Board Paper of Class 12Commerce 2015 Maths (SET 2)  Solutions
General Instructions :
(i) All questions are compulsory.
(ii) Please check that this Question Paper contains 26 Questions.
(iii) Marks for each question are indicated against it.
(iv) Questions 1 to 6 in SectionA are Very Short Answer Type Questions carrying one mark each.
(v) Questions 7 to 19 in SectionB are Long Answer I Type Questions carrying 4 marks each.
(vi) Questions 20 to 26 in SectionC are Long Answer II Type Questions carrying 6 marks each.
(vii) Please write down the serial number of the Question before attempting it.
* Kindly update your browser if you are unable to view the equations.
(i) All questions are compulsory.
(ii) Please check that this Question Paper contains 26 Questions.
(iii) Marks for each question are indicated against it.
(iv) Questions 1 to 6 in SectionA are Very Short Answer Type Questions carrying one mark each.
(v) Questions 7 to 19 in SectionB are Long Answer I Type Questions carrying 4 marks each.
(vi) Questions 20 to 26 in SectionC are Long Answer II Type Questions carrying 6 marks each.
(vii) Please write down the serial number of the Question before attempting it.
* Kindly update your browser if you are unable to view the equations.
 Question 1
If a line makes angles 90°, 60° and θ with x, y and zaxis respectively, where θ is acute, then find θ. VIEW SOLUTION
 Question 2
Write the element a_{23}_{ }of a 3 ✕ 3 matrix A = (a_{ij}) whose elements a_{ij} are given by ${a}_{ij}=\frac{\leftij\right}{2}.$ VIEW SOLUTION
 Question 3
Find the differential equation representing the family of curves $\mathrm{v}=\frac{\mathrm{A}}{\mathrm{r}}$+ B, where A and B are arbitrary constants. VIEW SOLUTION
 Question 4
Find the integrating factor of the differential equation
$\left(\frac{{{e}^{2}}^{\sqrt{x}}}{\sqrt{x}}\frac{y}{\sqrt{x}}\right)\frac{dx}{dy}=1$. VIEW SOLUTION
 Question 5
If $\overrightarrow{\mathrm{a}}=7\hat{\mathrm{i}}+\hat{\mathrm{j}}4\hat{\mathrm{k}}\mathrm{and}\overrightarrow{\mathrm{b}}=2\hat{\mathrm{i}}+6\hat{\mathrm{j}}+3\hat{\mathrm{k}},$then find the projection of $\overrightarrow{\mathrm{a}}\mathrm{on}\overrightarrow{\mathrm{b}}$. VIEW SOLUTION
 Question 6
Find λ, if the vectors $\overrightarrow{a}=\hat{i}+3\hat{j}+\hat{k},\overrightarrow{b}=2\hat{i}\hat{j}\hat{k}\mathrm{and}\overrightarrow{c}=\lambda \hat{j}+3\hat{k}$ are coplanar. VIEW SOLUTION
 Question 7
A bag A contains 4 black and 6 red balls and bag B contains 7 black and 3 red balls. A die is thrown. If 1 or 2 appears on it, then bag A is chosen, otherwise bag B, If two balls are drawn at random (without replacement) from the selected bag, find the probability of one of them being red and another black.ORAn unbiased coin is tossed 4 times. Find the mean and variance of the number of heads obtained. VIEW SOLUTION
 Question 8
$\mathrm{If}\overrightarrow{\mathrm{r}}=\mathrm{x}\hat{\mathrm{i}}+\mathrm{y}\hat{\mathrm{j}}+\mathrm{z}\hat{\mathrm{k}},\mathrm{find}\left(\overrightarrow{\mathrm{r}}\times \hat{\mathrm{i}}\right)\xb7\left(\overrightarrow{\mathrm{r}}\times \mathrm{j}\right)+\mathrm{xy}$ VIEW SOLUTION
 Question 9
Find the distance between the point (−1, −5, −10) and the point of intersection of the line $\frac{x2}{3}=\frac{y+1}{4}=\frac{z2}{12}$ and the plane x − y + z = 5. VIEW SOLUTION
 Question 10
If sin [cot^{−1} (x+1)] = cos(tan^{−}^{1}x), then find x.
