STUPIDSID
1 (a) (i)
The value of \[ \lim_{x\rightarrow 0}x \sin \dfrac {1}{x}\ is \] (A) 1
(B) π
(C) 0
(D) ∞
(B) π
(C) 0
(D) ∞
1 M
1 (a) (ii)
\[ y=\log (\sin x )\ then \ value \ of \ \dfrac {d^2y}{dx^2}=\ \]
(A) sec2 x
(B) -cosec2 x
(C) -cosec xcotx
(D) secxtanx
(A) sec2 x
(B) -cosec2 x
(C) -cosec xcotx
(D) secxtanx
1 M
1 (a) (iii)
Curve: x=t2-1, y=t2-t. For which value of t tangents of the curve are parallel to the x-axis? \[ (A) \ t=0 \\ (B)\ t=\dfrac {1} {\sqrt{3}} \\ (C)\ t=\dfrac {1}{2}\\ (D) \ t=-\dfrac {1}{\sqrt{3}} \]
1 M
1 (a) (iv)
\[ \int^{1}_{-1}x|x| dx= ....... \] (A) 2 (B) 1 (C) 0 (D) None of these
1 M
1 (a) (v)
\[ \begin {align*} If \ f(x) &=4-2x,x<1 \\ &=6x-4, x \ge 1 \ then \ find \ \lim_{x\rightarrow 1} f(x) \end{align*} \] (A) -2 (B) 2 (C) 0 (D) doesn't exist.
1 M
1 (a) (vi)
If f:R → R, f(x) = 3x+2, then find f-1
(A) Not possible
(B) 3x-2
(C) 2x-3
\[ (D) \ \dfrac {x-2}{3} \]
(A) Not possible
(B) 3x-2
(C) 2x-3
\[ (D) \ \dfrac {x-2}{3} \]
1 M
1 (a) (vii)
What is the solution for following equations ?
2x+3y-5=0, 4x+6y-7=0
(A) No solution
(B) Infinitely many solution
(C) unique solution
(D) { (0,0), (-5, -7) }
2x+3y-5=0, 4x+6y-7=0
(A) No solution
(B) Infinitely many solution
(C) unique solution
(D) { (0,0), (-5, -7) }
1 M
1 (b) (i)
\[ \lim_{n\to \infty}\dfrac {1-n^2}{\sum n}= \] (A) -2 (B) 1 (C) 2 (D) doesn't exist
1 M
1 (b) (ii)
\[ \int \dfrac {1}{\sqrt{x}}\cos \sqrt{x}dx=....+c \\(A) \ \dfrac {2}{3}\cos^{\frac {3}{2}}x \\ (B) \ 2 \sqrt{x} \\ (C) 2 \sin \sqrt{x} \\ (D) \ \dfrac {2}{3} \cos \sqrt{x} \]
1 M
1 (b) (iii)
Find the area of the curve y=x2 + 1 bounded by x-axis and lines x=1 and x=2.
\[ (A) \ \dfrac {3}{10} \\ (B) \ \dfrac {10}{3} \\ (C) 6 \\ (D) \ \dfrac {1}{6} \]
\[ (A) \ \dfrac {3}{10} \\ (B) \ \dfrac {10}{3} \\ (C) 6 \\ (D) \ \dfrac {1}{6} \]
1 M
1 (b) (iv)
f(x) = [x], x ∈ R then f(x) is
(A) Continuous for all real numbers.
(B) Continuous for all integers
(C) discontinuous for all integers
(D) none of these
(A) Continuous for all real numbers.
(B) Continuous for all integers
(C) discontinuous for all integers
(D) none of these
1 M
1 (b) (v)
Find the vertical asymptotes of the curve \[ y=\dfrac {2x^z}{x^2-1} \] (A) x=1
(B) x=-1
(C) x= ± 1
(D) y=2
(B) x=-1
(C) x= ± 1
(D) y=2
1 M
1 (b) (vi)
f(x) =x3-3x2+3x-100 is ....... function.
(A) Increasing
(B) decreasing
(C) constant
(D) none
(A) Increasing
(B) decreasing
(C) constant
(D) none
1 M
1 (b) (vii)
Find the value of c using Lagrange's mean value theorem for f(x)=ex, x ∈ [0,1]
(A) log 1
(B) log c
(C) log (1-e)
(D) log (e-1)
(A) log 1
(B) log c
(C) log (1-e)
(D) log (e-1)
1 M
2 (a) (i)
\[ Evaluate \ \lim_{x\rightarrow 0}\dfrac {e^x+e^{-x}-x^2-2}{\sin^2 x-x^2} \]
2 M
2 (a) (ii)
Test for convergence the series \[ \dfrac {1}{1.2}+ \dfrac {1}{2.3}+ \dfrac {1}{3.4}+ ..... \]
2 M
2 (a) (iii)
Expand log sin x in powers of (x-2)
3 M
2 (b) (i)
Test for convergence the series \[ 2+\dfrac {3}{2}x+\dfrac {4}{3}x^2+\dfrac {5}{4}x^3+...... \]
4 M
2 (b) (ii)
Trace the curve r=a sin 3θ
3 M
3 (a) (i)
\[ (i) \ Evaluate \ \lim_{x\rightarrow 0}(a^x+x)^{\frac {1}{x}} \]
(ii) Test the convergence of \[ \sum^{\infty}_{n=0} \dfrac {3^{2n}}{2^{3n}} \]
(ii) Test the convergence of \[ \sum^{\infty}_{n=0} \dfrac {3^{2n}}{2^{3n}} \]
4 M
3 (a) (ii)
Trace the curve x3 + y3 = 3axy.
