if the seventh term from the beginning and end in the binomial expansion of
Question: if the seventh term from the beginning and end in the binomial expansion of $\left(\sqrt[3]{2}+\frac{1}{\sqrt[3]{3}}\right)^{n}$ are equal, find $n$. Solution: In the binomail expansion of $\left(\sqrt[3]{2}+\frac{1}{\sqrt[3]{3}}\right)^{n},[(n+1)-7+1]^{\text {th }}$ i.e., $(n-5)^{\text {th }}$ term from the beginning is the $7^{\text {th }}$ term from the end. Now, $T_{7}={ }^{n} C_{6}(\sqrt[3]{2})^{n-6}\left(\frac{1}{\sqrt[3]{3}}\right)^{6}={ }^{n} C_{6} \times 2^{\frac{n}{3}-2} \tim...
Read More →Which term of the sequence 114, 109, 104,
Question: Which term of the sequence 114, 109, 104, ... is the first negative term? Solution: Here, A.P is So, first term, Now, Common difference (d) = $=109-114$ $=-5$ Now, we need to find the first negative term, $a_{n}0$ $114+(n-1)(-5)0$ $114-5 n+50$ $119-5 n0$ $5 n119$ Further simplifying, we get, $n\frac{119}{5}$ $n23 \frac{4}{5}$ $n \geq 24$ (as $n$ is a natural number) Thus, $n=24$ Therefore, the first negative term is the $24^{\text {* term }}$ of the given A.P....
Read More →Find n in the binomial
Question: Find $n$ in the binomial $\left(\sqrt[3]{2}+\frac{1}{\sqrt[3]{3}}\right)^{n}$, if the ratio of $7^{\text {th }}$ term from the beginning to the $7^{\text {th }}$ term from the end is $\frac{1}{6}$. Solution: In the binomail expansion of $\left(\sqrt[3]{2}+\frac{1}{\sqrt[3]{3}}\right)^{n},[(n+1)-7+1]^{\text {th }}$ i.e., $(n-5)^{\text {th }}$ term from the beginning is the $7^{\text {th }}$ term from the end. Now, $T_{7}={ }^{n} C_{6}(\sqrt[3]{2})^{n-6}\left(\frac{1}{\sqrt[3]{3}}\right)...
Read More →Find n in the binomial
Question: Find $n$ in the binomial $\left(\sqrt[3]{2}+\frac{1}{\sqrt[3]{3}}\right)^{n}$, if the ratio of $7^{\text {th }}$ term from the beginning to the $7^{\text {th }}$ term from the end is $\frac{1}{6}$. Solution: In the binomail expansion of $\left(\sqrt[3]{2}+\frac{1}{\sqrt[3]{3}}\right)^{n},[(n+1)-7+1]^{\text {th }}$ i.e., $(n-5)^{\text {th }}$ term from the beginning is the $7^{\text {th }}$ term from the end. Now, $T_{7}={ }^{n} C_{6}(\sqrt[3]{2})^{n-6}\left(\frac{1}{\sqrt[3]{3}}\right)...
Read More →In the given figure, AD divides ∠BAC in the ratio 1 : 3 and AD = DB.
Question: In the given figure,ADdivides BACin the ratio 1 : 3 andAD=DB. Determine the value ofx. Solution: $\angle B A C+\angle C A E=180^{\circ} \quad[\because B E$ is a straight line $]$ $\Rightarrow \angle B A C+108^{\circ}=180^{\circ}$ $\Rightarrow \angle B A C=72^{\circ}$ Now, divide $72^{\circ}$ in the ratio $1: 3$. $\therefore a+3 a=72^{\circ}$ $\Rightarrow a=18^{\circ}$ $\therefore a=18^{\circ}$ and $3 a=54^{\circ}$ Hence, the angles are $18^{\circ}$ and $54^{\circ}$ $\therefore \angle B...
Read More →Write the common difference of an A.P.
Question: Write the common difference of an A.P. whosenth term is an= 3n + 7. Solution: In the given problem,nth term is given by, . To find the common difference of the A.P., we need two consecutive terms of the A.P. So, let us find the first and the second term of the given A.P. First term, $a_{1}=3(1)+7$ $=3+7$ $=10$ Second term $(n=2)$ $a_{2}=3(2)+7$ $=6+7$ $=13$ Now, the common difference of the A.P. $(d)=a_{2}-a_{1}$ $=13-10$ $=3$ Therefore, the common difference is $d=3$....
