Board Paper of Class 12 2019 Mathematics - Solutions

Note:
(1) All questions are compulsory.
(2) Figures to the right indicate full marks.
(3) The Question paper consists of 30 questions divided into FOUR sections A, B, C, D.

• Section A contains 6 questions of 1 mark each.
• Section B contains 8 questions of 2 marks each.(One of them has internal option)
• Section C contains 6 questions of 3 marks each.(Two of them have internal options)
• Section D contains 10 questions of 4 marks each.(Three of them has internal options)

(4) For each MCQ, correct answer must be written along with its alphabet, e.g., (A) ……. / (B) ……. / (C) ……. / (D) ……. etc.

In case of MCQs, (Q. No. 1 to 6) evaluation would be done for the first attempt only.

(5) Use of logarithmic table is allowed. Use of calculator is not allowed.
(6) In L.P.P. only rough sketch of graph is expected. Graph paper is not necessary.
(7) Start each section on new page only.

Question 1
The principal solutions of cot $x = - \sqrt{3}$ are ________.

(A) $\frac{π}{6}, \frac{5 π}{6}$

(B) $\frac{5 π}{6}, \frac{7 π}{6}$

(C) $\frac{5 π}{6}, \frac{11 π}{6}$

(D) $\frac{π}{6}, \frac{11 π}{6}$ VIEW SOLUTION

Question 2
The acute angle between the two planes x + y + 2z = 3 and 3x – 2y + 2z = 7 is _______.

(A) $\sin^{- 1} (\frac{5}{\sqrt{102}})$

(B) $\cos^{- 1} (\frac{5}{\sqrt{102}})$

(C) $\sin^{- 1} (\frac{15}{\sqrt{102}})$

(D) $\cos^{- 1} (\frac{15}{\sqrt{102}})$ VIEW SOLUTION

Question 3
The direction ratios of the line which is perpendicular to the lines with direction ratios –1, 2, 2 and 0, 2, 1 are ________.
(A) –2, –1, –2
(B) 2, 1, 2
(C) 2, –1, – 2
(D) –2, 1, –2 VIEW SOLUTION

Question 4
If $f (x) = {(1 + 2 x)}^{\frac{1}{x}},$ for x ≠ 0 is continuous at x = 0, then f(0) =________.

(A) e
(B) e²
(C) 0
(D) 2 VIEW SOLUTION

Question 5
$\int \frac{d x}{9 x^{2} + 1} =___________.$

(A) $\frac{1}{3} \tan^{- 1} (2 x) + c$

(B) $\frac{1}{3} \tan^{- 1} x + c$

(C) $\frac{1}{3} \tan^{- 1} (3 x) + c$

(D) $\frac{1}{3} \tan^{- 1} (6 x) + c$ VIEW SOLUTION

Question 6
If y = ae^5x + be^–5x, then the differential equation is________.

(A) $\frac{d^{2} y}{d x^{2}} = 25 y$

(B) $\frac{d^{2} y}{d x^{2}} = - 25 y$

(C) $\frac{d^{2} y}{d x^{2}} = - 5 y$

(D) $\frac{d^{2} y}{d x^{2}} = 5 y$ VIEW SOLUTION

Question 7
Write the truth values of the following statements:

i. 2 is a rational number and $\sqrt{2}$ is an irrational number.

ii. 2 + 3 = 5 or $\sqrt{2} + \sqrt{3} = \sqrt{5}$ VIEW SOLUTION

Question 8
Find the volume of the parallelopiped, if the coterminous edges are given by the vectors $2 \hat{i} + 5 \hat{j} - 4 \hat{k}, 5 \hat{i} + 7 \hat{j} + 5 \hat{k}, 4 \hat{i} + 5 \hat{j} - 2 \hat{k}$ .
OR

Find the value of p, if the vectors $\hat{i} - 2 \hat{j} + \hat{k}, 2 \hat{i} - 5 \hat{j} + p \hat{k}$ and $5 \hat{i} - 9 \hat{j} + 4 \hat{k}$ are coplanar. VIEW SOLUTION

Question 9
Show that the points A(–7, 4, –2), B(–2, 1, 0) and C (3, –2, 2) are collinear. VIEW SOLUTION

Question 10
Write the equation of the plane 3x + 4y – 2z = 5 in the vector form VIEW SOLUTION

Question 11
If y = x^x , find $\frac{d y}{d x} .$ VIEW SOLUTION

Question 12
Find the equation of tangent to the curve y = x² + 4x + 1 at (–1, –2). VIEW SOLUTION

