Let f : R − {−1} → R − {1} be given by
Question: Let $f: R-\{-1\} \rightarrow R-\{1\}$ be given by $f(x)=\frac{x}{x+1} .$ Write $f^{-1}(x)$. Solution: Let $f^{-1}(x)=y$ $\ldots(1)$ $\Rightarrow f(y)=x$ $\Rightarrow \frac{y}{y+1}=x$ $\Rightarrow y=x y+x$ $\Rightarrow y-x y=x$ $\Rightarrow y(1-x)=x$ $\Rightarrow y=\frac{x}{1-x}$ $\Rightarrow f^{-1}(x)=\frac{x}{1-x}$ $[$ from (1) $]$...
Read More →(i) How many terms of the sequence 18, 16, 14, ... should be taken so that their sum is zero?
Question: (i) How many terms of the sequence 18, 16, 14, ... should be taken so that their sum is zero?(ii) How many terms are there in the A.P. whose first and fifth terms are 14 and 2 respectively and the sum of the terms is 40?(iii) How many terms of the A.P. 9, 17, 25,... must be taken so that their sum is 636?(iv) How many terms of the A.P. 63, 60, 57, ... must be taken so that their sum is 693? Solution: In the given problem, we have the sum of the certain number of terms of an A.P. and we...
Read More →Let f : R →
Question: Let $f: R \rightarrow R^{+}$be defined by $f(x)=a^{x}, a0$ and $a \neq 1$. Write $f^{-1}(x)$. Solution: Let $f^{-1}(x)=y$ $\ldots(1)$ $\Rightarrow f(y)=x$ $\Rightarrow a^{y}=x$ $\Rightarrow y=\log _{a} x$ $\Rightarrow f^{-1}(x)=\log _{a} x$ $[$ from (1)]...
Read More →Let f :
Question: Let $f:\left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \rightarrow A$ be defined by $f(x)=\sin x$. If $f$ is a bijection, write set $A$. Solution: $\because f$ is a bijection, co-domain of $f=$ range of $f$ As $-1 \leq \sin x \leq 1$ $-1 \leq y \leq 1$ So, $A=[-1,1]$...
Read More →Let A = {x ∈ R : −4 ≤ x ≤ 4 and x ≠ 0} and f : A → R be defined by
Question: Let $A=\{x \in R:-4 \leq x \leq 4$ and $x \neq 0\}$ and $f: A \rightarrow R$ be defined by $f(x)=\frac{|x|}{x}$. Write the range of $f$. Solution: $\because f(x)=\frac{|x|}{x}=\frac{\pm x}{x}=\pm 1 \forall x \in A$, range of $f=\{-1,1\}$...
Read More →In the given figure, AB || CD. Find the value of x.
Question: In the given figure,AB||CD. Find the value ofx. Solution: $A B \| C D$ and $A C$ is the transversal. Then, $\angle B A C+\angle A C D=180^{\circ} \quad[$ Consecutive Interior Angles $]$ $\Rightarrow 75+\angle A C D=180$ $\Rightarrow \angle A C D=105^{\circ}$ And, $\angle A C D=\angle E C F \quad[$ Vertically-Opposite Angles $]$ $\Rightarrow \angle E C F=105^{\circ}$ We know that the sum of the angles of a triangle is180"180180. $\angle E C F+\angle C F E+\angle C E F=180^{\circ}$ $\Rig...
Read More →If f : R → R, g : R → are given by f(x)
Question: If $f: R \rightarrow R, g: R \rightarrow$ are given by $f(x)=(x+1)^{2}$ and $g(x)=x^{2}+1$, then write the value of fog $(-3)$. Solution: $(f o g)(-3)=f(g(-3))$ $=f\left((-3)^{2}+1\right)$ $=f(10)$ $=(10+1)^{2}$ $=121$...
Read More →If f : R → R defined by f(x) = 3x − 4 is invertible,
Question: If $f: R \rightarrow R$ defined by $f(x)=3 x-4$ is invertible, then write $f^{-1}(x)$. Solution: Let $f^{-1}(x)=y$ $\ldots(1)$ $\Rightarrow f(y)=x$ $\Rightarrow 3 y-4=x$ $\Rightarrow 3 y=x+4$ $\Rightarrow y=\frac{x+4}{3}$ $\Rightarrow f^{-1}(x)=\frac{x+4}{3} \quad[$ from (1) $]$...
Read More →In the given figure, AB || PQ. Find the values of x and y.
