Time: 3 Hrs.
Max. Marks: 80
General instructions:
- Section A:
Q. 1 contains Eight multiple choice type ofquestions, each carrying Two marks.
Q. 2 contains Four very short answer type questions, each carrying one mark. - Section B:
Q. 3 to Q. 14 contain Twelve short answer type questions, each carryingTwo marks. (Attempt any Eight) - Section C:
Q. 15 to Q. 26 contain Twelve short answer type questions, each carryingThree marks. (Attempt any Eight) - Section D:
Q. 27 to Q. 34 contain Eight long answer type questions, each carrying Four marks.(Attempt any Five) - Use of log table is allowed. Use of calculator is not allowed.
- Figures to the right indicate full marks.
- Use of graph paper is not necessary. Only rough sketch of graph is expected.
- For each multiple choice type of question, it is mandatory to write the correct answer along with its alphabet, e.g. (a)……/ (b)……./ (c)………etc. No marks shall be given, if ONLY the correct answer or the alphabet of correct answer is written. Only the first attempt will be considered for evaluation.
- Start answer to each section on a new page.
SECTION-A
Q.1 Select and write the correct answer for the following multiple choice type of questions: (2M each: 16M)
i. If p\wedge q is F, p\rightarrow q is F then the truth values of p and q are _______ respectively.
(a) T, T
(b) T, F
(c) F, T
(d) F, F
ii. In \triangleABC, if c^2\;+\;a^2\;\;–\;b^2\;\;=\;ac, then \angleB = _______.
(a) 4 \frac{\mathrm\pi}4
(b) 3 \frac{\mathrm\pi}3
(c) 2 \frac{\mathrm\pi}2
(d) 6 \frac{\mathrm\pi}6
iii. The area of the triangle with vertices (1, 2, 0), (1, 0, 2) and (0, 3, 1) in sq. unit is _______.
(a) \sqrt5
(b) \sqrt7
(c) \sqrt6
(d) \sqrt3
iv. If the corner points of the feasible solution are (0, 10), (2, 2) and (4, 0) then the point of minimum z\;=\;3x\;+\;2y\; is______.
(a) (2, 2)
(b) (0, 10)
(c) (4, 0)
(d) (3, 4)
v. If y is a function of x and log (x + y) = 2xy, then the value of y'(0) = _______.
(a) 2
(b) 0
(c) –1
(d) 1
vi. \int\cos^3\;x\;dx\;=\_\_\_\_\_\_.
(a) \frac1{12}\;\sin\;3x\;+\;\frac34\;\sin\;x\;+\;c
(b) \frac1{12}\;\sin\;3x\;+\;\frac14\;\sin\;x\;+\;c
(c) \frac1{12}\;\sin\;3x\;-\;\frac34\;\sin\;x\;+\;c
(d) \frac1{12}\;\sin\;3x\;-\;\frac14\;\sin\;x\;+\;c
vii. The solution of the differential equation \frac{\operatorname dx}{\operatorname dt}=\frac{x\log x}t is____.
(a) x\;=\;e^{ct}
(b) x\;+\;e^{ct}\;=\;0
(c) x\;=\;e^t\;\;+\;t
(d) xe^{ct}\;=\;0
viii. Let the probability mass function (p.m.f.) of a random variable X be P(X = x){}^4C_x\begin{pmatrix}\frac59\end{pmatrix}^x\times\begin{pmatrix}\frac49\end{pmatrix}^{4-x}, for x = 0, 1, 2, 3, 4 then E(X) is equal to _______
(a) \frac{20}9
(b) \frac9{20}
(c) \frac{12}9
(d) \frac9{25}
Q.2. Answer the following questions: (1M each: 4M)
i. Write the joint equation of co-ordinate axes.
ii. Find the values of c which satisfy \left|c\overline u\right|=3 where \overline u\;=\widehat{i\;}+2\widehat j+3\widehat k .
iii. Write \int\cot\;x\;dx.
iv. Write the degree of the differential equation {}_e\frac{dy}{dx}_{+\frac{dy}{dx}=x}.
SECTION-B
Attempt any EIGHT of the following questions: (2M each: 16M)
Q.3. Write inverse and contrapositive of the following statement:If x < y then x^2<y^2
Q.4. If A=\begin{bmatrix}x&0&0\\0&y&0\\0&0&z\end{bmatrix} is a non singular matrix, then find A^{-1} by elementary row transformations. Hence write the inverse of\begin{bmatrix}2&0&0\\0&1&0\\0&0&-1\end{bmatrix}
Q.5. Find the cartesian co-ordinates of the point whose polar co-ordinates are \left(\sqrt2\frac{\mathrm\pi}4\right) .
Q.6. If ax^2+2hxy+by^2=0 represents a pair of lines and h^2=ab\neq0 then find the ratio of their slopes.
Q.7. If \overline a,\;\overline b\;,\overline c are the position vectors of the points A, B, C respectively and 5\overline a+\;3\overline b\;-8\overline c\;=\;\overline0 then find the
ratio in which the point C divides the line segment AB.
