B.Sc. 1st year Mathematics
705 videos • 1,177 views • by Doctor of Mathematics
1
Prove that the inverse of a non-singular matrix is unique.
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2
prove that the necessary and sufficient conditions for the existence of the inverse of a square matr
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3
Show that cosπ/7, cos3π/7 and cos5π/7 are the roots of the equation,8x^3-4x^2-4x+1=0 and deduce the
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4
Separate cos-1(cosθ+isinθ) into real and imaginary parts
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5
Solve the equation x^9+1=0
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6
Prove that the equation cos2θ+a.cosθ+b.sinθ+c=0 has the sum of the four roots is a multiple of 2π.
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7
(1-1/2^2)(1-1/3^2)(1-1/5^2)(1-1/7^2).......=6/π^2,where 2,3,5,7,.....are all prime numbers.
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8
Prove that 1/1^2+1/2^2+1/3^2+......=π^2/6
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9
Angle between two planes in vector form
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10
sin(α)+sin(α+β)+sin(α+2β)+sin(α+3β)+....upto n terms
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11
Expand tan-1x (tan inverse x ) in the power of (x-π/4) by Taylor's theorem.
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12
√1+sin2α +√1+sin2(α+β) +√1+sin2(α+2β)+.........upto nterms. Find the sum
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13
Find the divergence of r/r
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14
Prove that 1-1/3^3+1/5^3+..........=π^3/32
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15
2-Distribution Law for vectors:Vecor Analysis
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16
If cos(α+iβ)=e^(iθ),prove that sin^2(α)=±sinθ
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17
Separate cosech(x+iy) into real and imaginary parts
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18
Separate sech(x+iy) into real and imaginary parts
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19
Find the asymptotes of curve x^2.y^2-x^2.y-x.y^2+x+y+1=0
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20
Prove that the asymptotes of the curve (x^2-y^2).y-2ay^2+5x-7=0 form a triangle of area a^2.
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21
condition that xcosα+ysinα=p should touch x^m.y^n=a^(m+n) is p^(m+n).m^m.n^n=(m+n)^(m+n).a^m+n.sin^α
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22
If cos(θ+iϕ).cos (α+iβ)=1,prove that tanh^2(ϕ).cosh^2(β)=sin^2(α) and tanh^2(β).cosh^2(ϕ)=sin^2(θ).
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23
State and prove De Moivre's Theorem
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24
Prove that tan−1n+cot−1 (n+1) = tan−1 (n 2 +n+1) and if tan−1 x+tan−1 y+tan−1 z = π/2
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25
State and prove Euler's theorem.If u=sin-1(x+y)/(√x+√y) then show that xdelu/delx+ydelu/dely=1/2sinu
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26
Separate Cosh(x+iy) into real and imaginary parts.
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27
Separate sin(x+iy) into real and imaginary parts
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28
Find the area of the loop of the curve ay^2=x^2.(a-x) [B.Sc. Mathematics:Quadrature in hindi/urdu]
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29
Find the equation of the cone whose vertex is (1,1,0)and the guiding curve is y=0,x^2+z^2=4.
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30
If y=e^m.cos-1x,then show that (1-x^2)yn+2-(2n+1)x.yn+1-(n^2+m^2)yn=0
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31
If tan(θ+iϕ)=cosα+isinα,prove that θ=nπ/2+π/4 and ϕ=1/2.logtan(π/4+α/2)
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32
If (1+i)^p+iq/(1-i)^p+iq=a+ib,prove that one value of tan-1(b/a) is pi.p/2+q.log2
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33
Find the necessary and sufficient condition that the cone has set of three mutually perpendicular ge
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34
If coshu=secθ,show that u=logtan(π/4+θ/2) and tanh^2(u/2)=tan^2(θ/2)
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35
Lt x tends to 0 logx^2/logcot^2x
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36
Find the asymptotes of curve x^3-2x^2y+xy^2+x^2-xy+2=0
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37
Find the asymptotes of the curve (x^2-y^2)(x+2y+1) +x+y+1=0.
