If the coefficients of (2r + 4)th and (r−2)th terms in the expansion of
Question: If the coefficients of $(2 r+4)$ th and $(r-2)$ th terms in the expansion of $(1+x)^{18}$ are equal, find $r$. Solution: Given: $(1+x)^{18}$ We know that the coefficient of the $r$ th term in the expansion of $(1+x)^{n}$ is ${ }^{n} C_{r-1}$ Therefore, the coefficients of the $(2 r+4)$ th and $(r-2)$ th term $s$ in the given expansion are ${ }^{18} C_{2 r+4-1}$ and ${ }^{18} C_{r-2-1}$ For these coefficients to be equal, we must have ${ }^{18} C_{2 r+3}={ }^{18} C_{r-3}$ $\Rightarrow 2...
Read More →The first term of an A.P. is 5, the last term is 45 and the sum is 400. Find the number of terms and the common difference.
Question: The first term of an A.P. is 5, the last term is 45 and the sum is 400. Find the number of terms and the common difference. 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 number of terms and the common difference of the A.P. Here, The first term of the A.P (a) = 5 The last term of the A.P (l) = 45 Sum of all the terms Let the common difference of the A.P. bed. So, let us first find the...
Read More →Two angles of a triangle are equal and the third angle is greater than each one of them by 18°.
Question: Two angles of a triangle are equal and the third angle is greater than each one of them by 18. Find the angles. Solution: Let $\angle A=\angle B$ and $\angle C=\angle A+18^{\circ}$. Then, $\angle A+\angle B+\angle C=180^{\circ} \quad[$ Sum of the angles of a triangle $]$ $\angle A+\angle A+\angle A+18^{\circ}=180^{\circ}$ $\Rightarrow 3 \angle A=162^{\circ}$ $\Rightarrow \angle A=54^{\circ}$ Since, $\angle A=\angle B$ $\Rightarrow \angle B=54^{\circ}$ $\therefore \angle C=\angle A+18^{...
Read More →If the sum of 7 terms of an A.P. is 49 and that of 17 terms is 289,
Question: If the sum of 7 terms of an A.P. is 49 and that of 17 terms is 289, find the sum ofn terms. Solution: In the given problem, we need to find the sum ofnterms of an A.P. Let us take the first term asaand the common difference asd. Here, we are given that, $S_{7}=49$..............(1) $S_{17}=289$.............(2) So, as we know the formula for the sum ofnterms of an A.P. is given by, $S_{n}=\frac{n}{2}[2 a+(n-1) d]$ Where;a= first term for the given A.P. d= common difference of the given A...
Read More →Solve the following equations for x:
Question: Solve the following equations forx: (i) $\tan ^{-1} \frac{1}{4}+2 \tan ^{-1} \frac{1}{5}+\tan ^{-1} \frac{1}{6}+\tan ^{-1} \frac{1}{x}=\frac{\pi}{4}$ (ii) $3 \sin ^{-1} \frac{2 x}{1+x^{2}}-4 \cos ^{-1} \frac{1-x^{2}}{1+x^{2}}+2 \tan ^{-1} \frac{2 x}{1-x^{2}}=\frac{\pi}{3}$ (iii) $\tan ^{-1}\left(\frac{2 x}{1-x^{2}}\right)+\cot ^{-1}\left(\frac{1-x^{2}}{2 x}\right)=\frac{2 \pi}{3}, x0$ (iv) $2 \tan ^{-1}(\sin x)=\tan ^{-1}(2 \sin x), x \neq \frac{\pi}{2}$. (v) $\cos ^{-1}\left(\frac{x^{...
Read More →The sum of two angles of a triangle is 116° and their difference is 24°.
