Board Paper of Class 12Science 2016 Maths Abroad(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 $\mathrm{A}=\left(\begin{array}{cc}3& 5\\ 7& 9\end{array}\right)$ is written as A = P + Q, where P is a symmetric matrix and Q is skew symmetric matrix, then write the matrix P. VIEW SOLUTION
 Question 2
If $\overrightarrow{\mathrm{a}},\overrightarrow{\mathrm{b}},\overrightarrow{\mathrm{c}}$ are unit vectors such that $\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{c}}=\overrightarrow{\mathrm{0}},$ then write the value of $\overrightarrow{\mathrm{a}}\xb7\overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{b}}\xb7\overrightarrow{\mathrm{c}}+\overrightarrow{\mathrm{c}}\xb7\overrightarrow{\mathrm{a}}.$ VIEW SOLUTION
 Question 3
If ${\left\overrightarrow{\mathrm{a}}\times \overrightarrow{\mathrm{b}}\right}^{2}+{\left\overrightarrow{\mathrm{a}}\xb7\overrightarrow{\mathrm{b}}\right}^{2}=400$ and $\left\overrightarrow{\mathrm{a}}\right=5,$ then write the value of $\left\overrightarrow{\mathrm{b}}\right.$ VIEW SOLUTION
 Question 4
Write the equation of a plane which is at a distance of $5\sqrt{3}$ units from origin and the normal to which is equally inclined to coordinate axes. VIEW SOLUTION
 Question 5
If $\left(213\right)\left(\begin{array}{ccc}1& 0& 1\\ 1& 1& 0\\ 0& 1& 1\end{array}\right)\left(\begin{array}{c}1\\ 0\\ 1\end{array}\right)=\mathrm{A}$, then write the order of matrix A. VIEW SOLUTION
 Question 6
If $\left\begin{array}{ccc}x& \mathrm{sin}\mathrm{\theta}& \mathrm{cos}\mathrm{\theta}\\ \mathrm{sin}\mathrm{\theta}& x& 1\\ \mathrm{cos}\mathrm{\theta}& 1& x\end{array}\right=8$, write the value of x. VIEW SOLUTION
 Question 7
Find the values of a and b, if the function f defined by $f\left(x\right)=\left\{\begin{array}{ccc}{x}^{2}+3x+a& ,& x\u2a7d1\\ bx+2& ,& x>1\end{array}\right.$ is differentiable at x = 1. VIEW SOLUTION
 Question 8
Differentiate ${\mathrm{tan}}^{1}\left(\frac{\sqrt{1+{x}^{2}}1}{x}\right)\mathrm{w}.\mathrm{r}.\mathrm{t}.{\mathrm{sin}}^{1}\frac{2x}{1+{x}^{2}},$if x ∈ (–1, 1)
OR
If x = sin t and y = sin pt, prove that $\left(1{x}^{2}\right)\frac{{d}^{2}y}{d{x}^{2}}x\frac{dy}{dx}+{p}^{2}y=0$ VIEW SOLUTION
 Question 9
Find the angle of intersection of the curves ${y}^{2}=4ax\mathrm{and}{x}^{2}=4by$. VIEW SOLUTION
 Question 10
Evaluate :$\underset{0}{\overset{\pi}{\int}}\frac{x}{1+\mathrm{sin}\alpha \mathrm{sin}x}dx$ VIEW SOLUTION
 Question 11
Find : $\int \left(2x+5\right)\sqrt{104x3{x}^{2}}dx$OR
Find : $\int \frac{\left({x}^{2}+1\right)\left({x}^{2}+4\right)}{\left({x}^{2}+3\right)\left({x}^{2}5\right)}dx$ VIEW SOLUTION
 Question 12
Find : $\int \frac{x{\mathrm{sin}}^{1}x}{\sqrt{1{x}^{2}}}dx$ VIEW SOLUTION
 Question 13
Solve the following differential equation :
${y}^{2}dx+\left({x}^{2}xy+{y}^{2}\right)dy=0$ VIEW SOLUTION
 Question 14
Solve the following differential equation:
$\left({\mathrm{cot}}^{1}y+x\right)dy=\left(1+{y}^{2}\right)dx$ VIEW SOLUTION
 Question 15
If $\overrightarrow{a}\times \overrightarrow{b}=\overrightarrow{c}\times \overrightarrow{d}\mathrm{and}\overrightarrow{a}\times \overrightarrow{c}=\overrightarrow{b}\times \overrightarrow{d}$, show that $\overrightarrow{a}\overrightarrow{d}$ is parallel to $\overrightarrow{b}\overrightarrow{c}$, where $\overrightarrow{a}\ne \overrightarrow{d}\mathrm{and}\overrightarrow{b}\ne \overrightarrow{c}$. VIEW SOLUTION
 Question 16
Prove that the line through A(0, –1, –1) and B(4, 5, 1) intersects the line through C(3, 9, 4) and D(–4, 4, 4). VIEW SOLUTION
 Question 17
A box has 20 pens of which 2 are defective. Calculate the probability that out of 5 pens drawn one by one with replacement, at most 2 are defective.
