Question Paper-Delhi (2013)

General Instructions:

1. All questions are compulsory.

1. The question paper consists of 29 questions divided into three Sections A, B and C, Section A comprises of 10 questions of one mark each, Section B comprises of 12 questions of four marks each and Section C comprises of 7 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.
2. There is no overall choice. However, internal choice has been provided in 4 questions of four marks each and 2 questions of six marks each. You have to attempt only one of the alternatives in all such questions.
3. Use of calculators is not permitted.

SECTION-A

Question numbers 1 to 10 carry 1 mark each.

Q1. Write the principal value of <math>\tan^{-1}(1) + \cos^{-1}\left(-\frac{1}{2}\right)</math>

Q2. Write the value of <math>\tan \left(2 \tan^{-1} \frac{1}{5}\right)</math>.

Q3. Find the value of a if <math display="block">\begin{bmatrix} a-b & 2a+c \\ 2a-b & 3c+d \end{bmatrix} = \begin{bmatrix} -1 & 5 \\ 0 & 13 \end{bmatrix}</math>

Q4. If <math>\begin{vmatrix} x+1 & x-1 \\ x-3 & x+3 \end{vmatrix} = \begin{vmatrix} 4 & -1 \\ 1 & 3 \end{vmatrix}</math>, then write the value of x.

Q5. If <math>\begin{vmatrix} 9 & -1 & 4 \\ -2 & 1 & 3 \end{vmatrix} = A + \begin{vmatrix} 1 & 2 & -1 \\ 0 & 4 & 9 \end{vmatrix}</math>, then find the matrix A.

Q6. Write the degree of the differential equation <math>x^3 \left( \frac{d^2 y}{dx^2} \right)^2 + x \left( \frac{dy}{dx} \right)^4 = 0</math>.

Q7. If <math>\vec{a} = x\hat{i} + 2\hat{j} - z\hat{k}</math> and <math>\vec{b} = 3\hat{i} - y\hat{j} + \hat{k}</math> are two equal vectors, then write the value of <math>x + y + z</math>.

Q8. If a unit vector <math>\vec{a}</math> makes angles <math>\frac{\pi}{3}</math> with <math>\hat{i}, \frac{\pi}{4}</math>, with <math>\hat{j}</math> and an acute angle <math>\theta</math> with <math>\hat{k}</math>, then find the value of <math>\theta</math>.

Q9. Find the Cartesian equation of the line which passes through the point (-2, 4, -5) and is parallel to the line <math>\frac{x+3}{3} = \frac{4-y}{5} = \frac{z+8}{6}</math>.

Q10. The amount of pollution content added in air in a city due to x-diesel vehicles is given by <math>P(x) = 0.005x^3 + 0.02x^2 + 30x</math>. Find the marginal increase in pollution content when 3 diesel vehicles are added and write which value is indicated in the above question.

SECTION-B

Question numbers 11 to 22 carry 4 marks each.

Q11. Show that the function f in A = IR <math>-\left\{\frac{2}{3}\right\}</math> defined as <math>f(x) = \frac{4x+3}{6x-4}</math> is one-one and onto. Hence find <math>f^{-1}</math>.

Q12. Find the value of the following:

<math>\tan \frac{1}{2} \left| \sin^{-1} \frac{2x}{1+x^2} + \cos^{-1} \frac{1-y^2}{1+y^2} \right|, |x| < 1, y > 0 \text{ and } xy < 1.</math> OR Prove that: <math>\tan^{-1} \left( \frac{1}{2} \right) + \tan^{-1} \left( \frac{1}{5} \right) + \tan^{-1} \left( \frac{1}{8} \right) = \frac{\pi}{4}</math>

Q13. Using properties of determinants, prove the following

<math display="block">\begin{vmatrix} 1 & x & x^2 \\ x^2 & 1 & x \\ x & x^2 & 1 \end{vmatrix} = (1 - x^3)^2.</math>

Q14. Differentiate the following function with respect to x: <math>(\log x)^x + x^{\log x}</math>

Q15. If <math>y = \log[x + \sqrt{x^2 + a^2}]</math>, show that <math>(x^2 + a^2) \frac{d^2y}{dx^2} + x \frac{dy}{dx} = 0</math>. 2