Question: The sum of two angles of a triangle is 116 and their difference is 24. Find the measure of each angle of the triangle. Solution: Let $\angle A+\angle B=116^{\circ}$ and $\angle A-\angle B=24^{\circ}$ Then, $\therefore \angle A+\angle B+\angle A-\angle B=(116+24)^{\circ}$ $\Rightarrow 2 \angle A=140^{\circ}$ $\Rightarrow \angle A=70^{\circ}$ $\therefore \angle B=116^{\circ}-\angle A$ $=(116-70)^{\circ}$ $=46^{\circ}$ Also, in ∆ABC: $\angle A+\angle B+\angle C=180^{\circ} \quad[$ Sum of ...
Read More →Find the term independent of x in the expansion of the following expressions:
Question: Find the term independent ofxin the expansion of the following expressions: (i) $\left(\frac{3}{2} x^{2}-\frac{1}{3 x}\right)^{9}$ (ii) $\left(2 x+\frac{1}{3 x^{2}}\right)^{9}$ (iii) $\left(2 x^{2}-\frac{3}{x^{3}}\right)^{25}$ (iv) $\left(3 x-\frac{2}{x^{2}}\right)^{15}$ (v) $\left(\frac{\sqrt{x}}{3}+\frac{3}{2 x^{2}}\right)^{10}$ (vi) $\left(x-\frac{1}{x^{2}}\right)^{3 n}$ (vii) $\left(\frac{1}{2} x^{1 / 3}+x^{-1 / 5}\right)^{8}$ (viii) $\left(1+x+2 x^{3}\right)\left(\frac{3}{2} x^{2}...
Read More →Find the term independent of x in the expansion of the following expressions:
Question: Find the term independent ofxin the expansion of the following expressions: (i) $\left(\frac{3}{2} x^{2}-\frac{1}{3 x}\right)^{9}$ (ii) $\left(2 x+\frac{1}{3 x^{2}}\right)^{9}$ (iii) $\left(2 x^{2}-\frac{3}{x^{3}}\right)^{25}$ (iv) $\left(3 x-\frac{2}{x^{2}}\right)^{15}$ (v) $\left(\frac{\sqrt{x}}{3}+\frac{3}{2 x^{2}}\right)^{10}$ (vi) $\left(x-\frac{1}{x^{2}}\right)^{3 n}$ (vii) $\left(\frac{1}{2} x^{1 / 3}+x^{-1 / 5}\right)^{8}$ (viii) $\left(1+x+2 x^{3}\right)\left(\frac{3}{2} x^{2}...
Read More →Find the sum of first 51 terms of an A.P. whose second
Question: Find the sum of first 51 terms of an A.P. whose second and third terms are 14 and 18 respectively. Solution: In the given problem, let us take the first term asaand the common difference asd. Here, we are given that, $a_{2}=14$........(1) $a_{3}=18$........(2) Also, we know, $a_{n}=a+(n-1) d$ For the $2^{\text {nd }}$ term $(n=2)$, $a_{2}=a+(2-1) d$ $14=a+d$(Using 1) $a=14-d$....(3) Similarly, for the 3rdterm (n =3), $a_{3}=a+(3-1) d$ $18=a+2 d$ (Using 2) $a=18-2 d$.......(4) Subtracti...
Read More →In ∆PQR, if ∠P − ∠Q = 42° and ∠Q − ∠R = 21°, find ∠P, ∠Q and ∠R.
Question: In ∆PQR, if P Q= 42 and Q R= 21, find P, Qand R. Solution: Given : $\angle P-\angle Q=42^{\circ}$ and $\angle Q-\angle R=21^{\circ}$ Then, $\angle P=42^{\circ}+\angle Q$ and $\angle R=\angle Q-21^{\circ}$ $\therefore 42^{\circ}+\angle Q+\angle Q+\angle Q-21^{\circ}=180^{\circ} \quad[$ Sum of the angles of a triangle $]$ $\Rightarrow 3 \angle Q=159^{\circ}$ $\Rightarrow \angle Q=53^{\circ}$ $\therefore \angle P=42^{\circ}+\angle Q$ $=(42+53)^{\circ}$ $=95^{\circ}$ $\therefore \angle R=\...
Read More →In ∆ABC, ∠A + ∠B = 125° and ∠A + ∠C = 113°. Find ∠A, ∠B and ∠C.
