Answers posted by yamini.v

0 votes

Natural numbers are divided into groups $\;(1)\;,\;(2,3,4)\;,\;(5,6,7,8,9)\;.....$ Sum of first and last term of $n^{th}\;$ group will be :

answered Jan 21, 2014

Toolbox: Sum of $n$ terms of an $A.P.$ with first term $a$ and common difference, $d$ is $...

0 votes

If in an AP $m$ times $\;m^{th}\;$ term equals $n$ times $\;n^{th}$ term , then $\;(m+n)^{th}$ term will be

answered Jan 21, 2014

Answer : (b) 0Explanation : $\;m*a_{m}=n*a_{n}$$m*[a+(m-1)d]=n*[a+(n-1)d]$$(m-n)\;(a-d)+(m^2-n^2)\;...

0 votes

Let $a_{n}\;$ be $\;n^{th}$ term of AP. If $\;\displaystyle\sum_{r=1}^{50}\;a_{2r}=p\;and \;\displaystyle\sum_{r=1}^{50}\;a_{2r-1}=q\;$ the common difference is :

answered Jan 21, 2014

Answer : (d) $\frac{p-q}{50}$Let d be common difference $p-q=\displaystyle\sum_{r=1}^{50}\;(a_{2r}-...

0 votes

If $\;x=\sqrt{2+\sqrt{2+\sqrt{2+------\;\infty}}}\;and\;y=\sqrt{2\;\sqrt{2\;\sqrt{2\;\sqrt{2\;----\;\infty}}}}$ then $xy$ equals

answered Jan 20, 2014

Answer : (b) 4Explanation : $x^2=x+2\quad\;x=2$$y^2=2y\quad\;y=2$$xy=4.$

0 votes

Sum of series $\;S=\frac{1}{2\sqrt{1}+1\sqrt{2}}+\frac{1}{3\sqrt{2}+2\sqrt{3}}+\;....\;+\frac{1}{100\sqrt{99}+99\sqrt{100}} \;is$

answered Jan 20, 2014

Answer : $(b)\;\frac{9}{10}$Explanation : $a_{k}=\frac{1}{\sqrt{k+1}\;k+\sqrt{k}\;(k+1)}$$=\frac{1}{...

0 votes

If $1^4+2^4+3^4+.....n^4$ = $an^5+bn^4+cn^3+dn^2+en+f$, find $a$

answered Jan 20, 2014

Answer : (d) $\frac{1}{5}$Explanation : $1^4+2^4+3^4+\;----\;n^4=an^5+bn^4+cn^3dn^2+en+f$$1^4+2^4+...

0 votes

Sum of series $(n)(n)$ + $(n-1)(n+1)$ + $(n-2)(n+2)$ +....+ $1\;(2n-1)$ is

answered Jan 20, 2014

Answer : (d) NoneExplanation : $S_{n}=\sum_{r=0}^{r=n-1}\;(n-r)(n+r)=\sum\;n^2-r^2$$=n^3-\frac{1}{...

0 votes

Sum of series $\;\sum_{r=1}^{n}\;r\;log\;\frac{r+1}{r}$ is :

answered Jan 20, 2014

Answer : (c) $n\;log(n+1)-log\;n!$Explanation : $a_{r}=r\;log\;\frac{r+1}{r}$$=r\;[log(r+1)-log\;r...

0 votes

If $a_{1},a_{2},...a_{n}$ are in HP. , Then $\;a_{1}a_{2}+a_{2}a_{3}+.....+a_{n}\;a_{n-1}$ is equal to

answered Jan 20, 2014

Answer : $(c)\;(n-1)a_{1}a_{n}$Explanation : $a_{1}\;,a_{2}\;....\;a_{n}\;$ are in HP$\frac{1}{a_{1}...

0 votes

Sum of $n$ terms of series $S$ = $1$ + $2 \;(1+\frac{1}{n})$ + $3(1+\frac{1}{n})^2$ + ....is given by

answered Jan 20, 2014

Answer : (c) $n^2$Explanation : Let $\;1+\frac{1}{n}=x$$S=1+2x+3x^2+4x^3+....$$x\;S=x+2x^2+3x^3+4x^...

0 votes

answered Jan 20, 2014

Answer : (b) $\frac{\pi}{3}\;,\frac{2\pi}{3}$Explanation : $|cos\;x|\;<\;1$$2^{2\;(\frac{1}{1-|...

0 votes

If $f\;(x)$ is a two degree polynomial such that $f\;(3)=f\;(-3)$ and $a, b, c$ are in $AP$, then $f'(a)$, $f'(b)$ and $f'(c)$ are in

answered Jan 20, 2014

Answer : (a) APExplanation : $f\;(x)=Ax^2+Bx+C$$f\;(3)=f(-3)$$9A+3B+C=9A-3B+C$$B=0$$f\;(x)=Ax^2+C$...

