First Year (FE)
- Expand \(\sin^5\theta\cos^3\theta\) in a series of sines of multiples of \(\theta\). Let x=cosθ+isinθ : x-1/x=2isinθ,\ x+1/x=2cosθ,\ xⁿ-frac1xⁿ=2isin nθ . (2isinθ)⁵(2cosθ)³=(x-1/x)⁵(x+1/x)³=(x-1/x)²(x²-frac1x²)³ =(x²-2+frac1x... 1
- If \(\cos\alpha+\cos\beta+\cos\gamma=0,\ \sin\alpha+\sin\beta+\sin\gamma=0\), prove that (i) \(\cos3\alpha+\cos3\beta+\cos3\gamma=3\cos(\alpha+\beta+\gamma)\) (ii) \(\sin3\alpha+\sin3\beta+\sin3\gamma=3\sin(\alpha+\beta+\gamma)\) Let a=cosα+isinα,\ b=cosβ+isinβ,\ c=cosγ+isinγ . Then a+b+c=(cosα+cosβ+cosγ)+i(sinα+sinβ+sinγ)=0 . Identity: if a+b+c=0 then a³+b³+c³=3abc (... 1
- Find all the roots of \(\left(\frac12+i\frac{\sqrt3}{2}\right)^{3/4}\), hence find the continued product of the roots. Step 1 (polar form): 1/2+isqrt3/2=cosπ/3+isinπ/3 . Including the general angle, 1/2+isqrt3/2=cos(2kπ+π/3)+isin(2kπ+π/3) . Step 2 (De Moivre)... 1
- If \(\alpha+i\beta=\tanh\left(x+\frac{i\pi}{4}\right)\), prove that \(\alpha^2+\beta^2=1\). Separate real and imaginary parts: tanh(x+iy)=sinh(x+iy)cosh(x-iy)/cosh(x+iy)cosh(x-iy) . Numerator =1/2[sinh2x+sinh2iy]=1/2[sinh2x+isin2y]... 1
- Investigate for what values of \(\lambda\) and \(\mu\) the equations \(x+y+z=6,\; x+2y+3z=10,\; x+2y+\lambda z=\mu\) will have (i) No Solution (ii) Unique Solution (iii) Infinitely Many Solutions. Augmented matrix: [Amid B]=[beginarrayccc|c1&1&1&6\\1&2&3&10\\1&2&λ&μendarray] R₂→ R₂-R₁, R₃→ R₃-R₁ : [beginarrayccc|c1&1&1&6\\0&1&2&4\\0&1&... 1
- Find non-singular matrices P & Q such that PAQ is in normal form & hence find rank of the matrix A and obtain \(A^{-1}\). \(A=\begin{bmatrix}1&2&-2\\-1&3&0\\0&-2&1\end{bmatrix}\) Write A=I₃ A I₃ . R₂→ R₂+R₁ : beginbmatrix1&2&-2\\0&5&-2\\0&-2&1endbmatrix=beginbmatrix1&0&0\\1&1&0\\0&0&1endbmatrixA I₃ C₂→ C₂-2C₁,\ C₃→ C₃... 1
- Apply Gauss-Seidel iteration method to solve the following linear equations up to the three iterations. \(20x+y-2z=17,\ 3x+20y-z=-18,\ 2x-3y+20z=25\). The system is diagonally dominant. x=frac120(17-y+2z), y=frac120(-18-3x+z), z=frac120(25-2x+3y) . Start y=z=0 , using latest values. Iterati... 1
- If \(u=\sin^{-1}\left(\frac{x+y}{\sqrt x+\sqrt y}\right)\), prove that \(x^2\frac{\partial^2u}{\partial x^2}+2xy\frac{\partial^2u}{\partial x\partial y}+y^2\frac{\partial^2u}{\partial y^2}=-\frac{\sin u\cos2u}{4\cos^3u}\) Given: u=sin⁻¹(x+y/sqrt x+sqrt y)⇒ z=sin u=x+y/sqrt x+sqrt y . Step 1 (homogeneity): z(tx,ty)=t(x+y)/sqrt t(sqrt x+sqrt y)=t^(1/2)z , so z=s... 1
- Solve \(x^6+1=0\). x⁶=-1=cosπ+isinπ=cos(2k+1)π+isin(2k+1)π . By De Moivre: x=cos(2k+1)π/6+isin(2k+1)π/6,\ k=0,1,dots,5 . k=0:\ fracsqrt32+i/2 ; k=1:\ i ; k=2:\... 1
- If \(5\sinh x-\cosh x=5\), find \(\tanh x\). 5·frace^x-e^(-x)2-frace^x+e^(-x)2=5⇒4e^x-6e^(-x)=10 . Multiply by e^x and put e^x=t : 4t²-10t-6=0⇒2t²-5t-3=0⇒(2t+1)(t-3)=0 . t=e^x>0⇒ e^x=3... 1
- Using Newton-Raphson Method, find the real root of \(x^3-2x-5=0\) correct to three decimal places. f(x)=x³-2x-5,\ f'(x)=3x²-2 . f(2)=-1<0,\ f(3)=16>0 : root lies in (2, 3), nearer 2. Take x₀=2 . x_(n+1)=x_n-x_n³-2x_n-5/3x_n²-2=2x_n³+5/3x_n... 1
