STUPIDSID
Attempt the following.
1 (a) 1
Maclaurin series of sin x is
\[a) \sum_{n=0}^{\infty }\dfrac{x^{2n+1}}{(2n+1)!}\
b)\sum_{n=0}^{\infty }\dfrac{x^{2n+1}}{(2n+1)!}\
c)\sum_{n=0}^{\infty }\dfrac{x^{2n}}{(2n)!}\
d)\sum_{n=0}^{\infty }(-1)^{n}\dfrac{x^{2n}}{(2n)!}\].
\[a) \sum_{n=0}^{\infty }\dfrac{x^{2n+1}}{(2n+1)!}\
b)\sum_{n=0}^{\infty }\dfrac{x^{2n+1}}{(2n+1)!}\
c)\sum_{n=0}^{\infty }\dfrac{x^{2n}}{(2n)!}\
d)\sum_{n=0}^{\infty }(-1)^{n}\dfrac{x^{2n}}{(2n)!}\].
1 M
1 (a) 2
The total area enclosed between y=sin x and X-axis in [0,π] is
a) 0
b) 4
c) 2
d) 1.
a) 0
b) 4
c) 2
d) 1.
1 M
1 (a) 3
The value of \[\lim_{x\to 2}{x^{1/x}}\]
a) ∞
(b) -∞
(c) 1
(d) 0,
a) ∞
(b) -∞
(c) 1
(d) 0,
1 M
1 (a) 4
How many leaves r=sin5 θ has?
a) 6
(b) 2
(c) 3
(d) 5.
a) 6
(b) 2
(c) 3
(d) 5.
1 M
1 (a) 5
The sum of the series \\sum_{n=1}^{\infty }\dfrac{1}{2^{n}}\] is
a) 0
(b) 1
(c) -1
(d) 1/2.
a) 0
(b) 1
(c) -1
(d) 1/2.
1 M
1 (a) 6
The values of x at which the curves y=x& y=4x-x2 intersects each other are
a) 0 & 4
b) -4 & 4
c) 0 & 3
d) -3 & 3.
a) 0 & 4
b) -4 & 4
c) 0 & 3
d) -3 & 3.
1 M
1 (a) 7
The area enclosed between the curve r=f(θ) and two rays θ=α & θ=β is
\[a) \int_{\alpha }^{\beta }\limits r^{2}d\theta \ b)\int_{\alpha }^{\beta }\limits r^{3}d\theta \ c)\dfrac{4\pi}{3} \int_{\alpha }^{\beta }\limits r^{3}d\theta \ d) \dfrac{1}{2} \int_{\alpha }^{\beta }\limits r^{2}d\theta\].
\[a) \int_{\alpha }^{\beta }\limits r^{2}d\theta \ b)\int_{\alpha }^{\beta }\limits r^{3}d\theta \ c)\dfrac{4\pi}{3} \int_{\alpha }^{\beta }\limits r^{3}d\theta \ d) \dfrac{1}{2} \int_{\alpha }^{\beta }\limits r^{2}d\theta\].
1 M
Attempt the following.
1 (b) 1
If \[u=\sin^{-1}\dfrac{x}{y}+\tan^{-1}\dfrac{y}{x}\] the the value of xux+yuy is
a) u
(b) -u
(c) 0
(d) 1.
a) u
(b) -u
(c) 0
(d) 1.
1 M
1 (b) 2
For an implicit function f(x,y)=c the value of \[\dfrac{dy}{dx}\] is
\[a) \dfrac{f_{x}}{f_{y}} \ b)\dfrac{f_{y}}{f_{x}} \ c)-\dfrac{f_{x}}{f_{y}} \ d)-\dfrac{f_{y}}{f_{x}}\].
\[a) \dfrac{f_{x}}{f_{y}} \ b)\dfrac{f_{y}}{f_{x}} \ c)-\dfrac{f_{x}}{f_{y}} \ d)-\dfrac{f_{y}}{f_{x}}\].
