Board Paper of Class 12-Science 2019 Math All India(Set 1) - Solutions

General Instructions:

(i) All questions are compulsory.

(ii) This question paper contains 29 questions divided into four sections A, B, C and D. Section A comprises of 4 questions of one mark each, Section B comprises of 8 questions of two marks each, Section C comprises of 11 questions of four marks each and Section D comprises of 6 questions of six marks each.

(iii) All questions in Section A are to be answered in one word, one sentence or as per the exact requirement of the question.

(iv) There is no overall choice. However, internal choice has been provided in 1 question of Section A, 3 questions of Section B, 3 questions of Section C and 3 questions of Section D. You have to attempt only one of the alternatives in all such questions.

(v) Use of calculators is not permitted. You may ask logarithmic tables, if required.

Question 1
If A is a square matrix of order 3 with $|A| = 4$ , then write the value of $|- 2 A|$ . VIEW SOLUTION

Question 2
If $y = \sin^{- 1} x + \cos^{- 1} x$ , find $\frac{d y}{d x}$ . VIEW SOLUTION

Question 3
Write the order and the degree of the differential equation ${(\frac{d^{4} y}{d x^{4}})}^{2} = {[x + {(\frac{d y}{d x})}^{2}]}^{3}$ . VIEW SOLUTION

Question 4
If a line has the direction ratios −18, 12, −4, then what are its direction cosines?
OR
Find the cartesian equation of the line which passes through the point (−2, 4, −5) and is parallel to the line $\frac{x + 3}{3} = \frac{4 - y}{5} = \frac{z + 8}{6}$ . VIEW SOLUTION

Question 5

If * is defined on the set ℝ of all real numbers by * : $a * b = \sqrt{a^{2} + b^{2}}$ , find the identity element, if it exists in ℝ with respect to *.
VIEW SOLUTION

Question 6
If $A = [\begin{matrix} 0 & 2 \\ 3 & - 4 \end{matrix}]$ and $k A = [\begin{matrix} 0 & 3 a \\ 2 b & 24 \end{matrix}]$ , then find the values of k, a and b. VIEW SOLUTION

Question 7
Find: $\int \frac{\sin x - \cos x}{\sqrt{1 + \sin 2 x}} d x, 0 < x < π / 2$ VIEW SOLUTION

Question 8
Find: $\int \frac{\sin (x - a)}{\sin (x + a)} d x$

OR

Find: $\int {(\log x)}^{2} d x$ VIEW SOLUTION

Question 9
Form the differential equation representing the family of curves y² = m (a² − x²) by eliminating the arbitrary constants 'm' and 'a'. VIEW SOLUTION

Question 10
Find a unit vector perpendicular to both the vectors $\vec{a}$ and $\vec{b}$ , where $\vec{a} = \hat{i} - 7 \hat{j} + 7 \hat{k}$ and $\vec{b} = 3 \hat{i} - 2 \hat{j} + 2 \hat{k}$ .
OR
Show that the vectors $\hat{i} - 2 \hat{j} + 3 \hat{k}, - 2 \hat{i} + 3 \hat{j} - 4 \hat{k}$ and $\hat{i} - 3 \hat{j} + 5 \hat{k}$ are coplanar. VIEW SOLUTION

Question 11
Mother, father and son line up at random for a family photo. If A and B are two events given by A = Son on one end, B = Father in the middle, find P(B/A). VIEW SOLUTION

Question 12
Let X be a random variable which assumes values x₁, x₂, x₃, x₄ such that 2P(X = x₁) = 3P(X = x₂) = P(X = x₃) = 5P(X = x₄).
Find the probability distribution of X.
OR
A coin is tossed 5 times. Find the probability of getting (i) at least 4 heads, and (ii) at most 4 heads. VIEW SOLUTION

Question 13
Show that the relation R on the set Z of all integers, given by R = {(a, b) : 2 divides (a − b)} is an equivalence relation.
OR
If $f (x) = \frac{4 x + 3}{6 x - 4}, x \neq \frac{2}{3},$ show that fof(x) = x for all $x \neq \frac{2}{3}$ . Also, find the inverse of f. VIEW SOLUTION

Question 14
If $\tan^{- 1} x - \cot^{- 1} x = \tan^{- 1} (\frac{1}{\sqrt{3}}), x > 0$ , find the value of x and hence find the value of $\sec^{- 1} (\frac{2}{x})$ . VIEW SOLUTION

Question 15
Using properties of determinants, prove that
$|\begin{matrix} b + c & a & a \\ b & c + a & b \\ c & c & a + b \end{matrix}| = 4 a b c$
VIEW SOLUTION