0 votes

If $\frac{1}{1^2}+\frac{1}{2^2}+\frac{1}{3^2}+$......upto $\infty$ = $\frac{\phi^2}{g}$, then $\frac{1}{1^2}+\frac{1}{3^2}+\frac{1}{5^2}+$.....upto $\infty$ will be

answered Jan 20, 2014

Answer : (c) $\frac{\phi^2}{12}$Explanation : $\;\frac{1}{1^2}+\frac{1}{3^2}+\frac{1}{5^2}+\;...\;u...

0 votes

If $S_{n}=\frac{1}{6}\;n\;(2n^2+9n+13)\;$, then $\sum_{r=1}^n\;\sqrt{a_{r}}\;$ equals.

answered Jan 20, 2014

Answer : (a) $\frac{1}{2}n(n+3)$Explanation : $a_{n}=S_{n}-S_{n-1}$$=\frac{1}{6}n(2n^2+9n+13)-\frac{...

0 votes

Sum of series $\;\frac{1}{2}+\frac{3}{4}+\frac{7}{8}+\frac{15}{16}+.....\;is:$

answered Jan 20, 2014

Answer : $\;{n-1+\frac{1}{2^n}}$Explanation : $t_{r}=1-\frac{1}{2^r}$$\sum_{r=1}^n\;(1-\frac{1}{2^r}...

0 votes

If $f\;(xy)=f\;(x+y)\;\forall\;x,y\;\in\;R$ and $\;f\;(2009)=2009$, then $f\;(-2009)$ equals

answered Jan 20, 2014

Answer : (b) 2009Explanation : $f\;(2009)=f\;(2009+0)=f\;(2009*0)$$=f\;(0)=2009$$f\;(-2009)=f\;(-2...

0 votes

Sum of n terms of series $\;\frac{1}{1^3}\;+\frac{1+2}{1^3+2^3}\;+\frac{1+2+3}{1^3+2^3+3^3}\;+....\;is$

answered Jan 20, 2014

Answer : (b) $\;\frac{2n}{n+1}$Explanation : $t_{n}=\frac{1+2+3+\;....\;+n}{1^3+2^3+3^3+\;....\;+n^...

0 votes

If x, y, z > 1 are in GP, then $\frac{1}{1+lnx}$, $\frac{1}{lny}$, $\frac{1}{lnz}$ are in

answered Jan 20, 2014

Answer : (c) HPExplanation : x , y , z in GP$y^2=xz$$ln\;y^2\;=\;ln\;xz$$2\;ln\;y=ln\;x+ln\;z$$ln...

0 votes

If in an AP , $\;m^{th}$ term is $\;\frac{1}{n}$ and $\;n^{th}\;term\;is\frac{1}{m}$ , then $\;mn^{th}$ term is :

answered Jan 20, 2014

Answer : (d) 1Explanation : $a_{m}-a_{n}=\frac{1}{n}-\frac{1}{m}$$(m-n)\;d=\frac{m-n}{mn}\quad\;d=...

0 votes

Let $S_{1}, S_{2}$....be the squares such that for each $n\geq\;1$, the length of side of $S_{n}$ = length of diagonal of $S_{n+1}$. If side of $S_{1}$ is 20 cm, the smallest value of n such that area of $S_{n}\;<\;2$ is

answered Jan 20, 2014

Answer : (c) 9Explanation : Let $\;a_{n}\;be\;the\;length\;of\;side$ $a_{n}=\sqrt{2}\;a_{n+1}$$\f...

0 votes

If $a, b, c$ are in AP and $a^2$, $b^2$, $c^2$ in GP, and $a+b+c=\frac{3}{2}$, then value of $c$ is

answered Jan 20, 2014

Answer : (c) $\;\frac{1}{2}+\frac{1}{\sqrt{2}}$Explanation : $\;a+b+c=\frac{3}{2}$$3b=\;\frac{3}{2...

0 votes

If $(1+3+5+....+p)$+$(1+3+5+....+q)$ = the smallest value of $p+q+r, (p>6)$ is

answered Jan 20, 2014

Answer : (b) 21Explanation : $1+3+5\;.......\;+k=(\frac{k+1}{2})^{2}$So , $\;(\frac{p+1}{2})^2\;+(\...

