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# Board Paper of Class 12-Commerce 2018 Maths (SET 1) - Solutions

General Instructions:
(i) All questions are compulsory.
(ii) This question paper contains 29 questions.
(iii) Questions 1- 4 in Section A are very short-answer type questions carrying 1 mark each.
(iv) Questions 5-12 in Section B are short-answer type questions carrying 2 marks each.
(v) Questions 13-23 in Section C are long-answer I type questions carrying 4 marks each.
(vi) Questions 24-29 in Section D are long-answer II type questions carrying 6 marks each.

• Question 1
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 2
Find the magnitude of each of two vectors $\stackrel{\to }{a}$ and $\stackrel{\to }{b}$, having the same magnitude such that the angle between them is 60° and their scalar product is $\frac{9}{2}$. VIEW SOLUTION

• Question 3
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 4
Find the value of ${\mathrm{tan}}^{-1}\sqrt{3}-{\mathrm{cot}}^{-1}\left(-\sqrt{3}\right).$ VIEW SOLUTION

• Question 5
The total cost C(x) associated with the production of x units of an item is given by C(x) = 0.005x3 – 0.02x2 + 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 7
Give , compute A–1 and show that 2A–1 = 9I – A. VIEW SOLUTION

• Question 9
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 10
If θ is the angle between two vectors , find sin θ. VIEW SOLUTION

• Question 11
Find the differential equation representing the family of curves y = a ebx+5, where a and b are arbitrary constants. VIEW SOLUTION

• Question 14
Find the particular solution of the differential equation ex tan y dx + (2 – ex) sec2 y dy = 0, give that $y=\frac{\pi }{4}$ when x = 0.

OR

Find the particular solution of the differential equation given that y = 0 when $x=\frac{\pi }{3}$. VIEW SOLUTION

• Question 15
Find the shortest distance between the lines $\stackrel{\to }{r}=\left(4\stackrel{^}{i}-\stackrel{^}{j}\right)+\lambda \left(\stackrel{^}{i}+2\stackrel{^}{j}-3\stackrel{^}{k}\right)$ and $\stackrel{\to }{r}=\left(\stackrel{^}{i}-\stackrel{^}{j}+2\stackrel{^}{k}\right)+µ\left(2\stackrel{^}{i}+4\stackrel{^}{j}-5\stackrel{^}{k}\right).$ VIEW SOLUTION

• Question 16
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 17
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 18

Find the equations of the tangent and the normal, to the curve 16x2 + 9y2 = 145 at the point (x1, y1), where x1 = 2 and y1 > 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 20
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 21
Let Find a vector $\stackrel{\to }{d}$ which is perpendicular to both . VIEW SOLUTION

• Question 22
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 23
If .
OR
If x = a (2θ – sin 2θ) and y = a (1 – cos 2θ), find $\frac{dy}{dx}$ when $\mathrm{\theta }=\frac{\pi }{3}$. VIEW SOLUTION

• Question 24
Evaluate :

OR

Evaluate :
$\underset{1}{\overset{3}{\int }}\left({x}^{2}+3x+{e}^{x}\right)dx,$
as the limit of the sum. VIEW SOLUTION

• Question 25
A factory manufactures two types of screws A and B, each type requiring the use of two machines, an automatic and a hand-operated. It takes 4 minutes on the automatic and 6 minutes on the hand-operated machines to manufacture a packet of screws 'A' while it takes 6 minutes on the automatic and 3 minutes on the hand-operated 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 26
Let A = {x ∈ Z : 0 ≤ x ≤ 12}. Show that

R = {(a, b) : a, b ∈ A, |ab| 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 $f\left(x\right)=\frac{x}{{x}^{2}+1},\forall x\in \mathrm{ℝ}$ is neither one-one nor onto. Also, if g : ℝ → ℝ is defined as g(x) = 2x – 1, find fog(x). VIEW SOLUTION

• Question 27
Using integration, 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

• Question 28
If , find A–1. Use it to solve the system of equations

2x – 3y + 5z = 11

3x + 2y – 4z = –5

xy – 2z = –3.

OR

Using elementary row transformations, find the inverse of the matrix . VIEW SOLUTION

• Question 29
Find the distance of the point (–1, –5, –10) from the point of intersection of the line and the plane $\stackrel{\to }{r}·\left(\stackrel{^}{i}-\stackrel{^}{j}+\stackrel{^}{k}\right)=5$. VIEW SOLUTION
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