Question: In ∆ABC, A+ B= 125 and A+ C= 113. Find A, Band C. Solution: Let $\angle A+\angle B=125^{\circ}$ and $\angle A+\angle C=113^{\circ}$. Then, $\angle A+\angle B+\angle A+\angle C=(125+113)^{\circ}$ $\Rightarrow(\angle A+\angle B+\angle C)+\angle A=238^{\circ}$ $\Rightarrow 180^{\circ}+\angle A=238^{\circ}$ $\Rightarrow \angle A=58^{\circ}$ $\therefore \angle B=125^{\circ}-\angle A$ $=(125-58)^{\circ}$ $=67^{\circ}$ $\therefore \angle C=113^{\circ}-\angle A$ $=(113-58)^{\circ}$ $=55^{\circ...
Read More →In ∆ABC, if ∠A + ∠B = 108° and ∠B + ∠C = 130°, find ∠A, ∠B and ∠C.
Question: In ∆ABC, if A+ B= 108 and B+ C= 130, find A, Band C. Solution: Let $\angle A+\angle B=108^{\circ}$ and $\angle B+\angle C=130^{\circ}$. $\Rightarrow \angle A+\angle B+\angle B+\angle C=(108+130)^{\circ}$ $\Rightarrow(\angle A+\angle B+\angle C)+\angle B=238^{\circ} \quad\left[\because \angle A+\angle B+\angle C=180^{\circ}\right]$ $\Rightarrow 180^{\circ}+\angle B=238^{\circ}$ $\Rightarrow \angle B=58^{\circ}$ $\therefore \angle C=130^{\circ}-\angle B$ $=(130-58)^{\circ}$ $=72^{\circ}$...
Read More →Find the sum of the first 25 terms of an A.P.
Question: Find the sum of the first 25 terms of an A.P. whosenth term is given by an= 7 3n. Solution: Here, we are given an A.P. whose $n^{\text {th }}$ term is given by the following expression, $a_{n}=7-3 n$. We need to find the sum of first 25 terms. So, here we can find the sum of the $n$ terms of the given A.P., using the formula, $S_{n}=\left(\frac{n}{2}\right)(a+l)$ Where,a= the first term l= the last term So, for the given A.P, The first term (a) will be calculated usingin the given equa...
Read More →Find the sum of the first 25 terms of an A.P. whose nth term is given by an = 2 − 3n.
Question: Find the sum of the first 25 terms of an A.P. whosenth term is given by an= 2 3n. Solution: Here, we are given an A.P. whose $n^{\text {th }}$ term is given by the following expression, $a_{n}=2-3 n$. We need to find the sum of first 25 terms. So, here we can find the sum of the $n$ terms of the given A.P., using the formula, $S_{n}=\left(\frac{n}{2}\right)(a+l)$ Where,a= the first term l= the last term So, for the given A.P, The first term (a) will be calculated usingin the given equa...
Read More →Find the values of each of the following:
Question: Find the values of each of the following: (i) $\tan ^{-1}\left\{2 \cos \left(2 \sin ^{-1} \frac{1}{2}\right)\right\}$ (ii) $\cos \left(\sec ^{-1} x+\operatorname{cosec}^{-1} x\right),|x| \geq 1$ Solution: (i) Let $\sin ^{-1} \frac{1}{2}=y$ Then, $\sin y=\frac{1}{2}$ $\therefore \tan ^{-1}\left\{2 \cos \left(2 \sin ^{-1} \frac{1}{2}\right)\right\}=\tan ^{-1}\{2 \cos 2 y\}$ $=\tan ^{-1}\left(2\left(1-2 \sin ^{2} y\right)\right)$ $=\tan ^{-1}\left\{2\left(1-2 \times \frac{1}{4}\right)\rig...
Read More →Find the sum of first 20 terms of the sequence whose nth term is an = An + B.