Question: In the given figure,AB||PQ. Find the values ofxandy. Solution: Given, $A B \| P Q$. Let CD be the transversal cutting AB and PQ at E and F, respectively.Then, $\angle C E B+\angle B E G+\angle G E F=180^{\circ} \quad$ [Since CD is a straight line] $\Rightarrow 75^{\circ}+20^{\circ}+\angle G E F=180^{\circ}$ $\Rightarrow \angle G E F=85^{\circ}$ We know that the sum of angles of a triangle is180180. $\therefore \angle G E F+\angle E G F+\angle E F G=180$ $\Rightarrow 85^{\circ}+x+25^{\c...
Read More →Let f :
Question: Let $f:\left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \rightarrow R$ be a function defined by $f(x)=\cos [x]$. Write range (f). Solution: Domain $=\left(\frac{-\pi}{2}, \frac{\pi}{2}\right)=(-1.57,1.57) \quad\left(\right.$ as $\left.\pi=\frac{22}{7}\right)$ So, $\cos [x]=\cos (-2)=\cos 2 \quad \forall x \in(-1.57,0)$ Also, $\cos 0=1$ for $x=0$ And $\cos [x]=\cos 1 \forall x \in(0,1.57)$ $\therefore$ Range $=\{1, \cos 1, \cos 2\}$...
Read More →If f : R → R is defined by f(x) = 10 x − 7,
Question: If $f: R \rightarrow R$ is defined by $f(x)=10 x-7$, then write $f^{-1}(x)$. Solution: Let $f^{-1}(x)=y$ $\ldots(1)$\ $\Rightarrow f(y)=x$ $\Rightarrow 10 y-7=x$ $\Rightarrow 10 y=x+7$ $\Rightarrow y=\frac{x+7}{10}$ $\Rightarrow f^{-1}(x)=\frac{x+7}{10}$ (From (1)...
Read More →If f : C → C is defined by
Question: If $f: C \rightarrow C$ is defined by $f(x)=(x-2)^{3}$, write $f^{-1}(-1)$. Solution: Let $f^{-1}(-1)=x$ $\ldots(1)$ $\Rightarrow f(x)=-1$ $\Rightarrow(x-2)^{3}=-1$ $\Rightarrow x-2=-1$ or $-\omega$ or $-\omega^{2}$ (as the roots of $(-1)^{\frac{1}{3}}$ are $-1,-\omega$ and $-\omega^{2}$, where $\left.\omega=\frac{1+i \sqrt{3}}{2}\right)$ $\Rightarrow x=-1+2$ or $2-\omega$ or $2-\omega^{2}=1,2-\omega, 2-\omega$ $\Rightarrow f^{-1}(-1)=\left\{1,2-\omega, 2-\omega^{2}\right\} \quad[$ fro...
Read More →Let r and n be positive integers such that 1 ≤ r ≤ n.
Question: Letrandnbe positive integers such that 1 rn. Then prove the following: (a) $\frac{{ }^{n} C_{r}}{{ }^{n} C_{r-1}}=\frac{n-r+1}{r}$ (b) $n \cdot n-1 C_{r-1}=(n-r+1)^{n} C_{r-1}$ (c) $\frac{{ }^{n} C_{r}}{{ }^{n-1} C_{r-1}}=\frac{n}{r}$ (d) ${ }^{n} C_{r}+2 \cdot{ }^{n} C_{r-1}+{ }^{n} C_{r-2}={ }^{n+2} C_{r}$. Solution: (a) $\frac{{ }^{n} C_{r}}{{ }^{n} C_{r-1}}=\frac{n-r+1}{r}$ $\mathrm{LHS}=\frac{{ }^{n} C_{r}}{{ }^{n} C_{r-1}}$ $=\frac{n !}{r !(n-r) !} \times \frac{(r-1) !(n-r+1) !}{...
Read More →Let $f(x)=x^{2}$ and $g(x)=2^{x}$. Then, the solution set of the equation $f o g(x)=g o f(x)$ is
[question] Question. Let $f(x)=x^{2}$ and $g(x)=2^{x}$. Then, the solution set of the equation $f o g(x)=g o f(x)$ is [/question] [solution] Solution: Since $(f o g)(x)=(g o f)(x)$, $f(g(x))=g(f(x))$ $\Rightarrow f\left(2^{x}\right)=g\left(x^{2}\right)$ $\Rightarrow\left(2^{x}\right)^{2}=2^{x^{2}}$ $\Rightarrow 2^{2 x}=2^{x^{2}}$ $\Rightarrow x^{2}=2 x$ $\Rightarrow x^{2}-2 x=0$ $\Rightarrow x(x-2)=0$ $\Rightarrow x=0,2$ $\Rightarrow x \in\{0,2\}$ So, the answer is (c). [/solution]...
