Answers posted by rvidyagovindarajan_1

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Find the value of $a-2b$ if $\begin{bmatrix}a+4 & 3b\\8 & -6\end{bmatrix}=\begin{bmatrix}2a+2 & b+2\\8 & a-8b\end{bmatrix}\qquad$

answered Mar 21, 2014

Toolbox:If the order of 2 matrices are equal, their corresponding elements are equal, i.e, if $A_{ij...

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Find the projection of the vector $ \hat i + \hat j + \hat k$ on the vector $\hat j$.

answered Mar 20, 2014

Toolbox: Projection of $\overrightarrow a$ on $\overrightarrow b$ is $\large\frac{\overrightarrow...

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Find the value of the following:\[cos^{-1} \big(- \frac {1} {2} \big)+2 sin^{-1}\frac {1} {2} \]

answered Mar 20, 2014

<div class="clay6-toolbox"><b>Toolbox:</b><ul><li class="c...

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Let $R=\{(a,a^3):\:a\: is\: a\: prime\:number\: less\: than\: 5\}$ be a relation. Find the Range of $R$.

answered Mar 20, 2014

Given:$R=\{(a,a^3):\:a\: is\: a\: prime\:number\: less\: than\: 5\}$ $\Rightarrow\:R=\{(2,2^3),(3,3...

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If $A$ is a $3\times 3$ matrix, and $ |A|\neq0$ and $|3A|=k.|A|,$ then write the value of $k$

answered Mar 20, 2014

Toolbox:If $ |A|=x$, then $ |p.A|=p^n.|A|$ where $n$ is order of the matrix $A$.Given: $|3A|=k...

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Find the angle between the pairs of lines: $ \overrightarrow r = 2\hat i -5 \hat j +\hat k + \lambda (3\hat i +2 \hat j +6\hat k)\: and \: \overrightarrow r = 7\hat i - 6\hat k + \mu (\hat i +2\hat j +2\hat k) $

answered Mar 20, 2014

Given equations of the lines are$ \overrightarrow r = 2\hat i -5 \hat j +\hat k + \lambda (3\hat i ...

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Use elementary column operation $C_2\rightarrow C_2-2C_1$ in the matrix equation $\begin{bmatrix}4 & 2\\3 & 3\end{bmatrix}=\begin{bmatrix}1 & 2\\0 & 3\end{bmatrix}\:\begin{bmatrix}2 & 0\\1 & 1\end{bmatrix}$

answered Mar 20, 2014

Toolbox:Column operation is to be done on $2^{nd}$ matrix.Given $\begin{bmatrix}4 & 2\\3 & ...

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Evaluate $\int\large\frac{dx}{sin^2x\:cos^2x}$

answered Mar 20, 2014

$\int\large\frac{dx}{sin^2x\:cos^2x}$Multiplying and dividing by 4=4 $\int\large\frac{dx}{4sin^2x\:...

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Find the sum of $n$ terms of the series $(3\times 8)+(6\times 11)+(9\times 14).............$

answered Mar 13, 2014

Toolbox:$n^{th}$ term of an A.P$=a+(n-1)d$Sum of $n$ terms of any series, $S_n=\sum t_n$$\sum ( A+...

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Find the sum of 20 terms of the series. $5^2+6^2+7^2+.....20^2$

answered Mar 12, 2014

Toolbox:$n^{th}$ term of an A.P$=a+(n-1)d$ $\sum n=\large\frac{n(n+1)}{2}$$\sum n^2=\large\frac{n(n...

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Find the sum to $n$ terms of the series $\large\frac{1}{1\times 2}+\frac{1}{2\times 3}+\frac{1}{3\times 4}+..........$

answered Mar 10, 2014

The given series is $\large\frac{1}{1\times 2}+\frac{1}{2\times 3}+\frac{1}{3\times 4}+..........$S...

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Find the sum of $n$ terms of the series $3\times 1^2+5\times 2^2+7\times 3^2+.........$

answered Mar 7, 2014

Toolbox:$n^{th}$ term of an A.P.$=a+(n-1)d$Sum of $n$ terms of any series with $n^{th}$ term $t_n...