1 M
1 (b) 3
For the function u=(x2+y2)1/3 the value of x2uxx+2xyuxy+y2uyy is
\[a) -2u\
(b)-\dfrac{2u}{3} \
(c)\dfrac{u}{9} \
(d)-\dfrac{2u}{9}\].
\[a) -2u\
(b)-\dfrac{2u}{3} \
(c)\dfrac{u}{9} \
(d)-\dfrac{2u}{9}\].
1 M
1 (b) 4
\[If\ x=r \cos \theta \ and\ y=r \sin \theta \ then\ J=\dfrac{\partial (r,\theta)}{\partial (x,y)}\] is
\[a) r\ b) -r\
(c) \dfrac{1}{r}\
(d)-\dfrac{1}{r}\].
\[a) r\ b) -r\
(c) \dfrac{1}{r}\
(d)-\dfrac{1}{r}\].
1 M
1 (b) 5
The fundamental period of sin 2x is
a) π
(b) 2π
(c) 4π
(d) π/2.
a) π
(b) 2π
(c) 4π
(d) π/2.
1 M
1 (b) 6
The focus pf parabola y24ax is (ae,0) if
a) e<1
(b) e>1
(c) e=1
(d) e≠1.
a) e<1
(b) e>1
(c) e=1
(d) e≠1.
1 M
1 (b) 7
The function f(x)=2x3+3x2-12x+7 is decreasing in
a) [-2,1]
(b) R-[-2,1]
(c) [0,2]
(d) [1,3].
a) [-2,1]
(b) R-[-2,1]
(c) [0,2]
(d) [1,3].
1 M
Objective Question (MCQ)
1 (c) 1
\[ \lim_{x\to 2}\dfrac{x^{2}-4}{x^{2}+4}=\]
a) 1
(b) 0
(c) -1/2
(d) None of these.
a) 1
(b) 0
(c) -1/2
(d) None of these.
1 M
1 (c) 10
Use matrices to solve the following simultaneous equations
x+2 y= 3
2x+3 y=5
(a) x= 2 ; y=1
(b)x=1 ; y=1 (
(c) x=2 ; y=2
(d)None of these.
2x+3 y=5
(a) x= 2 ; y=1
(b)x=1 ; y=1 (
(c) x=2 ; y=2
(d)None of these.
1 M
1 (c) 11
Determine the area bounded by the x axis, the curve y=2x2+x-6, and the line x=4 and x=6
a) 298
(b) 126 \[c)\ 99\dfrac{1}{3}\
(d)\ 26\dfrac{2}{3}\].
a) 298
(b) 126 \[c)\ 99\dfrac{1}{3}\
(d)\ 26\dfrac{2}{3}\].
1 M
1 (c) 12
Given the function f ( x )=x2 which value of c satisfies the conclusion of mean value
theorem on the interval [-4, 5]?
(a) 0
(b)1
(c)1/2
(d)None of these.
(a) 0
(b)1
(c)1/2
(d)None of these.
1 M
1 (c) 13
Eliminate the parameter in the equations x=t2; y=t4
a) y=x2 for x≥0
(b) y=√(x) for x≥0
(c) y=2x2 for x≥0
(d) None of these.
a) y=x2 for x≥0
(b) y=√(x) for x≥0
(c) y=2x2 for x≥0
(d) None of these.
1 M
1 (c) 14
At how many places does the curve x=cos 3 t ; y=sin t cross the x-axis?
a) 2
(b) 1
(c) 3
(d) None of these.
a) 2
(b) 1
(c) 3
(d) None of these.
1 M
1 (c) 2
\[f(x)\dfrac{x^{2}-x}{2x};x\ne 0 \\ f(0)=k\]
If
and if f is continuous on x=0 then k=
a) -1
(b) -1/2
(c) 0
(d) None of these.
If
and if f is continuous on x=0 then k=
a) -1
(b) -1/2
(c) 0
(d) None of these.
1 M
1 (c) 3
What is the slope of the tangent line to the curve x+y=xy at point (2, 2)
a) -1
(b) -2
(c) -3
(d) -4
a) -1
(b) -2
(c) -3
(d) -4
1 M
1 (c) 4
Determine the second derivative of the function f(x)=x2. 1n 2x
a) 21n2x+3
(b)\[21n2x+\dfrac{3}{2}\]
(c) 0
(d) None of these.
a) 21n2x+3
(b)\[21n2x+\dfrac{3}{2}\]
(c) 0
(d) None of these.
