Board Paper of Class 12-Science 2019 Math Delhi(Set 2) - 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
Find the order and the degree of the differential equation $x^{2} \frac{d^{2} y}{d x^{2}} = {\{1 + {(\frac{d y}{d x})}^{2}\}}^{4}$ . VIEW SOLUTION

Question 2
If f(x) = x + 7 and g(x) = x − 7, x ∊ R, then find $\frac{d}{d x} (f o g) (x)$ . VIEW SOLUTION

Question 3
Find the value of x − y, if
$2 [\begin{matrix} 1 & 3 \\ 0 & x \end{matrix}] + [\begin{matrix} y & 0 \\ 1 & 2 \end{matrix}] = [\begin{matrix} 5 & 6 \\ 1 & 8 \end{matrix}]$ . VIEW SOLUTION

Question 4
If a line makes angles 90°, 135°, 45° with the x, y and z axes respectively, find its direction cosines.
OR
Find the vector equation of the line which passes through the point (3, 4, 5) and is parallel to the vector $2 \hat{i} + 2 \hat{j} - 3 \hat{k}$ . VIEW SOLUTION

Question 5
Examine whether the operation *defined on R by a * b = ab + 1 is (i) a binary or not. (ii) if a binary operation, is it associative or not ? VIEW SOLUTION

Question 6
If $A = [\begin{matrix} 2 & 0 & 1 \\ 2 & 1 & 3 \\ 1 & - 1 & 0 \end{matrix}]$ , then find (A² − 5A). VIEW SOLUTION

Question 7
$Find : \int \sqrt{1 - \sin 2 x} d x, \frac{π}{4} < x < \frac{π}{2}$

OR

Find : ∫ sin⁻¹ (2x) dx. VIEW SOLUTION

Question 8
Form the differential equation representing the family of curves y = e^2x (a + bx), where 'a' and 'b' are arbitrary constants. VIEW SOLUTION

Question 9
A die is thrown 6 times. If "getting an odd number" is a "success", what is the probability of (i) 5 successes? (ii) atmost 5 successes?
OR
The random variable X has a probability distribution P(X) of the following form, where 'k' is some number.
$P (X = x) = \{\begin{cases} k & , & if x = 0 \\ 2 k & , & if x = 1 \\ 3 k & , & if x = 2 \\ 0 & , & otherwise \end{cases}$
Determine the value of 'k'. VIEW SOLUTION

Question 10
A die marked 1, 2, 3 in red and 4, 5, 6 in green is tossed. Let A be the event "number is even" and B be the event "number is marked red". Find whether the events A and B are independent or not. VIEW SOLUTION

Question 11
If the sum of two unit vectors is a unit vector, prove that the magnitude of their difference is $\sqrt{3}$ .
OR
If $\vec{a} = 2 \hat{i} + 3 \hat{j} + \hat{k}, \vec{b} = \hat{i} - 2 \hat{j} + \hat{k} and \vec{c} = - 3 \hat{i} + \hat{j} + 2 \hat{k}, find [\vec{a} \vec{b} \vec{c}]$ . VIEW SOLUTION

Question 12
Find: $\int \frac{\tan^{2} x \sec^{2} x}{1 - \tan^{6} x} d x$ . VIEW SOLUTION

Question 13
Solve for x : $\tan^{- 1} (2 x) + \tan^{- 1} (3 x) = \frac{π}{4}$ . VIEW SOLUTION

Question 14
If log (x² + y²) = $2 \tan^{- 1} (\frac{y}{x})$ , show that $\frac{d y}{d x} = \frac{x + y}{x - y}$ .
OR
If x^y − y^x = a^b, find $\frac{d y}{d x}$ . VIEW SOLUTION

Question 15
Find: $\int \frac{3 x + 5}{x^{2} + 3 x - 18} d x$ . VIEW SOLUTION

Question 16
Prove that $\int_{0}^{a} f (x) d x = \int_{0}^{a} f (a - x) d x$ , hence evaluate $\int_{0}^{π} \frac{x \sin x}{1 + \cos^{2} x} d x$ . VIEW SOLUTION

Question 17
If $\hat{i} + \hat{j} + \hat{k}, 2 \hat{i} + 5 \hat{j}, 3 \hat{i} + 2 \hat{j} - 3 \hat{k}$ and $\hat{i} - 6 \hat{j} - \hat{k}$ respectively are the position vectors A, B, C and D, then find the angle between the straight lines AB and CD. Find whether $\vec{AB}$ and $\vec{CD}$ are collinear or not. VIEW SOLUTION

