If f : {5, 6} → {2, 3} and g : {2, 3} → {5, 6} are given by f = {(5, 2), (6, 3)} and g = {(2, 5), (3, 6)}, then find fog.
Question: If $f:\{5,6\} \rightarrow\{2,3\}$ and $g:\{2,3\} \rightarrow\{5,6\}$ are given by $f=\{(5,2),(6,3)\}$ and $g=\{(2,5),(3,6)\}$, then find fog. [NCERT EXEMPLAR] [NCE Solution: We have, $f:\{5,6\} \rightarrow\{2,3\}$ and $g:\{2,3\} \rightarrow\{5,6\}$ are given by $f=\{(5,2),(6,3)\}$ and $g=\{(2,5),(3,6)\}$ As,fog(2) =f(g(2)) =f(5) = 2,fog(3) =f(g(3)) =f(6) = 3, So, fog : $\{2,3\} \rightarrow\{2,3\}$ is defined as $f \circ g=\{(2,2),(3,3)\}$...
Read More →Two lines are respectively perpendicular to two parallel lines.
Question: Two lines are respectively perpendicular to two parallel lines. Show that they are parallel to each other. Solution: Let the two linesmandnbe respectively perpendicular to the two parallel linespandq.To prove:mis parallel ton.Proof: Since,mis perpendicular top $\therefore \angle 1=90^{\circ}$ Also, nis perpendicular toq $\therefore \angle 3=90^{\circ}$ Sincepandqare parallel andmis a transversal line $\therefore \angle 2=\angle 1=90^{\circ} \quad$ [Corresponding angles] Also, $\angle ...
Read More →Using binomial theorem determine which number is larger (1.2)
Question: Using binomial theorem determine which number is larger (1.2)4000or 800? Solution: We have: $(1.2)^{4000}=(1+0.2)^{4000}$ $={ }^{4000} C_{0}+{ }^{4000} C_{1} \times(0.2)^{1}+{ }^{4000} C_{2} \times(0.2)^{2}+\ldots{ }^{4000} C_{4000} \times(0.2)^{4000}$ $=1+4000 \times 0.2+$ other positive terms $=1+800+$ other positive terms $=801+$ other positive terms $\because 801800$ Hence, (1.2)4000is greater than 800...
Read More →Show that the sum of all odd integers between 1 and 1000 which are divisible by 3 is 83667.
Question: Show that the sum of all odd integers between 1 and 1000 which are divisible by 3 is 83667. Solution: In this problem, we need to prove that the sum of all odd numbers lying between 1 and 1000 which are divisible by 3 is 83667. So, we know that the first odd number after 1 which is divisible by 3 is 3, the next odd number divisible by 3 is 9 and the last odd number before 1000 is 999. So, all these terms will form an A.P. 3, 9, 15, 21 with the common difference of 6 So here, First term...
Read More →Let A = {1, 2, 3}, B = {4, 5, 6, 7} and let f = {(1, 4), (2, 5), (3, 6)} be a function from A to B. State whether f is one-one or not.
Question: Let $A=\{1,2,3\}, B=\{4,5,6,7\}$ and let $f=\{(1,4),(2,5),(3,6)\}$ be a function from $A$ to $B$. State whether $f$ is one-one or not. Solution: f= {(1, 4), (2, 5), (3, 6)}Here, different elements of the domain have different images in the co-domain.So,fis one-one....
Read More →Using binomial theorem, indicate which is larger (1.1)
Question: Using binomial theorem, indicate which is larger (1.1)10000or 1000. Solution: We have: (1.1)10000 $=(1+0.1)^{10000}$ $={ }^{10000} C_{0} \times(0.1)^{0}+{ }^{10000} C_{1} \times(0.1)^{1}+{ }^{10000} C_{2} \times(0.1)^{2}+\ldots{ }^{10000} C_{10000} \times(0.1)^{10000}$ $=1+10000 \times 0.1+$ other positive terms $=1+10000+$ other positive terms $=10001+$ other positive terms $\because 100011000$ $\therefore(1.1)^{10000}1000$...
Read More →If f : R → R is defined by f(x) = 3x + 2, find f (f (x)).
