Board Paper of Class 12-Humanities 2019 Math Delhi(Set 3) - 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 $3 A - B = [\begin{matrix} 5 & 0 \\ 1 & 1 \end{matrix}] and B = [\begin{matrix} 4 & 3 \\ 2 & 5 \end{matrix}]$ , then find the matrix A. VIEW SOLUTION

Question 2
Write the order and the degree of the following differential equation:
$x^{3} {(\frac{d^{2} y}{d x^{2}})}^{2} + x {(\frac{d y}{d x})}^{4} = 0$
VIEW SOLUTION

Question 3
If f(x) = x + 1, find $\frac{d}{d x} (f o f) (x)$ . 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
Find : ∫ sin x ⋅ log cos x dx VIEW SOLUTION

Question 6
Evaluate : $\int_{- π}^{π} (1 - x^{2}) \sin x \cos^{2} x d x .$

OR

Evaluate : $\int_{- 1}^{2} \frac{|x|}{x} d x .$ VIEW SOLUTION

Question 7
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 8
Find a matrix A such that 2A − 3B + 5C = O, where $B = [\begin{matrix} - 2 & 2 & 0 \\ 3 & 1 & 4 \end{matrix}] and C = [\begin{matrix} 2 & 0 & - 2 \\ 7 & 1 & 6 \end{matrix}] .$ VIEW SOLUTION

Question 9
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 10
Form the differential equation representing the family of curves y = e^2x (a + bx), where 'a' and 'b' are arbitrary constants. VIEW SOLUTION

Question 11
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 12
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 13
Using properties of determinants, prove the following:
$|\begin{matrix} a & b & c \\ a - b & b - c & c - a \\ b + c & c + a & a + b \end{matrix}| = a^{3} + b^{3} + c^{3} - 3 a b c .$ VIEW SOLUTION

Question 14
Solve: tan⁻¹ 4x + tan⁻¹ 6x = $\frac{π}{4}$ . VIEW SOLUTION

Question 15
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 16
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 17
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 18
If y = (sin⁻¹x)², prove that $(1 - x^{2}) \frac{d^{2} y}{d x^{2}} - x \frac{d y}{d x} - 2 = 0$ . VIEW SOLUTION

Question 19
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 20
Find: $\int \frac{\cos x}{(1 + \sin x) (2 + \sin x)} d x$ VIEW SOLUTION

Question 21
Solve the differential equation: $\frac{d y}{d x} - \frac{2 x}{1 + x^{2}} y = x^{2} + 2$
OR
Solve the differential equation: $(x + 1) \frac{d y}{d x} = 2 e^{- y} - 1; y (0) = 0$ . VIEW SOLUTION

Question 22
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 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
A tank with rectangular base and rectangular sides, open at the top is to be constructed so that its depth is 2 m and volume is 8 m³. If building of tank costs ₹ 70 per square metre for the base and ₹ 45 per square metre for the sides, what is the cost of least expensive tank? VIEW SOLUTION

Question 25
If $A = [\begin{matrix} 1 & 1 & 1 \\ 1 & 0 & 2 \\ 3 & 1 & 1 \end{matrix}],$ find A^–1. Hence, solve the system of equations x + y + z = 6, x + 2z = 7, 3x + y + z = 12.
OR

Find the inverse of the following matrix using elementary operations.
$A = [\begin{matrix} 1 & 2 & - 2 \\ - 1 & 3 & 0 \\ 0 & - 2 & 1 \end{matrix}]$ VIEW SOLUTION

Question 26
Prove that the curves y² = 4x and x² = 4y divide the area of the square bounded by sides x = 0, x = 4, y = 4 and y = 0 into three equal parts.
OR
Using integration, find the area of the triangle whose vertices are (2, 3), (3, 5) and (4, 4). VIEW SOLUTION

Question 27
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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