0 votes

Sum of series $S$ = $\frac{1}{2}$+$\frac{1}{3}\;(1+2)$+$\frac{1}{4}\;(1+2+3)$+$\frac{1}{5}\;(1+2+3+4)$+......upto 20 terms is

answered Jan 20, 2014

Answer : (b) 105Explanation : $k^{th}\;term\;t_{k}=\frac{1}{k+1}\;[1+2+\;.....\;+k]$$=\frac{1}{k+1...

0 votes

Sum of series $1^2$-$2^2$+$3^2$-$4^2$+$5^2$+.....+$1001^2$ is

answered Jan 20, 2014

Answer : (a) 501501Explanation : $S=1^2-2^2+3^2-4^2+\;.....\;+1001^2$$S=(1-2)(1+2)\;+(3-4)(3+4)\;+....

0 votes

If the $\;10^{th}\;$ term of a $GP$ is $9$ and the $\;4^{th}\;$ term is $4$, then the $\;7^{th} \;$term is

answered Jan 19, 2014

Answer : (c) 6Explanation : $\;a_{4}=4=ar^3$$a_{10}=9=ar^9$Method 1 : $a_{7}=\sqrt{(a_{4}*a_{10})...

0 votes

In an AP, the $\; 4^{th}\;$ term is $36$. The$\; 21^{st}\;$ term is $108$ more than the $\;9^{th}\;$ term. What is the first term a and the common difference $d$?

answered Jan 19, 2014

Answer : (b) a=9 , d=9Explanation : $The\;4^{th}\;term\;a_{4}=a+3d=36$$The\;9^{th}\;term\;a_{9}=a...

0 votes

The sum of 0.2, 0.22, 0.222, 0.2222.....till n terms is given by

answered Jan 18, 2014

Explanation : $\;S_{n}=0.2+0.22+0.222+0.2222\;-----\;n\;terms $$= 2 (0.1 + 0.11+ 0.111+ ------ n\; ...

0 votes

The sum of $n$ terms of an arithmetic progression is $\;n\;(2n-1)$. What is the $m^{th}$ term of the series?

answered Jan 18, 2014

Answer : (c) 4m-3Explanation : $T_{m}=S_{m}-S_{m-1}=m\;(2m-1)-(m-1)\;[2(m-1)-1]$$=2m^2-m-\;(2m^2-...

0 votes

If a, g and h are the AM, GM and HM respectively of two positive numbers x and y, then identify the correct statement.

answered Jan 18, 2014

Answer : (c) $g\;is\;the\;GM\;between\;a\;and\;h$Explanation : By definition , $a=\frac{x+y}{2}\;,...

0 votes

The sum of n terms of two APs is in the ratio$\; 5n + 1: 4n+10$. Find the ratio of their $\;5^{th}$ terms.

answered Jan 18, 2014

Answer : (b) 1Explanation : $\frac{S_{1}}{S_{2}}=\frac{n/2\;[2a_{1}+(n-1)d_{1}]}{n/2\;[2a_{2}+(n-1...

0 votes

The number of terms in the AP a, b,....c is

answered Jan 17, 2014

Answer : (c) $\;\frac{(b+c-2a)}{(b-a)}$Explanation : Let d be the common differance of the APThen...

0 votes

A train travels from A to B at speed of x km/hour and from B to A at speed of y km/hour. The average speed of the train from A to B and back to A, would be calculated by the

answered Jan 17, 2014

Answer : (a) Arithmetic meanExplanation : Average speed of the train is given by $\frac{x+y}{2}$ ...

0 votes

If three numbers are in GP, then their logarithms will be in

answered Jan 17, 2014

Answer : (a) APExplanation : a,b,c are in GP $\frac{b}{a}=\frac{c}{b}$$b^2=ac$logarithm on both...

0 votes

The sum of three decreasing numbers in AP is 27. If -1, -1, and 3 are added to them respectively the resulting series is in GP. The numbers are

answered Jan 17, 2014

Answer : (a) 17,9,1Explanation : Let the 3 numbers in AP be a+d,a and a-d (as the numbers are dec...

0 votes

The $n^{th}$ term of series is $\frac{1}{1}$, $\frac{(1+2)}{2}$, $\frac{(1+2+3)}{3}$......is

answered Jan 17, 2014

Answer : $\frac{n+1}{2}$Explanation : The series is of the form $\sum\;\frac{n}{n},$$\sum\;n=\fra...

0 votes

Sum of squares of first n natural numbers exceeds their sum by 330, then n =

answered Jan 17, 2014

Answer : (b) 10Explanation : sum of sqaures of first natural numbers $\;\frac{n}{6(n+1)(2n+1)}$Su...

0 votes

If a, b and c are in H.P, then a/(b+c), b/(c+a), c/(a+b) are in

answered Jan 17, 2014

Answer : (b) HPExplanation : Given that a,b,c are in HPTherefore 1/a,1/b,1/c are in APMultiplyi...

