Board Paper of Class 12Humanities 2016 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
Write the number of vectors of unit length perpendicular to both the vectors $\overrightarrow{a}=2\hat{i}+\hat{j}+2\hat{k}\mathrm{and}\overrightarrow{b}=\hat{j}+\hat{k}$. VIEW SOLUTION
 Question 2
Write the number of all possible matrices of order 2 × 2 with each entry 1, 2 or 3. VIEW SOLUTION
 Question 3
If x ∈ N and $\left\begin{array}{cc}x+3& 2\\ 3x& 2x\end{array}\right$ = 8, then find the value of x. VIEW SOLUTION
 Question 4
Write the position vector of the point which divides the join of points with position vectors $3\overrightarrow{a}2\overrightarrow{b}\mathrm{and}2\overrightarrow{a}+3\overrightarrow{b}$ in the ratio 2 : 1. VIEW SOLUTION
 Question 5
Find the vector equation of the plane with intercepts 3, –4 and 2 on x, y and zaxis respectively. VIEW SOLUTION
 Question 6
Use elementary column operation C_{2} → C_{2} + 2C_{1} in the following matrix equation :
$\left(\begin{array}{cc}2& 1\\ 2& 0\end{array}\right)=\left(\begin{array}{cc}3& 1\\ 2& 0\end{array}\right)\left(\begin{array}{cc}1& 0\\ 1& 1\end{array}\right)$ VIEW SOLUTION
 Question 7
The equation of tangent at (2, 3) on the curve y^{2} = ax^{3} + b is y = 4x – 5. Find the values of a and b. VIEW SOLUTION
 Question 8
Find the coordinates of the point where the line through the points A(3, 4, 1) and B(5, 1, 6) crosses the XZ plane. Also find the angle which this line makes with the XZ plane. VIEW SOLUTION
 Question 9
Find : $\int \left(3x+1\right)\sqrt{43x2{x}^{2}}dx$ VIEW SOLUTION
 Question 10
The two adjacent sides of a parallelogram are $2\hat{i}4\hat{j}5\hat{k}\mathrm{and}2\hat{i}+2\hat{j}+3\hat{k}$. Find the two unit vectors parallel to its diagonals. Using the diagonal vectors, find the area of the parallelogram. VIEW SOLUTION
 Question 11
Form the differential equation of the family of circles in the second quadrant and touching the coordinate axes. VIEW SOLUTION
 Question 12
In a game, a man wins Rs 5 for getting a number greater than 4 and loses Rs 1 otherwise, when a fair die is thrown. The man decided to thrown a die thrice but to quit as and when he gets a number greater than 4. Find the expected value of the amount he wins/loses.
OR
A bag contains 4 balls. Two balls are drawn at random (without replacement) and are found to be white. What is the probability that all balls in the bag are white? VIEW SOLUTION
 Question 13
A trust invested some money in two type of bonds. The first bond pays 10% interest and second bond pays 12% interest. The trust received Rs 2,800 as interest. However, if trust had interchanged money in bonds, they would have got Rs 100 less as interest. Using matrix method, find the amount invested by the trust. Interest received on this amount will be given to Helpage India as donation. Which value is reflected in this question? VIEW SOLUTION
 Question 14
Differentiate ${x}^{\mathrm{sin}x}+{\left(\mathrm{sin}x\right)}^{\mathrm{cos}x}$ with respect to x.
