Answers posted by thanvigandhi_1

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Solve the following non-homogeneous system of linear equation by determinant method : $3x+y-z=2\;,2x-y+2z=6\;,2x+y-2z=-2 $

answered May 28, 2013

Toolbox: Consistency of a system of 2 ( or 3 ) linear equations in 2 ( or 3 ) unknowns. Case 1 : If ...

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Solve the following non-homogeneous system of linear equation by determinant method : $ 2x+y-z=4\;,x+y-2z=0\;,3x+2y-3z=4 $

answered May 28, 2013

Toolbox:Consistency of a system of 2 ( or 3 ) linear equations in 2 ( or 3 ) unknowns. Case 1 : If $...

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Solve the following non-homogeneous system of linear equation by determinant method : $x+y+z=4\;,x-y+z=2\;,2x+y-z=1 $

answered May 27, 2013

Toolbox: Cramer's rule : If $Ax = B$ is a system of x linear equations in x unknowns such ...

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Solve the following non-homogeneous system of linear equation by determinant method : $4x+5y=9\;,8x+10y=18 $

answered May 27, 2013

Toolbox:Consistency of a system of 2 ( or 3 ) linear equations in 2 ( or 3 ) unknowns. Case 1 : If $...

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Solve the following non-homogeneous system of linear equation by determinant method: $2x+3y=5\;,4x+6y=12 $

answered May 27, 2013

Toolbox:Consistency of a system of 2 ( or 3 ) linear equations in 2 ( or 3 ) unknowns. Case 1 : If $...

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Solve the following non-homogeneous system of linear equation by determinant method : $3x+2y=5\;,x+3y=4$

answered May 27, 2013

Toolbox:Cramer's rule : If $Ax = B$ is a system of x linear equations in x unknowns such that $|A| \...

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Find the rank of the following matrix :$\begin{bmatrix} 1 & -2 & 3 &4 \\-2 & 4 & -1 &-3 \\-1 & 2 & 7 &6 \end{bmatrix}$

answered May 27, 2013

Toolbox: The rank of a matrix A is equal to $r$ if (i) A has atleast one minor of order r ...

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Find the rank of the following matrix :$\begin{bmatrix} 1 & 2 & -1 &3 \\2 & 4 & 1 &-2 \\3 & 6 & 3 &-7 \end{bmatrix}$

answered May 26, 2013

Toolbox: The rank of a matrix A is equal to $r$ if (i) A has atleast one minor of order r ...

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Find the rank of the following matrix :$\begin{bmatrix} 0 & 1 & 2 &1 \\2 & -3 & 0 &-1 \\1 & 1 & -1 & 0 \end{bmatrix}$

answered May 26, 2013

Toolbox: The rank of a matrix A is equal to $r$ if (i) A has atleast one minor of order r ...

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Find the rank of the following matrix :$\begin{bmatrix} 3 & 1 & 2 &0 \\1 & 0 & -1 &0 \\2 & 1 & 3 &0 \end{bmatrix}$

answered May 25, 2013

Toolbox: The rank of a matrix A is equal to $r$ if (i) A has atleast one minor of order r ...

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Find the rank of the following matrix :$\begin{bmatrix} 6 & 12 & 6 \\1 & 2 & 1 \\4 & 8 & 4 \end{bmatrix}$

answered May 25, 2013

Toolbox: The rank of a matrix A is equal to $r$ if (i) A has atleast one minor of order r ...

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Find the rank of the following matrix :$\begin{bmatrix} 1 & 1 & -1 \\3 & -2 & 3 \\2 & -3 & 4 \end{bmatrix}$

answered May 25, 2013

Toolbox: The rank of a matrix A is equal to $r$ if (i) A has atleast one minor of order r ...

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Solve by matrix inversion method following system of linear equation$\;x-3y-8z+10=0\;,3x+y=4\;,2x+5y+6z=13$

answered May 25, 2013

Toolbox: Let $ A = [ a_{ij} ] $ be a square matrix x. Let $ A_{ij}$ be the cofactor of $ a...

