Answers posted by rvidyagovindarajan_1

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$150$ workers were engaged to finish a job in certain no. of days. $4$ workers dropped out on second day. $4$ more workers dropped out on third day and so on. It took $8$ more days to finish the work. Find the no. of days in which the work was completed.

answered Apr 12, 2014

Toolbox:Sum of $n$ terms of an A.P. $=\large\frac{n}{2}$$\big[2a+(n-1)d\big]$Let the no. of days the...

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What will Rs. 500 amounts to in $10$ years after its deposit in a bank which pays annual interest of $10%$ compounded annually?

answered Apr 12, 2014

Given: the initial deposit $=Rs.500$The interest rate $=10$%After one year the amount $=500+\large\f...

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A manufacturer reckons that the value of a machine which costs him $Rs.15,625$, will depreciate each year by $20\%$. find the estimated value at the end of $5$ years.

answered Apr 11, 2014

Toolbox:$n^{th}$ term of a G.P. $=a.r^{n-1}$Given that the cost of the machine $=Rs.15,625$.The depr...

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A man deposited $Rs. 10,000$ in a bank at the rate of $5\%$ simple interest annually. Find the amount in $15^{th}$ year since he deposited the amount and also find the total amount after $20$ ears.

answered Apr 10, 2014

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

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A person writes a letter to four of his friends. He asks each one of them to copy the letter and mail to four different persons with the instruction that they move the chain similarly. Assuming that the chain is not broken, and that it costs 50 paise to mail one letter, find the amount spent on the postage when $8^{th}$ set of letter is mailed.

answered Apr 9, 2014

Toolbox:The $n^{th}$ term of a G.P. $=t_n=a.r^{n-1}$We have to find the number of letters mailed in...

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Shamshad Ali buys a scooter for Rs. 22,000. He pays Rs. 4,000 cash and agrees to pay the balance in annual installment of Rs. 1,000 plus 10% interest on the unpaid amount. How much will the scooter costs him?

answered Apr 9, 2014

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

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A farmer buys a used tractor for Rs. 12,000. He pays Rs. 6,000 cash and agrees to pay the balance in annual installments of Rs. 500 plus 12% interest on the unpaid amount. How much will the tractor cost him?

answered Apr 8, 2014

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

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Show that $\large\frac{1\times2^2+2\times 3^2+........n\times (n+1)^2}{1^2\times 2+2^2\times 3+.........n^2\times (n+1)}=\frac{3n+5}{3n+1}$

answered Apr 7, 2014

Toolbox:Sum of $n$ terms of any series $=S_n=\sum t_n$$\sum n^3=\large\frac{n^2(n+1)^2}{4}$$\sum n^...

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Find the sum of $n$ terms of the series $\large\frac{1^3}{1}+\frac{1^3+2^3}{1+3}+\frac{1^3+2^3+3^3}{1+3+5}+.........$

answered Apr 6, 2014

Toolbox:$\sum n^3=\large\frac{n^2(n+1)^2}{4}$Sum of $n$ terms of an A.P. $=\large\frac{n}{2}$$\big[...

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If $S_1,S_2,S_3$ are the sum of $n$ natural numbers,sum of their squares and sum of their cubes respectively then Show that $9S_2^2=S_3(1+8S_1)$

answered Apr 3, 2014

Toolbox:$\sum n=\large\frac{n(n+1)}{2}$$\sum n^2=\large\frac{n(n+1)(2n+1)}{6}$$\sum n^3=\large\frac{...

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Find the sum of $n$ terms of the series $0.6+0.66+0.666+......$

answered Apr 3, 2014

Toolbox:Sum of $n$ terms of a G.P is $a.\large\frac{1-r^n}{1-r}$Given series is $S_n=0.6+0.66+0....

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Find the sum of $n$ terms of the series $3+7+13+21+31+.............$

answered Apr 2, 2014

Toolbox:Sum of $n$ terms of ant series is given by $S_n=\sum t_n$um of $n-1$ terms of an A.P. $=\la...

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Find the $20^{th}$ term of the series $2\times 4+4\times 6+6\times 8+..........$

answered Apr 2, 2014

Given series is $(2\times 4)+(4\times 6)+(6\times 8)+..........$Each bracket consists of product of ...