3 M
3 (b) (i)
Test the convergence of \[ \sum^{\infty}_{n=1} \dfrac {[(n+1)x]^n}{n^n+1} \]
4 M
3 (b) (ii)
Evaluate improper integral \[ \int^{3}_0 \dfrac {1}{\sqrt{3-x}}dx \]
3 M
4 (a) (i)
\[ (i)\ If \ u=ln \left ( \dfrac {x^7+y^7+z^7}{x+y+z}\right ) \ show \ that \ x\dfrac {\partial u}{\partial x}+ y\dfrac {\partial u}{\partial y}+z\dfrac {\partial u}{\partial z}=6\]
(ii) State modified Euler's theorem and find the degree of homogeneous function \[ u(x,y)=\dfrac {1}{x^2}+\dfrac {1}{xy}+\dfrac {\log x-\log y}{x^2} \]
(ii) State modified Euler's theorem and find the degree of homogeneous function \[ u(x,y)=\dfrac {1}{x^2}+\dfrac {1}{xy}+\dfrac {\log x-\log y}{x^2} \]
4 M
4 (a) (ii)
z=f(x,y), xeu + e-v, y=e-u - ev then prove that
\[ \dfrac {\partial z}{\partial z}-\dfrac {\partial z}{\partial v}=x\dfrac {\partial z}{\partial x} -y\dfrac {\partial u}{\partial y} \]
\[ \dfrac {\partial z}{\partial z}-\dfrac {\partial z}{\partial v}=x\dfrac {\partial z}{\partial x} -y\dfrac {\partial u}{\partial y} \]
3 M
4 (b) (i)
\[ If \ u=e^{3xyz} \ show \ that \ \dfrac {\partial^3 u}{\partial x \partial y \partial z}= (3+27xyz+27x^2y^2z^2)e^{3xyz}\]
4 M
4 (b) (ii)
\[ \begin {align*} show \ that \ f(x,y)&=\dfrac {xy}{\sqrt{x^2+y^2}}; \ &(x,y)e (0,0) \\ &=0, \ &(x,y)=(0,0) \end{align*} \] is continuous at the origin.
3 M
5 (a) (i)
Find the extreme values of the function
f(x,y) = x3+y3-3x-12y+20.
f(x,y) = x3+y3-3x-12y+20.
4 M
5 (a) (ii)
Show that the surface z=xy-2 and x2+y2+z2 =3 have the same tangent plane at (1,1,1).
3 M
5 (b) (i)
Find the minimum value of x2 y z3 subject to the condition 2x+y+3z=a using Lagrange's method of undetermined multipliers.
4 M
5 (b) (ii)
Expand sin xy in powers of (x-1) and (y- π/2) upto second degree terms.
3 M
6 (a) (i)
Sketch the region of integration, reverse the order of integration and evaluate \[ \int^{2}_0\int^{4-x^2}_0 \dfrac {xe^{2y}}{4-y}dydx. \]
4 M
6 (a) (ii)
Evaluate the integral \[ \int^{\frac {\pi}{2}}_0\int^{1-\sin\theta}_0 r^2 \cos\theta drd\theta \]
3 M
6 (b) (i)
Evaluate the integral \[ \int^2_0\int^2_1\int^{yz}_0 xyz \ dxdydx. \]
4 M
6 (b) (ii)
Evaluate the integral \[ \int^a_0\int^{\sqrt{a^2-y^2}}_0 y^2\sqrt{x^2+y^2}dydx.\] by changing into polar co-ordinates
3 M
7 (a) (i)
Find the volume bounded by the cylinder x2 + y2 = 4 and the planes y+z=3, z=0.
4 M
7 (a) (ii)
Evaluate ∬R x2+y2 dA by changing the variables ,where R is the region lying in first quadrant and bounded by the hyperbola
x2 - y2=1, x2 - y2=9, xy=2 and xy=4.
x2 - y2=1, x2 - y2=9, xy=2 and xy=4.
3 M
7 (b) (i)
The region between the curve y=√x, 0≤x≤4 and the x-axis is revolved about the x-axis to generate a solid. Find its volume.
4 M
7 (b) (ii)
Test the convergence of the series \[ \sum^{\infty}_{n=1}\dfrac {(-1)^{n+1}}{\log n+1}\]
3 M
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