Read More →If p is a real number and if the middle term in the expansion of
Question: If $p$ is a real number and if the middle term in the expansion of $\left(\frac{p}{2}+2\right)^{8}$ is 1120, find $p$. Solution: In the binomial expansion of $\left(\frac{p}{2}+2\right)^{8}$, we observe that $\left(\frac{8}{2}+1\right)^{\text {th }}$ i.e., $5^{\text {th }}$ term is the middle term. It is given that the middle term is 1120. $\therefore T_{5}=1120$ $\Rightarrow^{8} C_{4}\left(\frac{p}{2}\right)^{8-4}(2)^{4}=1120$ $\Rightarrow p^{4}=16$ $\Rightarrow p=\pm 2$ Hence, the re...
Read More →Find the sixth term in the expansion
Question: Find the sixth term in the expansion $\left(y^{\frac{1}{2}}+x^{\frac{1}{3}}\right)^{n}$, if the binomial coefficient of the third term from the end is 45 . Solution: In the binomial expansion of $\left(y^{\frac{1}{2}}+x^{\frac{1}{3}}\right)^{n}$, there are $(n+1)$ terms. The third term from the end in the expansion of $\left(y^{\frac{1}{2}}+x^{\frac{1}{3}}\right)^{n}$, is the third term from the beginning in the expansion of $\left(x^{\frac{1}{3}}+y^{\frac{1}{2}}\right)^{n}$. $\therefo...
Read More →Find the sixth term in the expansion
Question: Find the sixth term in the expansion $\left(y^{\frac{1}{2}}+x^{\frac{1}{3}}\right)^{n}$, if the binomial coefficient of the third term from the end is 45 . Solution: In the binomial expansion of $\left(y^{\frac{1}{2}}+x^{\frac{1}{3}}\right)^{n}$, there are $(n+1)$ terms. The third term from the end in the expansion of $\left(y^{\frac{1}{2}}+x^{\frac{1}{3}}\right)^{n}$, is the third term from the beginning in the expansion of $\left(x^{\frac{1}{3}}+y^{\frac{1}{2}}\right)^{n}$. $\therefo...
Read More →Calculate the value of x in the given figure.
Question: Calculate the value ofxin the given figure. Solution: Join A and D to produce AD to E. Then, $\angle C A D+\angle D A B=55^{\circ}$ and $\angle C D E+\angle E D B=x^{\circ}$ Side AD of triangle ACD is produced to E. $\therefore \angle C D E=\angle C A D+\angle A C D \quad \ldots(i)$ (Exterior angle property) Side AD of triangle ABD is produced to E. $\therefore \angle E D B=\angle D A B+\angle A B D \ldots(i i)$ (Exterior angle property) Adding (i) and (ii) we get, $\angle C D E+\angle...
Read More →Define an arithmetic progression.
Question: Define an arithmetic progression. Solution: An arithmetic progression is a sequence of terms such that the difference between any two consecutive terms of the sequence is always same. Suppose we have a sequence So, if these terms are in A.P., then, $a_{2}-a_{1}=d$ $a_{3}-a_{2}=d$ $a_{4}-a_{3}=d$ And so on Here,dis the common difference of the A.P. Example: 1, 3, 5, 7, 9 is an A.P. with common difference (d) as 2....
Read More →In an A.P., the sum of first ten terms is −150
Question: In an A.P., the sum of first ten terms is 150 and the sum of its next ten terms is 550. Find the A.P. Solution: Here, we are givenand sum of the next ten terms is 550. Let us take the first term of the A.P. asaand the common difference asd. So, let us first finda10. For the sum of first 10 terms of this A.P, First term =a Last term =a10 So, we know, $a_{n}=a+(n-1) d$ For the 10thterm (n =10), $a_{10}=a+(10-1) d$ $=a+9 d$ So, here we can find the sum of the $n$ terms of the given A.P., ...
Read More →If the term free from x in the expansion of
Question: If the term free from $x$ in the expansion of $\left(\sqrt{x}-\frac{k}{x^{2}}\right)^{10}$ is 405, find the value of $k$. Solution: Let $(r+1)^{\text {th }}$ term, in the expansion of $\left(\sqrt{x}-\frac{k}{x^{2}}\right)^{10}$, be free from $x$ and be equal to $T_{r+1}$. Then, $T_{r+1}={ }^{10} C_{r}(\sqrt{x})^{10-r}\left(\frac{-k}{x^{2}}\right)^{r}={ }^{10} C_{r} x^{5-\frac{5 r}{2}}(-k)^{r} \quad \ldots(1)$ If $T_{r+1}$ is independent of $x$, then $5-\frac{5 r}{2}=0 \Rightarrow r=2$...