Question: If $f: R \rightarrow R$ is defined by $f(x)=3 x+2$, find $f(f(x))$. Solution: $f(f(x))=f(3 x+2)$ $=3(3 x+2)+2$ $=9 x+6+2$ $=9 x+8$...
Read More →In the figure given below, state which lines are parallel and why?
Question: In the figure given below, state which lines are parallel and why? Solution: Here, BAC =ACD = 110Thus, lines AB aand CD are intersected by a transversal AC such that the pair of alternate angles are equal. AB || CD (If a transversal intersects two lines such that a pair of alternate interior angles are equal, then the two lines are parallel)Thus, line AB is parallel to line CD.Also,ACD +CDE = 110 +80 =190180If a transversal intersects two lines such that a pair of interior angles on th...
Read More →If f : R → R be defined by
Question: If $f: R \rightarrow R$ be defined by $f(x)=\left(3-x^{3}\right) 1 / 3$, then find $f \circ f(x)$. Solution: $(f o f)(x)=f(f(x))$ $=f\left(\left(3-x^{3}\right)^{\frac{1}{3}}\right)$ $=\left[3-\left(\left(3-x^{3}\right)^{\frac{1}{3}}\right)^{3}\right]^{\frac{1}{3}}$ $=\left[3-\left(3-x^{3}\right)\right]^{\frac{1}{3}}$ $=\left(x^{3}\right)^{\frac{1}{3}}$ $=x$...
Read More →If n is a positive integer, prove that
Question: If $n$ is a positive integer, prove that $3^{3 n}-26 n-1$ is divisible by 676 . Solution: $3^{3 n}-26 n-1=27^{n}-26 n-1 \quad \ldots(1)$ Now, we have : $27^{n}=(1+26)^{n}$ On expanding, we get $(1+26)^{n}={ }^{n} C_{0} \times 26^{0}+{ }^{n} C_{1} \times 26^{1}+{ }^{n} C_{2} \times 26^{2}+{ }^{n} C_{3} \times 26^{3}+{ }^{n} C_{4} \times 26^{4}+\ldots{ }^{n} C_{n} \times 26^{n}$ $\Rightarrow 27^{n}=1+26 n+26^{2}\left[{ }^{n} C_{2}+{ }^{n} C_{3} \times 26^{1}+{ }^{n} C_{4} \times 26^{2}+\...
Read More →Find the sum of all odd numbers between (i) 0 and 50 (ii) 100 and 200.
Question: Find the sum of all odd numbers between (i) 0 and 50 (ii) 100 and 200. Solution: (i) In this problem, we need to find the sum of all odd numbers lying between 0 and 50. So, we know that the first odd number after 0 is 1 and the last odd number before 50 is 49. Also, all these terms will form an A.P. with the common difference of 2. So here, First term (a) = 1 Last term (l) = 49 Common difference (d) = 2 So, here the first step is to find the total number of terms. Let us take the numbe...
Read More →In the given figure, m and n are two plane mirrors perpendicular to each other.
Question: In the given figure,m and nare two plane mirrors perpendicular to each other. Show that the incident ray CA is parallel to the reflected ray BD. Solution: AP is normal to the plane mirror OA and BP is normal to the plane mirror OB.It is given that the two plane mirrors are perpendicular to each other.Therefore, BP || OA and AP || OB.So, BP AP (OA OB)⇒APB = 90 .....(1)In∆APB,2 +3 +APB =180 (Angle sum property)2 +3 +90 =180 [Using (1)]⇒2 +3 =18090=90⇒ 22 + 23 = 2 90 =180 .....(2)Bylaw o...
Read More →What is the range of the function
Question: What is the range of the function $f(x)=\frac{|x-1|}{x-1} ?$ Solution: $f(x)=\frac{|x-1|}{x-1}=\frac{\pm(x-1)}{x-1}=\pm 1$ Range of $f=\{-1,1\}$...
