STUPIDSID
Objective Question (MCQ)
1(a)1
The value of \(\displaystyle \lim_{n\rightarrow \infty }\left ( 1+\frac{1}{n} \right )^n\) is
(A) 1
(B) -1
(C) e
(d) 1/e
(A) 1
(B) -1
(C) e
(d) 1/e
1 M
1(a)2
The value of \(\displaystyle \lim_{x\rightarrow 0 }x^x\) is
(A) 1
(B) -1
(C) e
(D) 1/e
(A) 1
(B) -1
(C) e
(D) 1/e
1 M
1(a)3
Maclaurin's series of e-x is
(A) \[\sum {^\infty _{n=0}}\dfrac{x^n}{n!}\]
(B) \[\sum {^\infty _{n=0}}\dfrac{(-1)^nx^n)}{n!}\]
(c) \[\sum {^\infty _{n=1}}\dfrac{x^n}{n!}\] (d) \[\sum {^\infty _{n=1}}\dfrac{(-1)^nx^n}{n!}\]
(A) \[\sum {^\infty _{n=0}}\dfrac{x^n}{n!}\]
(B) \[\sum {^\infty _{n=0}}\dfrac{(-1)^nx^n)}{n!}\]
(c) \[\sum {^\infty _{n=1}}\dfrac{x^n}{n!}\] (d) \[\sum {^\infty _{n=1}}\dfrac{(-1)^nx^n}{n!}\]
1 M
1(a)4
\(If f(x, y, z, w) =3\dfrac{\cos (xw)\sin y^5}{e^y+(1+y^2)/xyw}+5xzw\), then \(\displaystyle \dfrac{\partial f}{\partial z}\) at (1, 2, 3, 4) is
(A) 20
(B) 200
(C) 0
(D) 1
(A) 20
(B) 200
(C) 0
(D) 1
1 M
1(a)5
Minimum value of f(x, y) = x2y2 is
(A) 1
(B) 2
(C) 4
(D) 0
(A) 1
(B) 2
(C) 4
(D) 0
1 M
1(a)6
The sum of the series \[\sum {^\infty _{n=1}}\dfrac{1}{2^n}\] is
(A) 0
(B) \[\dfrac{3}{4}\]
(C) 1
(D) 2
(A) 0
(B) \[\dfrac{3}{4}\]
(C) 1
(D) 2
1 M
1(a)7
Value of the \[\lim_{_(x,y)}\rightarrow (0,0)\dfrac{x^2-xy}{\sqrt{x}-\sqrt{y}}\] is
(A) 1
(B) 0
(C) -1 (D) linit does not exist
(A) 1
(B) 0
(C) -1 (D) linit does not exist
1 M
1(b)1
The value of \[\int \int 3y \ dxdy\] over the triangle with vertices (-1, 1), (0, 0), and (1, 1) is
(A) 0
(B) 1
(C) 2
(D) 3
(A) 0
(B) 1
(C) 2
(D) 3
1 M
1(b)2
The series \[\sum {^\infty _{n=1}}\dfrac{(-1)^n}{n}\] is
(A) divergent
(B) absolutely convergent
(C) conditionally convergent
(D) nothing can be said
(A) divergent
(B) absolutely convergent
(C) conditionally convergent
(D) nothing can be said
1 M
1(b)3
If \[l=\dfrac{\partial (x,y)}{\partial (r,\theta)}\] and \[J^*=\dfrac{\partial (r,\theta)}{\partial (x,y)}\] , then the value of lJ* is
(A) 0
(B) 1
(C) -1
(D) always l = J*
(A) 0
(B) 1
(C) -1
(D) always l = J*
1 M
1(b)4
What does the polar equation r = a, a>0 represent?
(A) line
(B) rectangle
(C) circle
(D) parabola
(A) line
(B) rectangle
(C) circle
(D) parabola
1 M
1(b)5
The curve x3+y3=3axy is symmetric about
(A) X-axis
(B) Y-axis
(C) origin
(D) the line y = x
(A) X-axis
(B) Y-axis
(C) origin
(D) the line y = x
1 M
1(b)6
The value of \[\int ^\infty _1 \dfrac{1}{x^2}dx\] is
(A) 1
(B) 0
(C) -1
(D) does not exist
(A) 1
(B) 0
(C) -1
(D) does not exist
1 M
1(b)7
If f(x, y) = \[\dfrac{x}{y}+\sin \left ( \dfrac{y}{x} \right )e^{y/x}\], then the value of \[x\dfrac{\partial f}{\partial x}+y\dfrac{\partial f}{\partial x}\] is
(A) 1
(B) 0
(C) -1
(D) f
(A) 1
(B) 0
(C) -1
(D) f
1 M
2(a)
Find the equations of tangent plane and normal line to the surface cosπx - x2y+e xz+yz = 4 at the point P(0, 1, 2).