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Find the sum of $n$ terms of the series $(1\times 2\times 3)+(2\times 3\times 4)+(3\times 4\times 5)+.......$

answered Mar 7, 2014

Toolbox:Sum of $n$ terms of any series $S_n=\sum t_n$$\sum n=\large\frac{n(n+1)}{2}$$\sum n^2=\larg...

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Find the sum of $n$ terms of the series $1\times 2+2\times 3+3\times 4+4\times 5+.............$

answered Mar 7, 2014

Toolbox:For any series $\sum t_n=Sum =S_n$$\sum n=1+2+3+.......n=\large\frac{n(n+1)}{2}$$\sum n^2=1^...

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If A.M. and G.M. of roots of a quadratic equation are $8$ and $5$ respectively. Find the quadratic equation.

answered Mar 7, 2014

Toolbox:A.M. between two numbers $a$ and $b= \large\frac{a+b}{2}$G.M. between the two numbers$= ...

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The number of bacteria in certain culture doubles every hour. If there were $30$ bacteria present in the culture originally, how many bacteria will be present at the end of $2^{nd}$ hour, $4^{th}$ hour and $n^{th}$ hour?

answered Mar 6, 2014

Given that there were $30$ bacteria present originally. $\therefore a=30$Also given that the number ...

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$ the A.M. and $G$ the G.M. between two positive numbers then show that the numbers are $A\pm \sqrt {(A+G)(A-G)}$

answered Mar 6, 2014

Toolbox:The A.M. between $a$ and $b$ $=\large\frac{a+b}{2}$The G.M. between $a$ and $b$ $=\s...

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The difference between any two consecutive interior angles of a polygon is $5^\circ$. If the smallest angle is $120^ \circ$, then find the number of sides of the polygon.

answered Mar 5, 2014

Toolbox:The sum of the interior angles of $n$ sided polygon = $(n-2).180^{\circ}$The sum of $n$ t...

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The sum of two numbers is $6$ times their geometric mean.Show that the ratio of the numbers is $3+2\sqrt 2\::\:3-2\sqrt 2$

answered Mar 5, 2014

Toolbox:The G.M. between two numbers $a$ and $b$ = $\sqrt {ab}$$(a+b)^2=a^2+b^2+2ab$The solutions ...

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Find the value of $n$ so that $\large\frac{a^{n+1}+b^{n+1}}{a^n+b^n}$ may be the geometric mean between $a$ and $b$.

answered Mar 4, 2014

Toolbox:G.M. $'g'$ between $a$ and $b$ is $\sqrt {ab}$$(x+y)^2=x^2+y^2+2xy$$\large\frac{x^m}{y...

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Insert $2$ numbers between $3$ and $81$ so that the resulting sequence is a G.P. Find the two numbers.

answered Mar 4, 2014

Toolbox:$a^{mn}=(a^m)^n$Given that $2$ numbers are inserted between $3$ and $81$ so that the re...

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Insert $2$ numbers between $3$ and $81$ so that the resulting sequence is a G.P. Find the two numbers.

answered Mar 3, 2014

Toolbox:$a^{mn}=(a^m)^n$Given that $2$ terms are inserted between $3$ and $81$ to form a G.P.Le...

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If $a,b,c,d$ are in G.P, then show that $(a^2+b^2+c^2)(b^2+c^2+d^2)=(ab+bc+cd)^2$

answered Mar 3, 2014

Toolbox:The first four terms of a G.P. are assumed as $a,ar,ar^2,ar^3$$(ab)^m=a^m.b^m$$(a^m)^n=a^...

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Show that the ratio of the first $n$ terms of a G.P. to the sum of the terms from $(n+1)^{th}$ to $(2n)^{th}$ term is $1:r^n$

answered Mar 2, 2014

Toolbox:Sum of first $n$ terms of a G.P$=a.\large\frac{1-r^n}{1-r}$Step 1Given that the G.P. has $...

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If the first and $n^{th}$ term of a G.P. are $a$ and $b$ respectively and if $P$ is the product of the $n$ terms then prove that $P^2=(ab)^n$

answered Mar 2, 2014

Toolbox:$t_n=a.r^{n-1}$General G.P is $a,ar,ar^2,................ar^{n-1}$$1+2+3+.........n=\large\...