Q.8. Solve the following inequations graphically and write the corner points of the feasible region:
2x\;+\;3y\leq6,\;x+y\geq2,\;x\geq0,\;y\geq0
Q.9. Show that the function f(x)\;=\;x^3\;+\;10x\;+\;7,\;x\in R is strictly increasing.
Q.10. Evaluate: \int_0^\frac{\mathrm\pi}2\sqrt{1-\cos4x}dx
Q.11. Find the area of the region bounded by the curve y^{2\;}=\;4x , the X-axis and the lines x = 1, x = 4 for y\geq0 .
Q.12. Solve the differential equation
cos x cos y dy – sin x sin y dx = 0
Q.13. Find the mean of number randomly selected from 1 to 15.
Q.14. Find the area of the region bounded by the curve y=x^2 and the line y = 4.
SECTION-C
Attempt any EIGHT of the following questions: (3M each: 24M)
Q.15. Find the general solution of \sin\theta\;+\;\sin\;3\theta\;+\;\sin\;5\theta\;=\;0\;
Q.16. If -1\leq x,\leq x\;1 , the prove that \sin^{-1}x+\;\cos^{-1}x\;=\;\frac{\mathrm\pi}2
Q.17. If \theta is the acute angle between the lines represented by ax^2\;+\;2hxy\;+\;by^2\;=\;0 then prove that \tan\;\theta=\left|\frac{2\sqrt{h^2-ab}}{a+b}\right|
Q.18. Find the direction ratios of a vector perpendicular to the two lines whose direction ratios are
–2, 1, –1 and –3, –4, 1.
Q.19. Find the shortest distance between lines \frac{x-1}2=\frac{y-2}3=\frac{z-3}4\;and\;\frac{x-2}3=\frac{y-4}4=\frac{z-5}5\;.
Q.20. Lines \overline r=\left(\widehat i+\widehat j-\widehat k\right)+\lambda\left(2\widehat i-2\widehat j+\widehat k\right)\;and\;\overline r=\left(4\widehat i-3\widehat j+2\widehat k\right)+\mu\left(\widehat i-2\widehat j+2\widehat k\right) are coplanar. Find the equation of the plane determined by them.
Q.21. If y\;=\;\sqrt{\tan x+\sqrt{\tan x+\sqrt{\tan x+.....+\infty}}} ,then
show that \frac{\operatorname dy}{\operatorname dx}=\frac{sec^2x}{2y-1} . find \frac{\operatorname dy}{\operatorname dx}\;at\;x=0.
Q.22. Find the approximate value of \sin(30^\circ30'). give that 1^\circ=0.0175^c\;and\;\cos\;30^\circ\;=\;0.866
Q.23. Evaluate \int x\;\tan^{-1}\;xdx
Q.24. Find the particular solution of the differential equation \frac{dy}{dx}=e^{2y}\;\cos\;x,\;when\;x=\frac{\mathrm\pi}6,\;y=0
Q.25. For the following probability density function of a random variable X, find (a) P(X < 1) and (b) P(| X| < 1).
f(x)=\frac{x+2}{18}\;;for -2<x<4
=0, otherwise
Q.26. A die is thrown 6 times. If ‘getting an odd number’ is a success, find the probability of at least 5
successes .
SECTION-D
Attempt any FIVE of the following questions: (4M each: 20M)
Q.27. Simplify the given circuit by writing its logical expression. Also write your conclusion.
Q.28. If A\;=\;\begin{vmatrix}1&2\\3&4\end{vmatrix}\;verify\;that\;A(adjA)\;=\;(adjA)A=\left|A\right|I
Q.29. Prove that the volume of a tetrahedron with coterminus edges \overline a,\;\overline b\;and\;\overline c\;is\;\frac16\left[\overline a\;\overline b\;\overline c\right]\;.
Hence, find the volume of tetrahedron whose coterminus edges are \overline a=\widehat i\;+\;2\widehat j+\;3\widehat k,\;\overline b=\;\widehat{-i}+\widehat{\;j}+\;\;2\widehat k\;and\;\;\overline c=\;2\widehat i\;+\widehat{\;j}+4\widehat k\;.
Q.30. Find the length of the perpendicular drawn from the point P(3, 2, 1) to the line \overline r\;=\;\left(7\widehat i\;+\;7\widehat j\;+\;6\widehat k\right)\;+\lambda\;\left(-2\widehat i\;+\;2\widehat j+\;3\widehat k\right)\;
Q.31. If y\;=\;\cos\left(m\;\cos^{-1}\;x\right) then show that \left(1\;-\;x^2\right)\frac{d^2y}{dx^2}\;-\;x\frac{dy}{dx}\;+\;m^2y=\;0
Q.32. Verify Lagrange’s mean value theorem for the function f(x)\;=\;\sqrt{x+4}\;on\;the\;interval\;\lbrack0,5\rbrack.
Q.33. Evaluate: \int\frac{2x^2-3}{\left(x^2-5\right)\left(x^2+4\right)}dx
Q.34. Prove that: \int_0^{2a}f(x)\;dx=\int_0^af(x)\;dx+\int_0^af(2a-x)dx
