STUPIDSID
Choose the most appropriate answer out of the following options given for each part of the question:
1(a)(i)
The sum of the series \( 1-\dfrac{1}{2}+\dfrac{1}{2^2}-\dfrac{1}{2^3}+\cdots \) is _______
(A) 2/3 (B) 3/2 (C) 1/2 (D) None of these
(A) 2/3 (B) 3/2 (C) 1/2 (D) None of these
1 M
1(a)(ii)
The series \( x+\dfrac{x^3}{3!}+\dfrac{x^5}{5!}+\cdots \) represent expansion of _______
(A) sin x (B) cos x (C) sinh x (D) coshx
(A) sin x (B) cos x (C) sinh x (D) coshx
1 M
1(a)(iii)
The value of the \( \lim_{x \rightarrow \infty }\dfrac{\sin x}{x} \) is _______
(A) 1 (B) -1 (C) 0 (D) None of these
(A) 1 (B) -1 (C) 0 (D) None of these
1 M
1(a)(iv)
The curve r=cos2θ has _______ petals.
(A) 2 (B) 4 (C) 3 (D) None of these
(A) 2 (B) 4 (C) 3 (D) None of these
1 M
1(a)(v)
If x=rcosθ, y=rsinθ, then \( \dfrac{\partial (r,\theta)}{\partial (x,y)} \) = _______
(A) r (B) 1/r (C) 3 (D) None of these
(A) r (B) 1/r (C) 3 (D) None of these
1 M
1(a)(vi)
If \( u=x^3\cos\left ( y/x \right ),\text{then}\ x\dfrac{\partial u}{\partial x}+y\dfrac{\partial u}{\partial y} \) = _______
(A) u (B) 2u (C) 3u (D) None of these
(A) u (B) 2u (C) 3u (D) None of these
1 M
1(a)(vii)
The equation of a cylindrical surface x2+y2=9 becomes _______ when
converted to cylindrical polar coordinates.
(A) r=9 (B) r2 = 9 (C) \( r=\pm 3 \) (D) r=3
(A) r=9 (B) r2 = 9 (C) \( r=\pm 3 \) (D) r=3
1 M
1(b)(i)
The series \( \sum ^{\infty}_{n=1}\dfrac{1}{(\log n)^n} \) is _______
(A) oscillatory (B) divergent (C) convergent (D) None of these
(A) oscillatory (B) divergent (C) convergent (D) None of these
1 M
1(b)(ii)
If in the equation of a curve, x occurs only as an even power then the curve is symmetrical about
(A) x-axis (B) y-axis (C) Origin (D) None of these
(A) x-axis (B) y-axis (C) Origin (D) None of these
1 M
1(b)(iii)
The value of the \( \lim_{x\rightarrow 0}(\cos x)^{\cot x} \) is _______
(A) 0 (B) 1 (C) &infin (D) None of these
(A) 0 (B) 1 (C) &infin (D) None of these
1 M
1(b)(iv)
The integral \( \int ^a_{-a}f(x)dx=2\int ^a_0f(x)dx \) if _______
(A) f is an odd function (B) f is neither even nor odd function (C) f is a even function (D) None of these
(A) f is an odd function (B) f is neither even nor odd function (C) f is a even function (D) None of these
1 M
1(b)(v)
The volume of the solid generated by revolving the region between the y - axis and the curve (x=2\sqrt{y},0\leq y\leq 4, \) about the y ' axis.
(A) 2π (B) 32π (C) 16π (D) None of these
(A) 2π (B) 32π (C) 16π (D) None of these
1 M
1(b)(vi)
\( \int ^2_0 \int ^{x^2}_0e^{y/x}dydx \) is equal to _______ (A) e2-1 (B) e2 (C) e2+1 (D) e-2
1 M
1(b)(vii)
A point ( a , b ) is said to be a saddle point if at ( a , b )
(A) rt-S2 >0 (B) rt-S2 = 0 (C) rt-S2 < 0 (D) rt-S2 ≥ 0
(A) rt-S2 >0 (B) rt-S2 = 0 (C) rt-S2 < 0 (D) rt-S2 ≥ 0
1 M
2(a)
Expand \( \left (\dfrac{\pi}{4}+x \right) \) in powers of x by using the Taylor's series. Also, find the value of tan 44°.