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38
If sin(θ+iϕ)=cosα+isinα,prove thatcos^2(θ)=±sinα and sinh^2(ϕ)=±sinα
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39
Prove that AUB=BUA (commutatuve law in union of sets)
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40
❌Lt x tends to 0 sinx.logx[I committed a mistake so differentiation of cosec x =-cosecx.cotx]
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41
Find the centre,vertices,foci,eccentricity,directrix and asymtotes of hyperbila x^2-4y^2-2x+8y-2=0
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42
logsin(x+iy)
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43
Lt x tends to 0. [1/2x^2-1/2xtanx]
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44
Expand tan-1x in terms of (x-1) upto non-zero three terms||Top Mathematics channel on YouTube||Best
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45
Find the radius of curvature of curve x^3+y^3=3axy
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46
Find the volume of solid formed by revolution of ellipse x^2/a^2+y^2/b^2=1 about x-axis.
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47
part 2:Length of arc along parabola y^2=4ax from vertex to on extrimity of latus rectum part2/2
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48
part 1:Length of arc along parabola y^2=4ax from vertex to on extrimity of latus rectum part1/2
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49
Trace thw curve y^2=x.(x-1)^2 and find the area of the loop.
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50
If y=sin(sinx); prove that d^2y/dx^2+tanx.dy/dx+y.cos^2x=0
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51
Analyse the equation 2x^2-3x+5y+4=0 and sketch the curve
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52
Find the orthogonal trajectory of the family of curve r=a+cosnθ.
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53
Find (x^2.e^x.cosx)n.
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54
Show that the set of matrices A alpha [cos -sin ,sin cos] where is a real number ,form a group unde
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55
Prove that the set of all integer I together with binary operation + forms an abelian group.
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56
limit x tends to a log(x-a)/log(e^x-e^a
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57
Show that the area included between curve y^2=x^2(a+x)/(a-x) and its asymptote is 1/2(pi+4)a^2.
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58
Show that the identity element in a group is unique.
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59
Find the whole area of the curve r=a(1-cosθ).
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60
whether the function f(x,y)=xy/√(x^2+y^2); (x,y)≠(0,0); 0,(x,y)=(0,0) is continuous at the origin?J
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61
The loop of the curve 2ay^2=x(x-a)^2 revlved about the x-axis .Find the volume of the solid so gener
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62
Part1:Sketch the curve y^2(4-x)x^3 and find the area bounded by the curve and its asymptote.
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63
Find the length of an arc of the parabola y^2=4ax measured from the vertex to one extremity of the
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64
Solve the initial value problem d^2x/dt^2-3dx/dt+2x=0 ;x(0)=2;x'(0)=0
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65
Solve p-2q=3x^2sin(y-2x)
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66
Show that the entire area of x^2/3+y^2/3=a^2/3 is 3/8.πa^2.
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67
The sum of three positive number is unity.What is the maximum value of their product.
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68
Integrate w.r.t x (1+x^2)/(1-x^2).√(1+x^2+x^4)
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69
cosπ/3+1/3cos2π/3+1/5cos3π/3+1/7cos4π/3+..........to infinity.(summation of series)
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70
Evaluate the double integral √x^2+y^2 dy dx x=0 to1 and y=0 to x
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71
Evaluate the area of the region that lies inside circle r=acosθ and outside cardioid r=a(1-cosθ)
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72
Find the condition that plane ax+by+cz+d=0 touches the sphere x^2+y^2+z^2+2ux+2vy+2wz+d1=0.
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73
If α =log[tan(π/4+θ/3)] prove rhat sinhα=tanθ
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74
Prove that tan-1(tanhy.cotx)=tan-1y/x-sigma n=1 to infinity tan-1[2xy/n^2pi^2-x^2+y^2]
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75
If u+iv=cot(x+iy); v=-sinh2y/(cosh2y-cos2x)
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76
Show that the vector (siny+z,xcosy-z,x-y)irrotational
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77
Prove that : div.[rgradr^(-4)]=3r^(-4) or div.[r grad r^-3]=3r^-4
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78
Check for the linear dependence of the following system of vectors:u=(1,-1,1),v=(2,1,1),w=(3,0,2).if
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79
Let f(x,y,z)=x^3+y^3+z^3-3xyz;a point at which the gradient of the function f is equal to zero is:
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80
Separate real and imaginary parts (1+c.cosα+ic.sinα)^1/2
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81
Separate into real and imaginarg parts e^iϕ/1-c.e^iϕ
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82
Find the radius of curvature at point (r,θ) of the curve r(1+cosθ)=a
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83
Find the condition that the line lx+my+n=0 is tangent to the conic ax^2+2hxh+by^2+2gx+2fy+c=0
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84
Show that f(x)=cos(1/x) when x≠0,f(0)=0 is discontinuous at origin.
