Board Paper of Class 12-Science 2014 Maths Delhi(SET 1) - Solutions

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

Prove that ${\cot }^{-1}\left(\frac{\sqrt{1+\sin x}+\sqrt{1-\sin x}}{\sqrt{1+\sin }x-\sqrt{1-\sin x}}\right)=\frac{x}{2}; x \in \left(0, \frac{\mathrm{\pi }}{4}\right)$. Prove that $2 {\tan }^{-1} \left(\frac{1}{5}\right)+ {\sec }^{-1}\left(\frac{5\sqrt{2}}{7}\right)+2 {\tan }^{-1} \left(\frac{1}{8}\right)=\frac{\mathrm{\pi }}{4}$. VIEW SOLUTION

Using properties of determinants, prove that

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

Differentiate tan⁻¹ $\left(\frac{\sqrt{1-x^2}}{x}\right)$ with respect to cos⁻¹ $\left(2x\sqrt{1-x^2}\right)$, when x ≠ 0. VIEW SOLUTION

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

Find the intervals in which the function f(x) = 3x⁴ − 4x³ − 12x² + 5 is
(a) strictly increasing
(b) strictly decreasing
Find the equations of the tangent and normal to the curve x = a sin³θ and y = a cos³θ at $\mathrm{\theta }=\frac{\mathrm{\pi }}{4}.$ VIEW SOLUTION

Evaluate : $\int \frac{{\sin }^{6}x+{\cos }^{6}x}{{\sin }^{2}x . {\cos }^{2}x}dx$
OR
Evaluate : $\int \left(x-3\right)\sqrt{x^2+3x-18}dx$ VIEW SOLUTION

Find the particular solution of the differential equation ${\mathrm{e}}^x\sqrt{1-{\mathrm{y}}^2} \mathrm{d}x+\frac{\mathrm{y}}{x} \mathrm{dy}=0,$ given that y = 1 when x = 0. VIEW SOLUTION

Solve the following differential equation:
$\left(x^2-1\right)\frac{dy}{dx}+2xy=\frac{2}{x^2-1}$ VIEW SOLUTION

Prove that, for any three vectors $\overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{b}}, \overrightarrow{\mathrm{c}}$
$\left[\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}, \overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{c}}, \overrightarrow{\mathrm{c}}+\overrightarrow{\mathrm{a}}\right]=2 \left[\overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{b}}, \overrightarrow{\mathrm{c}}\right]$
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

Show that the lines$\frac{x+1}{3}=\frac{y+3}{5}=\frac{z+5}{7} \mathrm{and} \frac{x-2}{1}=\frac{y-4}{3}=\frac{z-6}{5}$intersect. Also find their point of intersection. VIEW SOLUTION

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

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

Show that the semi-vertical angle of the cone of the maximum volume and of given slant height is ${\cos }^{-1}\frac{1}{\sqrt{3}}.$ VIEW SOLUTION

Evaluate : $\underset{\mathrm{\pi }/6}{\overset{\mathrm{\pi }/3}{\int }}\frac{dx}{1+\sqrt{\cot x}}$ VIEW SOLUTION

Find the area of the region in the first quadrant enclosed by the x-axis, the line y = x and the circle x² + y² = 32. VIEW SOLUTION

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).
  Find 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

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

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.
 
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