Board Paper of Class 12Commerce 2015 Maths (SET 2)  Solutions
General Instructions :
(i) All questions are compulsory.
(ii) Please check that this Question Paper contains 26 Questions.
(iii) Marks for each question are indicated against it.
(iv) Questions 1 to 6 in SectionA are Very Short Answer Type Questions carrying one mark each.
(v) Questions 7 to 19 in SectionB are Long Answer I Type Questions carrying 4 marks each.
(vi) Questions 20 to 26 in SectionC are Long Answer II Type Questions carrying 6 marks each.
(vii) Please write down the serial number of the Question before attempting it.
* Kindly update your browser if you are unable to view the equations.
(i) All questions are compulsory.
(ii) Please check that this Question Paper contains 26 Questions.
(iii) Marks for each question are indicated against it.
(iv) Questions 1 to 6 in SectionA are Very Short Answer Type Questions carrying one mark each.
(v) Questions 7 to 19 in SectionB are Long Answer I Type Questions carrying 4 marks each.
(vi) Questions 20 to 26 in SectionC are Long Answer II Type Questions carrying 6 marks each.
(vii) Please write down the serial number of the Question before attempting it.
* Kindly update your browser if you are unable to view the equations.
 Question 1
If $\mathrm{A}=\left[\begin{array}{ccc}5& 6& 3\\ 4& 3& 2\\ 4& 7& 3\end{array}\right]$, then write the cofactor of the element a_{21} of its 2^{nd} row. VIEW SOLUTION
 Question 2
Write the sum of the order and degree of the differential equation
${\left(\frac{{\mathrm{d}}^{2}\mathrm{y}}{{\mathrm{dx}}^{2}}\right)}^{2}+{\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)}^{3}+{\mathrm{x}}^{4}=0.$ VIEW SOLUTION
 Question 3
Write the solution of the differential equation
$\frac{\mathrm{dy}}{\mathrm{dx}}={2}^{\mathrm{y}}$ VIEW SOLUTION
 Question 4
Find the unit vector in the direction of the sum of the vectors $2\hat{i}+3\hat{j}\hat{k}\mathrm{and}4\hat{i}3\hat{j}+2\hat{k}.$ VIEW SOLUTION
 Question 5
Find the area of a parallelogram whose adjacent sides are represented by the vectors $2\hat{i}3\hat{k}\mathrm{and}4\hat{j}+2\hat{k}.$ VIEW SOLUTION
 Question 6
Find the sum of the intercepts cut off by the plane $2x+yz=5,$ on the coordinate axes. VIEW SOLUTION
 Question 7
Evaluate:
$\underset{\pi /2}{\overset{\pi /2}{\int}}\frac{\mathrm{cos}x}{1+{e}^{x}}dx$ VIEW SOLUTION
 Question 8
Three machines E_{1}, E_{2} and E_{3} in a certain factory producing electric bulbs, produce 50%, 25% and 25% respectively, of the total daily output of electric bulbs. It is known that 4% of the bulbs produced by each of machines E_{1} and E_{2} are defective and that 5% of those produced by machine E_{3} are defective. If one bulb is picked up at random from a day's production, calculate the probability that it is defective.
OR
Two numbers are selected at random (without replacement) from positive integers 2, 3, 4, 5, 6 and 7. Let X denote the larger of the two numbers obtained. Find the mean and variance of the probability distribution of X. VIEW SOLUTION
 Question 9
The two vectors $\hat{j}+\hat{k}\mathrm{and}3\hat{i}\hat{j}+4\hat{k}$ represent the two sides vectors $\overrightarrow{\mathrm{AB}}\mathrm{and}\overrightarrow{\mathrm{AC}}$ respectively of triangle ABC. Find the length of the median through A. VIEW SOLUTION
 Question 10
Find the equation of a plane which passes through the point (3, 2, 0) and contains the line $\frac{x3}{1}=\frac{y6}{5}=\frac{z4}{4}$. VIEW SOLUTION
 Question 11
If 2 tan^{−1} (cos θ) = tan^{−1} (2 cosec θ), (θ ≠ 0), then find the value of θ.