ORIf $y=2\mathrm{cos}\left(\mathrm{log}x\right)+3\mathrm{sin}\left(\mathrm{log}x\right)$, prove that ${x}^{2}\frac{{\mathrm{d}}^{2}y}{\mathrm{d}{x}^{2}}+x\frac{\mathrm{d}y}{\mathrm{d}x}+y=0$. VIEW SOLUTION
 Question 15
Solve the equation for $x:{\mathrm{sin}}^{1}x+{\mathrm{sin}}^{1}\left(1x\right)={\mathrm{cos}}^{1}x$If ${\mathrm{cos}}^{1}\frac{x}{a}+{\mathrm{cos}}^{1}\frac{y}{\mathrm{b}}=\alpha $, prove that $\frac{{x}^{2}}{{a}^{2}}2\frac{xy}{\mathrm{ab}}\mathrm{cos}\alpha +\frac{{y}^{2}}{{\mathrm{b}}^{2}}={\mathrm{sin}}^{2}\alpha $ VIEW SOLUTION
OR
 Question 16
If $x=a\mathrm{sin}2\mathrm{t}\left(1+\mathrm{cos}2\mathrm{t}\right)\mathrm{and}y=\mathrm{b}\mathrm{cos}2\mathrm{t}\left(1\mathrm{cos}2\mathrm{t}\right),\mathrm{find}\frac{\mathrm{d}y}{\mathrm{d}x}\mathrm{at}\mathrm{t}=\frac{\mathrm{\pi}}{4}$. VIEW SOLUTION
 Question 17
Solve the differential equation :
$y+x\frac{\mathrm{d}y}{\mathrm{d}x}=xy\frac{\mathrm{d}y}{\mathrm{d}x}$ VIEW SOLUTION
 Question 18
Evaluate : $\underset{0}{\overset{\frac{\mathrm{\pi}}{2}}{\int}}\frac{{\mathrm{sin}}^{2}x}{\mathrm{sin}x+\mathrm{cos}x}\mathrm{d}x$
OR
Evaluate : $\underset{0}{\overset{\frac{3}{2}}{\int}}\leftx\mathrm{cos\pi}x\right\mathrm{d}x$ VIEW SOLUTION
 Question 19
Find : $\int \frac{{x}^{2}}{{x}^{4}+{x}^{2}2}\mathrm{d}x$ VIEW SOLUTION
 Question 20
Using properties of determinants, show that ΔABC is isosceles if:
$\left\begin{array}{ccc}1& 1& 1\\ 1+\mathrm{cosA}& 1+\mathrm{cosB}& 1+\mathrm{cosC}\\ {\mathrm{cos}}^{2}\mathrm{A}+\mathrm{cosA}& {\mathrm{cos}}^{2}\mathrm{B}+\mathrm{cosB}& {\mathrm{cos}}^{2}\mathrm{C}+\mathrm{cosC}\end{array}\right=0$
OR
A shopkeeper has 3 varieties of pens 'A', 'B' and 'C'. Meenu purchased 1 pen of each variety for a total of Rs 21. Jeevan purchased 4 pens of 'A' variety 3 pens of 'B' variety and 2 pens of 'C' variety for Rs 60. While Shikha purchased 6 pens of 'A' variety, 2 pens of 'B' variety and 3 pens of 'C' variety for Rs 70. Using matrix method, find cost of each variety of pen. VIEW SOLUTION
 Question 21
There are two types of fertilisers 'A' and 'B'. 'A' consists of 12% nitrogen and 5% phosphoric acid whereas 'B' consists of 4% nitrogen and 5% phosphoric acid. After testing the soil conditions, farmer finds that he needs at least 12 kg of nitrogen and 12 kg of phosphoric acid for his crops. If 'A' costs Rs 10 per kg and 'B' cost Rs 8 per kg, then graphically determine how much of each type of fertiliser should be used so that nutrient requirements are met at a minimum cost. VIEW SOLUTION
 Question 22
Prove that the least perimeter of an isosceles triangle in which a circle of radius r can be inscribed is $6\sqrt{3}$r.
OR
If the sum of lengths of hypotenuse and a side of a right angled triangle is given, show that area of triangle is maximum, when the angle between them is $\frac{\mathrm{\pi}}{3}.$ VIEW SOLUTION
 Question 23
Five bad oranges are accidently mixed with 20 good ones. If four oranges are drawn one by one successively with replacement, then find the probability distribution of number of bad oranges drawn. Hence find the mean and variance of the distribution. VIEW SOLUTION
 Question 24
Prove that the curves y^{2} = 4x and x^{2} = 4y divide the area of square bounded by x = 0, x = 4, y = 4 and y = 0 into three equal parts. VIEW SOLUTION
 Question 25
Show that the binary operation * on A = R – { – 1} defined as a*b = a + b + ab for all a, b ∈ A is commutative and associative on A. Also find the identity element of * in A and prove that every element of A is invertible. VIEW SOLUTION
 Question 26
Find the position vector of the foot of perpendicular and the perpendicular distance from the point P with position vector $2\hat{i}+3\hat{j}+4\hat{k}$ to the plane $\overrightarrow{r}.\left(2\hat{i}+\hat{j}+3\hat{k}\right)26=0$. Also find image of P in the plane. VIEW SOLUTION
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