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Solve by matrix inversion method following system of linear equation $2x-y+z=7\;,3x+y-5z=13\;,x+y+z=5$

answered May 25, 2013

Toolbox: Let $ A = [ a_{ij} ] $ be a square matrix x. Let $ A_{ij}$ be the cofactor of $ a...

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Solve by matrix inversion method following system of linear equation $\;x+y+z=9\;,\;2x+5y+7z=52\;,2x+y-z=0$

answered May 25, 2013

Toolbox: Let $ A = [ a_{ij} ] $ be a square matrix x. Let $ A_{ij}$ be the cofactor of $ a...

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Solve by matrix inversion method following system of linear equation $\;7x\;+\;3y\;=\;-1\;,\;2x\;+\;y\;=\;0$

answered May 25, 2013

Toolbox: Let A be a square matrix of order 2. $ A = \begin{bmatrix} a & b \\ c & d...

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Solve by matrix inversion method following system of linear equation:$\;2x\;-\;y\;=7\;,\;3x\;-2y\;=11$

answered May 24, 2013

Toolbox: Let A be a square matrix of order 2. $ A = \begin{bmatrix} a & b \\ c & d...

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For $A=\begin{bmatrix} -1 & 2 & -2 \\4 & -3 & 4 \\4 & -4 & 5 \end{bmatrix}$ show that $A=A^{-1}$

answered May 24, 2013

Toolbox:Multiplication of two matrices A and B is possible when the number of columns in the first m...

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If $A=\Large\frac{1}{3}$$\begin{bmatrix} 2 & 2 & 1 \\-2 & 1 & 2 \\1 & -2 & 2 \end{bmatrix}$ prove that $A^{-1}=A^T$.

answered May 23, 2013

Toolbox:Multiplication of two matrices A and B is possible when the number of columns in the first m...

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Show that the adjoint of $A=\begin{bmatrix} -4 & -3 & -3 \\1 & 0 & 1 \\4 & 4 & 3 \end{bmatrix}$ is$\;A\;$ it self

answered May 23, 2013

Toolbox: Let $ A = [ a_{ij} ] $ be a square matrix x. Let $ A_{ij}$ be the cofactor of $ a...

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Show that the adjoint of $A=\begin{bmatrix} -1 & -2 & -2 \\2 & 1 & -2 \\2 & -2 & 1 \end{bmatrix}$ is$\; 3A^T.$

answered May 23, 2013

Toolbox:Let $ A = [ a_{ij} ] $ be a square matrix x. Let $ A_{ij}$ be the cofactor of $ a_{ij}$. The...

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Find the inverse of the matrix $A=\begin{bmatrix} 3 & -3 & 4 \\2 & -3 & 4 \\0 & -1 & 1 \end{bmatrix}$ and verify that $A^3=A^{-1}$

answered May 23, 2013

Toolbox:Let $ A = [ a_{ij} ] $ be a square matrix x. Let $ A_{ij}$ be the cofactor of $ a_{ij}$. The...

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If$A=\begin{bmatrix} 5 & 2 \\7 & 3 \end{bmatrix}$ and$B=\begin{bmatrix} 2 & -1 \\-1 & 1 \end{bmatrix}$ verify that $(AB)^{-1}=B^{-1}A^{-1}$

answered May 23, 2013

Toolbox: Let $ A = [ a_{ij} ] $ be a square matrix x. Let $ A_{ij}$ be the cofactor of $ a...

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Find the inverse of following matrix : $\begin{bmatrix} 2 & 2 & 1 \\1 & 3 & 1 \\1 & 2 & 2 \end{bmatrix}$

answered May 23, 2013

Toolbox: Let $ A = [ a_{ij} ] $ be a square matrix x. Let $ A_{ij}$ be the cofactor of $ a...

