Answers posted by meena.p

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Evaluate the following $\large\int\frac{(1+\cos x)}{x+\sin x}dx$

answered Apr 11, 2013

Toolbox:If $f(x)$ is substituted by t, then $f'(x)dx=f'(t)dt.$ Hence $\int f(x)dx=\int t.dt$$\large\...

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Evaluate the following$\displaystyle\int\frac{e^{6log x}-e^{5log x}}{e^{4log x}-e^{3log x}}$$dx$

answered Apr 11, 2013

Toolbox:$\large e^{logx}=x$$\int x^n dx=\large\frac{x^{n+1}}{n+1}+c$Let $I=\Large\int\frac{e^{6log x...

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Evaluate the following $ \displaystyle\int\frac{(x^2+2)}{x+1}dx$

answered Apr 11, 2013

Toolbox:A rational expression is said to be improper if the degree of the numerator is greater than ...

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Verify the following $ \large\int\frac{2x+3}{x^2+3x}dx=log \mid x^2+3x\mid+C$

answered Apr 11, 2013

Toolbox:Proper rational expression of the form $\large\frac{1}{(x+a)(x+b)}$ can be resolved into par...

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Verify the following$ \int\frac{2x-1}{2x+1}dx=x-log \mid (2x+3)^2\mid+C$

answered Apr 10, 2013

Toolbox:If the degree of the numerator of rational expression is equal to or greater than the degree...

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Using matrices, solve the following system of linear equations: $$x + y + z = 4$$ $$2x - y + z = -1 $$ $$2x + y - 3z = -9$$

answered Apr 10, 2013

Toolbox:If the value of determinant of a $ 3 \times 3$ square matrix is not equal to zero, then it i...

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Using properties of determinants, prove that $ \begin{bmatrix} 2ab & a^2 & b^2 \\ a^2 & b^2 & 2ab \\ b^2 & 2ab & a^2 \end{bmatrix} = -(a^3+b^3)^2$

answered Apr 10, 2013

Toolbox:Elementary transformation in a determinant can be made bya)by interchanging two rows or colu...

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Show that the given system of equations are consistent and non infinite number of a solutions $ 2x+3y-z=7 \\ x+2y-z=4 \\ 3x-y+4z=5$

answered Apr 10, 2013

Toolbox:If the value of the determinant of square matrix is zero, then it is a non-singular matrixIf...

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Solve by matrix method : $ x+y+z=20, \: 2x+y-z=23, \: 3x+y+z=46 $

answered Apr 10, 2013

Toolbox:If the value of a determinant of a $3 \times 3$ square matrix is not equal to zero, then it ...

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Using properties of determinants, prove that : $ \begin{vmatrix} a+b+nc & (n-1)a & (n-1)b \\ (n-1)c & b+c+na & (n-1)b \\ (n-1)c & (n-1)a & c+a+nb \end{vmatrix} = n(a+b+c)^3$

answered Apr 9, 2013

Toolbox:Elementary transformations can be made in a determinanta)by interchanging two rows or column...

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Using matrices, solve the following system of equations:$x+\frac{2}{y}+3xz=-1,2x-\frac{4}{y}-3xz=3,3x+\frac{6}{y}-2xz=4$

answered Apr 9, 2013

Toolbox:If the value of the determinant of a $3 \times 3$ square matrix is not equal to zero, then i...

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Using matrices, solve the following system, of equations : $ \large\frac{1}{x}-\frac{1}{y}+\frac{1}{z}$$=4: \: \large\frac{2}{x}+\frac{1}{y}-\frac{3}{z}$$=0: \: \large\frac{1}{x}+\frac{1}{y}+\frac{1}{z}$$=2 \qquad x \neq 0, y \neq 0, z \neq 0 $

answered Apr 8, 2013

Toolbox: If the value of the determinant of a $3 \times 3$ square matrix is not equal to z...

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Evaluate the determinant $ \begin{bmatrix} cos \theta & -sin \theta \\ sin \theta & cos \theta \end{bmatrix} $

answered Apr 8, 2013

Toolbox:The value of the determinant value of a $2 \times 2$ square matrix is $|A|=a_{11} \times a_...

