Board Paper of Class 12Commerce 2018 Maths  Solutions
General Instructions:
(i) All questions are compulsory.
(ii) This question paper contains 29 questions.
(iii) Questions 1 4 in Section A are very shortanswer type questions carrying 1 mark each.
(iv) Questions 512 in Section B are shortanswer type questions carrying 2 marks each.
(v) Questions 1323 in Section C are longanswer I type questions carrying 4 marks each.
(vi) Questions 2429 in Section D are longanswer II type questions carrying 6 marks each.
(i) All questions are compulsory.
(ii) This question paper contains 29 questions.
(iii) Questions 1 4 in Section A are very shortanswer type questions carrying 1 mark each.
(iv) Questions 512 in Section B are shortanswer type questions carrying 2 marks each.
(v) Questions 1323 in Section C are longanswer I type questions carrying 4 marks each.
(vi) Questions 2429 in Section D are longanswer II type questions carrying 6 marks each.
 Question 1
Find the magnitude of each of two vectors $\overrightarrow{a}$ and $\overrightarrow{b}$, having the same magnitude such that the angle between them is 60° and their scalar product is $\frac{9}{2}$. VIEW SOLUTION
 Question 2
Find the value of ${\mathrm{tan}}^{1}\sqrt{3}{\mathrm{cot}}^{1}\left(\sqrt{3}\right).$ VIEW SOLUTION
 Question 3
If a * b denotes the larger of 'a' and 'b' and if $a\circ b$ = (a * b) + 3, then write the value of $\left(5\right)\circ \left(10\right)$, where * and $\circ $ are binary operations. VIEW SOLUTION
 Question 4
If the matrix $A=\left[\begin{array}{ccc}0& a& 3\\ 2& 0& 1\\ b& 1& 0\end{array}\right]$ is skew symmetric, find the values of 'a' and 'b'. VIEW SOLUTION
 Question 5
A black and a red die are rolled together. Find the conditional probability of obtaining the sum 8, given that the red die resulted in a number less than 4. VIEW SOLUTION
 Question 6
If θ is the angle between two vectors $\hat{i}2\hat{j}+3\hat{k}\mathrm{and}3\hat{i}2\hat{j}+\hat{k}$, find sin θ. VIEW SOLUTION
 Question 7
Find the differential equation representing the family of curves y = a e^{bx+5}, where a and b are arbitrary constants. VIEW SOLUTION
 Question 8
Evaluate: $\int \frac{\mathrm{cos}2x+2{\mathrm{sin}}^{2}x}{{\mathrm{cos}}^{2}x}dx$ VIEW SOLUTION
 Question 9
The total cost C(x) associated with the production of x units of an item is given by C(x) = 0.005x^{3} – 0.02x^{2} + 30x + 5000. Find the marginal cost when 3 units are produced, where by marginal cost we mean the instantaneous rate of change of total cost at any level of output. VIEW SOLUTION
 Question 10
Differentiate ${\mathrm{tan}}^{1}\left(\frac{1+\mathrm{cos}x}{\mathrm{sin}x}\right)$ with respect to x. VIEW SOLUTION
 Question 11
Give $A=\left[\begin{array}{cc}2& 3\\ 4& 7\end{array}\right]$, compute A^{–1} and show that 2A^{–1} = 9I – A. VIEW SOLUTION
 Question 12
Prove that :
$3{\mathrm{sin}}^{1}x={\mathrm{sin}}^{1}\left(3x4{x}^{3}\right),x\in \left[\frac{1}{2},\frac{1}{2}\right]$ VIEW SOLUTION
 Question 13
Two numbers are selected at random (without replacement) from the first five positive integers. Let X denote the larger of the two numbers obtained. Find the mean and variance of X. VIEW SOLUTION
 Question 14
An open tank with a square base and vertical sides is to be constructed from a metal sheet so as to hold a given quantity of water. Show that the cost of material will be least when depth of the tank is half of its width. If the cost is to be borne by nearby settled lower income families, for whom water will be provided, what kind of value is hidden in this question? VIEW SOLUTION
 Question 15
Find the equations of the tangent and the normal, to the curve 16x^{2} + 9y^{2}^{ }= 145 at the point (x_{1}, y_{1}), where x_{1} = 2 and y_{1} > 0.
OR
Find the intervals in which the function $f\left(x\right)=\frac{{x}^{4}}{4}{x}^{3}5{x}^{2}+24x+12$ is (a) strictly increasing, (b) strictly decreasing.
VIEW SOLUTION
 Question 16
If ${\left({x}^{2}+{y}^{2}\right)}^{2}=xy,\mathrm{find}\frac{dy}{dx}$.ORIf x = a (2θ – sin 2θ) and y = a (1 – cos 2θ), find $\frac{dy}{dx}$ when $\mathrm{\theta}=\frac{\pi}{3}$. VIEW SOLUTION
 Question 17
If y = sin (sin x), prove that $\frac{{d}^{2}y}{d{x}^{2}}+\mathrm{tan}x\frac{dy}{dx}+y{\mathrm{cos}}^{2}x=0.$ VIEW SOLUTION
 Question 18
Find the particular solution of the differential equation e^{x} tan y dx + (2 – e^{x}) sec^{2} y dy = 0, give that $y=\frac{\pi}{4}$ when x = 0.