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Find the inverse of following matrix : $\begin{bmatrix} 8 & -1 &- 3 \\-5 & 1 & 2 \\10 & -1 & -4 \end{bmatrix}$

answered May 21, 2013

Toolbox: Let $ A = [ a_{ij} ] $ be a square matrix x. Let $ A_{ij}$ be the cofactor of $ a...

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Find the inverse of the following matrix : $\begin{bmatrix} 1 & 2 & -2 \\-1 & 3 & 0 \\0 & -2 & 1 \end{bmatrix}$

answered May 21, 2013

Toolbox: Let $ A = [ a_{ij} ] $ be a square matrix x. Let $ A_{ij}$ be the cofactor of $ a...

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Find the inverse of following matrix: $\begin{bmatrix} 1 & 3 & 7 \\4 & 2 & 3 \\1 & 2 & 1 \end{bmatrix}$

answered May 17, 2013

Toolbox: Let $ A = [ a_{ij} ] $ be a square matrix x. Let $ A_{ij}$ be the cofactor of $ a...

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Find the inverse of following matrix :$\begin{bmatrix} 1 & 0 & 3 \\2 & 1 & -1 \\1 & -1 & 1 \end{bmatrix}$

answered May 16, 2013

Toolbox: Let $ A = [ a_{ij} ] $ be a square matrix x. Let $ A_{ij}$ be the cofactor of $ a...

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Find the adjoint of matrix A=$\begin{bmatrix} 3 & -3 & 4 \\2 & -3 & 4 \\0 & -1 & 1 \end{bmatrix}$ and verify the result $A\;(adj A)=(adj A)\;A=$|$A$|$.I$

answered May 16, 2013

Toolbox: Let $ A = [ a_{ij} ] $ be a square matrix x. Let $ A_{ij}$ be the cofactor of $ a...

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Find the adjoint of the matrix A =$\begin{bmatrix} 1 & 2 \\3 & -5 \end{bmatrix}$ and verify the result $ A\;(adj\; A)=(adj\;A)\;A=$|$A$|$.I$

answered May 16, 2013

Toolbox:Let $ A = [ a_{ij} ] $ be a square matrix x. Let $ A_{ij}$ be the cofactor of $ a_{ij}$. The...

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Find the adjoint of the following matrics;$\begin{bmatrix} 2& 5 & 3 \\3 & 1 & 2 \\1 & 2 & 1 \end{bmatrix}$

answered May 16, 2013

Toolbox:Let $ A = [ a_{ij} ] $ be a square matrix x. Let $ A_{ij}$ be the cofactor of $ a_{ij}$. The...

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Find the adjoint of the following matrices:$\begin{bmatrix} 1 & 2 & 3 \\0 & 5 & 0 \\2 & 4 & 3 \end{bmatrix}$

answered May 16, 2013

Toolbox:Let $ A = [ a_{ij} ] $ be a square matrix x. Let $ A_{ij}$ be the cofactor of $ a_{ij}$. The...

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Without using derivatives find the maximum and minimum value of $ f(x) = \sin2x+7 $

answered Apr 13, 2013

Ans : $-1 \leq \sin2x \leq 1$So maximum value of $ \sin2x=1$$ \Rightarrow $ maximum value of $ f(x) ...

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FInd whether $ f(x)=e^x$ has maxima or minima.

answered Apr 13, 2013

Toolbox:For maximum or minimum $ f\: '(x) = 0$Ans : $ f\: '(x) = e^x$ cannot be 0 for any $ x $. Hen...

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At what point in the interval $ [0,2\pi] $ does the function $\sin2x$ attain its maximum value?

answered Apr 13, 2013

Toolbox: At maxima $ f\: ' (x) = 0$ Step 1 $ 2cos\: 2x = 0\: \Right...

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Find the intervals in which f(x) = logcosx is strictly decreasing and strictly increasing.

answered Apr 13, 2013

Toolbox:For strictly decreasing $ f ' (x) < 0$For strictly increasing $ f ' (x) > 0$$ f'(x) = ...