4 M
2(b)
Evaluate: \( \lim_{x\rightarrow 0}\left ( \dfrac{1}{x} \right )^{\tan x}. \)
3 M
2(c)(i)
Test for convergence the series \( x-\dfrac{x^3}{3}+\dfrac{x^5}{5}-\cdots,\ \ x>0. \)
3 M
2(c)(ii)
Trace the curve r2 = a2sin 2θ.
4 M
3(a)
If \( \theta=t^n e^{-\dfrac{r^2}{4t}}\text {then find n so that }\ \frac{1}{r^2}\frac{\partial }{\partial r}\left ( r^2\frac{\partial \theta }{\partial r} \right )=\frac{\partial \theta}{\partial t} \).
4 M
3(b)
Determine the continuity of the function \( f(x,y)=\left\{\begin{matrix}
(x^2+y^2)\sin \left ( \dfrac{1}{x^2+y^2} \right ) , & (x,y) \neq (0,0)\\
0 , & (x,y)=(0,0)
\end{matrix}\right. \ \ \ \ \text {at origin}. \)
3 M
3(c)(i)
If u = tan-1(x2+2y2), show that x2uxx + 2xy uxy + y2uyy = 2sin u.cos 3u
4 M
3(c)(ii)
If \( u=f\left ( \dfrac{y-x}{xy},\dfrac{z-x}{xz} \right ), \text{show that}\ x^2\dfrac{\partial u}{\partial x}+y^2\frac{\partial u}{\partial y}+z^2\dfrac{\partial u}{\partial z}=0 \)
3 M
4(a)
Find the extreme values of the function f(x, y) = x3+3xy2-15x2-15y2+72x.
4 M
4(b)
Find the equation of the tangent plane and the normal line of the surface \[\dfrac{x^2}{4}+y^2+\dfrac{z^2}{9}=3\ \text{at}\ (-2,1,-3)\].
3 M
4(c)(i)
Show that the rectangular solid of maximum volume that can be inscribed in a sphere is cube.
4 M
4(c)(ii)
Expand xy2+xy+3 in powers of (x-1) and (y+2) using Taylor's series expansion.
3 M
5(a)
Find the radius of convergence and interval of convergence of the series \[\sum ^{\infty}_{n=0}\dfrac{(-3)^n x^n}{\sqrt{n+1}}\]
4 M
5(b)
Test the convergence of the series \( \sum ^{\infty}_{n=1}\sqrt{n^4+1}-\sqrt{n^4-1} \).
3 M
5(c)(i)
Define geometric series and show that the geometric series is :
(i) convergent if |r| < 1, (ii) divergent if r ≥ 1
(i) convergent if |r| < 1, (ii) divergent if r ≥ 1
4 M
5(c)(ii)
Express (x-1)4+2(x-1)3+5(x-1)+2 in ascending power of x.
3 M
6(a)
Evaluate \( \iint _R (x+y)dy\ dx \) , where R is the region bounded by x=0, x=2, y=x, y=x+2.
4 M
6(b)
Evaluate \( \iint _R r^3 \sin 2\theta \ dr\ d\theta \) over the area bounded in the first quadrant between the circles r=2 and r=4.
3 M
6(c)(i)
Evaluate \( \int ^e_1 \int ^{\log y}_1 \int ^{e^x}_1 \log z\ dz\ dx\ dy \)
4 M
6(c)(ii)
Evaluate \( \lim_{x\rightarrow \frac{\pi}{2}}(\cos)^{\frac{\pi}{2}-x}. \)
3 M
7(a)
Find the volume of the solid generated by revolving the curve x2/3+y2/3=a2/3 about the x-axis.
4 M
7(b)
Find the area lying inside the circle r = a sinθ and outside the cardioid r = a (1 ' cosθ).
3 M
7(c)(i)
Evaluate \( \int _0^1 \int _x^{\sqrt{2-x^2}}\dfrac{x}{\sqrt{x^2+y^2}}dy\ dx \) by changing the order of integration.
4 M
7(c)(ii)
Check the convergence of \( \int ^5_0 \dfrac{1}{x^2}dx. \)
3 M
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