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. The negation of p \wedge(q \rightarrow r) is _______.
(a) \sim p\wedge (\sim q\rightarrow \sim r)
(b) \mathrm{p} \vee (\sim \mathrm{q} \vee \mathrm{r})
(c) \sim \mathrm{p} \wedge (\sim \mathrm{q} \rightarrow \mathrm{r})
(d) \mathrm{p} \rightarrow (\mathrm{q} \wedge \sim \mathrm{r})
ii. In \triangle \mathrm{ABC} if \mathrm{c}^2+\mathrm{a}^2-\mathrm{b}^2=\mathrm{ac}, then \angle \mathrm{B}=
(a) \frac{\pi}{4}
(b) \frac{\pi}{3}
(c) \frac{\pi}{2}
(d) \frac{\pi}{6}
iii. Equation of line passing through the points (0,0,0) and (2,1,-3) is __ .
(a) \frac{x}{2}=\frac{y}{1}=\frac{z}{-3}
(b) \frac{x}{2}=\frac{y}{-1}=\frac{z}{-3}
(c) \frac{x}{1}=\frac{y}{2}=\frac{z}{3}
(d) \frac{x}{3}=\frac{y}{1}=\frac{z}{2}
iv.The value of \hat{i} \cdot(\hat{j} \times \hat{k})+\hat{j} \cdot(\hat{k} \times \hat{i})+\hat{k} \cdot(\hat{i} \times \hat{j}) is __ .
(a) 0
(b) -1
(c) 1
(d) 3
v.If \mathrm{f}(x)=x^5+2 x-3, then \left(\mathrm{f}^{-1}\right)^{\prime}(-3)= _______.
(a) 0
(b) -3
(c) -1/3
(d) 1/2
vi. The maximum value of the function f(x)=\frac{log\; x}{x} is _______.
(a) e
(b) 1/e
(c) e2
(d) ) \frac{1}{e^{2}}
vii. If \int \frac{dx}{4x^{2}-1}=A\; log\left ( \frac{2x-1}{2x+1} \right )+c, then A = ________.
(a) 1
(b) 1/2
(c) 1/3
(d)1/4
viii. If the p.m.f of a r.v.X is
P(x)=\frac{c}{x^{3}} , for x=1,2,3
=0, otherwise,
then E(X)= _______
(a) \frac{216}{251}
(b) \frac{294}{251}
(c) \frac{297}{294}
(d) \frac{294}{297}
Q.2. Answer the following questions: (1M each: 4M)
i. Find the principal value of cot^{-1}\left ( \frac{-1}{\sqrt{3}} \right )
ii. Write the seperate equations of lines represented by the equation 5x2-9y2=0
iii. If f'(x)=x-1 , then find f(x)
iv. Write the degree of the differential equation
(y”’)2+3(y”)+3xy’ + 5y = 0
SECTION-B
Attempt any EIGHT of the following questions: (2M each: 16M)
Q.3. Using truth table verify that :
(p\wedge q)\vee \sim q \equiv p\vee \sim q
Q.4. Find the cofactors of the elements of the matrix \begin{bmatrix}-1 & 2 \\-3 & 4 \\\end{bmatrix}
Q.5. Find the principal solutions of cot\; \theta =0
Q.6. Find the value of k, if 2x+y = 0 is one of the lines represented by 3x2+kxy+2y2 = 0 .
Q.7. Find the cartesian equation of the plane passing through A(1,2,3) and the direction ratios of whose normal are 3,2,5.
Q.8. Find the cartesian co-ordinates of the point whose polar co-ordinates are \left ( \frac{1}{2} , \frac{\pi }{3}\right )
Q.9. Find the equation of tangent to the curve y=2x3-x2 +2 at \left ( \frac{1}{2} , 2\right )
Q.10. Evaluate: \int_0^\frac{\mathrm\pi}4sec^{4}x.dx
Q.11. Solve the differential equation y\frac{dy}{dx}+x=0
Q.12. Show that function f(x) = tan x is increasing in \left ( 0 , \frac{\pi }{2}\right )
Q.13. From the differential equation of all lines which makes intercept 3 on x-axis.