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using matrices, solve the following system of equations $ x+2y-3z=-4;2x+3y+2z=2 ; 3x-3y-4z=11 $

answered Apr 8, 2013

Toolbox: If the determinant value of a $3 \times 3$ square matrix is not equal to zero, th...

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Write the adjoint of the following matrix $ \begin{bmatrix} 2 & -1 \\ 4 & 3 \end{bmatrix} $

answered Apr 8, 2013

Toolbox: The determinant value of a $ 2\times 2$ square matrix is $|A|=a_{12} \times a_{22...

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Find the area of the triangle where vertices are $A(11,7),B(5,5)\; and\; C(-1,3)$

answered Apr 8, 2013

Toolbox:Area of the triangle where veritices are $A(x_1,y_1), B(x_2,y_2),C(x_3,y_3)$ is given by$\De...

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Solve for $x$ if $\begin{vmatrix} 1 & 4 & 4 \\ 1 & -2 & 1 \\ 1 & 2x & x^2 \end{vmatrix} =0 $

answered Apr 8, 2013

Toolbox:The deteminant value can be found by expanding along any of its rows or columnsLet $\Delta=\...

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Show that $A=\begin{bmatrix} 1 & log_x\;y & log_x\;z \\ log_y\;x & 1 & log_y\;z \\ log_z\;x & log_z\;y & 1 \end{bmatrix}=0$

answered Apr 6, 2013

Toolbox:Elementary transformation can be made bya)interchanging two columns or rowsb)adding or subtr...

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Show that $A=\begin{bmatrix} xc_{r} & xc_{r+1} & xc_{r+2} \\ yc_r & yc_{r+1} & yc_{r+2} \\ zc_r & zc_{r+1} & zc_{r+2} \end{bmatrix}=\begin{bmatrix} xc_{r} & x+1c_{r+1} & x+2c_{r+2} \\ yc_r & y+1c_{r+1} & y+2c_{r+2} \\ zc_r & z+1c_{r+1} & z+2c_{r+2} \end{bmatrix}$

answered Apr 5, 2013

Toolbox:(i) Elementary transformation can be made bya)interchanging two columns or rowsb)By adding o...

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Without expanding, show that : $ \begin{vmatrix} 42 & 1 & 6 \\ 28 & 7 & 4 \\ 14 & 3 & 2 \end{vmatrix} = 0 $

answered Apr 5, 2013

Toolbox:If two rows or columns of a determinant are identical, then its value is zeroIf a row or col...

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write $ A^{-1} $ for $ A = \begin{bmatrix} 2 & 5 \\ 1 & 3 \end{bmatrix} $

answered Apr 5, 2013

Toolbox:The determinant value of $2 \times 2$ matrix is$|A|=a_{11} \times a_{22}- a_{12} \times a_{2...

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Prove that $\begin{vmatrix} (x-2)^2 & (x-1)^2 & x^2 \\ (x-1)^2 & x^2 & (x+1)^2 \\ x^2 & (x+1)^2 & (x+2)^2 \end{vmatrix}=-8$

answered Apr 5, 2013

Toolbox:(i) If two rows or columns are identical then the value of the determinant is zero(ii)Elemen...

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Using matrices, solve the following system of equations: $ 4x+3y+3z=60, x+2y+3z=45 \: and \: 6x+2y+3z=70 $

answered Apr 5, 2013

Toolbox:If determinant value of the given matrix is not equal to zero, then it is a non-singular mat...

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Using properties of determinants, solve the following for x : $ \begin{vmatrix} x-2 & 2x-3 & 3x-4 \\ x-4 & 2x-9 & 3x-16 \\ x-8 & 2x-27 & 3x-64 \end{vmatrix} = 0 $

answered Apr 4, 2013

Toolbox: (i) If two rows or columns are identical then the value of the determinant is zero (i...

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Evaluate : $\begin{vmatrix} cos 15^{\circ} & sin 15^{\circ} \\ sin 75^{\circ} & cos 75^{\circ} \end{vmatrix} $

answered Apr 4, 2013

Toolbox:The determinant of a square matrix of order $2 \times 2$ is $|A|=a_{11} \times a_{22} - a_{1...

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Using matrices, solve the following system of equations : $ x+2y+z=7, x+3z=11 \: and \: 2x-3y=1 $

answered Apr 4, 2013

Toolbox:If |A| of a matrix A is non zero, then it is a non-singular matrixIf it is a nonsingular mat...