Question 16
If sin y = x sin (a + y), prove that $\frac{d y}{d x} = \frac{\sin^{2} (a + y)}{\sin a}$
OR
If (sin x)^y = x + y, find $\frac{d y}{d x}$ . VIEW SOLUTION

Question 17
If y = (sec⁻¹ x)², x > 0, show that $x^{2} (x^{2} - 1) \frac{d^{2} y}{d x^{2}} + (2 x^{3} - x) \frac{d y}{d x} - 2 = 0$ VIEW SOLUTION

Question 18
Find the equations of the tangent and the normal to the curve $y = \frac{(x - 7)}{(x - 2) (x - 3)}$ at the point where it cuts the x-axis. VIEW SOLUTION

Question 19
Find: $\int \frac{\sin 2 x}{(\sin^{2} x + 1) (\sin^{2} x + 3)} d x$ VIEW SOLUTION

Question 20
Prove that $\int_{a}^{b} f (x) d x = \int_{a}^{b} f (a + b - x) d x$ and hence evaluate $\int_{\frac{π}{6}}^{\frac{π}{3}} \frac{d x}{1 + \sqrt{\tan x}}$ . VIEW SOLUTION

Question 21
Solve the differential equation:
$\frac{d y}{d x} = \frac{x + y}{x - y}$

OR

Solve the differential equation:
(1 + x²) dy + 2xy dx = cot x dx

VIEW SOLUTION

Question 22
Let $\vec{a}$ , $\vec{b}$ and $\vec{c}$ be three vectors such that $|\vec{a}| = 1$ , $|\vec{b}| = 2$ and $|\vec{c}| = 3$ . If the projection of $\vec{b}$ along $\vec{a}$ is equal to the projection of $\vec{c}$ along $\vec{a}$ ; and $\vec{b}$ , $\vec{c}$ are perpendicular to each other, then find $|3 \vec{a} - 2 \vec{b} + 2 \vec{c}|$ . VIEW SOLUTION

Question 23
Find the value of λ for which the following lines are perpendicular to each other:
$\frac{x - 5}{5 λ + 2} = \frac{2 - y}{5} = \frac{1 - z}{- 1}; \frac{x}{1} = \frac{y + \frac{1}{2}}{2 λ} = \frac{z - 1}{3}$
Hence, find whether the lines intersect or not. VIEW SOLUTION

Question 24
If $A = [\begin{matrix} 1 & 1 & 1 \\ 0 & 1 & 3 \\ 1 & - 2 & 1 \end{matrix}]$ , find A⁻¹.
Hence, solve the following system of equations:
x + y + z = 6,
y + 3z = 11
and x − 2y + z = 0
OR
Find the inverse of the following matrix, using elementary transformations:
$A = [\begin{matrix} 2 & 3 & 1 \\ 2 & 4 & 1 \\ 3 & 7 & 2 \end{matrix}]$ VIEW SOLUTION

Question 25
Show that the height of a cylinder, which is open at the top, having a given surface area and greatest volume, is equal to the radius of its base. VIEW SOLUTION

Question 26
Find the area of the triangle whose vertices are (−1, 1), (0, 5) and (3, 2), using integration.
OR
Find the area of the region bounded by the curves (x −1)² + y² = 1 and x² + y² = 1, using integration. VIEW SOLUTION

Question 27
Find the vector and cartesian equations of the plane passing through the points (2, 5, −3), (−2, −3, 5) and (5, 3, −3). Also, find the point of intersection of this plane with the line passing through the points (3, 1, 5) and (−1, −3, −1).
OR
Find the equation of the plane passing through the intersection of the planes $\vec{r} \cdot (\hat{i} + \hat{j} + \hat{k}) = 1$ and $\vec{r} \cdot (2 \hat{i} + 3 \hat{j} - \hat{k}) + 4 = 0$ and parallel to x-axis. Hence, find the distance of the plane from x-axis. VIEW SOLUTION

Question 28
There are two boxes I and II. Box I contains 3 red and 6 Black balls. Box II contains 5 red and 'n' black balls. One of the two boxes, box I and box II is selected at random and a ball is drawn at random. The ball drawn is found to be red. If the probability that this red ball comes out from box II is $\frac{3}{5}$ , find the value of 'n'. VIEW SOLUTION

Question 29
A company manufactures two types of novelty souvenirs made of plywood. Souvenirs of type A require 5 minutes each for cutting and 10 minutes each for assembling. Souvenirs of type B require 8 minutes each for cutting and 8 minutes each for assembling. Their are 3 hours and 20 minutes available for cutting and 4 hours available for assembling. The profit is ₹ 50 each for type A and ₹ 60 each for type B souvenirs. How many souvenirs of each type should the company manufacture in order to maximize profit? Formulate the above LPP and solve it graphically and also find the maximum profit. VIEW SOLUTION

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