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If the $p^{th},q^{th}\:\:and\:\:r^{th}$ terms of G.P.are $a,b\:\:and\:\:c$ respectively then prove that $a^{q-r}.b^{r-p}.c^{p-q}=1$

answered Mar 1, 2014

Toolbox:$n^{th}$ term of a G.P.$=a.r^{n-1}$Given that $p^{th}$ term of a G.P.$=a$$q^{th}$ term$=b...

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Find the four numbers forming a G.P. in which the $3^{rd}$ term is greater than the $1^{st}$ term by $9$ and the $2^{nd}$ term is greater than the $4^{th}$ term by $18$.

answered Mar 1, 2014

Toolbox:$a^2-b^2=(a-b)(a+b)$$a^3-b^3=(a-b)(a^2+ab+b^2)$Let the four terms in G,P, be $a,ar,ar^2\:\...

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Show that the product of the corresponding terms of the sequences $a,ar,ar^2.........ar^n$ and $A,AR,AR^2.............AR^{n-1}$ form a G.P. and find the common ratio.

answered Mar 1, 2014

Given two sequences are$a,ar,ar^2.........ar^n$ ...........(i) and $A,AR,AR^2.............AR^{n-...

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Find the sum of the products of the corresponding terms of the sequences $2,4,8,.......32$ and $128,32,8,......\large\frac{1}{2}$

answered Mar 1, 2014

Given sequences are$2,4,8,16,32$.........(i) and$128,32,8,2,\large\frac{1}{2}$............(ii)Bot...

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Find the sum to $n$ terms of the sequence $8,88,888,.............$

answered Feb 28, 2014

Toolbox:Sum of $n$ terms of a G.P=$a.\large\frac{r^n-1}{r-1}$Given series $S=8+88+888+........$Taki...

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If the $4^{th},10^{th}\:\;and\:\:16^{th}$ terms of a G.P. are $x,y\:\;and\:\:z$ respectively. Prove that $x,y,z$ are in G.P.

answered Feb 27, 2014

Toolbox:$n^{th}$ term of a G.P. =$t_n=a.r^{n-1}$To prove $x,y,z$ are in G.P. we have to prove that...

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Find the G.P. the sum of whose first two terms is $-4$ and the $5^{th}$ term is $4$ times the $3^{rd}$ term.

answered Feb 27, 2014

Toolbox:$n^{th}$ term of a G.P=$t_n=a.r^{n-1}$Let the G.P. be $a,ar,ar^2........$Given that the su...

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Given a G.P. with $a=729$ and $7^{th}$ term is $64$. Find $S_7.$

answered Feb 27, 2014

Toolbox:$n^{th}$ term of a G.P.= $t_n=a.r^{n-1}$Given that in a G.P.first term $a=729$,and $7^{th}$...

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The sum of first three terms of a G.P. is $16$ and the sum of next three terms is $128$. Find the $1^{st}$ term, common ratio and the sum of $n$ terms of the G.P.

answered Feb 26, 2014

Let the first $6$ terms of the G.P. be $a,ar,ar^2,ar^3,ar^4,ar^5$It is given that sum of first 3 ter...

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How many terms of the G.P. $3,3^2,3^3.......$ are needed to give the sum $120$?

answered Feb 26, 2014

Toolbox:Sum of $n$ terms of a G.P.=$S_n=a.\large\frac{r^n-1}{r-1}$Given that sum of $n$ terms of th...

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The sum of first $3$ terms of a G.P is $\large\frac{39}{10}$ and their product is $1$. Find the common ratio and the terms of the G.P.

answered Feb 26, 2014

Toolbox:Assume any three terms of a G.P as $\large\frac{a}{r},$$a,ar$Sum of $n$ terms of a G.P.=$S...

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Evaluate $\sum\limits_{k=1}^{11} (2+3^k)$

answered Feb 26, 2014

Toolbox:$\sum (A+B)=\sum A+\sum B$Sum of $n$ terms of a G.P.$=a.\large\frac{r^n-1}{r-1}$$\sum\limits...