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85
Show that the following function is discontinuous at the origin: f(x)=1/1+e^1/x
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86
Examine the following function for continuity at x=2, f(x)=(x^2-4)/(x-2),f(2)=0
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87
With usual notations,for any curve prove d^2r/ds^2=sin^2(ϕ)/r-sin(ϕ)/ρ ;d^2r/ds^2=sin^2ϕ/r-sinϕ/ρ
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88
Prove that the lines x-1/2=y-2/3=z-3/4;x-2/3=y-3/4=z-4/5 are coplanar,find their point of intersect
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89
Find the shortest distance between two skew lines given by x-α/l=y-β/=z-/n and x-α1/l1=y-β/m1=z-/n1.
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90
Find the conditions that the line x-α/l=y-β/m=z-γ/n,may lie in the plane ax+by+cz+d=0
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91
e^ax.sinbx=bx+abx^2+(3a^2b-b^3)x^3/3!+....
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92
IF u=(x^2+y^2+z^2)^-1 then prove that del^2u/delx^2+del^2u/del^2y+del^2u/delz^2=
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93
Evaluate Double integration of 1/√x^2+y^2+1 w.r.t. y and x with limits
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94
Double integration of 1/√x^2+y^2+1 w.r.t. y and x with limits
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95
Find the radius of curvature at the point (r,θ) on the curve θ=√r^2-a^2/a-cos-1(a/r)
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96
Show that the functions u=x+y-z,v=x-y+z,w=x^2+y^2+z^2-2yz are not independentof one another.Also fin
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97
prove that cos[i.ln(a+ib)/(a-ib)]=(a^2-b^2)/(a^2+b^2)
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98
Prove that ln(1+r.e^iθ)=1/2ln(1+2r.cosθ+r^2)+i.tan-1(r.sinθ/1+r.cosθ)
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99
Prove that Sin-1(cosecθ)={2n+(-1)^n}π/2+i.(-1)^n.logcotθ/2
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100
Prove that the set of number form a+b√2 where a and b are rational number,is a field under addition
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101
Using (,) epsilen-delta definition show that lim x tends to x.sin(1/x)=0
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102
Prove that tan-1(tan2θ+tanh2ϕ/tan2θ-tanh2ϕ)+tan-1(tanθ-tanhϕ/tanθ+tanϕ)=tan-1(cotθ.cothϕ)
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103
Find all the asymptotes of the curve given by:x^2y-xy^2+xy+y^2+x-y=0
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104
n.sinα+n(n+1)sin2α/1.2+n(n+1)(n+2)sin3α/1.2.3.+.......sum upto infinity
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105
If coshu=secθ then prove that sinhu=tanθ
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106
If y=x^2.sinx,prove that D^n(y)=(x^2-n^2+n)sin(x+n.π/2)-2nxcos(x+n.π/2).
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107
Trace the curve (Cissoid) r=a.sin^2θ/cosθ
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108
B.Sc. Mathematics : Show that the whole length of the cardiod r=a (1+cos@) is 8a.
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109
If sin(x+iy)=tanA+i.secA then prove that cos2x.cosh2y=3.
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110
Find the area of circle of radius 'a' using Green's theorem.||Best Mathematics Channel on YouTube||
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111
If y=sin-1x/√1-x^2, prove that (1-x^2)yn+1-(2n+1)xyn-n^2yn=0
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112
Find the surface of the solid generated by revolving the arc of the parabola y^2=4ax bounded by its
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113
Integrate (Sin-1x)^4 with respect to x
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114
If tan-1(tanx+i.secx)=a+ib, prove that a=(2n+1)π/4+x/2 and b=1/2logcotx/2
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115
Show that the principal value of (x+iy) ^(a+ib) is wholly real and imaginary according as 1/2b.log(
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116
Apply Maclaurin theorem to obtain the term upto x^4 in the expansion of log(1+sin^2)
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117
Solve:(D+D'-1) (D+D'-3) (D+D')z =e^x+y.sin(2x+y) non-homogeneous Partial Differential Equation.