Question 18
Using properties of determinants, prove the following:
$|\begin{matrix} a + b + c & - c & - b \\ - c & a + b + c & - a \\ - b & - a & a + b + c \end{matrix}| = 2 (a + b) (b + c) (c + a)$ . VIEW SOLUTION

Question 19
If $x = \cos t + \log \tan (\frac{t}{2})$ , y = sin t, then find the values of $\frac{d^{2} y}{d t^{2}}$ and $\frac{d^{2} y}{d x^{2}}$ at $t = \frac{π}{4}$ . VIEW SOLUTION

Question 20
Show that the relation R on ℝ defined as R = {(a, b) : a ≤ b}, is reflexive, and transitive but not symmetric.
OR
Prove that the function f : N → N, defined by f(x) = x² + x + 1 is one-one but not onto. Find inverse of f : N → S, where S is range of f. VIEW SOLUTION

Question 21
Find the equation of tangent to the curve $y = \sqrt{3 x - 2}$ which is parallel to the line 4x − 2y + 5 = 0. Also, write the equation of normal to the curve at the point of contact. VIEW SOLUTION

Question 22
Solve the differential equation: x dy − y dx = $\sqrt{x^{2} + y^{2}} d x$ , given that y = 0 when x = 1.
OR
Solve the differential equation: $(1 + x^{2}) \frac{d y}{d x} + 2 x y - 4 x^{2} = 0$ , subject to the initial condition y(0) = 0. VIEW SOLUTION

Question 23
Find the value of λ, so that the lines $\frac{1 - x}{3} = \frac{7 y - 14}{λ} = \frac{z - 3}{2}$ and $\frac{7 - 7 x}{3 λ} = \frac{y - 5}{1} = \frac{6 - z}{5}$ are at right angles. Also, find whether the lines are intersecting or not. VIEW SOLUTION

Question 24
Show that the altitude of the right circular cone of maximum volume that can be inscribed in a sphere of radius r is $\frac{4 r}{3}$ . Also find the maximum volume of cone. VIEW SOLUTION

Question 25
If $A = [\begin{matrix} 2 & - 3 & 5 \\ 3 & 2 & - 4 \\ 1 & 1 & - 2 \end{matrix}]$ , then find A⁻¹. Hence solve the following system of equations:
2x − 3y + 5z = 11, 3x + 2y − 4z = −5, x + y − 2z = −3.
OR
Obtain the inverse of the following matrix using elementary operations:
$A = [\begin{matrix} - 1 & 1 & 2 \\ 1 & 2 & 3 \\ 3 & 1 & 1 \end{matrix}]$ VIEW SOLUTION

Question 26
A manufacture has three machine operators A, B and C. The first operator A produces 1% of defective items, whereas the other two operators B and C produces 5% and 7% defective items respectively. A is on the job for 50% of the time, B on the job 30% of the time and C on the job for 20% of the time. All the items are put into one stockpile and then one item is chosen at random from this and is found to be defective. What is the probability that it was produced by A? VIEW SOLUTION

Question 27
Find the vector and Cartesian equations of the plane passing through the points (2, 2 –1), (3, 4, 2) and (7, 0, 6). Also find the vector equation of a plane passing through (4, 3, 1) and parallel to the plane obtained above.
OR
Find the vector equation of the plane that contains the lines $\vec{r} = (\hat{i} + \hat{j}) + λ (\hat{i} + 2 \hat{j} - \hat{k})$ and the point (–1, 3, –4). Also, find the length of the perpendicular drawn from the point (2, 1, 4) to the plane thus obtained. VIEW SOLUTION

Question 28
Using integration, find the area of triangle ABC, whose vertices are A(2, 5), B(4, 7) and C(6, 2).
OR
Find the area of the region lying above x-axis and included between the circle x² + y²= 8x and inside of the parabola $y^{2} = 4 x$ . VIEW SOLUTION

Question 29
A manufacturer has employed 5 skilled men and 10 semi-skilled men and makes two models A and B of an article. The making of one item of model A requires 2 hours work by a skilled man and 2 hours work by a semi-skilled man. One item of model B requires 1 hour by a skilled man and 3 hours by a semi-skilled man. No man is expected to work more than 8 hours per day. The manufacturer's profit on an item of model A is ₹ 15 and on an item of model B is ₹ 10. How many of items of each model should be made per day in order to maximize daily profit? Formulate the above LPP and solve it graphically and find the maximum profit. VIEW SOLUTION

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