Board Paper of Class 12-Humanities 2016 Maths Delhi(SET 1) - Solutions

Prove that:

${\tan }^{-1}\frac{1}{5}+{\tan }^{-1}\frac{1}{7}+{\tan }^{-1}\frac{1}{3}+{\tan }^{-1}\frac{1}{8}=\frac{\mathrm{\pi }}{4}$
Solve for x:

$2 {\tan }^{-1}\left(\cos x\right)={\tan }^{-1}\left(2\mathrm{cosec} x\right)$ VIEW SOLUTION

The monthly incomes of Aryan and Babban are in the ratio 3 : 4 and their monthly expenditures are in the ratio 5 : 7. If each saves Rs 15,000 per month, find their monthly incomes using matrix method. This problem reflects which value? VIEW SOLUTION

If x = a sin 2t (1 + cos 2t) and y = b cos 2t (1 – cos 2t), find the values of $\frac{dy}{dx}$ at $\mathrm{t}=\frac{\mathrm{\pi }}{4}$ and $\mathrm{t}=\frac{\mathrm{\pi }}{3}.$
 
If y = x^x, prove that $\frac{{\mathrm{d}}^{2}y}{\mathrm{d}{x}^2}-\frac{1}{y}{\left(\frac{\mathrm{d}y}{\mathrm{d}x}\right)}^2-\frac{y}{x}=0.$ VIEW SOLUTION

Find the values of p and q for which

$\mathrm{f}\left(x\right)=\left\{\begin{array}{ll}\frac{1-{\sin }^{3}x}{3 {\cos }^{2}x}& , \mathrm{if} x<\frac{x}{2}\\ \mathrm{p}& , \mathrm{if} x=\pi /2\\ \frac{\mathrm{q}\left(1-\sin x\right)}{{\left(\pi -2x\right)}^2}& , \mathrm{if} x>\pi /2\end{array}\right.$

is continuous at x = π/2. VIEW SOLUTION

Show that the equation of normal at any point t on the curve x = 3 cos t – cos³t and y = 3 sin t – sin³t is

4 (y cos³t – sin³t) = 3 sin 4t. VIEW SOLUTION

Find $\int \frac{\left(3 \sin \mathrm{\theta }-2\right)\cos \mathrm{\theta }}{5-{\cos }^2 \mathrm{\theta }-4 \sin \mathrm{\theta }}\mathrm{d\theta }$ .

                     OR

Evaluate ${\int }_0^{\mathrm{\pi }}{e}^{2x}· \sin \left(\frac{\mathrm{\pi }}{4}+x\right) dx$ VIEW SOLUTION

Find $\int \frac{\sqrt{x}}{\sqrt{a^3-x^3}}dx$. VIEW SOLUTION

Evaluate $\underset{-1}{\overset{2}{\int }}\left|x^3-x\right| dx.$ VIEW SOLUTION

Find the particular solution of the differential equation
(1 – y²) (1 + log x) dx + 2xy dy = 0, given that y = 0 when x = 1. VIEW SOLUTION

Find the general solution of the following differential equation :
$\left(1+y^2\right)+\left(x-e^{{\tan }^{-1}y}\right) \frac{dy}{dx}=0$ VIEW SOLUTION

Show that the vectors $\overrightarrow{a}, \overrightarrow{b} \mathrm{and} \overrightarrow{c}$ are coplanar if $\overrightarrow{a}+\overrightarrow{b}, \overrightarrow{b}+\overrightarrow{c} \mathrm{and} \overrightarrow{c}+\overrightarrow{a}$ are coplanar. VIEW SOLUTION

Find the vector and Cartesian equations of the line through the point (1, 2, −4) and perpendicular to the two lines.

$\overrightarrow{\mathrm{r}}=\left(8\hat{\mathrm{i}}-19\hat{\mathrm{j}}+10\hat{\mathrm{k}}\right)+\lambda \left(3\hat{\mathrm{i}}-16\hat{\mathrm{j}}+7\hat{\mathrm{k}}\right)$ and $\overrightarrow{\mathrm{r}}=\left(15\hat{\mathrm{i}}+29\hat{\mathrm{j}}+5\hat{\mathrm{k}}\right)+\mathrm{\mu }\left(3\hat{\mathrm{i}}+8\hat{\mathrm{j}}-5\hat{\mathrm{k}}\right)$. VIEW SOLUTION

Three persons A, B and C apply for a job of Manager in a Private Company. Chances of their selection (A, B and C) are in the ratio 1 : 2 :4. The probabilities that A, B and C can introduce changes to improve profits of the company are 0.8, 0.5 and 0.3, respectively. If the change does not take place, find the probability that it is due to the appointment of C.
A and B throw a pair of dice alternately. A wins the game if he gets a total of 7 and B wins the game if he gets a total of 10. If A starts the game, then find the probability that B wins. VIEW SOLUTION

Let $\mathrm{f} : \mathrm{N}\to \mathrm{N}$ be a function defined as $f\left(x\right)=9x^2+6x-5$. Show that $\mathrm{f} : \mathrm{N}\to \mathrm{S}$, where S is the range of f, is invertible. Find the inverse of f and hence find ${\mathrm{f}}^{-1}\left(43\right) \mathrm{and} {\mathrm{f}}^{-1}\left(163\right)$. VIEW SOLUTION

Prove that $\left|\begin{array}{ccc}yz-x^2& zx-y^2& xy-z^2\\ zx-y^2& xy-z^2& yz-x^2\\ xy-z^2& yz-x^2& zx-y^2\end{array}\right|$ is divisible by (x + y + z) and hence find the quotient.
 
Using elementary transformations, find the inverse of the matrix $\mathrm{A}=\left(\begin{array}{ccc}8& 4& 3\\ 2& 1& 1\\ 1& 2& 2\end{array}\right)$ and use it to solve the following system of linear equations :

8x + 4y + 3z = 19

2x + y + z = 5

x + 2y + 2z = 7 VIEW SOLUTION

Show that the altitude of the right circular cone of maximum volume that can be inscribed in a sphere of radius r is $\frac{4\mathrm{r}}{3}.$ Also find maximum volume in terms of volume of the sphere.
 
Find the intervals in which f(x) = sin 3x – cos 3x, 0 < x < π, is strictly increasing or strictly decreasing. VIEW SOLUTION

Using integration find the area of the region $\left\{\left(x, y\right) : x^2+y^2⩽ 2ax, y^2⩾ ax, x, y ⩾ 0\right\}$. VIEW SOLUTION

Find the coordinate of the point P where the line through A(3, –4, –5) and B(2, –3, 1) crosses the plane passing through three points L(2, 2, 1), M(3, 0, 1) and N(4, –1, 0).
Also, find the ratio in which P divides the line segment AB. VIEW SOLUTION

An urn contains 3 white and 6 red balls. Four balls are drawn one by one with replacement from the urn. Find the probability distribution of the number of red balls drawn. Also find mean and variance of the distribution. VIEW SOLUTION

A manufacturer produces two products A and B. Both the products are processed on two different machines. The available capacity of first machine is 12 hours and that of second machine is 9 hours per day. Each unit of product A requires 3 hours on both machines and each unit of product B requires 2 hours on first machine and 1 hour on second machine. Each unit of product A is sold at Rs 7 profit and  B at a profit of Rs 4. Find the production level per day for maximum profit graphically. VIEW SOLUTION