Q.14. If X\sim B(n,p) and E(X)=6 and Var(X) = 4.2, then find n and p.
SECTION-C
Attempt any EIGHT of the following questions: (3M each: 24M)
Q.15. If 2 tan-1 (cos x) = tan-1 (2 cosec x), then find the value of x.
Q.16. If angle between the lines represented by ax2+2hxy+by2=0 is equal to the angle between the lines represented by 2x2-5xy+3y2=0, then show that 100(h2-ab)=(a+b)2.
Q.17. Find the distance between the parallel lines \frac{x}{2}=\frac{y}{-1}=\frac{z}{2} and \frac{x-1}{2}=\frac{y-1}{-1}=\frac{z-1}{2}
Q.18. If (A(5,1, p), B(1, q, p) and (C(1,-2,3) are vertices of a triangle and G\left(r, \frac{-4}{3}, \frac{1}{3}\right) is its centroid, then find the values of (p, q, r) by vector method.
Q.19. If A(\bar{a}) and B (\bar{b}) be any two points in the space and R(\overline{\mathrm{r}}) be a point on the line segment AB dividing it internally in the ratio m: n then prove that \bar{r}=\frac{m \bar{b}+n \bar{a}}{m+n} .
Q.20. Find the vector equation of the plane passing through the point A(-1,2,-5) and parallel to the vectors 4 \hat{i}-\hat{j}+3 \hat{k} and \hat{i}+\hat{j}-\hat{k}.
Q.21. If y=\mathrm{e}^{\mathrm{m} \tan ^{-1} x}, then show that \left(1+x^2\right) \frac{\mathrm{d}^2 y}{\mathrm{~d} x^2}+(2 x-\mathrm{m}) \frac{\mathrm{d} y}{\mathrm{~d} x}=0
Q.22. Evaluate: \int \frac{d x}{2+\cos x-\sin x}
Q.23. Solve x+y \frac{\mathrm{~d} y}{\mathrm{~d} x}=\sec \left(x^2+y^2\right)
Q.24. A wire of length 36 meters is bent to form a rectangle. Find its dimensions if the area of the rectangle is maximum.
Q.25. Two dice are thrown simultaneously. If X denotes the number of sixes, find the expectation of X .
Q.26. If a fair coin is tossed 10 times. Find the probability of getting at most six heads.
SECTION-D
Attempt any FIVE of the following questions: (4M each: 20M)
Q.27. Without using truth table prove that
(p \wedge q) \vee(\sim p \wedge q) \vee(p \wedge \sim q) \equiv p \vee qQ.28.Solve the following system of equations by the method of inversion
x-y+z=4,2 x+y-3 z=0, x+y+z=2Q.29. Using vectors prove that the altitudes of a triangle are concurrent.
Q.30.Solve the L.P.P. by graphical method,
Minimize z=8x+10y
Subject to 2 x+y \geq 7,
\begin{aligned}& 2 x+3 y \geq 15 \\& y \geq 2, x \geq 0\end{aligned}
Q.31. If x=\mathrm{f}(\mathrm{t}) and y=\mathrm{g}(\mathrm{t}) are differentiable functions of t so that y is differentiable function of x and \frac{\mathrm{dx}}{\mathrm{dt}} \neq 0, then prove that:
\frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{\frac{\mathrm{d} y}{\mathrm{dt}}}{\frac{\mathrm{~d} x}{\mathrm{dt}}}Hence find \frac{\mathrm{d} y}{\mathrm{~d} x} if x=\sin \mathrm{t} and y=\cos \mathrm{t}.
Q.32. If u and v are differentiable function of x, then prove that:
\int \mathrm{uv} \mathrm{~d} x=\mathrm{u} \int \mathrm{vd} x-\int\left[\frac{\mathrm{du}}{\mathrm{~d} x} \int \mathrm{v} \mathrm{~d} x\right] \mathrm{d} xHence evaluate \int \log x \mathrm{~d} x
Q.33. Find the area of region between parabolas y^2=4 \mathrm{a} x and x^2=4 \mathrm{a} y.
Q.34. Show that: \int_0^{\frac{\pi}{4}} \log (1+\tan x) \mathrm{d} x=\frac{\pi}{8} \log 2