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118
grad(a.b)=ax curl b+bx curl a+(a.del)b +(b.del)a
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119
Del x(fxr) =2f-(del.f) r+(r.del) f
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120
If a is a constant vector, then prove that del(a.u) =(a.del)u +axcurl u
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121
del. (axu) =-a.curl u
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122
I/log(1+i) =A+iB, find values of A and B.
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123
The rank of matrix A and it's transpose are same.
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124
Find the radius of curvature at any point 't' on the curve x=a.cost, y=b.sint.
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125
If x=cosh(1/m.logy) , prove that (x^2-1) yn+2+(2n+1) xyn+1+(n^2-m^2) yn=0
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126
Explanation of Shab-e-miraj and Isra on basis on special theory of relativity
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127
Find the length of curve y=log[tanh(x/2) ] from x=1 to x=2.
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128
tan-1(4/1+3.4) +tan-1(6/1+8.9)+tan-1(8/1+15.16) +... n terms
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129
Find the length of the arc of the curve y=x(2-x) as x varies 0 to 2.
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130
Show that the functions u=x/y-z, v=y/z-x, w=z/x-y are not independent of one another. find relation
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131
Find the length of the arc of the curve x=t^2.cost, y=t^2.sint from origin to t I. e. 0 to t.
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132
Show that the functions u=x^3+x^2y+x^2z-z^2(x+y+z),v=z+x,w=x^2-z^2+xy-zy are not independent.find re
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133
Find maximum and minimum of x^3y^2(1-x-y)
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134
Sn sum of n term sinθ+sin2θ+sin3θ+... prove that Lt n tends to infinity S1+S2+S3+.. +Sn/n=1/2.cotθ/2
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135
The loop of curve 2ay^2=x(x-a) ^2 revolves about the straight line y=a.Find volume of the solid so
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136
Discuss the convergence of following series 1+( √ 2-1) /1! +( √ 2-1) ^2/2! +( √ 2-1) ^3/! +......
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137
Find the length of the curve y=x^3/2 from x=0 to x=4.
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138
Evaluate integration of coshx+sinhx.sinx/1+cosx
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139
G-74:prove that curl r=0 vector. del x r=0
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140
If the directional derivative Φ=axy^2+byz+cz^2x^3 at (-1,1,2) has maximum magnitude of 32 units
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141
Find the area included between the curve y^2(a-x) =x^3, and it's asymptote. cissoid
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142
To prove that a.cosbx+b.sinbx/a^2+b^2=cos(bx-tan-1b/a) /squ.roota^2+b^2=
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143
Find pedal equation of the curve x^2/3+y^2/3=a^2/3
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144
dr/dθ+r/=aθ^n
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145
Find the intervals in which curve y=(x^2+4x+5) e^-x is concave upward or downward.find inflexion poi
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146
Integrate e^x.(1-x/1+x) ^2dx
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147
Grad 1/r
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148
If tan-1(a+ib) =sin-1(x+iy), show that a^2+b^2=x^2+y^2/x^4+y^4+2x^2.y^2-2x^2+2y^2+1
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149
Find the orthogonal trajectories of the system of curves given by r=a+sin5θ.
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150
Show that [a+b b+c c+a]=2[a b c] all vectors (Vector Algebra)
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151
Prove that if (1+I.tanα) ^1+itanβ can have only real values, one of them is (secα) ^sec^2β
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152
ax(bxc) +bx(cxa) +cx(axb) =0 all vectors
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153
Evaluate integration of x. e^-1ax.cosbxdx
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154
Show that the equation l/r=1+e.cos and l/r=-1+e.cos represent the same conic.
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155
Using Taylor's Theorem show that tan(π/4+x) =1+2x+2x²+8/3x^3+....
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156
Using Taylor's theorem show that tan(π/4+x) =1+2x+2x^2+8/3.x^3+....
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157
If cos(α+iβ) =r(cosθ+isinθ), then find the β
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158
Integrate Cos^3/e^3x w.r.t. x
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159
Find the radius of curvature of curve the cardiod r=a(1-cosθ)
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160
Expand log[cos(x+π/4)]
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161
Show value of the solid by revolving area included between y^2=x^3 and x^2=y^3 about x- axis 5π/28.
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162
If p=cisθ and q=cisΦ, prove that (p+q) (pq-1) /(p-q) (p-q+1) =sinθ+sinΦ/sinθ-sinΦ
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163
Trace the curve a^2.y^=x^3.(2a-x)
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164
Find the volume of solid generated by revolution of one arc of the cycloid x=a(θ-sinθ);y=a(1-cosθ)
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165
Lt x tends to infinity x^n/e^x ; n being a positive integer.