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Find the interval in which $ y=x^2e^{-x}$ is increasing.

answered Apr 13, 2013

Toolbox:For increasing $ f ' (x) \geq 0$For decreasing $ f ' (x) \leq 0$Step 1$ \large\frac{dy}{dx} ...

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Find two positive numbers x and y such that their sum is 16 and sum of whose cubes is minimum.

answered Apr 12, 2013

Toolbox:For greatest area, maximise area.Working rule for maxima minima.Find the function to be maxi...

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Find the local maximum and local minimum value of the function. \[f(x)=\sin x+\frac{1}{2}\cos 2x\;in\;[0,\pi/2]\]

answered Apr 12, 2013

Toolbox:For local maximum or minimum $ f '(x)=0$For maximum $ f "(x)$ is negative and for minimum $ ...

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Two equal sides of an isosceles triangle with fixed base b are decreasing at the rate of 3cm/sec. How fast is the area decreasing when the two equal sides are equal to the base?

answered Apr 12, 2013

Toolbox:Let two equal sides be $ x $$ \perp$ from vertex of the isoceles $ \Delta$ will divide the b...

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A poster is to contain $50cm^2$ of matter with borders of 4 cm at top and bottom and of 2 cm on each side. Find the dimensions if the total area of the poster is minimum.

answered Apr 12, 2013

Toolbox:For greatest area, maximise area.Working rule for maxima minima.Find the function to be maxi...

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Find the sides of a rectangle of greatest area that can be inscribed in the ellipse $ x^2+4y^2=16$

answered Apr 12, 2013

Toolbox:For greatest area, maximise area.Working rule for maxima minima.Find the function to be maxi...

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Find the maximum profit that a company can make if the profit function is given by \[P(x)=41-24x-18x^2\]

answered Apr 11, 2013

Toolbox:Necessary condition for maximum or minimum 'p' is $ \large\frac{dp}{dx}=0$Condition for maxi...

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Differentiate the following w.r.t. $x: \large \frac {\cos x}{\log x}$,$\; x > 0 $

answered Apr 4, 2013

Toolbox:According to the Quotient Rule for differentiation, given two functions $u$ and $v,\; \large...

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Differentiate the following w.r.t. $ x: $ $ cos \: ( log\: x + e^x ), x > 0 $

answered Apr 4, 2013

Toolbox:According to the Chain Rule for differentiation, given two functions $f(x)$ and $g(x)$, and ...

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Let f be continuous function on [1,3]. If f takes only rational values for all x and f(2) = 10 then find the value of f(1.5)

answered Mar 12, 2013

Given $ f$ is continuous and $ f(x)$ is rational no.$ \forall \in [ 1, 3 ]$ and $ f(2) = 10. \R...

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Let f be a non zero continuous function satisfying f(x+y) = f(x).f(y) for all x, y $ \epsilon$ R. If f(2) = 9 then find the value of f(3).

answered Mar 12, 2013

Ans : $ f(2) = f(1+1)=f(1).f(1)$ $ \Rightarrow f(1)=3$ $ f(3) = f(2+1)=f(2).f(1)$ $...

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Let $y=y=\sqrt{e^\sqrt x },x>0\;find\;\large\frac{dy}{dx}.$

answered Mar 12, 2013

Toolbox: Use chain rule. $ \large\frac{dy}{dx}\: \large\frac{1}{2\sqrt{e^{\sqrt x}...

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Find $ \frac{dy}{dx}\;when\;y=\Large x^{x^{x^{x.....}}}$

answered Mar 12, 2013

Toolbox: Express $ y = x^y$ Take log before differentiating. $ y = x^y$ ...

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Find the set of all points where the function f(x) = 2x |x| is differentiable.

answered Mar 12, 2013

Toolbox: Write the function $ 2x | x |$ as $ f(x)= \left\{ \begin{array}{1 1} 2x^2, & ...

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