OR
If (tan^{−}^{1}x)^{2} + (cot^{−1}x)^{2} = $\frac{5{\mathrm{\pi}}^{2}}{8}$, then find x. VIEW SOLUTION
 Question 11
If $y={\mathrm{tan}}^{1}\left(\frac{\sqrt{1+{x}^{2}}+\sqrt{1{x}^{2}}}{\sqrt{1+{x}^{2}}\sqrt{1{x}^{2}}}\right),{x}^{2}\le 1,$ then find $\frac{dy}{dx}$. VIEW SOLUTION
 Question 12
If x = a cos θ + b sin θ, y = a sin θ − b cos θ, show that ${y}^{2}\frac{{d}^{2}y}{d{x}^{2}}x\frac{dy}{dx}+y=0.$ VIEW SOLUTION
 Question 13
The side of an equilateral triangle is increasing at the rate of 2 cm/s. At what rate is its area increasing when the side of the triangle is 20 cm ? VIEW SOLUTION
 Question 14
Find : $\int \left(x+3\right)\sqrt{34x{x}^{2}}\mathrm{dx}$ VIEW SOLUTION
 Question 15
Three schools A, B and C organized a mela for collecting funds for helping the rehabilitation of flood victims. They sold hand made fans, mats and plates from recycled material at a cost of Rs 25, Rs 100 and Rs 50 each. The number of articles sold are given below:
School Article A B C Handfans 40 25 35 Mats 50 40 50 Plates 20 30 40
Find the funds collected by each school separately by selling the above articles. Also find the total funds collected for the purpose.
Write one value generated by the above situation. VIEW SOLUTION
 Question 16
If $\mathrm{A}=\left(\begin{array}{ccc}2& 0& 1\\ 2& 1& 3\\ 1& 1& 0\end{array}\right)$ find ${\mathrm{A}}^{2}5\mathrm{A}+4\mathrm{I}$ and hence find a matrix X such that ${\mathrm{A}}^{2}5\mathrm{A}+4\mathrm{I}+\mathrm{X}=\mathrm{O}$
OR
If $\mathrm{A}=\left[\begin{array}{ccc}1& 2& 3\\ 0& 1& 4\\ 2& 2& 1\end{array}\right],\mathrm{find}{\left(\mathrm{A}\text{'}\right)}^{1}.$ VIEW SOLUTION
 Question 17
If $f\left(x\right)=\left\begin{array}{ccc}a& 1& 0\\ ax& a& 1\\ a{x}^{2}& ax& a\end{array}\right$, using properties of determinants find the value of f(2x) − f(x). VIEW SOLUTION
 Question 18
Find : $\int \frac{dx}{\mathrm{sin}x+\mathrm{sin}2x}$
OR
Integrate the following w.r.t. x
$\frac{{x}^{2}3x+1}{\sqrt{1{x}^{2}}}$ VIEW SOLUTION
 Question 19
Evaluate : $\underset{x}{\overset{x}{\int}}{\left(\mathrm{cos}ax\mathrm{sin}bx\right)}^{2}dx$ VIEW SOLUTION
 Question 20
Solve the differential equation :
$\left({\mathrm{tan}}^{1}yx\right)dy=\left(1+{y}^{2}\right)dx.$OR
Find the particular solution of the differential equation $\frac{dy}{dx}=\frac{xy}{{x}^{2}+{y}^{2}}$ given that y = 1, when x = 0. VIEW SOLUTION
 Question 21
If lines $\frac{x1}{2}=\frac{y+1}{3}=\frac{z1}{4}\mathrm{and}\frac{x3}{1}=\frac{yk}{2}=\frac{z}{1}$ intersect, then find the value of k and hence find the equation of the plane containing these lines. VIEW SOLUTION
 Question 22
If A and B are two independent events such that $\mathrm{P}\left(\overline{\mathrm{A}}\cap \mathrm{B}\right)=\frac{2}{15}\mathrm{and}\mathrm{P}\left(\mathrm{A}\cap \overline{\mathrm{B}}\right)=\frac{1}{6},$ then find P(A) and P(B). VIEW SOLUTION
 Question 23
Find the local maxima and local minima, of the function f(x) = sin x − cos x, 0 < x < 2π. VIEW SOLUTION
 Question 24
Find graphically, the maximum value of z = 2x + 5y, subject to constraints given below :
$2x+4y\le 8\phantom{\rule{0ex}{0ex}}3x+y\le 6\phantom{\rule{0ex}{0ex}}x+y\le 4\phantom{\rule{0ex}{0ex}}x\ge 0,y\ge 0\phantom{\rule{0ex}{0ex}}$ VIEW SOLUTION
 Question 25
Let N denote the set of all natural numbers and R be the relation on N × N defined by (a, b) R (c, d) if ad (b + c) = bc (a + d). Show that R is an equivalence relation. VIEW SOLUTION
 Question 26
Using integration find the area of the triangle formed by positive xaxis and tangent and normal of the circle ${x}^{2}+{y}^{2}=4\mathrm{at}\left(1,\sqrt{3}\right)$
OR
Evaluate $\underset{1}{\overset{3}{\int}}\left({e}^{23x}+{x}^{2}+1\right)dx$ as a limit of a sum.
VIEW SOLUTION
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