Answer the following questions by choosing the most appropriate answer:
1 (a) (i)
What is ∫x−3xdx?(a) 1−3 ln x+C(b)x−3 ln x+C(c) 1+3x2+C(d)x2−3xx2+C
1 M
1 (a) (ii)
if tany+x3=y2+1 and dxdt=−2,what is the value ofdydt at the point (1,0)
(A) -6
(B) -2.5
(C) 0
(D) 6
(A) -6
(B) -2.5
(C) 0
(D) 6
1 M
1 (a) (iii)
The values of x for which the graphs of y=x and y2 =4x intersect are
(A) 4 and 4 (B) -4 and 4 (C) 0 and 4 (D) 0 and -4
(A) 4 and 4 (B) -4 and 4 (C) 0 and 4 (D) 0 and -4
1 M
1 (a) (iv)
The value of the limit limx→0tanxx is
(A) 0
(B) 1
(C) π
(D) ∞
(A) 0
(B) 1
(C) π
(D) ∞
1 M
1 (a) (v)
if y=ex−e−xex+e−x then the derivative of the function y w.r.t. x is (A) 0(B) 1(C) 2(ex+e−x)2(D) 4(ex+e−x)2
1 M
1 (a) (vi)
if y=ln(√2 x) the derivative of the function y w.r.t. x is (A) √2x(B) 1√2 x(C) 12x(D) 1x
1 M
1 (a) (vii)
The sum of the squares of two positive numbers is 200; their minimum product is
(A) 200 (B) 25 √7 (C) 28 (D) none of these
(A) 200 (B) 25 √7 (C) 28 (D) none of these
1 M
Answer the following questions by choosing the most appropriate answer
1 (b) (i)
The value of the integral ∫sin√x√xdx is(A) −2cos1/2x+C(B) −cos√x+C(C) −2cos√x+C(D) 12cos√x+C
1 M
1 (b) (ii)
The value of the limit limx→0|x|x is
(A) 0 (B) 1
(C) π (D) ∞
(A) 0 (B) 1
(C) π (D) ∞
1 M
1 (b) (iii)
The value of the integral ∫ln yy2 is (A) 1y(1−ln y)+C(B) 12yln2 y+C(C) 13y3(4 ln y+1)(D) −1y(ln y+1)+C For
1 M
1 (b) (iv)
f(x) = x2 + 2x -1, 0 < x < 1 the value of C is mean value theorem is
(A) /2 (B) 0 (C) 1 (D) 1/3
(A) /2 (B) 0 (C) 1 (D) 1/3
1 M
1 (b) (v)
The total area bounded by x-axis and x y sin = is equal to (a) 4 (b) 1 (c) -1 (d) 0
1 M
1 (b) (vi)
if A=[−5−321] then A−1 is:(A) [−5−32−1](B) [−1−325](C) [13−2−5](D) [53−21]
1 M
1 (b) (vii)
The function 2x3 + 3x2 - 12x + 7 is decreasing in
(A) [-2,1] (B) -[-2,1] (C) [0,2] (D) [1,3]
(A) [-2,1] (B) -[-2,1] (C) [0,2] (D) [1,3]
1 M
2 (a) (i)
Test the convergence of the series 1.232.42+3.452.62+5.672.82+....
3 M
2 (a) (ii)
Find value of xfor which the given series 12√1+x23√2+x44√3+x65√4+.... converges.
4 M
2 (b) (i)
Determine convergence or divergence of series ∞∑n=1(2n2−1)13(3n3+2n+5)14
3 M
2 (b) (ii)
Determine absolute or conditional convergence of the series ∞∑n=1(−1)n⋅n2n3+1
4 M
3 (a) (i)
Find the expansion of tan(x+π4) in ascending powers of x upto terms in x4 and find approximately the value of tan 43°.
4 M
3 (a) (ii)
Prove that: tan−1(√1+x2−1x)=12(x−x33+x55−x77+....)
3 M
3 (b) (i)
Evaluate(i) limx→0(cosx)cotx(ii) limx→∞(a1x−1)x
4 M
3 (b) (ii)
Express 5+4(x-1)2-3(x-1)3+(x-1)4 in ascending powers of x.
3 M
4 (a) (i)
Evaluate the iterated integral ∫10∫1xsin(y2)dy dx
3 M
4 (a) (ii)
Evaluate the integral ∫4a0∫yy24ax2−y2x2+y2dx dy by transforming into Polar coordinates.
4 M
4 (b) (i)
Evaluate the triple integral ∫10∫π0∫π0ysinz dx dy dz.
3 M
4 (b) (ii)
Find the area common to both of the circles r=cos θ and r=sin θ
4 M
5 (a) (i)
Determine the set of points at which the given function is continuous f(x,y)={3x2yx2+y2if(x,y)e(0,0)0if(x,y)=(0,0)
4 M
5 (a) (ii)
If z=x2y+3xy4, where x= sin 2t and y=cos t, find dxdt when t=0
3 M
5 (b) (i)
If u=sin−1x+y√x+√y then prove that:(A) 2x∂u∂x+2x∂u∂y=tanu(B) x2∂2u∂x2+2xy∂2u∂x∂y+y2∂2u∂y2=14(tan−3u−tanu)
4 M
5 (b) (ii)
If z=x+yx, prove that ∂2z∂x∂y=∂2z∂y∂x
3 M
6 (a) (i)
Find the volume of the solid obtained by rotating the region enclosed by the curves y=x and y=x2 about the x-axis.
3 M
6 (a) (ii)
Trace the witch of agnessi xy2=4a2 (a-x)
4 M
6 (b) (i)
A rectangular box without a lid is to be made from 12m2 of cardboard. Find the maximum volume of such a box.
4 M
6 (b) (ii)
Find the equations of the tangent plane and normal line at the point (-z, 1, -3) to the ellipsoid x24+y2+x29=3.
3 M
7 (a) (i)
If u=f(xy,yx,zx), prove that x+∂u∂x+y∂u∂y+z∂u∂z=0
4 M
7 (a) (ii)
Evaluate the limit: limx→1(1x2)1log(1−x)
3 M
7 (b) (i)
Evaluate∫a0∫ayxx2+y2dxdy by transforming into polar coordinates.
3 M
7 (b) (iii)
Find the interval of convergence of the series ∞∑n=0(−3)nxn√n+1
4 M
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