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Using properties of determinants, solve the following for x : $ \begin{vmatrix} a+x & a-x & a-x \\ a-x & a+x & a-x \\ a-x & a-x & a+x \end{vmatrix} = 0 $

answered Apr 4, 2013

Toolbox: (i) If two rows or columns are identical then the value of the determinant is zer...

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If $A = \begin{bmatrix} 2 & 3 \\ 5 & -2 \end{bmatrix} $ show that $A^{-1} = \frac{1}{19}A.$

answered Apr 4, 2013

Toolbox:Inverse of a $2 \times 2 $ matrix is$A^{-1}=\frac{1}{ |A| } \begin{bmatrix} a_{22} & -a_...

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Use product of AB where $A= \begin{bmatrix} 1 & -1 & 2 \\ 0 & 2 & -3 \\ 3 & -2 & 4 \end{bmatrix} \; and \;B=\begin{bmatrix} -2 & 0 & 1 \\ 9 & 2 & -3 \\ 6 & 1 & -2 \end{bmatrix}$
to solve the system of equations : $ \\ x-y+2z=1 \\ 2y-3z=1 \\ 3x-2y+4z=2 $

answered Apr 4, 2013

Toolbox:If $|A| \neq 0, $ then $'A'$ is a non -singular matrix. Then inverse exists$A{-1}=\frac{1}{|...

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Evaluate $\left|\begin{array}{ccc} 1 & \omega^n & \omega^{2n} \\ \omega^{2n} & 1 & \omega^{n} \\ \omega^n & \omega^{2n} & 1 \end{array}\right|$

answered Apr 4, 2013

Toolbox: (i) If two rows or columns are identical then the value of the determinant is zer...

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Prove that $\begin{bmatrix} x+y & x & x \\ 5x+4y & 4x & 2x \\ 10x+8y & 8x & 3x \end{bmatrix}=x^3$

answered Apr 3, 2013

Toolbox:(i) If two rows or columns are identical then the value of the determinant is zero(ii)Elemen...

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Without expanding evaluate the determinent $\begin{vmatrix} (a^x+a^{-x})^2 & (a^x-a^{-x})^2 & 1 \\ (a^y+a^{-y})^2 & (a^y-a^{-y})^2 & 1 \\ (a^z+a^{-z})^2 & (a^z-a^{-z})^2 & 1 \end{vmatrix}$ where $a>0$ ,and $x,y,z \in R$

answered Apr 3, 2013

Toolbox:(i)If two rows or columns are identical, then the value of the determinant is zero(ii)Elemen...

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If $ A = \begin{bmatrix} 1 & 2 \\ 4 & 2 \end{bmatrix} $ then show that | 2A | = 4| A |

answered Apr 3, 2013

Toolbox:If $A= \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix}$Then $|A|=a_...

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Using properties of determinants prove that \[ \begin{vmatrix} -bc & b^2+bc & c^2+bc \\ a^2+ac & -ac & c^2+ac \\ a^2+ab & b^2+ab & -ab \end{vmatrix}= (ab+bc+ca)^3 \]

answered Apr 3, 2013

Toolbox: (i)Elementary transformation can be made by interchanging any two rows or two col...

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Evaluate : $ \begin{vmatrix} 1 & log_ba \\ log_ab & 1 \end{vmatrix}$

answered Apr 3, 2013

Toolbox: If $A= \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix}$ ...

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Evaluate:$\int \limits_0^ {2\pi} \frac{\sin 2\theta}{a-b \cos \theta}d\theta$

answered Apr 2, 2013

Toolbox:$\int \limits _a^b f(x)dx=F(b)-F(a)$$\int \limits_0^{2a} f(x)dx=0$ when $f(2a-x)=-f(x)$Given...

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answered Apr 2, 2013

Toolbox:$\int \limits _a^b f(x)dx=F(b)-F(a)$$\int \limits _a^b |x-c| dx$ Here we need to break the i...

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Evaluate:$\int \limits_3^9 \frac {\sqrt {12-x}}{\sqrt x+\sqrt {12-x}}dx$

answered Apr 2, 2013

Toolbox: (i)$\int \limits _a^b f(x)dx=F(b)-F(a)$ (ii) $ \int \limits_a^b f(x)dx=\int...