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Find the sum of the series $5+55+555+..........$ upto $n$ terms.

answered Apr 2, 2014

Toolbox:Sum of $n$ terms of a G.P $=a.\large\frac{r^n-1}{r-1}$Given series is $ S_n=5+55+555+.....

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If $a,b,c$ are in A.P., $b,c,d$ are in G.P. and $\large\frac{1}{c},\frac{1}{d},\frac{1}{e}$ are in A.P., then prove that $a,c,e$ are in G.P.$

answered Apr 2, 2014

Toolbox:If $x,y,z$ are in A.P. then $2y=x+z$If $x,y,z$ are in G.P., then $y^2=xz$Given $a,b,c$ ...

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The ratio of A.M. and G.M. of two positive numbers $a$ and $b$ is $m:n$. Show that $a:b=(m+\sqrt {m^2-n^2}):(m-\sqrt {m^2-n^2})$

answered Apr 1, 2014

Toolbox:A.M. between $a$ and $b$ is $\large\frac{a+b}{2}$G.M. between $a$ and $b$ is $\sqrt ...

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If $a$ and $b$ are roots of $x^2-3x+p=0$ and $c$ and $d$ are roots of $x^2-12x+q=0$, where $a,b,c,d$ form a G.P. then prove that $(q+p):(q-p)=17:15$

answered Mar 31, 2014

Toolbox:$\alpha$ and $\beta$ are roots of the equation $ax^2+bx+c=0$ then $\alpha+\beta=-\large\...

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If $a,b,c,d$ are in G.P. then show that $(a^n+b^n),(b^n+c^n),(c^n+d^n)$ are in G.P.

answered Mar 31, 2014

Toolbox:If $x,y,z$ are in G.P. then $y^2=xy$Given that $a,b,c,d$ are in G.P.Let $a,b,c,d$ repres...

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If $a\bigg(\large\frac{1}{b}+\frac{1}{c}\bigg)$$,\:b\bigg(\large\frac{1}{c}+\frac{1}{a}\bigg)$$,\:c\bigg(\large\frac{1}{a}+\frac{1}{b}\bigg)$ are in A.P., then prove that $a,b,c$ are in A.P.

answered Mar 31, 2014

Toolbox:If $a,b,c$ are in A.P. then $2b=a+c$Given: $a\bigg(\large\frac{1}{b}+\frac{1}{c}\bigg)$$...

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If $p^{th},\:q^{th}\:\:and\:\:r^{th}$ terms oa an A.P. are $a,\:b\:and\:c$ respectively, then show that $(q-r)a+(r-p)b+(p-q)c=0$

answered Mar 31, 2014

Given $p^{th}$ term of an A.P.$a$, $q^{th}$ term $=b$ and $r^{th}$ term $=c$$\Rightarrow\:t_...

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Let $S$ be the sum, $P$ the product and $R$ the sum of reciprocals of $n$ terms of a G.P. Prove that $P^2R^n=S^n$

answered Mar 31, 2014

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

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If $\large\frac{a+bx}{a-bx}=\frac{b+cx}{b-cx}=\frac{c+dx}{c-dx}$$\:(x\neq0)$, then show that $a,b,c\:and\:d$ are in G.P.

answered Mar 30, 2014

Toolbox:comoponendo and dividendo: if $\large\frac{a}{b}=\frac{c}{d}$ then $\large\frac{a+b}{a-b}...

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The sum of first four terms of an A.P. is $56$. The sum of last four terms is $112$. If its first term is $11$ then find the number of terms.

answered Mar 30, 2014

Toolbox:Sum of $n$ terms of an A.P. $=S_n=\large\frac{n}{2}$$(2a+(n-1)d)$Let the A.P. be $a,\:a+d...

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A G.P. consists of even number of terms. If the sum of all the numbers is $5$ times the sum of terms occupying the odd places, then find the common ratio.

answered Mar 30, 2014

Since it is given that the number of terms is even number,let the number of terms in the G.P be $2n...

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The sum of three numbers in G.P. is $56$. If we subtract $1,7,21$ from the numbers in that order, we obtain an A.P. Find the numbers.

answered Mar 30, 2014

Toolbox:If $a,b,c$ are in A.P. then $2b=a+c$Let the three numbers in G.P. be $a,ar,ar^2$It given...

