1(a)
Evaluate the following:
3 M
1(b)
Solve (D2-1)(D-1)2y=0
3 M
1(c)
Prove that E=1+Δ=ehD
3 M
1(d)
Solve the following:
3 M
1(e)
Change into polar co-ordinates and evaluate:
4 M
1(f)
Evaluate the following:
4 M
2(a)
Solve (x3y3-xy)dy=dx
6 M
2(b)
Change the order of Integration and evaluate:
6 M
2(c)(1)
Prove that:
4 M
2(c)(2)
Evaluate
, where a>0
4 M
3(a)
Evaluate the following:
6 M
3(b)
Find the area bounded by 9xy=4 and 2x+y=2
6 M
3(c)(1)
Solve the following:
4 M
3(c)(2)
Solve the equation by variation of parameters:
4 M
4(a)
Show that for the parabola
from θ=0 to θ=π/2, length of the arc is
6 M
4(b)
Solve the following:
6 M
4(c)
Apply Runge-Kutta method of fourth order to find an approximation value of y at x=0.2, if dy/dx=x+y2, given y=1 when x=0, in steps of h=0.1
8 M
5(a)
Solve: (2xy4ey+2xy3+y)dx+(x2y4ey-x2y2-3x)dy=0
6 M
5(b)
Solve dy/dx=2x+y with x0=0,y0=0 by Taylor’s method. Obtain y as a series in power of x. Find approximation value of y for x=0.2,0.4. Compare your result with exact values.
6 M
5(c)
Evaluate the following equation
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAHYAAABECAIAAADbd1bgAAAK9ElEQVR4nO2ce1BMXxzAv1ubTV5pGY/1ardiNo+iKBmsjMeUCcmQ8n7M5DlYYmnQMux41iSlFA1ZeUxMJExNHtHQEFZRu5VK2qRtd6Wtu/f3xxk7+0tp9+69G02f/5w9j+9+3D333Hu+J8C7oBjo6AA6P12KKadLMeV0KaacLsWU06WYcroUU46ZFKvV6pycnKamJvMM91dBuWIMw65fvz558uQJEyY0NzdTPdxfCLWKy8rKQkJCAgICWCyWs7MzhmGUDtcqjY2NWq3W/OPqoFZxVVVVZmamVqtdvXq1k5OT2RTfv39/3bp1mzZtWrZsGZ/P79hfj5nm4pUrV5pTsUwm4/F4AAAAS5cuNc+gbUGmYq1W29ZPcsWKFeZUjON4aGgoUnz27FmzDdoq5CiuqqpavXo1k8lks9lCofD3CuZX7OfnBwB0Ov3ly5dmG7RVSFAsl8snTJgwZcqUQ4cOAcDixYt/r2OeiaKoqOjy5cspKSmZmZkODg4AwGaz6+rq9Os8fvz4xo0bpaWlupLc3Nxr165VV1dTFBUJijdv3tyvX7/Kykocx2NiYlJTU3+vExQU5OjoaPpYbVFTU7N8+XIGg8HhcDw8PHr37k2j0QDA399f9/9aXV3t7+8PADQabfTo0chyXFwcnU4HgLt371IUm6mKS0pK+vTps2XLllY/1Wq1Dx8+FIlEdnZ2lpaWYWFh6enparXaxEFbIJVKp06dCgCenp6FhYVyudzd3R1NxEeOHNFVCw0NnT9//r59+9BHkZGRjx8/7tOnD5fLdXJykslk5Ealw1TFCQkJAPDgwYNWP21ubj537tyBAweEQuGRI0fCwsKioqJqampMHFQfDMOWL18OAD169NBNu4GBgWgivn37tq6mTCZTKpXfv38fMWIEAEycOJHD4ezevVupVBYXF1M3iZmqeNmyZRYWFvpTm5l5/vw5g8EAgCVLlugKvb29AYDFYn369KlF/aamJh8fH916zgx3YJMUYxg2cuTIAQMGyOVysgIylrCwMOQrKSkJlcjlcnt7ewDw8vJq9a0In88HAAaDkZ+fb4YITVKMJmIXFxelUklWQMaC5gRLS8vMzExUkpaWZmVlBQAhISG/19dqtYsWLQIACwuLtLQ0M0RokuLs7GwajTZ37twOfEL18PAAgMGDBxcUFKASNDUDQGJi4u/1Dxw4YGVl1a1bNwAQiURmiNAkxVevXgWAwMBAsqIhgJubGwBwOBx0F7148SLyS6PRCgoK6uvr0WxQWloql8vj4+P79u0bExMzdOhQAAgODsZxXCKRVFRUUBehSYrPnDkDANu2bSMrGgIEBwcDgJWVVWho6N69e318fFatWoUsHzx4kMfjbdmyBcOwGTNmcLlcOp1+/PhxHMenT58OACNGjBAKhaNHj3769Cl1EZqkeP/+/QCwd+9esqIhwKtXr0aOHIksz5w5s7CwsLy83NXVFQCGDh26ceNGhUJRXl6OpO/fvx+1Sk1N7devHwAMGTLk/PnzlL7tNEnxhg0bWizvO4Ty8vKMjIxnz541NjaiksrKyszMTN2KTaFQpKen5+bm6rfKz8+/d+9eUVER1eERV6y7NZ8+fZrEgDofxBU3NDSgd7Lnz58nMaDOB3HFdXV1aMpLTk4mMaDOB3HFX79+ZbPZAHDz5k0SA+p8EFf8+fNnJpNJo9Hu379PYkCdD+KKpVIpg8Gg0+mPHj0iMaDOB3HFEokEAHr16pWTk0NiQJ0P4opzcnIAoHfv3i9evCAxoM4HccUZGRkAwGQyyXolWFlZKfunQHtp7UJc8e3btwFg4MCBhYWFhDvRx8XFBf4pxo8fb8j3Iq44KSkJAIYNG1ZWVka4E33y8vKy/yny8vIM+V7EFV+4cAEAOBxOB255/BMQV3zq1CkA4HK5P378IDGgv5ba2tqqqioCDYkrPn78OAC4uLh0SL6l2Whqarp582ZQUFBgYKC/v39AQECLN3btQlwxyv2ZNGkS4R7aRavV1tbWUte/IZSUlHTr1m3hwoXFxcUfPnxwdHQcNWqUUalDxBXv3LkTAGbNmkW4hz/z8uVLX1/f8PBwivo3EKVSeevWLYVCgf6JLqw3b94Y3gNxxevXrweAoKAgwj20RXl5+fbt27t37w4AbeUZtUV2dnZ4ePi3b99Ijwrh4+Pj7OxsVP/EFS9ZsgQo2LhLTk6eNGnSmjVrfH19AWDHjh1GNQ8PD7e0tPz48SO5USEuXbpkZ2eXnp5uVCviiufMmQMAx44dI9xDq1y9evXevXv4r1Su7du3G9Uc5c8Zu11UUlLy4sUL3WyA43hFRUVOTo5+SVpa2uDBg69du2ZUz7gpir28vKCNXAVSiI2NNYNijUYjEAj69u1ra2s7bdo0tNuflZVlb2/fq1evW7duoWrp6emDBg2Kj483KhgEQcUajWb8+PEAgK44KoiJiTGDYpFI5O7uvnLlSvRMfPr0aalUOmzYMFtb2549e7569QrHcYlEwuVydRdTbm5ucXGx4SERVFxbWztq1CgAeP36NbEe2oWY4sjISCaTafiiKi8vTyaTKZXK4cOHAwCPx/P09FywYEFxcfGzZ880Gs2PHz94PB6LxYqOjo6IiBCJRIMGDTIqU4ug4tLSUhaL1b17d6lUSqyHdjFEcUVFhVAoDA0NFQgEAoFg37593t7e1tbWISEhgl/s2rWr3a0vDMPQwQUA8PDw0E+sf//+/bhx48aMGePk5OTg4MBms52dnd++fWv4FyGo+M2bNzY2Ns7OztQ9GhiiuKCgYNq0aS4uLq6urq6uru7u7iwWi06nc7lcNzc3VDh27NiDBw+2O5xAIAAAGo2WlZWlX97Y2FhXV1f/C4VCoVQqjcrhI6g4KysLAObNm0fd07MhijEMU6vVKpVKrVar1eqfP38ePnzYzs7u7du3DQ0NqFClUulSWP7A1q1b0VWsn/VNCgQVi8ViABAIBATafv/+3ZA3R0ixseviEydO2NnZlZSUGNUqOTnZ2tra0tISAA4fPmxU23YhqPjYsWMA0OrJmT9QXV0tEok8PT1lBhysiI+PBwA+n2/UEEatKOrr6xsbG7Oysvr3779nzx6Us4Cy7Wtra8maAwkqXrVqlY2NzZcvXwys//Pnz6SkpPnz5w8YMIBOpxuiGOWyz50716i5yHDFGIYFBwfPmjWLyWSuXbsWx/EZM2YAAJvNvnz58pQpU8g6w0REsUqlsre3X7BggeFNFApFREREfn7+0aNHbWxs/rAOKSoq4vP5fn5+NjY2AGBhYeHt7b1p0yYD97mFQiGDwTDkAbqmpgYl0wcEBDQ0NOA4LhaL0bERW1tbPp+PCk3HCMVlZWXowMHDhw8B4M6dOwTGi4qK+vNSr6KiIi4uLiEh4cqVKykpKWKxODExMTY2ViKRGNJ/SkpKQECAIRuXdXV1MTExYrFY/2aYkZERGRmpO9NACkYonj179tq1a6VSqZeXF4/H02g0+uGmpaWltsa7d+/0Ozlz5gylq+nm5uYO/+sILTBUsVarRZnSw4cPd3Nza/FLlEgkrq6uHA7H8f/Y29u3yD6mWvFfiBFXsVgs3rFjx8mTJ3/fwsIwTKVSKZVK1f9Bt2z9mhEREV2KqSU6Otra2pq6M91/IeZTXFVV9eTJE3RyMyoq6v3792Tdsv9yzKc4Li4uKCho3rx5fn5+Cxcu3Lp1q4GLhH8d8ynWaDT6x2U1Gk3nzg7Q0fWn8CinSzHldCmmnC7FlPMfrVSiZKPClp4AAAAASUVORK5CYII=)
by Trapezoidal method, Simpson's 1/3rd and 3/8th methods. Compare result with exact values.
by Trapezoidal method, Simpson's 1/3rd and 3/8th methods. Compare result with exact values.
8 M
6(a)
In a circuit containing inductance L, resistance R, voltage E, the current i is given by L(di/dt)+Ri=E. Find i at a time t if at t=0,i=0, and if L, R, E are constants.
6 M
6(b)
Evaluate ∫∫xy dxdy bounded by y=x, x2+y2-2x=0, and y2=2x
6 M
6(c)(1)
Find the volume of a tetrahedron bounded by the plane x=0,y=0,z=0 and x+y+z=a
4 M
6(c)(2)
Find the volume bounded by the cone z2=x2+y2 and paraboloid z=x2+y2.
4 M
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