1 M
1 (c) 5
At a minimum, the second differential function of the form y=axn+bxn-1+...... is
a) Positive
(b)Negative
(c)Zero
(d)Infinite.
a) Positive
(b)Negative
(c)Zero
(d)Infinite.
1 M
1 (c) 6
\[If y=\dfrac{3}{x^{4}}\ then \ \int \ y\ dx\\
a) \dfrac{15}{x^{5}}+c\
b) -\dfrac{1}{x^{3}}+c\
c)\dfrac{12}{x^{3}}+c\
d) None \ of \ these\].
1 M
1 (c) 7
Evaluate \[y=\int_{0}^{\dfrac{\pi}{2}}\limits \ \sin(4x) \sin(6x)dx\]
\[a) y=\dfrac{\pi}{2}\ b) y=\dfrac{\pi}{4}\ c) y=\dfrac{3\pi}{4}\ d y=0\].
\[a) y=\dfrac{\pi}{2}\ b) y=\dfrac{\pi}{4}\ c) y=\dfrac{3\pi}{4}\ d y=0\].
1 M
1 (c) 8
Use matrices to solve the following simultaneous equations
x+2z=9 4x+2y+z=14 x+3y+4 z=26
(a) x=3; y=-1/2;z=3
(b)x=-1;y=1;z=6 (
(c) x=1;y=3;z=4
(d)x=1;y=4;z=2.
x+2z=9 4x+2y+z=14 x+3y+4 z=26
(a) x=3; y=-1/2;z=3
(b)x=-1;y=1;z=6 (
(c) x=1;y=3;z=4
(d)x=1;y=4;z=2.
1 M
1 (c) 9
Determine the area bounded by The x axis, the curve y=sin 2 x ,and the line\[x=\dfrac{\pi}{4} \ and\ x=\pi/2\]
a) 0.5
(b) 1
(c) 0.25
(d) 1.5.
a) 0.5
(b) 1
(c) 0.25
(d) 1.5.
1 M
2 (a)
Show that the P-series \[\sum_{n=1}^{\infty }\dfrac{1}{(n)^{p}}\] (P- a real constant ) converges if P>1 and diverges if p≤1.
3 M
2 (b)
Define the Geometric series and find the sum of the following series \[\sum_{n=1}^{\infty }\dfrac{3^{n-1}-1}{6^{n-1}}\].
4 M
2 (c)
1)Investigate the convergence of the series \[\sum_{n=1}^{\infty }\dfrac{2^{n}+5}{3^{n}}\]
2) Is the series \[\sum_{n=1}^{\infty }\dfrac{2n+1}{(n+1)^{2}}\]converges or diverges?
2) Is the series \[\sum_{n=1}^{\infty }\dfrac{2n+1}{(n+1)^{2}}\]converges or diverges?
7 M
3 (a)
Define Taylor?s series for the function of one variable and using it show that \[\tan ^{-1}(x+h)= \tan^{-1}x+(h\sin \alpha)\dfrac{\sin \alpha}{1}-(h \sin \alpha)^{2}\dfrac{\sin 2 \alpha}{2}+(h \sin \alpha)^{3} \dfrac{\sin 3 \alpha}{3}+----\ where\ \alpha=\cot ^{-1}x\].
3 M
3 (b)
Trace the curve 9ay2=x(x-3a)2.
4 M
3 (c)
Attempt the following.
1)\[\lim_{x\to 2}\left [ \dfrac{a^{x}+b^{x}+c^{x}}{3} \right ]^{1/3x}\]
2)Define volume of solid of revolution by Washer's method and use it to find the volume of solid generated when the region between the graphs
\[f(x)=\dfrac{1}{2}+x^{2}\] and g(x)=x over the interval [0,2] is revolved about x-axis.