OR
Find the particular solution of the differential equation $\frac{dy}{dx}+2y\mathrm{tan}x=\mathrm{sin}x,$ given that y = 0 when $x=\frac{\pi}{3}$. VIEW SOLUTION
 Question 19
Find the shortest distance between the lines $\overrightarrow{r}=\left(4\hat{i}\hat{j}\right)+\lambda \left(\hat{i}+2\hat{j}3\hat{k}\right)$ and $\overrightarrow{r}=\left(\hat{i}\hat{j}+2\hat{k}\right)+\mu \left(2\hat{i}+4\hat{j}5\hat{k}\right).$ VIEW SOLUTION
 Question 20
Find :
$\int \frac{2\mathrm{cos}x}{\left(1\mathrm{sin}x\right)\left(1+{\mathrm{sin}}^{2}x\right)}dx$ VIEW SOLUTION
 Question 21
Suppose a girl throws a die. If she gets 1 or 2, she tosses a coin three times and notes the number of tails. If she gets 3, 4, 5 or 6, she tosses a coin once and notes whether a 'head' or 'tail' is obtained. If she obtained exactly one 'tail', what is the probability that she threw 3, 4, 5 or 6 with the die?VIEW SOLUTION
 Question 22
Let $\overrightarrow{a}=4\hat{i}+5\hat{j}\hat{k},\overrightarrow{b}=\hat{i}4\hat{j}+5\hat{k}\mathrm{and}\overrightarrow{c}=3\hat{i}+\hat{j}\hat{k}.$ Find a vector $\overrightarrow{d}$ which is perpendicular to both $\overrightarrow{c}\mathrm{and}\overrightarrow{b}\mathrm{and}\overrightarrow{d}\xb7\overrightarrow{a}=21$. VIEW SOLUTION
 Question 23
Using properties of determinants, prove that
$\left\begin{array}{ccc}1& 1& 1+3x\\ 1+3y& 1& 1\\ 1& 1+3z& 1\end{array}\right=9\left(3xyz+xy+yz+zx\right)$ VIEW SOLUTION
 Question 24
Using integration, find the area of the region in the first quadrant enclosed by the xaxis, the line y = x and the circle x^{2} + y^{2} = 32. VIEW SOLUTION
 Question 25
Let A = {x ∈ Z : 0 ≤ x ≤ 12}. Show that
R = {(a, b) : a, b ∈ A, a – b is divisible by 4}is an equivalence relation. Find the set of all elements related to 1. Also write the equivalence class [2].
OR
Show that the function f : ℝ → ℝ defined by $f\left(x\right)=\frac{x}{{x}^{2}+1},\forall x\in \mathrm{\mathbb{R}}$ is neither oneone nor onto. Also, if g : ℝ → ℝ is defined as g(x) = 2x – 1, find fog(x). VIEW SOLUTION
 Question 26
Find the distance of the point (–1, –5, –10) from the point of intersection of the line $\overrightarrow{r}=2\hat{i}\hat{j}+2k+\lambda \left(3\hat{i}+4\hat{j}+2\hat{k}\right)$ and the plane $\overrightarrow{r}\xb7\left(\hat{i}\hat{j}+\hat{k}\right)=5$. VIEW SOLUTION
 Question 27
A factory manufactures two types of screws A and B, each type requiring the use of two machines, an automatic and a handoperated. It takes 4 minutes on the automatic and 6 minutes on the handoperated machines to manufacture a packet of screws 'A' while it takes 6 minutes on the automatic and 3 minutes on the handoperated machine to manufacture a packet of screws 'B'. Each machine is available for at most 4 hours on any day. The manufacturer can sell a packet of screws 'A' at a profit of 70 paise and screws 'B' at a profit of Rs 1. Assuming that he can sell all the screws he manufactures, how many packets of each type should the factory owner produce in a day in order to maximize his profit? Formulate the above LPP and solve it graphically and find the maximum profit. VIEW SOLUTION
 Question 28
Evaluate :
$\underset{0}{\overset{\pi /4}{\int}}\frac{\mathrm{sin}x+\mathrm{cos}x}{16+9\mathrm{sin}2x}dx$
OR
Evaluate :
$\underset{1}{\overset{3}{\int}}\left({x}^{2}+3x+{e}^{x}\right)dx,$
as the limit of the sum. VIEW SOLUTION
 Question 29
If $A=\left[\begin{array}{ccc}2& 3& 5\\ 3& 2& 4\\ 1& 1& 2\end{array}\right]$, find A^{–1}. Use it to solve the system of equations
2x – 3y + 5z = 11
3x + 2y – 4z = –5
x + y – 2z = –3.
OR
Using elementary row transformations, find the inverse of the matrix $A=\left[\begin{array}{ccc}1& 2& 3\\ 2& 5& 7\\ 2& 4& 5\end{array}\right]$. VIEW SOLUTION
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