begin{array}{cc}6& 2\\ 7& 3\end{array}\right$, write the value of x. VIEW SOLUTION
 Question 7
If $\mathrm{sin}\left({\mathrm{sin}}^{1}\frac{1}{5}+{\mathrm{cos}}^{1}x\right)=1,$ then find the value of x. VIEW SOLUTION
 Question 8
Let * be a binary operation, on the set of all nonzero real numbers, given by $a*b=\frac{ab}{5}$for all a, b ∈ R – {0}. Find the value of x, given that 2 * (x * 5) = 10. VIEW SOLUTION
 Question 9
Evaluate : $\int {\mathrm{cos}}^{1}(\mathrm{sin}x)\mathrm{d}x$. VIEW SOLUTION
 Question 10
If vectors $\overrightarrow{\mathrm{a}}\mathrm{and}\overrightarrow{\mathrm{b}}$ are such that $\left\overrightarrow{\mathrm{a}}\right=3,\left\overrightarrow{\mathrm{b}}\right=\frac{2}{3}\mathrm{and}\overrightarrow{\mathrm{a}}\times \overrightarrow{\mathrm{b}}$ is a unit vector, then write the angle between $\overrightarrow{\mathrm{a}}\mathrm{and}\overrightarrow{\mathrm{b}}$. VIEW SOLUTION
 Question 11
Find the intervals in which the function f(x) = 3x^{4} − 4x^{3} − 12x^{2} + 5 is
(a) strictly increasing
(b) strictly decreasing
OR
Find the equations of the tangent and normal to the curve x = a sin^{3}θ and y = a cos^{3}θ at $\mathrm{\theta}=\frac{\mathrm{\pi}}{4}.$ VIEW SOLUTION
 Question 12
Evaluate : $\int \frac{{\mathrm{sin}}^{6}x+{\mathrm{cos}}^{6}x}{{\mathrm{sin}}^{2}x.{\mathrm{cos}}^{2}x}dx$
OR
Evaluate : $\int \left(x3\right)\sqrt{{x}^{2}+3x18}dx$ VIEW SOLUTION
 Question 13
Solve the following differential equation:
$\left({x}^{2}1\right)\frac{dy}{dx}+2xy=\frac{2}{{x}^{2}1}$ VIEW SOLUTION
 Question 14
If y = x^{x}, prove that $\frac{{\mathrm{d}}^{2}\mathrm{y}}{\mathrm{d}{x}^{2}}\frac{1}{\mathrm{y}}{\left(\frac{\mathrm{dy}}{\mathrm{d}x}\right)}^{2}\frac{\mathrm{y}}{x}=0.$ VIEW SOLUTION
 Question 15
Prove that, for any three vectors $\overrightarrow{\mathrm{a}},\overrightarrow{\mathrm{b}},\overrightarrow{\mathrm{c}}$
$\left[\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}},\overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{c}},\overrightarrow{\mathrm{c}}+\overrightarrow{\mathrm{a}}\right]=2\left[\overrightarrow{\mathrm{a}},\overrightarrow{\mathrm{b}},\overrightarrow{\mathrm{c}}\right]$OR
Vectors $\overrightarrow{\mathrm{a}},\overrightarrow{\mathrm{b}}$ and $\overrightarrow{\mathrm{c}}$ are such that $\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{c}}=\overrightarrow{0}\mathrm{and}\left\overrightarrow{\mathrm{a}}\right=3,\left\overrightarrow{\mathrm{b}}\right=5\mathrm{and}\left\overrightarrow{\mathrm{c}}\right=7$.
Find the angle between $\overrightarrow{\mathrm{a}}\mathrm{and}\overrightarrow{\mathrm{b}}$. VIEW SOLUTION
 Question 16
Prove that ${\mathrm{cot}}^{1}\left(\frac{\sqrt{1+\mathrm{sin}x}+\sqrt{1\mathrm{sin}x}}{\sqrt{1+\mathrm{sin}}x\sqrt{1\mathrm{sin}x}}\right)=\frac{x}{2};x\in \left(0,\frac{\mathrm{\pi}}{4}\right)$.Prove that $2{\mathrm{tan}}^{1}\left(\frac{1}{5}\right)+{\mathrm{sec}}^{1}\left(\frac{5\sqrt{2}}{7}\right)+2{\mathrm{tan}}^{1}\left(\frac{1}{8}\right)=\frac{\mathrm{\pi}}{4}$. VIEW SOLUTION
OR
 Question 17
Let A = {1, 2, 3,......, 9} and R be the relation in A × A defined by (a, b) R (c, d) if a + d = b + c for (a, b), (c, d) in A × A. Prove that R is an equivalence relation. Also, obtain the equivalence class [(2, 5)]. VIEW SOLUTION
 Question 18
Assume that each born child is equally likely to be a boy or a girl. If a family has two children, what is the conditional probability that both are girls? Give that
(i) the youngest is a girl.