Read More →Let r and n be positive integers such that 1 ≤ r ≤ n.
Question: Letrandnbe positive integers such that 1 rn. Then prove the following: (a) $\frac{{ }^{n} C_{r}}{{ }^{n} C_{r-1}}=\frac{n-r+1}{r}$ (b) $n \cdot n-1 C_{r-1}=(n-r+1)^{n} C_{r-1}$ (c) $\frac{{ }^{n} C_{r}}{{ }^{n-1} C_{r-1}}=\frac{n}{r}$ (iv) ${ }^{n} C_{r}+2 \cdot{ }^{n} C_{r-1}+{ }^{n} C_{r-2}={ }^{n+2} C_{r}$. Solution: (a) $\frac{{ }^{n} C_{r}}{{ }^{n} C_{r-1}}=\frac{n-r+1}{r}$ $\mathrm{LHS}=\frac{{ }^{n} C_{r}}{{ }^{n} C_{r-1}}$ $=\frac{n !}{r !(n-r) !} \times \frac{(r-1) !(n-r+1) !}...
Read More →If f : R → R is defined by
Question: If $f: R \rightarrow R$ is defined by $f(x)=x^{2}$, find $f^{-1}(-25)$ Solution: Let $f^{-1}(-25)=x$ $\Rightarrow f(x)=-25$ $\Rightarrow x^{2}=-25$ We cannot find $x \in R$, such that $x^{2}=-25$ (as $x^{2} \geq 0$ for all $x \in R$ ) So, $f^{-1}(-25)=\phi$...
Read More →If f : C → C is defined by f(x)
Question: If $f: C \rightarrow C$ is defined by $f(x)=x^{4}$, write $f^{-1}(1)$. Solution: Let $f^{-1}(1)=x \quad \ldots$ (1) $\Rightarrow f(x)=1$ $\Rightarrow x^{4}=1$ $\Rightarrow x^{4}-1=0$ $\Rightarrow\left(x^{2}-1\right)\left(x^{2}+1\right)=0$ [u sing identity : $a^{2}-b^{2}=(a-b)(a+b)$ ] $\Rightarrow(x-1)(x+1)(x-i)(x+i)=0$, where $i=\sqrt{-1}$ [u sing identity : $\left.a^{2}-b^{2}=(a-b)(a+b)\right]$ $\Rightarrow x=\pm 1, \pm i$ $\Rightarrow f^{-1}(1)=\{-1,1, i,-i\} \quad[$ from (1) $]$...
Read More →If the sum of a certain number of terms starting from first term
Question: If the sum of a certain number of terms starting from first term of an A.P. is 25, 22, 19, ..., is 116. Find the last term. Solution: In the given problem, we have the sum of the certain number of terms of an A.P. and we need to find the last term for that A.P. So here, let us first find the number of terms whose sum is 116. For that, we will use the formula, $S_{n}=\frac{n}{2}[2 a+(n-1) d]$ Where;a= first term for the given A.P. d= common difference of the given A.P. n= number of term...
Read More →If f : R → R be defined by
Question: If $f: R \rightarrow R$ be defined by $f(x)=x^{4}$, write $f^{-1}(1)$. Solution: Let $f^{-1}(1)=x$ $\ldots$ (1) $\Rightarrow f(x)=1$ $\Rightarrow x^{4}=1$ $\Rightarrow x^{4}-1=0$ $\Rightarrow\left(x^{2}-1\right)\left(x^{2}+1\right)=0$ $\Rightarrow\left(x^{2}-1\right)\left(x^{2}+1\right)=0$ $\left[\mathrm{u}\right.$ sing identity : $\left.a^{2}-b^{2}=(a-b)(a+b)\right]$ $\Rightarrow(x-1)(x+1)\left(x^{2}+1\right)=0$ $\left[\right.$ u sing identity : $\left.a^{2}-b^{2}=(a-b)(a+b)\right]$ $...
Read More →Solve the following
Question: Evaluate ${ }^{20} C_{5}+\sum_{r=2}^{5}{ }^{25-r} C_{4}$ Solution: Given: ${ }^{20} C_{5}+\sum_{r=2}^{5} 25-{ }^{r} C_{4}$ ${ }^{20} C_{5}+\sum_{r=2}^{5} 25-{ }^{r} C_{4}$ $={ }^{20} C_{5}+{ }^{23} C_{4}+{ }^{22} C_{4}+{ }^{21} C_{4}+{ }^{20} C_{4}$ $=\left({ }^{20} C_{4}+{ }^{20} C_{5}\right)+{ }^{21} C_{4}+{ }^{22} C_{4}+{ }^{23} C_{4}$ $={ }^{21} C_{5}+{ }^{21} C_{4}+{ }^{22} C_{4}+{ }^{23} C_{4} \quad\left[\because{ }^{n} C_{r-1}+{ }^{n} C_{r}={ }^{n+1} C_{r}\right]$ $=\left({ }^{2...