Question: Find the sum of first 20 terms of the sequence whosenth term isan= An + B. Solution: Here, we are given an A.P. whosenthterm is given by the following expression. We need to find the sum of first 20 terms. So, here we can find the sum of the $n$ terms of the given A.P., using the formula, $S_{n}=\left(\frac{n}{2}\right)(a+l)$ Where,a= the first term l= the last term So, for the given A.P, The first term (a) will be calculated usingin the given equation fornthterm of A.P. $a=A(1)+B$ $=A...
Read More →Show that
Question: Show that $2 \tan ^{-1} x+\sin ^{-1} \frac{2 x}{1+x^{2}}$ is constant for $x \geq 1$, find that constant. Solution: We have $2 \tan ^{-1} x+\sin ^{-1}\left(\frac{2 x}{1+x^{2}}\right)$ (1) For $x1$, $=2 \tan ^{-1} x+\sin ^{-1}\left(\frac{2 x}{1+x^{2}}\right)$ $=\pi-\sin ^{-1}\left(\frac{2 x}{1+x^{2}}\right)+\sin ^{-1}\left(\frac{2 x}{1+x^{2}}\right) \quad\left[\because 2 \tan ^{-1} x=\pi-\sin ^{-1}\left(\frac{2 x}{1+x^{2}}\right), x1\right]$ $=\pi$ (2) For $x=1$ $=2 \tan ^{-1} x+\sin ^{...
Read More →Find the sum of the first 15 terms of each of the following sequences having nth term as
Question: Find the sum of the first 15 terms of each of the following sequences having nth term as (i) $a_{n}=3+4 n$ (ii) $b_{n}=5+2 n$ (iii) $x_{n}=6-n$ (iv) $y_{n}=9-5 n$ Solution: (i) Here, we are given an A.P. whose $n^{\text {th }}$ term is given by the following expression, $a_{n}=3+4 n$. We need to find the sum of first 15 terms. So, here we can find the sum of the $n$ terms of the given A.P., using the formula, $S_{n}=\left(\frac{n}{2}\right)(a+l)$ Where,a= the first term l= the last ter...
Read More →Find the middle terms(s) in the expansion of:
Question: Find the middle terms(s) in the expansion of: (i) $\left(x-\frac{1}{x}\right)^{10}$ (ii) $\left(1-2 x+x^{2}\right)^{n}$ (iii) $\left(1+3 x+3 x^{2}+x^{3}\right)^{2 n}$ (iv) $\left(2 x-\frac{x^{2}}{4}\right)^{9}$ (v) $\left(x-\frac{1}{x}\right)^{2 n+1}$ (vi) $\left(\frac{x}{3}+9 y\right)^{10}$ (vii) $\left(3-\frac{x^{3}}{6}\right)^{7}$ (viii) $\left(2 a x-\frac{b}{x^{2}}\right)^{12}$ (ix) $\left(\frac{p}{x}+\frac{x}{p}\right)^{9}$ (X) $\left(\frac{x}{a}-\frac{a}{x}\right)^{10}$ Solution:...
Read More →If
Question: If $\sin ^{-1} \frac{2 a}{1+a^{2}}+\sin ^{-1} \frac{2 b}{1+b^{2}}=2 \tan ^{-1} x$, prove that $x=\frac{a+b}{1-a b}$ Solution: Let: $a=\tan z$ $b=\tan y$ Then, $\sin ^{-1} \frac{2 a}{1+a^{2}}+\sin ^{-1} \frac{2 b}{1+b^{2}}=2 \tan ^{-1} x$ $\Rightarrow \sin ^{-1} \frac{2 \tan z}{1+\tan ^{2} z}+\sin ^{-1} \frac{2 \tan y}{1+\tan ^{2} y}=2 \tan ^{-1} x$ $\Rightarrow \sin ^{-1}(\sin 2 z)+\sin ^{-1}(\sin 2 y)=2 \tan ^{-1} x$ $\left[\because \sin 2 x=\frac{2 \tan x}{1+\tan ^{2} x}\right]$ $\Ri...