3 M
2(b)
The amount of work done by the heart's main pumping chamber, the left ventricle, is given by the equation
\[W(P,V,\delta,v,g)=PV+\dfrac{V\delta v^2}{2g}\]
where W is the work per unit time, is the average blood pressure, V is the volume of blood pumped out during the unit of time, δ is the weight density of the blood, v is the average velocity of the existing blood, and g is the acceleration of the gravity which is constant. Find partial derivative of W with respect to each variable.
\[W(P,V,\delta,v,g)=PV+\dfrac{V\delta v^2}{2g}\]
where W is the work per unit time, is the average blood pressure, V is the volume of blood pumped out during the unit of time, δ is the weight density of the blood, v is the average velocity of the existing blood, and g is the acceleration of the gravity which is constant. Find partial derivative of W with respect to each variable.
4 M
2(c)
Give an example of a non-homogeneous function in and . If
\[u=\tan^{-1}(x^2+xy+2015y^2)\] then show that
(i) \[x\dfrac{\partial u}{\partial x}+y\dfrac{\partial u}{\partial x}=\sin 2u\]
(ii) \[x^2\dfrac{\partial^2 u}{\partial x^2}+2xy\dfrac{\partial^2 u}{\partial x \partial y}+y^2\dfrac{\partial^2 u}{\partial y^2}=2\sin u\ \cos 3u\]
\[u=\tan^{-1}(x^2+xy+2015y^2)\] then show that
(i) \[x\dfrac{\partial u}{\partial x}+y\dfrac{\partial u}{\partial x}=\sin 2u\]
(ii) \[x^2\dfrac{\partial^2 u}{\partial x^2}+2xy\dfrac{\partial^2 u}{\partial x \partial y}+y^2\dfrac{\partial^2 u}{\partial y^2}=2\sin u\ \cos 3u\]
7 M
3(a)
The figure shows the first seven of a sequence of squares. The outermost square has an area of 4m2 . Each of the other squares is obtained by joining the midpoints of the sides of the squares before it. Find the sum of areas of all squares in the infinite sequence.
3 M
3(b)
Test the convergence of
(i) \[\sum {^\infty _{n=3}}\dfrac{1}{n\log n\sqrt{\log^n-1}}\]
(ii) \[\sum {^\infty _{n=1}}\dfrac{1}{1+2^2+3^2+\cdots+n^2}\]
(i) \[\sum {^\infty _{n=3}}\dfrac{1}{n\log n\sqrt{\log^n-1}}\]
(ii) \[\sum {^\infty _{n=1}}\dfrac{1}{1+2^2+3^2+\cdots+n^2}\]
4 M
3(c)
You are to construct an open rectangular box from 12 ft2 of material. What dimensions will result in a box of maximum volume?
7 M
4(a)
For \[z=\tan ^{-1}\left ( \dfrac{x}{y} \right )\]; x = u cosv, y = u sinv evaluate \[\dfrac{\partial z}{\partial u}\] and \[\dfrac{\partial z}{\partial v}\] at the point \[\left ( 1.3, \dfrac{\pi}{6} \right )\].
3 M
4(b)
For the series \[\sum {^\infty _{n=1}}\dfrac{(-1)^n(x+2)^n}{n}\] find the series' radius and interval of convergence. For what values of x does the series converge absolutely, conditionally?
4 M
4(c)
Trace the curve \[y^2(a-x)=x^3, a > 0\] .
7 M
5(a)
Evaluate \[\lim_{_{x\rightarrow \frac{1}{2}}}\dfrac{\cos ^2 \pi x}{e^{2x}-2ex}\]
3 M
5(b)
Change the order of integration in \[\int ^\infty _0\int ^\infty _x \dfrac{e^{-y}}{y}dydx\] and hence evaluate the same.
4 M
5(c)
Expand \[e^y\log(1+x)\] in powers of x and y up to third degree terms in two different ways.
7 M
6(a)
By considering different paths of approach, show that the function \[f(x,y)=\dfrac{x^4-y^2}{x^4+y^2}\] has no limit as (x, y) → (0, 0).
3 M
6(b)
Evaluate \[\int ^3_0 \dfrac{dx}{(x-1)^{2/3}}\]
4 M
6(c)
Evaluate \[\int ^4_0 \int ^{\dfrac{y}{2}+1}_{\dfrac{y}{2}}\dfrac{2x-y}{2}dxdy\] by applying the transformations \[u=\dfrac{2x-y}{2},v=\dfrac{y}{2}\]. Draw both the regions.
7 M
7(a)
Find the volume of the solid that results when the region enclosed by the curves y = x2 and x = y2 is revolved about the Y-axis.
3 M
7(b)
Use the method of slicing to find the volume of solid with semicircular base defined by \[y=5\sqrt{\cos x}\] on the interval \[\left [ -\dfrac{\pi}{2},\dfrac{\pi}{2} \right ]\]. The cross-sections of the solid are squares perpendicular to the X- axis with bases running from the X-axis to the curve.
4 M
7(c)
Evaluate \[\int \int \int _E 2xdV\] where E is the region under the plane 2x+3y+x = 6 that lies in the first octant.
7 M
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