SET-2

EXAM DATE: 18/03/2019

Code No. 65/1/2

Class XII Mathematics (CBSE 2019)

Time: 3 Hrs.

Max. Marks: 100

GENERAL INSTRUCTIONS:

1. All questions are compulsory.

1. 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.

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

1. There is no overall choice. However, an 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.

1. Use of calculator is not permitted. You may ask logarithmic tables, if required.

Section-A

1. Find the order and the degree of the differential equation <math>x^2 \frac{d^2y}{dx^2} = \left\{1 + \left(\frac{dy}{dx}\right)^2\right\}^4</math>.

$[1]$

1. If <math>f(x) = x + 7</math> and <math>g(x) = x - 7</math>, <math>x \in R</math>, then find <math>\frac{d}{dx}(f \circ g)(x)</math>.

$[1]$

1. Find the value of x - y, if

<math display="block">2\begin{bmatrix} 1 & 3 \\ 0 & x \end{bmatrix} + \begin{bmatrix} y & 0 \\ 1 & 2 \end{bmatrix} = \begin{bmatrix} 5 & 6 \\ 1 & 8 \end{bmatrix}.</math> [1]

1. If a line makes angles 90°, 135°, 45° with the x, y and z axes respectively, find its direction cosines. [1]

Find the vector equation of the line which passes through the point (3, 4, 5) and is parallel to the vector <math>2\hat{i} + 2\hat{j} - 3\hat{k}</math>. OR

Section-B

1. 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? [2]
2. If <math>A = \begin{bmatrix} 2 & 0 & 1 \\ 2 & 1 & 3 \\ 1 & -1 & 0 \end{bmatrix}</math>, then find <math>(A^2 - 5A)</math>.

$[2]$

7. Find: <math>\int \sqrt{1-\sin 2x} dx</math>, <math>\frac{\pi}{4} < x < \frac{\pi}{2}</math>

[2] OR Find: <math>\int \sin^{-1}(2x) dx</math>.

1. Form the differential equation representing the family of curves <math>y = e^{2x}</math> (a + bx), where 'a' and 'b' are arbitrary constants. [2]
2. 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? [2]

OR The random variable X has a probability distribution P(X) of the following form, where 'k' is some number.

<math display="block">P(X = x) = \begin{cases} k, & \text{if } x = 0 \\ 2k, & \text{if } x = 1 \\ 3k, & \text{if } x = 2 \end{cases}</math> 0, otherwise

Determine the value of 'k'.

1. 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. [2]

1. If the sum of two unit vectors is a unit vector, prove that the magnitude of their difference is <math>\sqrt{3}</math>. [2]

OR If <math>\vec{a} = 2\hat{i} + 3\hat{j} + \hat{k}</math>, <math>\vec{b} = \hat{i} - 2\hat{j} + \hat{k}</math> and <math>\vec{c} = -3\hat{i} + \hat{j} + 2\hat{k}</math>, find <math>\vec{a}\vec{b}\vec{c}</math>.

12. Find: <math display="block">\int \frac{\tan^2 x \sec^2 x}{1 - \tan^6 x} dx.</math>

$[2]$

Section-C

13. Solve for x : <math>tan^{-1}(2x) + tan^{-1}(3x) = \frac{\pi}{4}</math>.

$[4]$

14. If <math>\log(x^2 + y^2) = 2\tan^{-1}\left(\frac{y}{x}\right)</math>, show that <math>\frac{dy}{dx} = \frac{x+y}{x-y}</math>.

$[4]$