If x + 1, 3x and 4x + 2 are in A.P., find the value of x.
Question: If x + 1, 3x and 4x + 2 are in A.P., find the value of x. Solution: Here, we are given three terms which are in A.P., First term (a1) = Second term (a2) = Third term (a3) = We need to find the value ofx. So, in an A.P. the difference of two adjacent terms is always constant. So, we get, $d=a_{2}-a_{1}$ $d=(3 x)-(x+1)$ $d=3 x-x-1$ $d=2 x-1 \quad \ldots \ldots(1)$ Also, $d=a_{2}-a_{2}$ $d=(4 x+2)-(3 x)$ $d=4 x-3 x+2$ $d=x+2 \ldots \ldots(2)$ Now, on equating (1) and (2), we get, $2 x-1=x...
Read More →Solve the following
Question: Ifn+2C8:n2P4= 57 : 16, findn. Solution: We have, ${ }^{n+2} C_{8}:{ }^{n-2} P_{4}=57: 16$ $\Rightarrow \frac{{ }^{n+2} C_{8}}{{ }^{n-2} P_{4}}=\frac{57}{16}$ $\Rightarrow \frac{(n+2) !}{8 !(n-6) !} \times \frac{(n-6) !}{(n-2) !}=\frac{57}{16}$ $\Rightarrow \frac{(n+2)(n+1) n(n-1)(n-2) !}{8 !} \times \frac{1}{(n-2) !}=\frac{57}{16}$ $\Rightarrow(n+2)(n+1) n(n-1)=\frac{57}{16} \times 8 !=\frac{19 \times 3}{16} \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1$ $\Rig...
Read More →Find the value of x for which (8x + 4), (6x − 2) and
Question: Find the value of x for which (8x + 4), (6x 2) and (2x + 7) are in A.P. Solution: Here, we are given three terms, First term $\left(a_{1}\right)=8 x+4$ Second term $\left(a_{2}\right)=6 x-2$ Third term $\left(a_{3}\right)=2 x+7$ We need to find the value ofxfor which these terms are in A.P. So, in an A.P. the difference of two adjacent terms is always constant. So, we get, $d=a_{2}-a_{1}$ $d=(6 x-2)-(8 x+4)$ $d=6 x-8 x-2-4$ $d=-2 x-6 \ldots \ldots(1)$ Also, $d=a_{3}-a_{2}$ $d=(2 x+7)-(...
Read More →Solve the following
Question: If15Cr:15Cr 1= 11 : 5, findr. Solution: Given: 15Cr:15Cr 1= 11 : 5 We have, $\frac{{ }^{15} C_{r}}{{ }^{15} C_{r-1}}=\frac{11}{5}$ $\Rightarrow \frac{15-r+1}{r}=\frac{11}{5} \quad\left[\because \frac{{ }^{n} C_{r}}{{ }^{n} C_{r-1}}=\frac{n-r+1}{r}\right]$ $\Rightarrow 75-5 r+5=11 r$ $\Rightarrow 16 r=80$ $\Rightarrow r=5$...
Read More →The sum of three numbers in A.P. is 12 and the
Question: The sum of three numbers in A.P. is 12 and the sum of their cubes is 288. Find the numbers. Solution: In the given problem, the sum of three terms of an A.P is 12 and the sum of their cubes is 288. We need to find the three terms. Here, Let the three terms bewhere,ais the first term anddis the common difference of the A.P So, $(a-d)+a+(a+d)=12$ $3 a=12$ $a=\frac{12}{3}$ $a=4$ Also, it is given that $(a-d)^{3}+a^{3}+(a+d)^{3}=288$ So, using the properties: $(a-b)^{3}=a^{3}-b^{3}+3 a b^{...
Read More →Solve the following
Question: If8Cr7C3=7C2, findr. Solution: Given: 8Cr7C3=7C2 We have, ${ }^{8} C_{r}={ }^{7} C_{2}+{ }^{7} C_{3}$ $\Rightarrow{ }^{8} C_{r}={ }^{8} C_{3} \quad\left[\because{ }^{n} C_{r}+{ }^{n} C_{r-1}={ }^{n+1} C_{r} ; r \leq n\right]$ $\Rightarrow r=3 \quad\left[\because{ }^{n} C_{x}={ }^{n} C_{y} \Rightarrow x=y\right.$ or, $\left.n=x+y\right]$ And $r+3=8$ $\Rightarrow r=5$...