Read More →In ∆ABC, if 3∠A = 4 ∠B = 6 ∠C, calculate ∠A, ∠B and ∠C.
Question: In ∆ABC, if 3A= 4 B= 6 C, calculate A, Band C. Solution: Let $3 \angle A=4 \angle B=6 \angle C=x^{\circ}$. Then, $\angle A=\left(\frac{x}{3}\right)^{\circ}, \angle B=\left(\frac{x}{4}\right)^{\circ}$ and $\angle C=\left(\frac{x}{6}\right)^{\circ}$ $\therefore \frac{x}{3}+\frac{x}{4}+\frac{x}{6}=180^{\circ} \quad[$ Sum of the angles of a triangle $]$ $\Rightarrow 4 x+3 x+2 x=2160^{\circ}$ $\Rightarrow 9 x=2160^{\circ}$ $\Rightarrow x=240^{\circ}$ Therefore, $\angle A=\left(\frac{240}{3}...
Read More →Prove that
Question: Prove that (i) $\tan ^{-1}\left(\frac{1-x^{2}}{2 x}\right)+\cot ^{-1}\left(\frac{1-x^{2}}{2 x}\right)=\frac{\pi}{2}$ (ii) $\sin \left\{\tan ^{-1} \frac{1-x^{2}}{2 x}+\cos ^{-1} \frac{1-x^{2}}{1+x^{2}}\right\}=1$ (iii) $\sin ^{-1}\left(2 x \sqrt{1-x^{2}}\right)=2 \cos ^{-1} x, \frac{1}{\sqrt{2}} \leq x \leq 1$ Solution: (i) $\tan ^{-1}\left(\frac{1-x^{2}}{2 x}\right)+\cot ^{-1}\left(\frac{1-x^{2}}{2 x}\right)=\frac{\pi}{2}$ $\mathrm{LHS}=\tan ^{-1}\left(\frac{1-x^{2}}{2 x}\right)+\cot ^...
Read More →Find the sum:
Question: Find the sum: (i) $2+4+6 \ldots+200$ (ii) $3+11+19+\ldots+803$ (iii) $(-5)+(-8)+(-11)+\ldots+(-230)$ (iv) $1+3+5+7+\ldots+199$ (v) $7+10 \frac{1}{2}+14+\ldots+84$ (vi) $34+32+30+\ldots+10$ (vii) $25+28+31+\ldots+100$ (viii) $18+15 \frac{1}{2}+13+\ldots+\left(-49 \frac{1}{2}\right)$ Solution: In the given problem, we need to find the sum of terms for different arithmetic progressions. So, here we use the following formula for the sum ofnterms of an A.P., $S_{n}=\frac{n}{2}[2 a+(n-1) d]$...
Read More →The angles of a triangle are in the ratio 2 : 3 : 4. Find the angles.
Question: The angles of a triangle are in the ratio 2 : 3 : 4. Find the angles. Solution: Let the angles of the given triangle measure $(2 x)^{\circ},(3 x)^{\circ}$ and $(4 x)^{\circ}$, respectively. Then, $2 x+3 x+4 x=180^{\circ} \quad$ [Sum of the angles of a triangle] $\Rightarrow 9 x=180^{\circ}$ $\Rightarrow x=20^{\circ}$ Hence, the measures of the angles are $2 \times 20^{\circ}=40^{\circ}, 3 \times 20^{\circ}=60^{\circ}$ and $4 \times 20^{\circ}=80^{\circ}$....
Read More →In ∆ABC, if ∠B = 76° and ∠C = 48°, find ∠A.
Question: In ∆ABC, if B= 76 and C= 48, find A. Solution: In $\Delta A B C$, $\angle A+\angle B+\angle C=180^{\circ} \quad$ [Sum of the angles of a triangle] $\Rightarrow \angle A+76^{\circ}+48^{\circ}=180^{\circ}$ $\Rightarrow \angle A+124^{\circ}=180^{\circ}$ $\Rightarrow \angle A=56^{\circ}$...
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