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The first term of a G.P. is $1$. The sum of third term and fifth term is $90$. Find the common ratio of the G.P.

answered Mar 30, 2014

Toolbox:$n^{th}$ term of a G.P. $=t_n=a.r^{n-1}$Given: $1^{st}$ term =1 and sum of $3^{rd}$ and...

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The sum of some terms of a G.P. is $315$ whose first term and the common ratio are $5$ and $2$ respectively. Find the last term and number of terms of the series.

answered Mar 30, 2014

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

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If $f$ is a function satisfying $f(x+y)=f(x).f(y)\:\:\forall x,y\in N$ such that $f(1)=3\:\:and\:\:\displaystyle\sum ^{n}_{x=1} f(x)=120$, then find the value of $n$.

answered Mar 29, 2014

Toolbox:Sum of $n$ terms of G.P. $=S_n=a\large\frac{r^n-1}{r-1}$Given: $f(x+y)=f(x).f(y),$$\qquad...

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Find the sum of all two digit numbers which when divided by $4$ yields $1$ as remainder.

answered Mar 29, 2014

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

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Find the sum of all the integers from $1$ and $100$ that are divisible by $2$ or $5$.

answered Mar 29, 2014

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

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Find the sum of all the numbers between $200$ and $400$ which are divisible by $7$.

answered Mar 28, 2014

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

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If the sum of $n,2n\:and\:3n$ terms of an A.P. are $S_1,S_2\:and\:S_3$ respectively, then show that $S_3=3(S_2-S_1)$

answered Mar 28, 2014

Toolbox:Sum of $n$ terms of an A.P $=\large\frac{n}{2}$$(2a+(n-1)d)$Given that $S_1=$ Sum of $n$ ...

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If the sum of three numbers in A.P. is $24$ and their product is $440$, then find the numbers.

answered Mar 28, 2014

Toolbox:Three numbers in A.P. are to be assumed as $a-d,\:a,\:a+d$Let the three numbers in A.P. be...

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Show that the sum of $(m+n)^{th}$ and $(m-n)^{th}$ terms of an A.P is twice its $m^{th}$ term.

answered Mar 27, 2014

In an A.P. $n^{th}$ term = $t_n=a+(n-1)d$$\Rightarrow\:t_{m+n}=a+(m+n-1)d$ ....(i) and$t_{m-n}=a+(...

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Find the sum of $n$ terms of the series whose $n^{th}$ term is $(2n-1)^2$

answered Mar 27, 2014

Toolbox:Sum of $n$ terms of any series $=S_n=\sum t_n$$\sum (A+B)=\sum A+\sum B$$\sum k.A=k.\sum A$...

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Find the sum of $n$ terms of the series whose $n^{th}$ term is given by $n^2+2^n$.

answered Mar 27, 2014

Toolbox:Sum of $n$ terms of any series is $=S_n=\sum\:t_n$$\sum (A+B)=\sum A+\sum B$$\sum n^2=1^2+2...

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Find the sum of $n$ terms of the series whose $n^{th}$ term is given by $n(n+1)(n+4)$

answered Mar 24, 2014

Toolbox:Sum of $n$ terms of any series =$S_n=\sum t_n$$\sum (A+B+C)=\sum A+\sum B+\sum C$$\sum kA=k...

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Find the sum to $n$ terms of the series $1^2+(1^2+2^2)+(1^2+2^2+3^2)+........$

answered Mar 24, 2014

Toolbox:$1^2+2^2+3^2+.......=\sum \:n^2=\large\frac{n(n+1)(2n+1)}{6}$Sum of $n$ terms of any series...

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Two schools $P$ and $Q$ want to award their selected students on the values of Tolerance,Kindness and Leadership. The school $P$ wants to award $Rs.x,\:Rs.y\:and\:\:Rs.z$ each for the three respective values to $3,2\:and\:1$ students respectively with a total award money of $ Rs.2,200.$ School $Q$ wants to spend $Rs.3,100$ to award its 4,1 and 3 students on the respective values (by giving the same award money to the three values as school $P$). If the total amount of award for one prize on each value is $Rs.1,200,$ using matrices, find the award money for each value.

answered Mar 23, 2014

3 equations are formed from the given statements.$i.e.,$ Given: $3x+2y+z=2200$$\qquad\:4x+y+3z=31...