Read More →Using binomial theorem, prove that
Question: Using binomial theorem, prove that $3^{2 n+2}-8 n-9$ is divisible by $64, n \in N$. Solution: $3^{2 n+2}-8 n-9=9^{n+1}-8 n-9 \quad \ldots(1)$ Consider $9^{n+1}=(1+8)^{n+1}$ $\Rightarrow 9^{n+1}={ }^{n+1} C_{0} \times 8^{0}+{ }^{n+1} C_{1} \times 8^{1}+{ }^{n+1} C_{2} \times 8^{2}+{ }^{n+1} C_{3} \times 8^{3}+\ldots+{ }^{n+1} C_{n+1} \times 8^{n+1}$ $\Rightarrow 9^{n+1}=1+8(n+1)+\left[{ }^{n+1} C_{2} \times 8^{2}+{ }^{n+1} C_{3} \times 8^{3}+\ldots+{ }^{n+1} C_{n+1} \times 8^{n+1}\right...
Read More →In the given figure, BA || ED and BC || EF. Show that ∠ABC + ∠DEF = 180°.
Question: In the given figure, BA || ED and BC || EF. Show that ABC + DEF = 180. Solution: It is given that,BA || ED and BC || EF.Construction: Extend ED such that it intersects BC at G. Now,BA || GE and BC is a transversal.ABC =EGC .....(1) (Pair of corresponding angles)Also,BC || EF and EG is a transversal.EGC +GEF = 180 .....(2) (Interior angles on the same side of the transversal are supplementary)From (1) and (2), we haveABC+GEF= 180 Or ABC +DEF = 180...
Read More →Using binomial theorem, prove that
Question: Using binomial theorem, prove that $3^{2 n+2}-8 n-9$ is divisible by $64, n \in N$. Solution: $3^{2 n+2}-8 n-9=9^{n+1}-8 n-9 \quad \ldots(1)$ Consider $9^{n+1}=(1+8)^{n+1}$ $\Rightarrow 9^{n+1}={ }^{n+1} C_{0} \times 8^{0}+{ }^{n+1} C_{1} \times 8^{1}+{ }^{n+1} C_{2} \times 8^{2}+{ }^{n+1} C_{3} \times 8^{3}+\ldots+{ }^{n+1} C_{n+1} \times 8^{n+1}$ $\Rightarrow 9^{n+1}=1+8(n+1)+\left[{ }^{n+1} C_{2} \times 8^{2}+{ }^{n+1} C_{3} \times 8^{3}+\ldots+{ }^{n+1} C_{n+1} \times 8^{n+1}\right...
Read More →If f(x)
Question: If $f(x)=x+7$ and $g(x)=x-7, x \in R$, write fog (7). Solution: $(f o g)(7)=f(g(7))$ $=f(7-7)$ $=f(0)$ $=0+7$ $=7$...
Read More →In the given figure, BA || ED and BC || EF. Show that ∠ABC = ∠DEF.
Question: In the given figure, BA || ED and BC || EF. Show that ABC = DEF. Solution: It is given that,BA || ED and BC || EF.Construction: Extend DE such that it intersects BC at J. Also, extend FE such that it intersects AB at H. Now,BA || JD and BC is a transversal.ABC =DJC .....(1) (Pair of corresponding angles)Also,BC || HF and DJ is a transversal.DJC=DEF .....(2) (Pair of corresponding angles)From (1) and (2), we haveABC = DEF...
Read More →Write whether f : R → R, given by
Question: Write whether $f: R \rightarrow R$, given by $f(x)=x+\sqrt{x^{2}}$, is one-one, many-one, onto or into. Solution: $f(x)=x+\sqrt{x^{2}}=x \pm x=0$ or $2 x$ So, each element $x$ in the domain may contain 2 images. For example, $f(0)=0+\sqrt{0^{2}}=0$ $f(-1)=-1+\sqrt{(-1)^{2}}=-1+\sqrt{1}=-1+1=0$ Here, the image of 0 and $-1$ is 0 . Hence, $f$ is may-one....
Read More →Find the sum of first n odd natural numbers.
Question: Find the sum of firstnodd natural numbers. Solution: In this problem, we need to find the sum of firstnodd natural numbers. So, we know that the first odd natural number is 1. Also, all the odd terms will form an A.P. with the common difference of 2. So here, First term (a) = 1 Common difference (d) = 2 So, let us take the number of terms asn Now, as we know, $S_{n}=\frac{n}{2}[2 a+(n-1) d]$ So, fornterms, $S_{n}=\frac{n}{2}[2(1)+(n-1) 2]$ $=\frac{n}{2}[2+2 n-2]$ $=\frac{n}{2}(2 n)$ $=...