1)\[\lim_{x\to 2}\left [ \dfrac{a^{x}+b^{x}+c^{x}}{3} \right ]^{1/3x}\]
2)Define volume of solid of revolution by Washer's method and use it to find the volume of solid generated when the region between the graphs
\[f(x)=\dfrac{1}{2}+x^{2}\] and g(x)=x over the interval [0,2] is revolved about x-axis.
7 M
4 (a)
Define Improper integral of both the kinds. Check the convergence of \[\int_{0}^{3}\limits \dfrac{dx}{\sqrt{9-x^{2}}}\].
3 M
4 (b)
Trace the curve r2=a2 cos 2θ.
4 M
Attempt the following.
4 (c)
1) Evaluate \[ \lim_{x\to 2}\left [ \dfrac{1}{x^{2}}-\dfrac{1}{\sin ^{2}x} \right ]\]
2)Define the volume of solid of revolution by disk and use it to find the volume of the solid that is obtained when the region under the curve y=√(x) over the Interval [1,4]is revolved about X- axis.
2)Define the volume of solid of revolution by disk and use it to find the volume of the solid that is obtained when the region under the curve y=√(x) over the Interval [1,4]is revolved about X- axis.
7 M
5 (a)
Define Homogeneous function of two variables x and y of degree n . Also prove the following Euler's theorem for this homogeneous function of degree n .
\[x\dfrac{\partial u}{\partial x}+ y\dfrac{\partial u}{\partial y}=nu\]
\[x\dfrac{\partial u}{\partial x}+ y\dfrac{\partial u}{\partial y}=nu\]
3 M
5 (b)
If \[u=\tan ^{-1}\left [ \dfrac{x^{3}+y^{3}}{x-y} \right ]\] then prove that \[x^{2}{\dfrac{\partial^2 u}{\partial x^{2}}}+2xy{\dfrac{\partial^2 u}{\partial x \partial y}}+y^{2}{\dfrac{\partial^2 u}{\partial y^{2}}}=2 \cos3u\sin u\].
4 M
Attempt the following.
5 (c)
1) If x=ρsinφcosφ, y=ρsinφsinφand z=ρcosφ then obtain the Jacobean
\[J=\dfrac{\partial (x,y,z) }{\partial (\rho, \phi, \theta)}\]
2) In a plane triangle ABC find the extreme values of cos A cos B cos C .
\[J=\dfrac{\partial (x,y,z) }{\partial (\rho, \phi, \theta)}\]
2) In a plane triangle ABC find the extreme values of cos A cos B cos C .
7 M
6 (a)
If u=sin-1(x-y),x=3t.y=4t3 then show that \[\dfrac{du}{dt}=\dfrac{3}{\sqrt{1-t^{2}}}\].
3 M
6 (b)
If u=f(r) and r2=x2+y2+z2 then show that \[\dfrac{\partial^2u }{\partial x^2}+\dfrac{\partial^2u }{\partial y^2}+\dfrac{\partial^2u }{\partial z^2}=f(r)+\dfrac{2}{r}f(r)\].
4 M
Attempt the following.
6 (c)
1) Find the equations of the tangent plane and normal line to the surface
2xz2-3xy-4x=7 at (1,-1,2).
2) Find the minimum value of x2+y2+z2, given that ax+by+cz=p.
2xz2-3xy-4x=7 at (1,-1,2).
2) Find the minimum value of x2+y2+z2, given that ax+by+cz=p.
7 M
7 (a)
Evaluate \[\iint_{R}\limits \left ( 2x-y^{2} \right )dA\] over the triangular region R enclosed between the lines y=-x+1,y=x+1& y=3.
3 M
7 (b)
Evaluate the integral \[\int_{0}^{2}\limits \int_{y/2}^{1}\limits e^{x^{2}}\] dxdy by changing the order of integration.
4 M
Attempt the following.
7 (c)
1) Use double integral in polar integral to find the area enclosed by three petalled rose r= sin 3θ.
2).Use triple integral to find the volume of the solid within the cylinder x2+ y2=9 between the planes z= 1 and x+z=1.
2).Use triple integral to find the volume of the solid within the cylinder x2+ y2=9 between the planes z= 1 and x+z=1.
7 M
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