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If $x=a\:cos\theta+b\:sin\theta$ and $y=a\:sin\theta-b\:cos\theta$, then show that $y^2.\large\frac{d^2y}{dx^2}$$-x\large\frac{dy}{dx}$$+y=0$

answered Mar 23, 2014

Given: $x=a\:cos\theta+b\:sin\theta$ and $y=a\:sin\theta-b\:cos\theta$Step 1Differentiating $y$ ...

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Three cards are drawn at random (without replacement) from a well shuffled pack of 52 playing cards. Find the probability distribution of number of red cards. Hence find the mean of the distribution.

answered Mar 23, 2014

Toolbox: Mean of the probability distribution $=\overline {X}= \sum (X_i \times P(X_i))$ ...

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Show that $cot^{-1}7+cos^{-1}8+cot^{-1}18=cot^{-1}3$

answered Mar 21, 2014

Toolbox:$cot^{-1}x=tan^{-1}\large\frac{1}{x}$$tan^{-1}x+tan^{-1}y=tan^{-1}\bigg(\large\frac{x+y}{1-x...

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Solve: $cos(tan^{-1}x)=sin(cot^{-1}(\large\frac{3}{4}))$

answered Mar 21, 2014

Let $cot^{-1}\large\frac{3}{4}$$=y$$\Rightarrow\:coty=\large\frac{3}{4}$$\therefore\:\:siny=\large\...

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Integrate the function $(3x-2)\sqrt{x^2+x+1}$

answered Mar 21, 2014

Toolbox: Method of substitution: Given $\int f(x)dx$ can be transformed into another form by c...

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Solve the differential equation $(x^2-yx^2)dy+(y^2+x^2y^2)dx=0$ Given $y=1$ when $x=1$

answered Mar 21, 2014

Given equation is $(x^2-yx^2)dy+(y^2+x^2y^2)dx=0$$\Rightarrow\:\large\frac{dy}{dx}=-\large\frac{y^...

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Find a unit vector perpendicular to each of the vector $ \overrightarrow a + \overrightarrow b$ and $ \overrightarrow a - \overrightarrow b$, where $ \overrightarrow a = \hat i + \hat j + \hat k$ and $ \overrightarrow b = \hat i + 2\hat j +3\hat k$

answered Mar 21, 2014

Toolbox: Unit vector $\perp$ to two vectors $\overrightarrow a$ and $\overrightarrow b$ is given ...

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Find the distance of the point $P(-1,-5,-10)$ from the point of intersection of the line joining the points $A(2,-1,2)$ and $B(5,3,4)$with the plane $x-y+z=5$

answered Mar 21, 2014

Given: $A(2,-1,2)$, $B(5,3,4)$Direction ratio of the line $AB=(5-2,3+1,4-2)$=(3,4,2)$Equation of t...

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Find the equation of the plane that contains the point $(1,-1,2)$ and is $\perp$ to the planes $2x+3y-2z=5$ and $x+2y-3z=8$. Hence find the distance of the point $(-2,5,5)$ from the plane obtained above.

answered Mar 21, 2014

Toolbox:Distance of the point $(x_1,y_1,z_1)$ from the plane $ax+by+cz+d=0$ is given by $\bigg|\l...

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Find the distance between the two lines $l_1$ and $l_2$ where $l_1: \overrightarrow r=(\hat i+2\hat j-4\hat k)+\lambda(2\hat i+3\hat j+6\hat k)$ and $l_2: \overrightarrow r=(3\hat i+3\hat j-5\hat k)+\mu(4\hat i+6\hat j+12\hat k)$

answered Mar 21, 2014

Toolbox:General vector equation of a line is $\overrightarrow r=\overrightarrow a+\lambda\overrighta...

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Find the value of $\hat i\times (\hat j+\hat k)+\hat j\times (\hat k+\hat i)+\hat k\times (\hat i+\hat j)$

answered Mar 21, 2014

Toolbox:$\hat i\times \hat i=\hat j\times\hat j=\hat k \times\hat k=0$$\hat i\times\hat j=\hat k,\:\...

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