Question 13
Evaluate: $\int \frac{e^{x} (1 + x)}{\cos^{2} (x e^{x})} d x$ VIEW SOLUTION

Question 14
Evaluate: $\int_{0}^{\frac{π}{2}} \sin^{2} x d x$ VIEW SOLUTION

Question 15
In ΔABC, prove that

$\sin (\frac{B - C}{2}) = (\frac{b - c}{a}) \cos (\frac{A}{2})$
OR

Show that $\sin^{- 1} (\frac{5}{13}) + \cos^{- 1} (\frac{3}{5}) = \tan^{- 1} (\frac{63}{16})$ VIEW SOLUTION

Question 16
If $A (\bar{a})$ and $B (\bar{b})$ are any two points in the space and $R (\bar{r})$ be a point on the line segment AB dividing it internally in the ratio m : n, then prove that $\bar{r} = \frac{m \bar{b} + n \bar{a}}{m + n}$ VIEW SOLUTION

Question 17
The equation of a line is 2x – 2 = 3y + 1 = 6z – 2, find its direction ratios and also find the vector equation of the line. VIEW SOLUTION

Question 18
Discuss the continuity of the function

$\begin{matrix} f (x) = \frac{\log (2 + x) - \log (2 - x)}{\tan x}, & for x \neq 0 \\ = 1 & for x = 0 \end{matrix}$

at the point x = 0 VIEW SOLUTION

Question 19
The probability distribution of a random variable X, the number of defects per 10 meters of fabric is given by

x 0 1 2 3 4

P (X = x) 0.45 0.35 0.15 0.03 0.02

Find the variance of X.
OR

For the following probability density function (p.d.f.) of X, find:
(i) P (X < 1),
(ii) P ( | X | < 1)

$\begin{matrix} if f (x) = \frac{x^{2}}{18}, & - 3 < x < 3 \\ 0, & otherwise \end{matrix}$ VIEW SOLUTION

Question 20
Given is X ~ B (n, p).
If E (X) = 6, Var. (X) = 4.2, find n and p. VIEW SOLUTION

Question 21
Find the symbolic form of the given switching circuit. Construct its switching table and interpret your result.
VIEW SOLUTION

Question 22
If three numbers are added, their sum is 2. If two times the second number is substracted from the sum of first and third numbers we get 8 and if three times the first number is added to the sum of second and third numbers we get 4. Find the numbers using matrices. VIEW SOLUTION

Question 23
In ∆ABC, with usual notations prove that

${(a - b)}^{2} \cos^{2} (\frac{C}{2}) + {(a + b)}^{2} \sin^{2} (\frac{C}{2}) = c^{2} .$
OR

In ∆ABC, with usual notations prove that b² = c² + a² – 2ca cos B VIEW SOLUTION

Question 24
Find ‘p’ and ‘q’ if the equation px² – 8xy + 3y² + 14x + 2y + q = 0 represents a pair of perpendicular lines. VIEW SOLUTION

Question 25
Maximize:

z = 3x + 5y Subject to

x + 4y ≤ 24, 3x + y ≤ 21,

x + y ≤ 9, x ≥ 0, y ≥ 0

VIEW SOLUTION

Question 26
If x = f(t) and y = g(t) are differentiable functions of t, then prove that y is a differentiable function of x and

$\frac{d y}{d x} = \frac{\frac{d y}{d t}}{\frac{d x}{d t}}, where \frac{d x}{d t} \neq 0$

Hence find $\frac{d y}{d x}$ if x = a cos² t and y = a sin² t. VIEW SOLUTION

Question 27
A rod of 108 meters long is bent to form a rectangle. Find its dimensions if the area is maximum.
OR

f(x) = (x – 1) (x – 2)(x – 3), x ∈[0, 4], find ‘c’ if LMVT can be applied. VIEW SOLUTION

Question 28
prove that : $\int \frac{d x}{\sqrt{x^{2} + a^{2}}} = \log |x + \sqrt{x^{2} + a^{2}}| + c$ VIEW SOLUTION

Question 29
Show that: $\int_{0}^{\frac{π}{4}} \log (1 + \tan x) d x = \frac{π}{8} \log 2$ VIEW SOLUTION

Question 30
Solve the differential equation:

$\frac{d y}{d x} + y \sec x = \tan x$
OR

Solve the differential equation: $(x + y) \frac{d y}{d x} = 1$ VIEW SOLUTION

z = 3x + 5y		Subject to
x + 4y ≤ 24,		3x + y ≤ 21,
x + y ≤ 9,		x ≥ 0, y ≥ 0

x	0	1	2	3	4
P (X = x)	0.45	0.35	0.15	0.03	0.02