OR
If ${\mathrm{tan}}^{1}\left(\frac{1}{1+1.2}\right)+{\mathrm{tan}}^{1}\left(\frac{1}{1+2.3}\right)+...+{\mathrm{tan}}^{1}\left(\frac{1}{1+n.\left(n+1\right)}\right)={\mathrm{tan}}^{1}\mathrm{\theta}$, then find the value of θ. VIEW SOLUTION
 Question 12
If $\mathrm{A}=\left[\begin{array}{cc}2& 1\\ 1& 2\end{array}\right]$ and I is the identity matrix of order 2, then show that A^{2}= 4 A − 3 I. Hence find A^{−1}.OR
If $\mathrm{A}=\left[\begin{array}{cc}1& 1\\ 2& 1\end{array}\right]\mathrm{and}\mathrm{B}=\left[\begin{array}{cc}\mathrm{a}& 1\\ \mathrm{b}& 1\end{array}\right]\mathrm{and}{\left(\mathrm{A}+\mathrm{B}\right)}^{2}={\mathrm{A}}^{2}+{\mathrm{B}}^{2}$, then find the values of a and b. VIEW SOLUTION
 Question 13
Using properties of determinants, prove the following :
$\left\begin{array}{ccc}1& a& {a}^{2}\\ {a}^{2}& 1& a\\ a& {a}^{2}& 1\end{array}\right={\left(1{a}^{3}\right)}^{2}$ VIEW SOLUTION
 Question 14
Evaluate :
$\int \frac{\mathrm{sin}\left(xa\right)}{\mathrm{sin}\left(x+a\right)}dx$
OR
Evaluate :
$\int \frac{{x}^{2}}{\left({x}^{2}+4\right)\left({x}^{2}+9\right)}dx$ VIEW SOLUTION
 Question 15
Find whether the following function is differentiable at x = 1 and x = 2 or not :
$f\left(x\right)=\left\{\begin{array}{ccc}x,& & x1\\ 2x,& & 1\le x\le 2\\ 2+3x{x}^{2},& & x2\end{array}\right.$ VIEW SOLUTION
 Question 16
In a parliament election, a political party hired a public relations firm to promote its candidates in three ways − telephone, house calls and letters. The cost per contact (in paise) is given in matrix A as
$\mathrm{A}=\left[\begin{array}{c}140\\ 200\\ 150\end{array}\right]\begin{array}{c}\mathrm{Telephone}\\ \mathrm{House}\mathrm{Call}\\ \mathrm{Letters}\end{array}$
The number of contacts of each type made in two cities X and Y is given in the matrix B as
$\begin{array}{ccc}\mathrm{Telephone}& \mathrm{House}\mathrm{Call}& \mathrm{Letters}\end{array}\phantom{\rule{0ex}{0ex}}\mathrm{B}=\left[\begin{array}{ccc}1000& 500& 5000\\ 3000& 1000& 10000\end{array}\right]\begin{array}{c}\mathrm{City}\mathrm{X}\\ \mathrm{City}\mathrm{Y}\end{array}$
Find the total amount spent by the party in the two cities.
What should one consider before casting his/her vote − party's promotional activity or their social activities ? VIEW SOLUTION
 Question 17
 Question 18
Find the point on the curve 9y^{2} = x^{3}, where the normal to the curve makes equal intercepts on the axes. VIEW SOLUTION
 Question 19
$\mathrm{If}y={\left(x+\sqrt{1+{x}^{2}}\right)}^{n},\mathrm{then}\mathrm{show}\mathrm{that}\phantom{\rule{0ex}{0ex}}\left(1+{x}^{2}\right)\frac{{d}^{2}y}{d{x}^{2}}+x\frac{dy}{dx}={n}^{2}y.$ VIEW SOLUTION
 Question 20
Find the minimum value of (ax + by), where xy = c^{2}.
OR
Find the coordinates of a point of the parabola y = x^{2} + 7x + 2 which is closest to the straight line y = 3x − 3. VIEW SOLUTION
 Question 21
Maximise z = 8x + 9y subject to the constraints given below :
2x + 3y ≤ 6
3x − 2y ≤6
y ≤ 1
x, y ≥ 0 VIEW SOLUTION
 Question 22
Find the distance of the point (1, −2, 3) from the plane x − y + z = 5 measured parallel to the line whose direction cosines are proportional to 2, 3, −6. VIEW SOLUTION
 Question 23
Let f : N → ℝ be a function defined as f(x) = 4x^{2} + 12x + 15. Show that f : N → S, where S is the range of f, is invertible. Also find the inverse of f. VIEW SOLUTION
 Question 24
Using integration, find the area of the region bounded by the line x – y + 2 = 0, the curve x = $\sqrt{y}$ and yaxis. VIEW SOLUTION
 Question 25
Find the probability distribution of the number of doublets in four throws of a pair of dice. Also find the mean and variance of this distribution. VIEW SOLUTION
 Question 26
Solve the following differential equation :
$\left[yx\mathrm{cos}\left(\frac{y}{x}\right)\right]dy+\left[y\mathrm{cos}\left(\frac{y}{x}\right)2x\mathrm{sin}\left(\frac{y}{x}\right)\right]dx=0$ORSolve the following differential equation :
$\left(\sqrt{1+{x}^{2}+{y}^{2}+{x}^{2}{y}^{2}}\right)dx+xydy=0$ VIEW SOLUTION
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