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2
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Show that the matrix $A = \begin{bmatrix} 8 & -8 & -2 \\ 4 & -3 & -2 \\ 3 & -4 & 1 \end{bmatrix}$ is similar to diagonal matrix. Find the diagonalising matrix and diagonal form.
engineering mathematics updated 3.9 years ago by phenjoisilab • 0
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Evaluate $\oint \frac{e^{kz}}{z}dz$ over the circle |z| = 1 and k is real. Hence prove that $\int\limits_{0}^{\pi} e^{kcos\theta} cos(ksin\theta)d\theta = \pi$
engineering mathematics updated 3.9 years ago by phenjoisilab • 0
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Using calculus of Residues, prove that $\int\limits_{0}^{2\pi} e^{cos\theta} cos(sin\theta - n\theta)d\theta = \frac{2\pi}{n!}$
engineering mathematics updated 3.9 years ago by phenjoisilab • 0
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Find the eigen value and eigen vectors of the matrix $A^3$ where $A = \begin{bmatrix} 4 & 6 & 6 \\ 1 & 3 & 2 \\ -1 & -5 & -2 \end{bmatrix}$
engineering mathematics updated 3.9 years ago by phenjoisilab • 0
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Show that eigen values of unitary matrix are of unit modulus.
engineering mathematics updated 3.9 years ago by phenjoisilab • 0
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Verify Caykey Hamilton theorem and find the value of $A^{64}$ for the matrix $A = \begin{bmatrix} 1 & 2 \\ 2 & -1 \end{bmatrix}$
engineering mathematics updated 3.9 years ago by krithikkm200 • 10
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Reduce the Quadratic form xy + yz + zx to diagonal form through congruent transformation.
engineering mathematics updated 3.9 years ago by krithikkm200 • 10
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Find the singular value decomposition of the matrix $\begin{bmatrix} 1 & 1 \\ 1 & 1 \\ 1 & -1 \end{bmatrix}$
engineering mathematics updated 3.9 years ago by krithikkm200 • 10
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Verify Greens theorem in the plane for $\oint (x^2 - y)dx + (2y^2 + x)dy$ around the bounded of the region defined by $y^2 = x$ and $y = 4$.
engineering mathematics updated 3.9 years ago by krithikkm200 • 10
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Evaluate by Stoke's theorem $\int_{c}xydx + xy^2 dy$ where C is the square in the xy plane with vertices (1, 0), (0,1) (-1, 0) and (0, -1)
engineering mathematics updated 3.9 years ago by krithikkm200 • 10
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Using the Kuhn-Tucker conditions solve the following N.L.P.P. Maximize $Z = -x_{1}^2 - x_{2}^2 - x_{3}^2 + 4x_{1} + 6x_{2}$. Subject to $x_{1}+x_{2} \leq 2, 2x_{1}+3x_{2} \leq 12, x_{1}, x_{2} \geq 0$
engineering mathematics updated 3.9 years ago by krithikkm200 • 10
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Find the relative maximum or minimum of the function. $z = x_{1}^2 + x_{2}^2 + x_{3}^2 - 6x_{1} - 10x_{2} - 14x_{3} + 103$
engineering mathematics updated 3.9 years ago by krithikkm200 • 10
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Evaluate $\int\int_{s}\overline{F} \cdot d\overline{s}$ where $\overline{F} = 4xi - 2y^2 j + z^2 k$ and S is the region bounded by $y^2 = 4x, x=1, z= 0, z=3$.
engineering mathematics updated 3.9 years ago by krithikkm200 • 10
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By Gauss divergence theorem evaluate $\int_{S} \overline{F} \cdot \overline{N}ds$ where $\overline{F} = 4x\overline{i} + 3y\overline{j} - 2z\overline{k}$ surface is bounded by x=0,y=0,z=0 & 2x+2y+z=4
engineering mathematics updated 3.9 years ago by krithikkm200 • 10
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Use stokes theorem to evaluate $\int_{c}\overline{F}\cdot\overline{dr}$ where $\overline{F} = x^2i + xyj$ and c is the boundary of the rectangle x=0, y=0, x=a, y=b.
engineering mathematics updated 3.9 years ago by krithikkm200 • 10
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By using Green theorem show that the area bounded by a simple closed curve c is given by $\frac{1}{2}\int xdy - ydx$
engineering mathematics updated 3.9 years ago by krithikkm200 • 10
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Prove that $\overline{F} = (y^2 cosx + z^3)\overline{i} + (2y sinx - 4)\overline{j} + (3x z^2 + 2)\overline{k}$ is a conservative field. Also find the following.
engineering mathematics updated 3.9 years ago by krithikkm200 • 10
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Verify Green's theorem in the plane for where $\int_{c} (xy + y^2)dx + x^2 dy$. C is closed curve of the region bounded by y = x and $y = x^2$.
engineering mathematics updated 3.9 years ago by krithikkm200 • 10
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Using the Kuhn Tucker conditions solve the following N.L.P.P. Maximize to Z = $x_{1}^{2}+x_{2}^{2}$, subjected to $x_{1}+x_{2}-4 \le$ 0 and $2x_{1}+x_{2}-5 \le 0, x_{1},x_{2}\ge$ 0.
engineering mathematics updated 3.9 years ago by krithikkm200 • 10
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The marks of 1000 students of university are found to be normally distributed with mean 70 and standard deviation 5. Estimate the number of students whose marks will be 1) between 60 & 75 2) < 75
engineering mathematics updated 3.9 years ago by krithikkm200 • 10
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In a laboratory experiment two samples gave the following results. Test the equality of sample variances at 5% level of significance.
engineering mathematics updated 3.9 years ago by krithikkm200 • 10
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The probability that at any moment one telephone line out of 10 will be busy is 0.2.
engineering mathematics updated 3.9 years ago by krithikkm200 • 10
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Seven dice are thrown 729 time. How many times do you expect at least four dice to show three or five?
engineering mathematics updated 3.9 years ago by krithikkm200 • 10
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A car hire firm has two cars which it hires out day by day. The number of demands for a car on each day is distributed as poisson variate with mean 1.5. Also do the following.
engineering mathematics updated 3.9 years ago by krithikkm200 • 10
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Find the total work done in moving a particle in the force field $\overline{F} = 3xyi - 5zj + 10xk$ along $x = t^2 + 1, y = 2t^2, z = t^3$ from t = 1 to t = 2.
engineering mathematics updated 3.9 years ago by krithikkm200 • 10
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