(ii) at least one is a girl. VIEW SOLUTION
 Question 19
Prove the following using properties of determinants :
$\left\begin{array}{ccc}\mathrm{a}+\mathrm{b}+2\mathrm{c}& \mathrm{a}& \mathrm{b}\\ \mathrm{c}& \mathrm{b}+\mathrm{c}+2\mathrm{a}& \mathrm{b}\\ \mathrm{c}& \mathrm{a}& \mathrm{c}+\mathrm{a}+2\mathrm{b}\end{array}\right=2\left(\mathrm{a}+\mathrm{b}+\mathrm{c}\right){}^{3}$ VIEW SOLUTION
 Question 20
Differentiate ${\mathrm{tan}}^{1}\left(\frac{x}{\sqrt{1{x}^{2}}}\right)$ with respect to ${\mathrm{sin}}^{1}\left(2x\sqrt{1{x}^{2}}\right)$. VIEW SOLUTION
 Question 21
Solve the following differential equation:
$\mathrm{cosec}x\mathrm{log}\mathrm{y}\frac{\mathrm{dy}}{\mathrm{d}x}+{x}^{2}{\mathrm{y}}^{2}=0$. VIEW SOLUTION
 Question 22
Show that the lines $\frac{5x}{4}=\frac{\mathrm{y}7}{4}=\frac{\mathrm{z}+3}{5}\mathrm{and}\frac{x8}{7}=\frac{2\mathrm{y}8}{2}=\frac{\mathrm{z}5}{3}$are coplanar. VIEW SOLUTION
 Question 23
A dealer in rural area wishes to purchase a number of sewing machines. He has only Rs 5,760 to invest and has space for at most 20 items for storage. An electronic sewing machine cost him Rs 360 and a manually operated sewing machine Rs 240. He can sell an electronic sewing machine at a profit of Rs 22 and a manually operated sewing machine at a profit of Rs 18. Assuming that he can sell all the items that he can buy, how should he invest his money in order to maximize his profit? Make it as a LPP and solve it graphically. VIEW SOLUTION
 Question 24
A card from a pack of 52 playing cards is lost. From the remaining cards of the pack three cards are drawn at random (without replacement) and are found to be all spades. Find the probability of the lost card being a spade.
OR
From a lot of 15 bulbs which include 5 defectives, a sample of 4 bulbs is drawn one by one with replacement. Find the probability distribution of number of defective bulbs. Hence find the mean of the distribution. VIEW SOLUTION
 Question 25
Two schools P and Q want to award their selected students on the values of discipline, politeness and punctuality. The school P wants to award Rs x each, Rs y each and Rs z each for the three respective values to its 3, 2 and 1 students with a total award money of Rs 1,000. School Q wants to spend Rs 1,500 to award its 4, 1 and 3 students on the respective values (by giving the same award money for the three values as before). If the total amount of awards for one prize on each value is Rs 600, using matrices, find the award money for each value.
Apart from the above three values, suggest one more value for awards. VIEW SOLUTION
 Question 26
Find the distance between the point (7, 2, 4) and the plane determined by the points A(2, 5, −3), B(−2, −3, 5) and C(5, 3, −3).
ORFind the distance of the point (−1, −5, −10) from the point of intersection of the line $\overrightarrow{\mathrm{r}}=2\hat{\mathrm{i}}\hat{\mathrm{j}}+2\hat{\mathrm{k}}+\lambda (3\hat{\mathrm{i}}+4\hat{\mathrm{j}}+2\hat{\mathrm{k}})$and the plane $\overrightarrow{\mathrm{r}}.(\hat{\mathrm{i}}\hat{\mathrm{j}}+\hat{\mathrm{k}})=5.$ VIEW SOLUTION
 Question 27
Find the area of the region in the first quadrant enclosed by the xaxis, the line y = x and the circle x^{2} + y^{2} = 32. VIEW SOLUTION
 Question 28
Evaluate :$\underset{0}{\overset{\mathrm{\pi}}{\int}}\frac{x\mathrm{tan}x}{\mathrm{sec}x\xb7\mathrm{cosec}x}\mathrm{d}x$. VIEW SOLUTION
 Question 29
Prove that the semivertical angle of the Right circular cone of given volume and least curved surface area is ${\mathrm{cot}}^{1}\sqrt{2}$. VIEW SOLUTION
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