Read More →If f(x) = 3x + 10 and g(x)
Question: If $f(x)=3 x+10$ and $g(x)=x^{2}-1$, then $(f \circ g)^{-1}$ is equal to Solution: $f o g(x)=f(g(x))$ $=f\left(x^{2}-1\right)$ $=3\left(x^{2}-1\right)+10$ $=3 x^{2}-3+10$ $=3 x^{2}+7$ Thus, $\operatorname{fog}(x)=3 x^{2}+7$ $\Rightarrow y=3 x^{2}+7$ $\Rightarrow 3 x^{2}=y-7$ $\Rightarrow x^{2}=\frac{y-7}{3}$ $\Rightarrow x=\pm \sqrt{\frac{y-7}{3}}$ $f o g^{-1}(x)=\pm \sqrt{\frac{x-7}{3}}$ Hence, $(\text { fog })^{-1}$ is equal to $\pm \sqrt{\frac{x-7}{3}}$....
Read More →Solve the following
Question: If15C3r=15Cr+ 3, findr. Solution: Given; ${ }^{15} C_{3 r}={ }^{15} C_{r+3}$ $15=3 r+r+3 . \quad\left[\because{ }^{n} C_{x}={ }^{n} C_{y} \Rightarrow x=y\right.$ or, $\left.n=x+y\right]$ $\Rightarrow 15=4 r+3$ $\Rightarrow 4 r=12$ $\Rightarrow r=3$...
Read More →If f (f(x)) = x + 1 for all x ∈ R and if f(0)
Question: If $f(f(x))=x+1$ for all $x \in R$ and if $f(0)=\frac{1}{2}$, then $f(1)=$____________. Solution: Given: $f(f(x))=x+1$ for all $x \in R$ and $f(0)=\frac{1}{2}$ $f(f(x))=x+1$ $\Rightarrow f(f(0))=0+1$ $\Rightarrow f\left(\frac{1}{2}\right)=1 \quad\left(\because f(0)=\frac{1}{2}\right) \quad \ldots(1)$ Now, $f\left(f\left(\frac{1}{2}\right)\right)=\frac{1}{2}+1$ $\Rightarrow f(1)=\frac{1+2}{2} \quad\left(\because f\left(\frac{1}{2}\right)=1\right)$ $\Rightarrow f(1)=\frac{3}{2}$ Hence, $...
Read More →In the given figure, AB || CD || EF. Find the value of x.
Question: In the given figure,AB||CD||EF. Find the value ofx. Solution: $E F \| C D$ and $\mathrm{CE}$ is the transversal Then $\angle E C D+\angle C E F=180^{\circ} \quad$ [Consecutive Interior Angles] $\Rightarrow \angle E C D+130^{\circ}=180^{\circ}$ $\Rightarrow \angle F C D=50^{\circ}$ Again, $A B \| C D$ and $\mathrm{BC}$ is the transversal. Then, $\angle A B C=\angle B C D \quad[$ Alternate Interior Angles $]$ $\Rightarrow 70^{\circ}=x+50^{\circ} \quad[\because \angle B C D=\angle B C E+\...
Read More →Solve the following
Question: If18Cx=18Cx+ 2, findx. Solution: Given: ${ }^{18} C_{X}={ }^{18} C_{X}+2$ By using ${ }^{n} C_{x}={ }^{n} C_{y} \Rightarrow x=y$ or $n=x+y$ we get, $18=x+x+2$ $\Rightarrow 2 x=16$ $\Rightarrow x=8$...
Read More →Let f(x)
Question: Let $f(x)=\frac{x}{x-1}$ and $\frac{f(\alpha)}{f(\alpha+1)}=f\left(\alpha^{k}\right)$, then $k=$______________. Solution: Given: $f(x)=\frac{x}{x-1}$ and $\frac{f(\alpha)}{f(\alpha+1)}=f\left(\alpha^{k}\right)$ $\frac{f(a)}{f(a+1)}=\frac{\frac{a}{a-1}}{\frac{a+1}{a+1-1}}$ $=\frac{\frac{a}{a-1}}{\frac{a+1}{a}}$ $=\frac{a^{2}}{(a-1)(a+1)}$ $=\frac{a^{2}}{a^{2}-1}$ ...(1) It is given that, $\frac{f(a)}{f(a+1)}=f\left(a^{k}\right)$ $\Rightarrow \frac{a^{2}}{a^{2}-1}=\frac{a^{k}}{a^{k}-1}$ ...
