estion Paper-Delhi (2012)

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

Questions numbers 1 to 10 carry 1 mark each.

Q1. If a line has direction ratios <math>2, -1, -2</math>, then what are its direction cosines? 1

Q2. Find '<math>\lambda</math>' when the projection of <math>\vec{a} = \lambda \hat{i} + \hat{j} + 4\hat{k}</math> on <math>\vec{b} = 2\hat{i} + 6\hat{j} + 3\hat{k}</math> is 4 units. 1

Q3. Find the sum of the vectors <math>\vec{a} = \hat{i} - 2\hat{j} + \hat{k}</math>, <math>\vec{b} = -2\hat{i} + 4\hat{j} + 5\hat{k}</math> and <math>\vec{c} = \hat{i} - 6\hat{j} - 7\hat{k}</math>. 1

Q4. Evaluate : <math display="block">\int_{-\infty}^{3} \frac{1}{x} dx</math> 1

Q5. Evaluate <math>\int (1-x)\sqrt{x} dx</math>. 1

Q6. If <math>\Delta = \begin{bmatrix} 5 & 3 & 8 \\ 2 & 0 & 1 \\ 1 & 2 & 3 \end{bmatrix}</math>, write the minor of the element <math>a_{23}</math>. 1

Q7. If <math>\begin{pmatrix} 2 & 3 \\ 5 & 7 \end{pmatrix} \begin{pmatrix} 1 & -3 \\ -2 & 4 \end{pmatrix} = \begin{pmatrix} -4 & 6 \\ -9 & x \end{pmatrix}</math>, write the value of x. 1

Q8. Simplify: <math>\cos \theta \begin{bmatrix} \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{bmatrix} + \sin \theta \begin{bmatrix} \sin \theta & -\cos \theta \\ \cos \theta & \sin \theta \end{bmatrix}</math> 1

Q9. Write the principal value of <math>\cos^{-1}\left(\frac{1}{2}\right) - 2\sin^{-1}\left(-\frac{1}{2}\right)</math>. 1

1 Q10. Let * be a 'binary' operation on N given by a * b = LCM (a, b) for all <math>a, b \in \mathbb{N}</math>. Find 5 * 7.

SECTION-B

Question numbers 11 to 22 carry 4 mark each.

4

Q11. If <math>(\cos x)^y = (\cos y)^x</math>, find <math>\frac{dy}{dx}</math>. OR

If <math>\sin y = x \sin (a + y)</math>, prove that <math>\frac{dy}{dx} = \frac{\sin^2(a + y)}{\sin a}</math>.

Q12. How many times must a man toss a fair coin, so that the probability of having at least one head is more than 80%? 4

Q13. Find the Vector and Cartesian equations of the line passing through the point (1, 2, -4) and perpendicular to the two lines <math>\frac{x-8}{3} = \frac{y+19}{-16} = \frac{z-10}{7}</math> and <math>\frac{x-15}{3} = \frac{y-29}{9} = \frac{z-5}{-5}</math>. 4

Q14. If <math>\vec{a}, \vec{b}, \vec{c}</math> are three vectors such that <math>|\vec{a}| = 5</math>, <math>|\vec{b}| = 12</math> and <math>|\vec{c}| = 13</math>, and <math>\vec{a} + \vec{b} + \vec{c} = \vec{0}</math>, find the value of <math>\vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} + \vec{c} \cdot \vec{a}</math>. 4

Q15. Solve the following differential equation : 4

<math>2x^2 \frac{dy}{dx} - 2xy + y^2 = 0.</math> 4 Q16. Find the particular solution of the following differential equation. <math>\frac{dy}{dx} = 1 + x^2 + y^2 + x^2 y^2</math>, given that <math>y = 1</math> where <math>x = 0</math>.

Q17. Evluate : <math>\int \sin x \sin 2x \sin 3x \, dx</math>

OR

Evaluate: <math display="block">\int \frac{2}{(1-x)(1+x^2)} dx</math>

Q18. Find the point on the curve <math>y = x^3 - 11x + 5</math> at which the equation of tangent is <math>y = x - 11</math>. OR Using differentials, find the approximate value of <math>\sqrt{49.5}</math>. 4 4 Q19. If <math>y = (\tan^{-1} x)^2</math>, show that

<math>(x^2+1)^2 \frac{d^2y}{dx^2} + 2x(x^2+1) \frac{dy}{dx} = 2</math>. 6 Q20. Using properties of determinants, prove that

<math display="block">\begin{vmatrix} b+c & q+r & y+z \\ c+a & r+p & z+x \\ a+b & p+q & x+y \end{vmatrix} = 2 \begin{vmatrix} a & p & x \\ b & q & y \\ c & r & z \end{vmatrix}</math> 2