Board Paper of Class 12Humanities 2014 Maths (SET 1)  Solutions
General Instructions:
i. All questions are compulsory.
ii. The question paper consists of 29 questions divided into three sections A, B and C. Section A comprises of 10 questions of one mark each, Section B comprises of 12 questions of four marks each, and Section C comprises of 7 questions of six marks each.
iii. All questions in section A are to be answered in one word, one sentence or as per the exact requirements of the question.
iv. There is no overall choice. However, internal choice has been provided in 4 questions of four marks each and 2 questions of six marks each. You have to attempt only one of the alternatives in all such questions.
v. Use of calculators is not permitted. You may ask for logarithmic tables, if required.
 Question 1
Let R = {(a, a^{3}) : a is a prime number less than 5} be a relation. Find the range of R. VIEW SOLUTION
 Question 2
Write the value of ${\mathrm{cos}}^{1}\left(\frac{1}{2}\right)+2{\mathrm{sin}}^{1}\left(\frac{1}{2}\right)$. VIEW SOLUTION
 Question 3
Use elementary column operations ${\mathrm{C}}_{2}\to {\mathrm{C}}_{2}2{\mathrm{C}}_{1}$ in the matrix equation $\left(\begin{array}{cc}4& 2\\ 3& 3\end{array}\right)=\left(\begin{array}{cc}1& 2\\ 0& 3\end{array}\right)\left(\begin{array}{cc}2& 0\\ 1& 1\end{array}\right)$. VIEW SOLUTION
 Question 4
If $\left(\begin{array}{cc}\mathrm{a}+4& 3\mathrm{b}\\ 8& 6\end{array}\right)=\left(\begin{array}{cc}2\mathrm{a}+2& \mathrm{b}+2\\ 8& \mathrm{a}8\mathrm{b}\end{array}\right),$write the value of a − 2b. VIEW SOLUTION
 Question 5
If A is a 3 × 3 matrix, $\left\mathrm{A}\right\ne 0\mathrm{and}\left3\mathrm{A}\right=\mathrm{k}\left\mathrm{A}\right$, then write the value of k. VIEW SOLUTION
 Question 6
Evaluate :
$\int \frac{\mathrm{dx}}{{\mathrm{sin}}^{2}\mathrm{x}{\mathrm{cos}}^{2}\mathrm{x}}$ VIEW SOLUTION
 Question 7
Evaluate :
$\underset{0}{\overset{\frac{\pi}{4}}{\int}}\mathrm{tan}\mathrm{x}\mathrm{dx}$ VIEW SOLUTION
 Question 8
Write the projection of vector $\hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}}$ along the vector $\hat{\mathrm{j}}$. VIEW SOLUTION
 Question 9
Find a vector in the direction of vector $2\hat{\mathrm{i}}3\hat{\mathrm{j}}+6\hat{\mathrm{k}}$ which has magnitude 21 units. VIEW SOLUTION
 Question 10
Find the angle between the lines $\overrightarrow{r}=2\hat{i}5\hat{j}+\hat{k}+\lambda \left(3\hat{i}+2\hat{j}+6\hat{k}\right)\mathrm{and}\overrightarrow{r}=7\hat{i}6\hat{k}+\mu \left(\hat{i}+2\hat{j}+2\hat{k}\right)$. VIEW SOLUTION
 Question 11
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
 Question 12
Solve for x :
$\mathrm{cos}\left({\mathrm{tan}}^{1}x\right)=\mathrm{sin}\left({\mathrm{cot}}^{1}\frac{3}{4}\right)$
OR
Prove that:
cot^{−1} 7 + cot^{−1} 8 + cot^{−1} 18 = cot^{−1} 3 VIEW SOLUTION
 Question 13
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
 Question 14
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
 Question 15
If x^{m}y^{n} = (x + y)^{m+n}, prove that $\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{\mathrm{y}}{\mathrm{x}}.$ VIEW SOLUTION
 Question 16
Find the approximate value of f(3.02), up to 2 places of decimal, where f(x) = 3x^{2} + 5x + 3.
OR
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
 Question 17
Evaluate :
$\int \frac{\mathrm{x}{\mathrm{cos}}^{1}\mathrm{x}}{\sqrt{1{\mathrm{x}}^{2}}}\mathrm{dx}$
OR
Evaluate :
$\int (3\mathrm{x}2)\sqrt{{\mathrm{x}}^{2}+\mathrm{x}+1}\mathrm{dx}$ VIEW SOLUTION
 Question 18
Solve the differential equation (x^{2} − yx^{2}) dy + (y^{2} + x^{2}y^{2}) dx = 0, given that y = 1, when x = 1. VIEW SOLUTION
 Question 19
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
 Question 20
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.
OR
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
 Question 21
Find the shortest distance between the lines whose vector equations are
$\overrightarrow{\mathrm{r}}=\hat{\mathrm{i}}+\hat{\mathrm{j}}+\mathrm{\lambda}(2\hat{\mathrm{i}}\hat{\mathrm{j}}+\hat{\mathrm{k}})$ and $\overrightarrow{\mathrm{r}}=2\hat{\mathrm{i}}+\hat{\mathrm{j}}\hat{\mathrm{k}}+\mathrm{\mu}(3\hat{\mathrm{i}}5\hat{\mathrm{j}}+2\hat{\mathrm{k}}).$ VIEW SOLUTION
 Question 22
Three cards are drawn at random (without replacement) from a wellshuffled pack of 52 playing cards. Find the probability distribution of the number of red cards. Hence, find the mean of the distribution. VIEW SOLUTION
 Question 23
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
 Question 24
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
 Question 25
Evaluate :
$\underset{0}{\overset{\mathrm{\pi}}{\int}}\frac{\mathrm{x}\mathrm{tan}\mathrm{x}}{\mathrm{sec}\mathrm{x}+\mathrm{tan}\mathrm{x}}\mathrm{dx}$ VIEW SOLUTION
 Question 26
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
 Question 27
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
OR
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
 Question 28
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
 Question 29
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?
OR
Five cards are drawn one by one, with replacement, from a wellshuffled 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
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