Read More →The angles of a quadrilateral are in A.P. whose common
Question: The angles of a quadrilateral are in A.P. whose common difference is 10. Find the numbers. Solution: Here, we are given that the angles of a quadrilateral are in A.P, such that the common difference is 10. So, let us take the angles as $a-d, a, a+d, a+2 d$ Now, we know that the sum of all angles of a quadrilateral is 360. So, we get, $(a-d)+(a)+(a+d)+(a+2 d)=360$ $a-d+a+a+d+a+2 d=360$ $4 a+2(10)=360$ $4 a=360-20$ On further simplifying fora, we get, $a=\frac{340}{4}$ $a=85$ So, the fir...
Read More →Solve the following
Question: IfnC10=nC12, find23Cn. Solution: Given: ${ }^{n} C_{10}={ }^{n} C_{12}$ We have: ${ }^{n} C_{10}={ }^{n} C_{12}$ $\Rightarrow n=12+10=22 \quad\left[\because{ }^{n} C_{x}={ }^{n} C_{y} \Rightarrow x=y\right.$ or, $\left.n=x+y\right]$ Now, ${ }^{23} C_{22}={ }^{23} C_{1} \quad\left[\because{ }^{n} C_{r}={ }^{n} C_{n-r}\right]$ $\Rightarrow{ }^{23} C_{22}={ }^{23} C_{1}=\frac{23}{1} \times{ }^{22} C_{0} \quad\left[\because{ }^{n} C_{r}=\frac{n}{r}{ }^{n-1} C_{r-1}\right]$ $\Rightarrow{ }^...
Read More →In the given figure, AB || CD and BC || ED. Find the value of x.
Question: In the given figure,AB||CDandBC||ED. Find the value ofx. Solution: $B C \| E D$ and $\mathrm{CD}$ is the transversal. Then, $\angle B C D+\angle C D E=180^{\circ} \quad$ [Angles on the same side of a transversal line are supplementary] $\Rightarrow \angle B C D+75=180$ $\Rightarrow \angle B C D=105^{\circ}$ $A B \| C D$ and $\mathrm{BC}$ is the transversal. $\angle A B C=\angle B C D$ (alternate angles) $\Rightarrow x^{\circ}=105^{\circ}$ $\Rightarrow x=105$...
Read More →If
Question: If $f(x)=\frac{1-x}{1+x}$, then fof $(\cos 2 \theta)=$______________. Solution: Given: $f(x)=\frac{1-x}{1+x}$ $f o f(\cos 2 \theta)=f(f(\cos 2 \theta))$ $=f\left(\frac{1-\cos 2 \theta}{1+\cos 2 \theta}\right)$ $=f\left(\frac{2 \sin ^{2} \theta}{2 \cos ^{2} \theta}\right)$ $=f\left(\tan ^{2} \theta\right)$ $=\frac{1-\tan ^{2} \theta}{1+\tan ^{2} \theta}$ $=\cos 2 \theta$ Hence, fof $(\cos 2 \theta)=\cos 2 \theta$....
Read More →Solve the following
Question: IfnC4=nC6, find12Cn. Solution: We have, ${ }^{n} C_{4}={ }^{n} C_{6}$ $\Rightarrow n=6+4=10 \quad\left[\because{ }^{n} C_{x}={ }^{n} C_{y} \Rightarrow x=y\right.$ or, $\left.n=x+y\right]$ Now, ${ }^{12} C_{10}={ }^{12} C_{2} \quad\left[\because{ }^{n} C_{r}={ }^{n} C_{n-r}\right]$ $\Rightarrow^{12} C_{10}={ }^{12} C_{2}=\frac{12}{2} \times \frac{11}{1} \times{ }^{10} C_{0} \quad\left[\because{ }^{n} C_{r}=\frac{n}{r}{ }^{n-1} C_{r-1}\right]$ $\Rightarrow{ }^{12} C_{10}=66 \quad\left[\b...
Read More →Find the four numbers in A.P., whose sum is 50
Question: Find the four numbers in A.P., whose sum is 50 and in which the greatest number is 4 times the least. Solution: Here, we are given that four number are in A.P., such that there sum is 50 and the greatest number is 4 times the smallest. So, let us take the four terms as $a-d, a, a+d, a+2 d$. Now, we are given that sum of these numbers is 50, so we get, $(a-d)+(a)+(a+d)+(a+2 d)=50$ $a-d+a+a+d+a+2 d=50$ $4 a+2 d=50$ $2 a+d=25 \ldots \ldots$ (1) Also, the greatest number is 4 times the sma...