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Evaluate:$\int \limits _0^{\pi/2} \frac{1}{2\cos x+4 \sin x}dx$

answered Apr 1, 2013

Toolbox:(i)$\int \limits _a^b f(x)dx=F(b)-F(a)$(ii)If we substitute a function f(x) as t, then $f'(x...

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Evaluate:$\int \limits_{-\pi}^{\pi} x^{20} \sin ^9x dx $

answered Apr 1, 2013

Toolbox:$\int \limits _a^b f(x)dx=F(b)-F(a)$$\int \limits _{-a}^a f(x)dx=0$ if f(x) is an odd functi...

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Evaluate:$\int \limits_0^{\pi/2} log (\cos \theta) d\theta$

answered Apr 1, 2013

Toolbox:(i)$\int \limits _a^b f(x)dx=F(b)-F(a)$(ii) $\int \limits_0^{2a} f(x)dx=2 \int \limits_0^a f...

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Let A = {1,2,3}, B = {4,5,6,7} and let f = {(1,4), (2,5), (3,6)} be a function from A to B. State whether f is one-one or not.

answered Mar 29, 2013

Toolbox: A function $f: X \rightarrow Y$ where for every $x1, x2 \in X, f(x1) = f(x2) \Rig...

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Consider $ f : R_+ \rightarrow [ 4, \infty ]$ given by $ f(x)=x^2+4$. Show that f is invertible with the inverse $ (f^{-1}) $ of f given by $ f^{-1}(y)=\sqrt{y-4}$ where $ R_+$ is the set of all non-negative real numbers.

answered Mar 29, 2013

Toolbox: To check if a function is invertible or not ,we see if the function is both one-o...

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Prove that the relation R on the set A = {1,2,3,4,5} given by R = { (a,b) : | a - b | is even } is an equivalence relation.

answered Mar 29, 2013

Toolbox:A relation R in a set A is an equivalence relation if R is reflexive, symmetric and transiti...

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If Matrix A = $\begin{bmatrix} 3 &-3 \\ -3 & 3 \end{bmatrix}$, and $A^2=\lambda A$, then write the value of $\lambda$.

answered Mar 26, 2013

$A=\begin{bmatrix} 3 & -3 \\ -3 & 3 \end{bmatrix}\; and \;A^2=\lambda A$$A^2=\begin{bmatrix}...

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Find the area of the region $\{(x,y):y^2\leq 6ax$ and $x^2+y^2\leq 16a^2\}$ using method of integration.

answered Mar 26, 2013

$x^2+y^2 \leq 16a^2$Let us consider $x^2+y^2=16a^2$This represents a circle with centre at the orgin...

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Find the particular solution of the differential equation $\large\frac{dx}{dy}$$+x\cot y=2y+y^2\cot y,(y\neq 0)$,given that $x=0$ when $y=\large \frac{\pi}{2}$.

answered Mar 25, 2013

$\frac{dx}{dy}+x \cot y =2y+y^2 \cot y$The solution for the given differential equation$xe^{\int pdy...

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Using properties of determinants,prove the following $\begin{vmatrix}3x& -x+y & -x+z\\x-y & 3y & z-y\\x-z & y-z & 3z\end{vmatrix}=3(x+y+z)(xy+yz+zx)$

answered Mar 24, 2013

$\bigtriangleup=\begin{vmatrix}3x& -x+y & -x+z\\x-y & 3y & z-y\\x-z & y-z & ...

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Using vectors,find the area of triangle ABC,whose vertices are $A(1,2,3),B(2,-1,4),C(4,5,-1).$

answered Mar 24, 2013

Let $\overrightarrow{a},\overrightarrow{b}\; and \;\overrightarrow{c}$ be the position vectors of th...

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L and M are two points with position vectors $2\overrightarrow{a}-\overrightarrow{b}$ and $\overrightarrow{a}+2\overrightarrow{b}$ respectively.Write the position vector of a point N which divides the line segment LM in the ratio 2 : 1 externally.

answered Mar 24, 2013

Let $\overrightarrow{OL}=2\overrightarrow{a}-\overrightarrow{b}$$\overrightarrow{OM}=\overrightarrow...

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