Mathematics 2019 Delhi Set-1

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, 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 calculators is not permitted. You may ask logarithmic tables, if required.

Question 1

If A and B are square matrices of the same order 3, such that |A| = 2 and AB = 21, write the value of [B].

SOLUTION:

We have, AB = 21

<math>|AB| = |2I|</math>

<math>\Rightarrow |\mathbf{A}| |\mathbf{B}| = 8</math>

(Given <math>|A| = 2</math>)

<math>\Rightarrow 2|\mathbf{B}| = 8</math>

<math>\Rightarrow |B| = 4</math>

Question 2

If <math>f(x) = x + 1</math>, find <math>\frac{d}{dx}(f \circ f)(x)</math>.

SOLUTION:

Given: <math>f(x) = x + 1</math>

<math>fof(x) = (x+1) + 1 = x + 2</math>

<math>\frac{d}{dx}(fof)(x) = \frac{d}{dx}(x+2) = 1</math>

Question 3

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\}^{\frac{1}{4}}</math>.

SOLUTION:

The highest order derivative present in the given differential equations is <math>\frac{d^2y}{dx^2}</math>, so its order is 2. It is a polynomial <math>\frac{d^2y}{dx^2}</math> and <math>\frac{dy}{dx}</math> and the highest power raised to <math>\frac{d^2y}{dx^2}</math> is 1, so its degree is 1.

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 <math>2\hat{i} + 2\hat{j} - 3\hat{k}</math>.

SOLUTION:

A line makes <math>90^{\circ}</math>, <math>135^{\circ}</math>, <math>45^{\circ}</math> with x, y and z axes respectively. Therefore, Direction cosines of the line are <math>\cos 90^\circ</math>, <math>\cos 135^\circ</math> and <math>\cos 45^\circ</math> <math>\Rightarrow</math> Direction cosines of the line are <math>0, -\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}</math>

OR

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

<math>\overrightarrow{r} = 3\hat{\mathbf{i}} + 4\hat{\mathbf{j}} + 5\hat{\mathbf{k}} + \mu \left(2\hat{\mathbf{i}} + 2\hat{\mathbf{j}} - 3\hat{\mathbf{k}}\right)</math>

Question 5

Examine whether the operation *defined on R by <math>a * b = ab + 1</math> is (i) a binary or not. (ii) if a binary operation, is it associative or not?