Board Paper of Class 12-Commerce 2015 Maths All India(SET 2) - Solutions

Using the properties of determinants, prove the following:

$\left|\begin{array}{ccc}1& x& x+1\\ 2x& x(x-1)& x(x+1)\\ 3x(1-x)& x(x-1) (x-2)& x(x+1) (x-1)\end{array}\right|=6x^2 \left(1-x^2\right)$ VIEW SOLUTION

If $x=\alpha \sin 2t (1 + \cos 2t) \mathrm{and} y=\beta \cos 2t (1-\cos 2t)$, show that $\frac{dy}{dx}=\frac{\beta }{\alpha }\tan t$. VIEW SOLUTION

Find : $\frac{d}{dx}{\cos }^{-1} \left(\frac{x-x^{-1}}{x+x^{-1}}\right)$ VIEW SOLUTION

Find the derivative of the following function f(x) w.r.t. x, at x = 1 :
${\cos }^{-1} \left[\sin \sqrt{\frac{1+x}{2}}\right]+x^x$ VIEW SOLUTION

Evaluate :
$\underset{0}{\overset{\frac{\mathrm{\pi }}{2}}{\int }}\frac{2^{\sin x}}{2^{\sin x}+2^{\cos x}}dx$
 
Evaluate :$\underset{0}{\overset{3/2}{\int }} \left|x·\cos \left(\mathrm{\pi }x\right)\right|dx$ VIEW SOLUTION

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, scrap-books 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 scrap-books and 34 pastel sheets. School B sold 22 paper bags, 15 scrap-books and 28 pastel sheets while School C sold 26 paper bags, 18 scrap-books 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

Prove that :

$2 {\tan }^{-1}\left(\sqrt{\frac{a-b}{a+b}}\tan \frac{\mathrm{x}}{2}\right)={\cos }^{-1}\left(\frac{a \cos x+b}{a+b \cos x}\right)$
Solve the following for x :

${\tan }^{-1}\left(\frac{x-2}{x-3}\right)+{\tan }^{-1}\left(\frac{x+2}{x+3}\right)=\frac{\mathrm{\pi }}{4}, \left|x\right|<1.$ VIEW SOLUTION

If A = $\left(\begin{array}{ccc}2& 0& 1\\ 2& 1& 3\\ 1& -1& 0\end{array}\right)$, find A² − 5 A + 16 I. VIEW SOLUTION

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

Show that the following two lines are coplanar:

$\frac{x-a+d}{\alpha -\delta }= \frac{y-a}{\alpha }=\frac{z-a-d}{\alpha +\delta } and \frac{x-b+c}{\beta -\gamma }=\frac{y-b}{\beta }=\frac{z-b-c}{\beta +\gamma }$
Find the acute angle between the plane 5x − 4y + 7z − 13 = 0 and the y-axis. VIEW SOLUTION

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? VIEW SOLUTION

Evaluate :
$\int \left(\sqrt{\cot x}+\sqrt{\tan x}\right) dx$ VIEW SOLUTION

Find :

$\int \frac{x^3-1}{x^3+x}dx$ VIEW SOLUTION

Using integration, find the area of the region bounded by the lines y = 2 + x, y = 2 – x and x = 2. VIEW SOLUTION

Find the the differential equation for all the straight lines, which are at a unit distance from the origin.
Show that the differential  equation $2xy\frac{dy}{dx}=x^2+3y^2$ is homogeneous and solve it. VIEW SOLUTION

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

If the function f : R → R be defined by f(x) = 2x − 3 and g : R → R by g(x) = x³ + 5, then find the value of (fog)⁻¹ (x).
 
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

If the function $f\left(x\right)=2x^3-9m{x}^2+12m^{2}x+1$, 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

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 pay-roll at a minimum ? Formulate an LPP and solve it graphically. VIEW SOLUTION

40% students of a college reside in hostel and the remaining reside outside. At the end of the year, 50% of the hostellers 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 hosteller ? VIEW SOLUTION