- If \(u=2(ax+by)^2-k(x^2+y^2)\) and \(a^2+b^2=k\), find \(\frac{\partial^2u}{\partial x^2}+\frac{\partial^2u}{\partial y^2}\). ∂ u/∂ x=4a(ax+by)-2kx⇒∂²u/∂ x²=4a²-2k ∂ u/∂ y=4b(ax+by)-2ky⇒∂²u/∂ y²=4b²-2k ∂²u/∂ x²+∂²u/∂ y²=4(a²+b²)-4k=4k-4k ∂²u/∂ x²+∂²u/∂ y²=0 1
- Find \(n^{th}\) derivative of \(\frac{x^2}{(x-1)(x-2)(x-3)}\). [The numerator is faint in the source scan; it appears to be \(x^2\).] (Solved taking the numerator as x² , as it appears in the scan.) Partial fractions: x²/(x-1)(x-2)(x-3)=frac Ax-1+frac Bx-2+frac Cx-3 . A=1/(... 1
- Examine the function \(f(x,y)=y^2+4xy+3x^2+x^3\) for extreme values. f_x=4y+6x+3x², f_y=2y+4x . Stationary points: f_y=0⇒ y=-2x . Then f_x=-8x+6x+3x²=3x²-2x=0⇒ x=0 or x=2/3 . Points: (0,0) and (2/3,-4/3) . r=f... 1
- Prove that \(\sqrt{1+\mathrm{cosec}(\theta/2)}=\left(1-e^{i\theta}\right)^{-1/2}+\left(1-e^{-i\theta}\right)^{-1/2}\). Step 1: 1-e^iθ=e^(iθ/2)(e^(-iθ/2)-e^(iθ/2))=e^(iθ/2)(-2isinfracθ2)=2sinfracθ2 e^(i(fracθ2-fracπ2)) (since -i=e^(-iπ/2) ). therefore(1-e^iθ)^... 1
- If \(\tan[\log(x+iy)]=a+ib\), prove that \(\tan[\log(x^2+y^2)]=\frac{2a}{1-a^2-b^2}\) when \(a^2+b^2\ne1\). Let log(x+iy)=α+iβ . Then α=log√(x²+y²)=1/2log(x²+y²) , so log(x²+y²)=2α . Given tan(α+iβ)=a+ib ; taking conjugates, tan(α-iβ)=a-ib . tan2α=... 1
- Reduce the following matrix to normal form and find the rank of matrix \(A=\begin{bmatrix}1&1&1\\1&-1&-1\\3&1&1\end{bmatrix}\). R₂→ R₂-R₁,\ R₃→ R₃-3R₁ : beginbmatrix1&1&1\\0&-2&-2\\0&-2&-2endbmatrix R₃→ R₃-R₂ : beginbmatrix1&1&1\\0&-2&-2\\0&0&0endbmatrix C₂→ C₂-C₁,\ C... 1
- If \(u=f\left(e^{x-y},e^{y-z},e^{z-x}\right)\), then prove that \(\frac{\partial u}{\partial x}+\frac{\partial u}{\partial y}+\frac{\partial u}{\partial z}=0\). Let X=e^(x-y),\ Y=e^(y-z),\ Z=e^(z-x) , so u=f(X,Y,Z) . ∂ X/∂ x=X,\ ∂ X/∂ y=-X,\ ∂ X/∂ z=0 ; ∂ Y/∂ x=0,\ ∂ Y/∂ y=Y,\ ∂ Y/∂ z=-Y ; ∂ Z/∂ x=-Z... 1
- Find \(n^{th}\) derivative of \(y=\sin2x\cos6x\). Using 2sin Acos B=sin(A+B)+sin(A-B) : y=1/2[sin8x+sin(-4x)]=1/2[sin8x-sin4x] . Using Dⁿsin(ax)=aⁿsin(ax+fracnπ2) : y_n=1/2[8ⁿsin(8x+fracnπ2)... 1
- If \(y=e^{m\sin^{-1}x}\), prove that \((1-x^2)y_{n+2}-(2n+1)xy_{n+1}-(n^2+m^2)y_n=0\) y₁=e^msin⁻¹x·frac m√(1-x²)⇒√(1-x²) y₁=my⇒(1-x²)y₁²=m²y² . Differentiate: (1-x²)2y₁y₂-2xy₁²=2m²yy₁ ; divide by 2y₁ : (1-x²)y₂-xy₁-m²y=0 . Lei... 1
- If \(z=\frac1{\sqrt2}+\frac i{\sqrt2}\) and \(\bar z\) is the conjugate of z, then prove that \((z)^{10}+(\bar z)^{10}=0\). z=cosfracπ4+isinfracπ4, bar z=cosfracπ4-isinfracπ4 . By De Moivre: z¹⁰=cosfrac10π4+isinfrac10π4 , bar z¹⁰=cosfrac10π4-isinfrac10π4 . z¹⁰+bar... 1
- Prove that \(\log\left(\frac{x+iy}{x-iy}\right)=2i\tan^{-1}\left(\frac yx\right)\). Let x+iy=re^iθ where r=√(x²+y²),\ θ=tan⁻¹y/x . Then x-iy=re^(-iθ) . log(x+iy/x-iy)=logfracre^iθre^(-iθ)=log e^2iθ=2iθ (principal value) =2it... 1
- For the matrix \(A=[a_{ij}]_{3\times3}\), where \(a_{ij}=i+j\), find the rank. A=beginbmatrix2&3&4\\3&4&5\\4&5&6endbmatrix R₂→ R₂-R₁,\ R₃→ R₃-R₁ : beginbmatrix2&3&4\\1&1&1\\2&2&2endbmatrix ; R₃→ R₃-2R₂ : beginbmatrix2&3... 1
- Solve \(10x+y+z=12,\ x+10y+z=12,\ x+y+10z=12\) using the Gauss-Jacobi's method by taking two iterations. Diagonally dominant. x=12-y-z/10,\ y=12-x-z/10,\ z=12-x-y/10 . Start x₀=y₀=z₀=0 . Iteration 1: x₁=y₁=z₁=12/10=1.2 Iteration 2: x₂=12-1.2-1.2... 1