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Find the sum of $n$ terms of the G,P, $x^2,x^4,x^6...........$ in terms of $x$

answered Feb 26, 2014

Toolbox:Sum of $n$ terms of a G.P.$=S_n=a.\large\frac{1-r^n}{1-r}$ where $a=$ first term and ...

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Find the sum up to $n$ terms of the G.P. $1,-a,a^2,-a^3.........$

answered Feb 26, 2014

Toolbox:Sum of $n$ terms of a G.P=$S_n=a.\large\frac{1-r^n}{1-r}$ where first term=$a$ and common...

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Find the sum upto $n$ terms of the G.P. $\sqrt 7,\sqrt {21},3\sqrt 7,.......$

answered Feb 26, 2014

Toolbox:Sum of $n$ terms of a G.P.=$S_n=a.\large\frac{r^n-1}{r-1}$ where $a=$first term and $r=$...

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Find the sum of $20$ terms of the G.P., $0.15,0.015,0.0015................$

answered Feb 25, 2014

Toolbox:Sum of $n$ terms of a G.P$=S_n=a.\large\frac{1-r^n}{1-r}$Given G.P. is $ 0.15+0.015+0.0015...

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For what value of $x$ the numbers $-\large\frac{2}{7},$$x,-\large\frac{7}{2}$ are in G.P.?

answered Feb 25, 2014

Toolbox:If $x,y,z$ are in G.P. then common ratio $r=\large\frac{y}{x}=\frac{z}{y}$Given that $-\la...

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Which term of the sequence $\large\frac{1}{3},\frac{1}{9},\frac{1}{27}.......$ is $\large\frac{1}{19683}$?

answered Feb 25, 2014

Toolbox:$n^{th}$ term of a G.P$=t_n=a.r^{n-1}$ where $a=1^{st}\:term\:and\:r=common\:ratio.$The gi...

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Which term of the sequence $\sqrt 3,3,3\sqrt 3.......$ is $729?$

answered Feb 25, 2014

Toolbox:$n^{th}$ term of a G.P$=t_n=a.r^{n-1}$ where $a=1^{st}\:term\:and\:r=common\:ratio.$The g...

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Which term of the sequence $2,2\sqrt 2,4,.......\:\:is\:\:128?$

answered Feb 25, 2014

Toolbox:$n^{th}\:term$ of a G.P.$=t_n=a.r^{n-1}$ where $a=1^{st}\:term\:and\:d=common \:ratio.$Giv...

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The $4^{th}$ term of a G.P. is square of its second term, and the $1^{st}$ term is $-3$. Find its $7^{th}$ term.

answered Feb 25, 2014

Toolbox:$n^{th}\:term$ of a G.P.$=a.r^{n-1}$ where $1^{st}\:term=a$ and $common\: ratio=r$Given: ...

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The $5^{th},8^{th}\:and\:11^{th}$ terms of a $G.P.$ are $p,q\:and\:s$ respectively. Show that $q^2=ps$

answered Feb 25, 2014

Toolbox:$n^{th}$ term of a $G.P.=t_n=a.r^{n-1}$ where $a=1^{st} term,\:\:r=common\:ratio.$Given th...

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Find the $12^{th}$ term of the $G.P.$ whose $8^{th}$ term is $192$ and common ratio=$2$

answered Feb 25, 2014

Toolbox:$n^{th}$ tem of a $G.P.=t_n=a.r^{n-1}$ where $1^{st}$ term$ =a$ and common ratio$=r$Give...

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Find the $20^{th}$ term of the $G.P.\:\:\large\frac{5}{2},\frac{5}{4},\frac{5}{8},..........$

answered Feb 25, 2014

Toolbox:$n^{th}\:term$ of a $G.P.=t_n=a.r^{n-1}$ where $r$ is common ratio and $a=first\:term$Giv...

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A man starts repaying a loan as first installment of $Rs.100$. If he increases the installment by $Rs.5$ every month, what amount he will pay in his $30^{th}$ installment?

answered Feb 24, 2014

Toolbox:$n^{th}$ term of an $A.P.=t_n=a+(n-1)d$Given that the first installment is $Rs.100$Each ins...

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