Read More →In the figure given alongside, AB || CD, EF || BC, ∠BAC = 60º and ∠DHF = 50º. Find ∠GCH and ∠AGH.
Question: In the figure given alongside, AB || CD, EF || BC, BAC = 60 and DHF = 50. Find GCH and AGH. Solution: In the given figure, AB || CD and AC is the transversal.ACD =BAC = 60 (Pair of alternate angles)OrGCH =60Now,GHC =DHF =50 (Vertically opposite angles)In∆GCH,AGH =GCH +GHC (Exterior angle of a triangle is equal to the sum of the two interior opposite angles)⇒ AGH = 60 + 50 = 110...
Read More →Find a, b and n in the expansion of
Question: Find a, b and n in the expansion of (a+b)n, if the first three terms in the expansion are 729, 7290 and 30375 respectively. Solution: We have : $T_{1}=729, T_{2}=7290$ and $T_{3}=30375$ Now, ${ }^{n} C_{0} a^{n} b^{0}=729$ $\Rightarrow a^{n}=729$ $\Rightarrow a^{n}=3^{6}$ ${ }^{n} C_{1} a^{n-1} b^{1}=7290$ ${ }^{n} C_{2} a^{n-2} b^{2}=30375$ Also, $\frac{{ }^{n} C_{2} a^{n-2} b^{2}}{{ }^{n} C_{1} a^{n-1} b^{1}}=\frac{30375}{7290}$ $\Rightarrow \frac{n-1}{2} \times \frac{b}{a}=\frac{25}...
Read More →In an A.P., the first term is 2, the last term is 29 and the
Question: In an A.P., the first term is 2, the last term is 29 and the sum of the terms is 155. Find the common difference of the A.P. Solution: In the given problem, we have the first and the last term of an A.P. along with the sum of all the terms of A.P. Here, we need to find the common difference of the A.P. Here, The first term of the A.P (a) = 2 The last term of the A.P (l) = 29 Sum of all the terms (Sn) = 155 Let the common difference of the A.P. bed. So, let us first find the number of t...
Read More →Calculate the value of x in each of the following figures.
Question: Calculate the value ofxin each of the following figures. Solution: (i) SideACof triangleABCis produced toE. $\therefore \angle E A B=\angle B+\angle C$ $\Rightarrow 110^{\circ}=x+\angle C \quad \ldots(i)$ Also, $\angle A C D+\angle A C B=180^{\circ} \quad$ [linear pair] $\Rightarrow 120^{\circ}+\angle A C B=180^{\circ}$ $\Rightarrow \angle A C B=60^{\circ}$ $\Rightarrow \angle C=60^{\circ}$ Substituting the value of $\angle C$ in $(i)$, we get $x=50$ (ii) From $\Delta A B C$ we have: $...
Read More →The first and the last term of an A.P. are 17 and 350 respectively.
Question: The first and the last term of an A.P. are 17 and 350 respectively. If the common difference is 9, how many terms are there and what is their sum? Solution: In the given problem, we have the first and the last term of an A.P. along with the common difference of the A.P. Here, we need to find the number of terms of the A.P. and the sum of all the terms. Here, The first term of the A.P (a) = 17 The last term of the A.P (l) = 350 The common difference of the A.P. = 9 Let the number of ter...
Read More →If the 2nd, 3rd and 4th terms in the expansion of
Question: If the 2nd, 3rd and 4th terms in the expansion of (x+a)nare 240, 720 and 1080 respectively, findx,a,n. Solution: In the expansion of $(x+a)^{n}$, the $2 \mathrm{nd}, 3$ rd and 4 th terms are ${ }^{n} \mathrm{C}_{1} x^{\mathrm{n}-1} a^{1},{ }^{\mathrm{n}} \mathrm{C}_{2} \mathrm{x}^{\mathrm{n}-2} \mathrm{a}^{2}$ and ${ }^{n} C_{3} x^{\mathrm{n}-3} a^{3}$, respectively. According to the question, ${ }^{n} C_{1} x^{n-1} a^{1}=240$ ${ }^{n} C_{2} x^{n-2} a^{2}=720$ ${ }^{n} C_{3} x^{n-3} a^...
