Board Paper of Class 12Commerce 2015 Maths All India(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
Write the value of $\overrightarrow{\mathrm{a}}.\left(\overrightarrow{\mathrm{b}}\times \overrightarrow{\mathrm{a}}\right).$ VIEW SOLUTION
 Question 2
If $\overrightarrow{\mathrm{a}}=\hat{\mathrm{i}}+2\hat{\mathrm{j}}\hat{\mathrm{k}},\overrightarrow{\mathrm{b}}=2\hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}}$ and $\overrightarrow{\mathrm{c}}=5\hat{\mathrm{i}}4\hat{\mathrm{j}}+3\hat{\mathrm{k}}$, then find the value of $\left(\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}\right).\overrightarrow{\mathrm{c}}.$ VIEW SOLUTION
 Question 3
Write the direction ratios of the following line :
$\mathrm{x}=3,\frac{\mathrm{y}4}{3}=\frac{2\mathrm{z}}{1}$ VIEW SOLUTION
 Question 4
If $\mathrm{A}=\left[\begin{array}{cc}2& 3\\ 5& 2\end{array}\right]$, then write A^{−1}. VIEW SOLUTION
 Question 5
Find the differential equation representing the curve y = cx + c^{2}. VIEW SOLUTION
 Question 6
Write the integrating factor of the following differential equation:
$(1+{y}^{2})dx({\mathrm{tan}}^{1}yx)dy=0$ VIEW SOLUTION
 Question 7
Using the properties of determinants, prove the following:
$\left\begin{array}{ccc}1& x& x+1\\ 2x& x(x1)& x(x+1)\\ 3x(1x)& x(x1)(x2)& x(x+1)(x1)\end{array}\right=6{x}^{2}\left(1{x}^{2}\right)$ VIEW SOLUTION
 Question 8
If $x=\alpha \mathrm{sin}2t(1+\mathrm{cos}2t)\mathrm{and}y=\beta \mathrm{cos}2t(1\mathrm{cos}2t)$, show that $\frac{dy}{dx}=\frac{\beta}{\alpha}\mathrm{tan}t$. VIEW SOLUTION
 Question 9
Find : $\frac{d}{dx}{\mathrm{cos}}^{1}\left(\frac{x{x}^{1}}{x+{x}^{1}}\right)$ VIEW SOLUTION
 Question 10
Find the derivative of the following function f(x) w.r.t. x, at x = 1 :
${\mathrm{cos}}^{1}\left[\mathrm{sin}\sqrt{\frac{1+x}{2}}\right]+{x}^{x}$ VIEW SOLUTION
 Question 11
Evaluate :
$\underset{0}{\overset{\frac{\mathrm{\pi}}{2}}{\int}}\frac{{2}^{\mathrm{sin}x}}{{2}^{\mathrm{sin}x}+{2}^{\mathrm{cos}x}}dx$
OR
Evaluate :$\underset{0}{\overset{3/2}{\int}}\leftx\xb7\mathrm{cos}\left(\mathrm{\pi}x\right)\rightdx$ VIEW SOLUTION
 Question 12
To raise money for an orphanage, students of three schools A, B and C organised an exhibition in their locality, where they sold paper bags, scrapbooks and pastel sheets made by them using recycled paper, at the rate of Rs 20, Rs 15 and Rs 5 per unit respectively. School A sold 25 paper bags, 12 scrapbooks and 34 pastel sheets. School B sold 22 paper bags, 15 scrapbooks and 28 pastel sheets while School C sold 26 paper bags, 18 scrapbooks and 36 pastel sheets. Using matrices, find the total amount raised by each school.