Read More →Let f be a function from C (set of all complex numbers) to itself given by
Question: Let $f$ be a function from $C$ (set of all complex numbers) to itself given by $f(x)=x^{3}$. Write $f^{-1}(-1)$. Solution: Let $f^{-1}(-1)=x$ $\ldots(1)$ $\Rightarrow f(x)=-1$ $\Rightarrow x^{3}=-1$ $\Rightarrow x^{3}+1=0$ $\Rightarrow(x+1)\left(x^{2}-x+1\right)=0 \quad\left[\right.$ u sing the identity : $\left.a^{3}+b^{3}=(a+b)\left(a^{2}-a b+b^{2}\right)\right]$ $\Rightarrow(x+1)(x+\omega)\left(x+\omega^{2}\right)=0$, where $\omega=\frac{1 \pm i \sqrt{3}}{2}$ $\Rightarrow x=-1,-\ome...
Read More →Let C denote the set of all complex numbers. A function f : C → C is defined by
Question: Let $C$ denote the set of all complex numbers. A function $f: C \rightarrow C$ is defined by $f(x)=x^{3}$. Write $f^{-1}(1)$. Solution: Let $f^{-1}(1)=x \quad \ldots(1)$ $\Rightarrow f(x)=1$ $\Rightarrow x^{3}=1$ $\Rightarrow x^{3}-1=0$ $\Rightarrow(x-1)\left(x^{2}+x+1\right)=0 \quad\left[\right.$ Using identity : $\left.a^{3}-b^{3}=(a-b)\left(a^{2}+a b+b^{2}\right)\right]$ $\Rightarrow(x-1)(x-\omega)\left(x-\omega^{2}\right)=0$, where $\omega=\frac{1+i \sqrt{3}}{2}$ $\Rightarrow x=1, ...
Read More →In the given figures, AB || CD. Find the value of x.
Question: In the given figures,AB||CD. Find the value ofx. Solution: Draw $E F\|A B\| C D$. $E F \| C D$ and CE is the transversal. Then, $\angle E C D+\angle C E F=180^{\circ} \quad$ [Angles on the same side of a transversal line are supplementary] $\Rightarrow 130^{\circ}+\angle C E F=180^{\circ}$ $\Rightarrow \angle C E F=50^{\circ}$ Again, $E F \| A B$ and $\mathrm{AE}$ is the transversal. Then, $\angle B A E+\angle A E F=180^{\circ}$ [Angles on the same side of a transversal line are supple...
Read More →If f : R → R is given by
Question: If $f: R \rightarrow R$ is given by $f(x)=x^{3}$, write $f^{-1}$ (1) Solution: Let $f^{-1}(1)=x \quad \ldots(1)$ $\Rightarrow f(x)=1$ $\Rightarrow x^{3}=1$ $\Rightarrow x^{3}-1=0$ $\Rightarrow(x-1)\left(x^{2}+x+1\right)=0 \quad\left[\mathrm{u}\right.$ sing the identity $\left.: a^{3}-b^{3}=(a-b)\left(a^{2}+a b+b^{2}\right)\right]$ $\Rightarrow x=1 \quad($ as $x \in R)$ $\Rightarrow f^{-1}(1)=\{1\} \quad[$ from $(1)]$...
Read More →If f : C → C is defined by f(x)
Question: If $f: C \rightarrow C$ is defined by $f(x)=x^{2}$, write $f^{-1}(-4)$. Here, $C$ denotes the set of all complex numbers. Solution: Let $f^{-1}(-4)=x$ $\ldots$ (1) $\Rightarrow f(x)=-4$ $\Rightarrow x^{2}=-4$ $\Rightarrow x^{2}+4=0$ $\Rightarrow(x+2 i)(x-2 i)=0 \quad\left[\mathrm{u} \operatorname{sing}\right.$ the identity : $\left.a^{2}+b^{2}=(a-i b)(a+i b)\right]$ $\Rightarrow x=\pm 2 i$ $[$ as $x \in C]$ $\Rightarrow f^{-1}(25)=\{-2 i, 2 i\}$ $[$ from (1)]...
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