Read More →If the sum of the first n terms of an A.P is 4n − n2,
Question: If the sum of the first $n$ terms of an A.P is $4 n-n^{2}$, What is the first term? What is the sum of first two terms? What is the second term? Similarly, find the third, the tenth and the $n$th terms. Solution: In the given problem, the sum ofnterms of an A.P. is given by the expression, $S_{n}=4 n-n^{2}$ So here, we can find the first term by substituting, $S_{n}=4 n-n^{2}$ $S_{1}=4(1)-(1)^{2}$ $=4-1$ $=3$ Similarly, the sum of first two terms can be given by, $S_{2}=4(2)-(2)^{2}$ $...
Read More →In an A.P., the first term is 22, nth term is −11 and the sum
Question: In an A.P., the first term is 22, nth term is 11 and the sum to first n terms is 66. Find n and d, the common difference Solution: In the given problem, we have the first and thenth term of an A.P. along with the sum of thenterms of A.P. Here, we need to find the number of terms and the common difference of the A.P. Here, The first term of the A.P (a) = 22 Thenth term of the A.P (l) = 11 Sum of all the terms Let the common difference of the A.P. bed. So, let us first find the number of...
Read More →If the 6th, 7th and 8th terms in the expansion of
Question: If the 6th, 7th and 8th terms in the expansion of (x+a)nare respectively 112, 7 and 1/4, findx,a,n. Solution: The 6 th, 7 th and 8 th terms in the expansion of $(x+a)^{n}$ are ${ }^{n} C_{5} x^{n-5} a^{5},{ }^{n} C_{6} x^{n-6} a^{6}$ and ${ }^{n} C_{7} x^{n-7} a^{7}$. According to the question, ${ }^{n} C_{5} x^{n-5} a^{5}=112$ ${ }^{n} C_{6} x^{n-6} a^{6}=7$ ${ }^{n} C_{7} x^{n-7} a^{7}=\frac{1}{4}$ Now, $\frac{{ }^{n} C_{6} x^{n-6} a^{6}}{{ }^{n} C_{5} x^{n-5} a^{5}}=\frac{7}{112}$ $...
Read More →In an A.P. the first term is 8, nth term is 33 and the sum to first n terms is 123.
Question: In an A.P. the first term is 8,nth term is 33 and the sum to firstnterms is 123. Findnandd, the common differences. Solution: In the given problem, we have the first and thenth term of an A.P. along with the sum of thenterms of A.P. Here, we need to find the number of terms and the common difference of the A.P. Here, The first term of the A.P (a) = 8 Thenth term of the A.P (l) = 33 Sum of all the terms Let the common difference of the A.P. bed. So, let us first find the number of the t...
Read More →If the coefficients of three consecutive terms in the expansion of
Question: If the coefficients of three consecutive terms in the expansion of (1 +x)nbe 76, 95 and 76, findn. Solution: Suppose $r,(r+1)$ and $(r+2)$ are three consecutive terms in the given expansion. The coefficients of these terms are ${ }^{n} C_{r-1},{ }^{n} C_{r}$ and ${ }^{\mathrm{n}} \mathrm{C}_{\mathrm{r}+1}$. According to the question, ${ }^{n} C_{r-1}=76$ ${ }^{n} C_{r}=95$ ${ }^{n} C_{r+1}=76$ $\Rightarrow{ }^{n} C_{r-1}={ }^{n} C_{r+1}$ $\Rightarrow r-1+r+1=n \quad\left[\right.$ If ${...
Read More →The number of real solutions of the equation
Question: The number of real solutions of the equation $\sqrt{1+\cos 2 x}=\sqrt{2} \sin ^{-1}(\sin x),-\pi \leq x \leq \pi$ is (a) 0(b) 1(c) 2(d) infinite Solution: (c) 2 For, $-\pi \leq x \leq \frac{-\pi}{2}$ $\sqrt{1+\cos 2 x}=\sqrt{2} \sin ^{-1}(\sin x)$ $\Rightarrow \sqrt{2}|\cos x|=\sqrt{2}(-\pi-\mathrm{x})$ $\Rightarrow \sqrt{2}(-\cos x)=\sqrt{2}(-\pi-\mathrm{x})$ $\Rightarrow \cos x=\pi+\mathrm{x}$ It does not satisfy for any value of $x$ in the interval $\left(-\pi, \frac{-\pi}{2}\right)...
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