0 votes

If arithmetic, geometric and harmonic means between two positive real numbers is A, G and H respectively, then, which of the following is true?

answered Jan 17, 2014

Answer : (a) A>G>HExplanation : Let the two real numbers be a,b$A=(a+b)/2\quad\;G=\sqrt...

0 votes

The sum of n terms of the infinite series $1.3^2$+$2.5^2$+$3.7^2$+-------$\infty$ is given by

answered Jan 17, 2014

Answer : (a) $\;\frac{n}{6(n+1)(6n^2+14n+7)}$The $\;n^{th}\;$ term of the series is given by $\...

0 votes

Integrate the rational functions$\;\large\frac{3x-1}{(x+2)^2}$

answered Jan 9, 2014

Toolbox: $(i)\;$Form of the rational function\[\frac{px+q}{(x+a)^2}\] $\;$Form of the partial ...

0 votes

Show that the height of the cylinder of maximum volume that can be inscribed in a sphere of radius R is $\large\frac{2R}{\sqrt 3}$ . Also find the maximum volume.

answered Jan 6, 2014

Toolbox:Volume =$\pi r^2h$https://clay6.com/mpaimg/ch6%20misc%20q17.jpgStep 1:Radius of the sphere$=...

0 votes

Find the maximum area of an isosceles triangle inscribed in the ellipse $\large\frac{x^2}{a^2} + \frac{y^2}{b^2} $$= 1 $ with its vertex at one end of the major axis.

answered Jan 6, 2014

Toolbox: Area of a triangle =$\large\frac{1}{2}$$\times l\times h$ $\large\frac{d}{d...

0 votes

Ram shares a photo link on facebook with 3 persons and requests each of them to share with 3 more persons and continue the chain. For every share of the photo link if Rs 100 is donated to the Adyar Cancer society, what is the total amount collected by the society till the $b^{th}$ set of shares.

answered Jan 6, 2014

Answer : (c) 36400Explanation : a=1 ,r=3 ,n=6$S_{6}\frac{a(r^n-1)}{r-1}=\frac{1(3^6-1)}{3-1}$$\fr...

0 votes

Given $4$ numbers, the first $3$ numbers are in $AP$ and the last $3$ numbers are in $GP$. The $1^{st}$ and $3^{rd}$ terms add upto $2$ and the $2^{nd}$ and $4^{th}$ terms add to $17$. Find the numbers.

answered Jan 6, 2014

Answer : (a) -2,1,4,16Explanation : Let the 4 numbers be a,b,c,d$a,b,c\;in\;AP\quad\;2b=a+c$$b,c,d...

0 votes

If $a, b, c$ are in $AP, a, x, b$ and $b, y, c$ in $GP$, what is the progression type for $x^2,b^2,y^2$?

answered Jan 6, 2014

Answer : (a) APExplanation : a,b,c in AP 2b=a+c a,x,b in GP $x^2=ab$b,y,c in GP $y^2=bc$$x...

0 votes

Find the sum of the series upto 20 term 1+(1+2)+(1+2+3)+--------

answered Jan 6, 2014

Answer : (a) 1540Explanation : $t_{n}=\frac{n(n+1)}{2}$$S_{n}=\sum\;\frac{n(n+1)}{2}=\frac{1}{2}\;...

0 votes

Find the sum of 8 terms of the series whose $n^{th}$ term is given by $n^2+2^n$

answered Jan 6, 2014

Answer : (c) 714Explanation : $t_{n}=n^2+2^n$$S_{n}=\frac{n(n+1)(2n+1)}{6}+\frac{2(2^n-1)}{2-1}$$...

0 votes

For what value of n , the $n^{th}$ term of the GPs 2560,1280,640 ----- and 10,20,40 ------ are equal.

answered Jan 6, 2014

Answer : (c) 5Explanation : In the GP 2560,1280,640 ------ $a=2560\;,r=1/2$$t_{n}=\frac{2560}{2^{n...

0 votes

The first term of a $GP$ is $4$ and the sum of its $3^{rd}$ and $5^{th}$ terms is $360$. Find the common ratio.

answered Jan 6, 2014

Answer : (c) 3Explanation : a=4$ar^2\;+\;ar^4=360$$r^4+r^2=90$$r^4+r^2-90=0$$r^4+10r^2-9r^2-90=0$$r...

0 votes

Find three numbers in AP whose sum is 33 and the sum of whose squares is 395

answered Jan 6, 2014

Answer : (c) 7,11,15Explanation : Let the numbers in AP be a-d,a,a+d$sum=33=a-d+a+a+d$$33=3a$$a...

Ask Questions, Get Answers