Read More →Using binomial theorem, prove that
Question: Using binomial theorem, prove that $2^{3 n}-7 n-1$ is divisible by 49 , where $n \in N$. Solution: $2^{3 n}-7 n-1=8^{n}-7 n-1$ ...(i) Now, $8^{n}=(1+7)^{n}$ $={ }^{n} C_{0}+{ }^{n} C_{1} \times 7^{1}+{ }^{n} C_{2} \times 7^{2}+{ }^{n} C_{3} \times 7^{3}+{ }^{n} C_{4} \times 7^{4}+\ldots+{ }^{n} C_{n} \times 7^{n}$ $\Rightarrow 8^{n}=1+7 n+49\left[{ }^{n} C_{2}+{ }^{n} C_{3} \times 7^{1}+{ }^{n} C_{4} \times 7^{2}+\ldots+{ }^{n} C_{n} \times 7^{n-2}\right]$ $\Rightarrow 8^{n}-1-7 n=49 \...
Read More →Let $f(x)=\frac{1}{1-x}$. Then, $\{f o(f o f)\}(x)$
[question] Question. Let $f(x)=\frac{1}{1-x}$. Then, $\{f o(f o f)\}(x)$ (a) $x$ for all $x \in R$ (b) $x$ for all $x \in R-\{1\}$ (c) $x$ for all $x \in R-\{0,1\}$ (d) none of these [/question] [solution] Solution: Domain of $f$ : $1-x \neq 0$ $\Rightarrow x \neq 1$ Domain of $f=R-\{1\}$ Range of $f$ : $y=\frac{1}{1-x}$ $\Rightarrow 1-x=\frac{1}{y}$ $\Rightarrow x=1-\frac{1}{y}$ $\Rightarrow x=\frac{y-1}{y}$ $\Rightarrow y \neq 0$ Range of $f=R-\{0\}$ So, $f: R-\{1\} \rightarrow R-\{0\}$ and $f...
Read More →In the given figure, AB || CD and a transversal t cuts them at E and F respectively.
Question: In the given figure, AB || CD and a transversaltcuts them at E and F respectively. If EP and FQ are the bisectors of AEF and EFD respectively, prove that EP || FQ . Solution: It is given that,AB || CD andtisa transversal.AEF = EFD .....(1) (Pair of alternate interior angles)EP is the bisectors of AEF. (Given) $\therefore \angle \mathrm{AEP}=\angle \mathrm{FEP}=\frac{1}{2} \angle \mathrm{AEF}$ ⇒AEF = 2FEP .....(2)Also, FQ is the bisectors of EFD. $\therefore \angle \mathrm{EFQ}=\angle \...
Read More →Find the sum of all natural numbers between 1 and 100, which are divisible by 3.
Question: Find the sum of all natural numbers between 1 and 100, which are divisible by 3. Solution: In this problem, we need to find the sum of all the multiples of 3 lying between 1 and 100. So, we know that the first multiple of 3 after 1 is 3 and the last multiple of 3 before 100 is 99. Also, all these terms will form an A.P. with the common difference of 3. So here, First term (a) = 3 Last term (l) = 99 Common difference (d) = 3 So, here the first step is to find the total number of terms. ...
Read More →Write the domain of the real function
Question: Write the domain of the real function $f(x)=\frac{1}{\sqrt{|x|-x}}$. Solution: Case-1: When $x0$ $|x|=x$ $\Rightarrow \frac{1}{\sqrt{|x|-x}}=\frac{1}{\sqrt{x-x}}=\frac{1}{0}=\infty$ Case-2: When $x0$ $|x|=-x$ $\Rightarrow \frac{1}{\sqrt{|x|-x}}=\frac{1}{\sqrt{-x-x}}=\frac{1}{\sqrt{-2 x}}$ (exists because when $x0,-2 x0$ ) $\Rightarrow f(x)$ is defined when $x0$ So, domain $=(-\infty, 0)$...
Read More →