Read More →Solve the following
Question: IfnC12=nC5, find the value ofn. Solution: We have, nC12=nC5 $\Rightarrow n=12+5=17 \quad\left[\because{ }^{n} C_{x}={ }^{n} C_{y} \Rightarrow x=y\right.$ or, $\left.n=x+y\right]$...
Read More →If f : R → R is given by f(x) = 2x + |x|, then f(2x) + f(–x) + 4x =
Question: If $f: R \rightarrow R$ is given by $f(x)=2 x+|x|$, then $f(2 x)+f(-x)+4 x=$_______________. Solution: Given:f(x) = 2x+ |x| $f(x)=2 x+|x|$ $f(2 x)=2(2 x)+|2 x|$ $\Rightarrow f(2 x)=4 x+2|x|$$\ldots(1)$ $f(-x)=2(-x)+|-x|$ $\Rightarrow f(-x)=-2 x+|x|$$\ldots(2)$ Now, $f(2 x)+f(-x)+4 x=4 x+2|x|-2 x+|x|+4 x$ $=6 x+3|x|$ $=3(2 x+|x|)$ $=3 f(x)$ Hence, $f(2 x)+f(-x)+4 x=\underline{3 f(x)}$....
Read More →Evaluate the following:
Question: Evaluate the following: (i)14C3 (ii)12C10 (iii)35C35 (iv)n+1Cn (v) $\sum_{r=1}^{5}{ }^{5} C_{r}$ Solution: (i) We have, ${ }^{14} C_{3}=\frac{14}{3} \times \frac{13}{2} \times \frac{12}{1} \times{ }^{11} C_{0} \quad\left[\because{ }^{n} C_{r}=\frac{n}{r}{ }^{n-1} C_{r-1}\right]$ $\Rightarrow^{14} C_{3}=364^{2} \quad\left[\because{ }^{n} C_{0}=1\right]$ (ii) We have, ${ }^{12} C_{10}={ }^{12} C_{2} \quad\left[\because{ }^{n} C_{r}={ }^{n} C_{n-r}\right]$ $\Rightarrow{ }^{12} C_{10}={ }^...
Read More →For what value of x will the lines l and m be parallel to each other?
Question: For what value of $x$ will the lines / and $m$ be parallel to each other? Solution: $\Leftrightarrow 3 x+5+4 x=180$ [Consecutive Interior Angles] $\Leftrightarrow 7 x=175$ $\Leftrightarrow x=25$...
Read More →For what value of x will the line l and m be parallel to each other?
Question: For what value ofxwill the linelandmbe parallel to each other? Solution: For the lines l and m to be parallel $\Leftrightarrow 3 x-20=2 x+10 \quad$ [Corresponding Angles] $\Leftrightarrow x=\mathbf{3 0}$...
Read More →Three numbers are in A.P. If the sum of these numbers be
Question: Three numbers are in A.P. If the sum of these numbers be 27 and the product 648, find the numbers. Solution: In the given problem, the sum of three terms of an A.P is 27 and the product of the three terms is 648. We need to find the three terms. Here, Let the three terms bewhereais the first term anddis the common difference of the A.P So, $(a-d)+a+(a+d)=27$ $3 a=27$ $a=9$.......(1) Also, $(a-d) a(a+d)=a+6$ $a\left(a^{2}-d^{2}\right)=648$ $\left[\right.$ Using $\left.a^{2}-b^{2}=(a+b)(...
Read More →If f : R → R, g : R → R are defined by
Question: If $f: R \rightarrow R, g: R \rightarrow R$ are defined by $f(x)=5 x-3, g(x)=x^{2}+3$, then $\left(g \circ f^{-1}\right)(3)=$ Solution: Given: $f(x)=5 x-3$ and $g(x)=x^{2}+3$ $f(x)=5 x-3$ $\Rightarrow y=5 x-3$ $\Rightarrow 5 x=y+3$ $\Rightarrow x=\frac{y+3}{5}$ Thus, $f^{-1}(y)=\frac{y+3}{5}$. Now, $g o f^{-1}(3)=g\left(f^{-1}(3)\right)$ $=g\left(\frac{3+3}{5}\right)$ $=g\left(\frac{6}{5}\right)$ $=\left(\frac{6}{5}\right)^{2}+3$ $=\frac{36}{25}+3$ $=\frac{36+75}{25}$ $=\frac{111}{25}$...
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