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166
1+1/3.y.cosα+1.4/3.4.y^2.cos2α+1.3.7/3.6.9.y^3.cos3α+.....sum the series.
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167
Sum the following series: sinhα+n.sinh2α+n(n-1)/2!. sinh3α+... to (n+1) terms, where n is a positive
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168
Test the convergence of the following series 1+3/7.x+3.6/7.10.x^2+3.6.9/7.10.13.x^3+...............
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169
#bsctrigonometry Separate ecp(e^iθ) into real and imaginary parts
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170
Separate exp{(x+iy)(α+iβ)} into real and imaginary psrts
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171
Separate tanh-1z into real and imaginary parts.
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172
Test the convergence of the following series 1+3/7.x+3.6/7.10.x^2+3.6.9/7.10.13.x^3+...............
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173
1+1/3.y.cosα+1.4/3.4.y^2.cos2α+1.3.7/3.6.9.y^3.cos3α+.....sum the series.
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174
Sinhα+nsinh2α+n(n-1)/2!sing3α+... to (n+1)terms, where n is a positive Integer. #summation_of_series
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175
Define commutative and associative binary operation on a set and give examples
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176
Separate sinai/(x+iy) into real and Imaginary parts
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177
Find the volume of the solid that result when the region enclosed by x=y^2and y=x is revolved about
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178
If logsin(θ+iΦ) =α+iβ ,prove cos(θ-β)=e^2Φ.cos(β+θ)
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179
Integrate(2x-2) /_/x^2-2x+5 with respect to x
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180
tan-1(e^ix) -tan-1(e^-ix) =tan-1(i)
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181
Find the sum to n terms the series sin^3(x/3) +3sin^3(x/9) +9sin^3(x/27)+...
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182
Sum to n terms, and to infinity, the series 1+c.cos+c^2cos2+....where c is less than one numerically
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183
If tan(α+iβ) =tanθ+isecθ, show that e^2β=cotθ/2, α=nπ/2+π/4+θ/2
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184
Express tanh-1(x+iy) in form A+iB and hence the value of tanh-1(iy)
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185
verify stokes'theorem for A=(y-z+2)I+(y-z+4)j -xzk,S is surface of cube x=0,y =0,z=0,x =2,y=2,z=2
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186
If u=(x^1/4+y^1/4)/(x^1/5+y^1/5) ;then the value of x∂u/∂x+y∂u/∂y
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187
If u=sin-1(√x-√y/√x+√y) , prove that ∂u/∂x=-y/x.u∂/∂y
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188
If u=cos-1(x+y/√x+√y) , prove that xdelu/delx+ydelu/dely+1/2.cotu=0
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189
If u=tan-1(x²+y²/x+y), prove that x∂u/∂x+y∂u/∂y=1/2.sin2u
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190
tan-1[(tan2x+tanh2y) /(tan2x-tanh2y)] +tan-1[(tanx-tanhy) /(tanx+tanhy)] =tan-1(cotxcithy)
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191
Verify Euler's theorem for the homogenous function:u=x^n.log(y/x)
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192
If u=ax²+2hxy+by², prove that x∂u/∂x+y∂u/∂y=2u
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193
If u=sin-1(x+y/√x+√y) , prove that xdelu/delx+ydelu/dely=1/2.tanu
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194
Verify Euler's theorem for the function u=1/√(x^2+y^2)
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195
If u=log[(x^3+y^3) /(x+y)] , prove that x.delu/x+y.delu/dely=2.
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196
If u=x^2y^2/(x^2+y^2), prove that
Doctor of Mathematics
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197
u=x²y²/(x²+y²), prove that x²∂²u/∂x²+2xy∂²u/∂x∂y+y²∂²u/∂y²=2u.
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198
If V=logsin {π(2x^2+y^+xz)^1/2/2(x^2+xy+2yz+z^2)^1/3}, find x∂V/∂x+y∂V/∂y+z∂V/∂z
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199
If u=tan-1y/x, prove that x∂u/∂x+y∂u/∂y=0
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200
Show that in the exponential curve y=be^x/a, the subtangent is of constant length and subnormal vari
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