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

Let f : W → W be defined as f(x) = x − 1 if x is odd and f(x) = x + 1 if x is even. Show that f is invertible. Find the inverse of f, where W is the set of all whole numbers. VIEW SOLUTION

Solve for x :
$\cos \left({\tan }^{-1}x\right)=\sin \left({\cot }^{-1}\frac{3}{4}\right)$
 
Prove that:
cot⁻¹ 7 + cot​⁻¹ 8 + cot​⁻¹ 18 = cot​⁻¹ 3 VIEW SOLUTION

Using properties of determinants, prove that

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

If x = a cos θ + b sin θ and y = a sin θ − b cos θ, show that

${\mathrm{y}}^2\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2}-\mathrm{x}\frac{\mathrm{dy}}{\mathrm{dx}}+\mathrm{y}=0.$ VIEW SOLUTION

If x^myⁿ = (x + y)^m+n, prove that $\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{\mathrm{y}}{\mathrm{x}}.$ VIEW SOLUTION

Find the approximate value of f(3.02), up to 2 places of decimal, where f(x) = 3x² + 5x + 3.
Find the intervals in which the function $\mathrm{f}\left(\mathrm{x}\right)=\frac{3}{2}{\mathrm{x}}^4-4{\mathrm{x}}^3-45{\mathrm{x}}^2+51$ is
(a) strictly increasing
(b) strictly decreasing VIEW SOLUTION

Evaluate :
$\int \frac{\mathrm{x} {\cos }^{-1} \mathrm{x}}{\sqrt{1-{\mathrm{x}}^2}}\mathrm{dx}$

OR

Evaluate :
$\int (3\mathrm{x}-2) \sqrt{{\mathrm{x}}^2+\mathrm{x}+1 }\mathrm{dx}$ VIEW SOLUTION

Solve the differential equation (x² − yx²) dy + (y² + x²y²) dx = 0, given that y = 1, when x = 1. VIEW SOLUTION

Solve the differential equation $\frac{\mathrm{dy}}{\mathrm{dx}}$ + y cot x = 2 cos x, given that y = 0 when x = $\frac{\mathrm{\pi }}{2}$. VIEW SOLUTION

Show that the vectors $\overrightarrow{\mathrm{a},} \overrightarrow{\mathrm{b},} \overrightarrow{\mathrm{c} }$ are coplanar if and only if $\overrightarrow{\mathrm{a} }+ \overrightarrow{\mathrm{b} }$, $\overrightarrow{\mathrm{b} }+ \overrightarrow{\mathrm{c} }$ and $\overrightarrow{\mathrm{c} }+ \overrightarrow{\mathrm{a} }$ are coplanar.
Find a unit vector perpendicular to both the vectors $\overrightarrow{\mathrm{a} }+ \overrightarrow{\mathrm{b} }\mathrm{and} \overrightarrow{\mathrm{a} }- \overrightarrow{\mathrm{b} }$, where $\overrightarrow{\mathrm{a} }= \hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}}, \overrightarrow{\mathrm{b} }= \hat{\mathrm{i}}+2\hat{\mathrm{j}}+3\hat{\mathrm{k}}$. VIEW SOLUTION

Find the shortest distance between the lines whose vector equations are
$\overrightarrow{r}=\hat{\mathit{i}}+\hat{\mathit{j}}+\mathrm{\lambda }(2\hat{\mathit{i}}-\hat{\mathit{j}}+\hat{\mathit{k}})$ and $\overrightarrow{r}=\left(2\hat{i}+\hat{j}-\hat{k}\right) + \mu \left(3\hat{i}-5\hat{j}+2\hat{k}\right)$. VIEW SOLUTION

Three cards are drawn at random (without replacement) from a well-shuffled pack of 52 playing cards. Find the probability distribution of the number of red cards. Hence, find the mean of the distribution. VIEW SOLUTION

Two schools P and Q want to award their selected students on the values of tolerance, kindness and leadership. School P wants to award Rs x each, Rs y each and Rs z each for the three respective values to 3, 2 and 1 students, respectively, with a total award money of Rs 2,200. School Q wants to spend Rs 3,100 to award 4, 1 and 3 students on the respective values (by giving the same award money to the three values as school P). If the total amount of award for one prize on each value is Rs 1,200, using matrices, find the award money for each value. VIEW SOLUTION

Show that a cylinder of a given volume, which is open at the top, has minimum total surface area when its height is equal to the radius of its base. VIEW SOLUTION

Evaluate :
$\underset{0}{\overset{\mathrm{\pi }}{\int }}\frac{\mathrm{x} \tan \mathrm{x}}{\sec \mathrm{x} + \tan \mathrm{x}}\mathrm{dx}$ VIEW SOLUTION

Find the area of the smaller region bounded by the ellipse $\frac{x^2}{9}+\frac{y^2}{4}=1$ and the line $\frac{x}{3}+\frac{y}{2}=1.$

VIEW SOLUTION

Find the equation of the plane that contains the point (1, –1, 2) and is perpendicular to both the planes 2x + 3y – 2z = 5 and x + 2y – 3z = 8. Hence, find the distance of point P (–2, 5, 5) from the plane obtained
 
Find the distance of the point P (–1, –5, –10) from the point of intersection of the line joining the points A (2, –1, 2) and B (5, 3, 4) with the plane x – y + z = 5. VIEW SOLUTION

A cottage industry manufactures pedestal lamps and wooden shades, each requiring the use of a grinding/cutting machine and a sprayer. It takes 2 hours on the grinding/cutting machine and 3 hours on the sprayer to manufacture a pedestal lamp. It takes 1 hour on the grinding/cutting machine and 2 hours on the sprayer to manufacture a shade. On any day, the sprayer is available for at the most 20 hours and the grinding/cutting machine for at most 12 hours. The profit from the sale of a lamp is Rs 25 and that from a shade is Rs 15. Assuming that the manufacturer can sell all the lamps and shades that he produces, how should he schedule his daily production in order to maximise his profit? Formulate an LPP and solve it graphically. VIEW SOLUTION

An insurance company insured 2000 scooter drivers, 4000 car drivers and 6000 truck drivers. The probabilities of an accident for them are 0.01, 0.03 and 0.15, respectively. One of the insured persons meets with an accident. What is the probability that he is a scooter driver or a car driver?
Five cards are drawn one by one, with replacement, from a well-shuffled deck of 52 cards. Find the probability that

(i) all the five cards diamonds
(ii) only 3 cards are diamonds
(iii) none is a diamond VIEW SOLUTION