OR
Let, X denote the number of colleges where you will apply after your results and P(X = x) denotes your probability of getting admission in x number of colleges. It is given that
$\mathrm{P}\left(\mathrm{X}=x\right)=\left\{\begin{array}{lll}kx& ,& \mathrm{if}x=0\mathrm{or}1\\ 2kx& ,& \mathrm{if}x=2\\ k\left(5x\right)& ,& \mathrm{if}x=3\mathrm{or}4\\ 0& ,& \mathrm{if}x4\end{array}\right.$
where k is a positive constant. Find the value of k. Also find the probability that you will get admission in (i) exactly one college (ii) at most 2 colleges (iii) at least 2 colleges. VIEW SOLUTION
 Question 18
Prove that :
${\mathrm{cot}}^{1}\frac{\sqrt{1+\mathrm{sin}x}+\sqrt{1\mathrm{sin}x}}{\sqrt{1+\mathrm{sin}x}\sqrt{1\mathrm{sin}x}}=\frac{\mathrm{x}}{2},0x\frac{\mathrm{\pi}}{2}$
OR
Solve for x :
${\mathrm{tan}}^{1}\left(\frac{x2}{x1}\right)+{\mathrm{tan}}^{1}\left(\frac{x+2}{x+1}\right)=\frac{\mathrm{\pi}}{4}$ VIEW SOLUTION
 Question 19
A coaching institute of English (subject) conducts classes in two batches I and II and fees for rich and poor children are different. In batch I, it has 20 poor and 5 rich children and total monthly collection is Rs 9,000, whereas in batch II, it has 5 poor and 25 rich children and total monthly collection is Rs 26,000. Using matrix method, find monthly fees paid by each child of two types. What values the coaching institute is inculcating in the society? VIEW SOLUTION
 Question 20
Using integration find the area of the region bounded by the curves $y=\sqrt{4{x}^{2}},{x}^{2}+{y}^{2}4x=0$ and the xaxis. VIEW SOLUTION
 Question 21
Find the equation of the plane which contains the line of intersection of the planes $x+2y+3z4=0\mathrm{and}2x+yz+5=0$ and whose xintercept is twice its zintercept. VIEW SOLUTION
 Question 22
Bag A contains 3 red and 5 black balls, while bag B contains 4 red and 4 black balls. Two balls are transferred at random from bag A to bag B and then a ball is drawn from bag B at random. If the ball drawn from bag B is found to be red find the probability that two red balls were transferred from A to B. VIEW SOLUTION
 Question 23
In order to supplement daily diet, a person wishes to take X and Y tablets. The contents (in milligrams per tablet) of iron, calcium and vitamins in X and Y are given as below :
Tablets Iron Calcium Vitamin X 6 3 2 Y 2 3 4
The person needs to supplement at least 18 milligrams of iron, 21 milligrams of calcium and 16 milligrams of vitamins. The price of each tablet of X and Y is Rs 2 and Rs 1 respectively. How many tablets of each type should the person take in order to satisfy the above requirement at the minimum cost? Make an LPP and solve graphically. VIEW SOLUTION
 Question 24
If f, g : R → R be two functions defined as f(x) = x + x and g(x) = x –$x,\forall x\in \mathrm{R}$. Then find fog and gof. Hence find fog(–3), fog(5) and gof (–2). VIEW SOLUTION
 Question 25
If a, b and c are all nonzero and $\left\begin{array}{ccc}1+\mathrm{a}& 1& 1\\ 1& 1+\mathrm{b}& 1\\ 1& 1& 1+\mathrm{c}\end{array}\right=0,$ then prove that $\frac{1}{\mathrm{a}}+\frac{1}{\mathrm{b}}+\frac{1}{\mathrm{c}}+1=0$
OR
If $\mathrm{A}=\left(\begin{array}{ccc}\mathrm{cos}\alpha & \mathrm{sin}\alpha & 0\\ \mathrm{sin}\alpha & \mathrm{cos}\alpha & 0\\ 0& 0& 1\end{array}\right),$ find adj·A and verify that A(adj·A) = (adj·A)A = A I_{3}. VIEW SOLUTION
 Question 26
The sum of the surface areas of a cuboid with sides x, 2x and $\frac{x}{3}$ and a sphere is given to be constant. Prove that the sum of their volumes is minimum, if x is equal to three times the radius of sphere. Also find the minimum value of the sum of their volumes.
OR
Find the equation of tangents to the curve y = cos(x + y), –2π ≤ x ≤ 2π that are parallel to the line x + 2y = 0. VIEW SOLUTION
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