By such exhibition, which values are generated in the students? VIEW SOLUTION
 Question 13
Prove that :
$2{\mathrm{tan}}^{1}\left(\sqrt{\frac{ab}{a+b}}\mathrm{tan}\frac{\mathrm{x}}{2}\right)={\mathrm{cos}}^{1}\left(\frac{a\mathrm{cos}x+b}{a+b\mathrm{cos}x}\right)$
OR
Solve the following for x :
${\mathrm{tan}}^{1}\left(\frac{x2}{x3}\right)+{\mathrm{tan}}^{1}\left(\frac{x+2}{x+3}\right)=\frac{\mathrm{\pi}}{4},\leftx\right1.$ VIEW SOLUTION
 Question 14
If A = $\left(\begin{array}{ccc}2& 0& 1\\ 2& 1& 3\\ 1& 1& 0\end{array}\right)\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}$, find A^{2} − 5 A + 16 I. VIEW SOLUTION
 Question 15
Show that four points A, B, C and D whose position vectors are $4\hat{\mathrm{i}}+5\hat{\mathrm{j}}+\hat{\mathrm{k}},\hat{\mathrm{j}}\hat{\mathrm{k}},3\hat{\mathrm{i}}+9\hat{\mathrm{j}}+4\hat{\mathrm{k}}\mathrm{and}4\left(\hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}}\right)$ respectively are coplanar. VIEW SOLUTION
 Question 16
Show that the following two lines are coplanar:
$\frac{xa+d}{\alpha \delta}=\frac{ya}{\alpha}=\frac{zad}{\alpha +\delta}and\frac{xb+c}{\beta \gamma}=\frac{yb}{\beta}=\frac{zbc}{\beta +\gamma}$OR
Find the acute angle between the plane 5x − 4y + 7z − 13 = 0 and the yaxis. VIEW SOLUTION
 Question 17
A and B throw a die alternatively till one of them gets a number greater than four and wins the game. If A starts the game, what is the probability of B winning?
OR
A die is thrown three times. Events A and B are defined as below:VIEW SOLUTION
A : 5 on the first and 6 on the second throw.
B: 3 or 4 on the third throw.
Find the probability of B, given that A has already occurred.
 Question 18
 Question 19
 Question 20
Using integration, find the area of the region bounded by the lines y = 2 + x, y = 2 – x and x = 2. VIEW SOLUTION
 Question 21
Find the the differential equation for all the straight lines, which are at a unit distance from the origin.
OR
Show that the differential equation $2xy\frac{dy}{dx}={x}^{2}+3{y}^{2}$ is homogeneous and solve it. VIEW SOLUTION
 Question 22
Find the direction ratios of the normal to the plane, which passes through the points (1, 0, 0) and (0, 1, 0) and makes angle $\frac{\mathrm{\pi}}{4}$ with the plane x + y = 3. Also find the equation of the plane. VIEW SOLUTION
 Question 23
If the function f : R → R be defined by f(x) = 2x − 3 and g : R → R by g(x) = x^{3} + 5, then find the value of (fog)^{−1} (x).
OR
Let A = Q ✕ Q, where Q is the set of all rational numbers, and * be a binary operation defined on A by
(a, b) * (c, d) = (ac, b + ad), for all (a, b) (c, d) ∈ A.
Find
(i) the identity element in A
(ii) the invertible element of A. VIEW SOLUTION
 Question 24
If the function $f\left(x\right)=2{x}^{3}9m{x}^{2}+12{m}^{2}x+1$, where $m>0$ attains its maximum and minimum at p and q respectively such that ${p}^{2}=q$, then find the value of m. VIEW SOLUTION
 Question 25
The postmaster of a local post office wishes to hire extra helpers during the Deepawali season, because of a large increase in the volume of mail handling and delivery. Because of the limited office space and the budgetary conditions, the number of temporary helpers must not exceed 10. According to past experience, a man can handle 300 letters and 80 packages per day, on the average, and a woman can handle 400 letters and 50 packets per day. The postmaster believes that the daily volume of extra mail and packages will be no less than 3400 and 680 respectively. A man receives Rs 225 a day and a woman receives Rs 200 a day. How many men and women helpers should be hired to keep the payroll at a minimum ? Formulate an LPP and solve it graphically. VIEW SOLUTION
 Question 26
40% students of a college reside in hostel and the remaining reside outside. At the end of the year, 50% of the hostelers got A grade while from outside students, only 30% got A grade in the examination. At the end of the year, a student of the college was chosen at random and was found to have gotten A grade. What is the probability that the selected student was a hosteler ? VIEW SOLUTION
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