Given: $f(x)=\sqrt{cos(sinx)}+sin^{-1}\bigg(\large\frac{1+x^2}{2x}\bigg)$For domain $ cos(sinx)\geq ...

Toolbox: A set is collection of well defined things. Since the no. of poor student...

$\phi$ means null set. Compliment of null set is universal set which is $U$

$f(x)=sin(log(\large\frac{\sqrt{4-x^2}}{1-x}))$$\Rightarrow\:\large\frac{\sqrt{4-x^2}}{1-x}>0$$\R...

Toolbox:$n(A\times B)=n(A).n(B)$$8=1\times 8,\:\:2\times 4,\:\:4\times 2,\:\:8\times 1$$\Rightarrow ...

Toolbox:$|x|=x,$ if $x\geq 0$ and $|x|=-x$ if $x<0$$\sqrt{x^2}=\pm x=|x|$

Since Munshi Premchand has written only a finite no. of Novels,this is finite set.

Given : $f(x+y)=f(x).f(y)$Put $ y=0$$\Rightarrow\:f(x)=f(x).f(0)$$\Rightarrow\:f(0)=1$

Put $x=0$ in $f(x)$, then we get$f(0)=\left|\begin {array}{ccc}-1 & 0 &1 \\0 & -1 &...

Toolbox:$loga^n=nlog a$Given: $ 2^x+2^{f(x)}=2$$\Rightarrow\:2^{f(x)}=2-2^x$Taking log on both the s...

Given $f(x) $ is a polynomial.$\therefore\:Let\:f(x)=a_0x^n+a_1x^{n-1}+a_2x^{n-2}+...............a_{...

Toolbox:$2cosA.cosB=cos(A+B)+cos(A-B)$$loga+logb=log(ab)$$loga-logb=log(\large\frac{a}{b})$Given $f(...

Toolbox:A Binary Operation on a non empty set $A$ is a function from $A\times A\rightarrow A$$Given:...

Toolbox:If a set has n elements then no. of binary operations in that set = $\large n^{n^2}$No. of b...

$f(x)=\large^{7-x}P_{x-3}$This is possible only if$x\in Z,\:\:7-x\geq 0,\:\:x-3\geq 0\:\:and\:\:7-x\...

Given: $gof(x)=4x^2-10x+4$ and$g(x)=x^2+x-2$$\Rightarrow\:g(f)=f^2+f-2=4x^2-10x+4$$\Rightarrow\:f^2...

Toolbox:$[x+n]=[x]+n\:\:if\:\:n\in Z$$f(x)=[x]^2+[x+1]-3$$=[x]^2+[x]+1-3$$=[x]^2+[x]-2$Factorising t...

$fof(x)=\left\{\begin {array}{c,c}f(|x|) & if\:x\leq 1\\f(2-x) & if\:x>1\end {array}\righ...

Toolbox:$sinAcosB+cosAsinB=sin(A+B)$Here A is domain of $f(x)$$f(x)=\sqrt 3sinx+cosx +4$$=2(\large\f...

Since $f(x)=ax+b$ is linear, it is either increasing or decreasing.$\Rightarrow\:f'(x)>0\:\:or\:\...

Let $f(x)=ax+b,$ which is linear. Since $f(x)$ is linear, it should be either increasing or decrea...

Toolbox:If $n(A)=m\:and\:n(B)=n$ then, no. of onto functions from $A\rightarrow B$ is given by the...

$f^{-1}(A)=\{x\in R:f(x)\in A\}$ $\Rightarrow\:|x|\in (0,1)$ $\Rightarrow\:0<|x|<...

Let $y=f(x)=x+\large\frac{1}{x}$$\Rightarrow x=f^{-1}(y)$$\Rightarrow\:xy=x^2+1$$\Rightarrow\:x^2-xy...

If $f^{-1}(x)=f(x)$, then $ fof=x$ and $ f^{-1}(x)=x$$\Rightarrow\:x^2-x+1=x$$\Rightarrow\:(x-1)^...

Toolbox:$logx^n=nlogx$Given $f(x)=x^n$$gof(x)=g(x^n)=ng(x)$We know that $logx^n=nlogx$$\therefore\:g...

Toolbox:$loga^n=nloga$Domain of $f(x)$ is set of all possible values of x for which $f$ is defined.G...

Since $f:Z\rightarrow Z$, $x\:and\:f(x)$ take only integer values.$\Rightarrow f(0)\:and\:f(1)$ ar...

Since $f$ is 1-1 and only one out of the statements is true,$f(z)\neq 2 $ is true.Remaining statemen...

Toolbox:For $\large^nP_r$ to exist $n\:\:and\:\:r\in W\:and\:\: r\leq n$For $\large^{7-x}P_{x-3}$ t...

Toolbox:$loga^n=nloga$$\large\frac{loga}{logb}$ $=log_b a$Let $y=f(x)=2^{x(x-1)}$$\Rightarrow x=f^{-...

Toolbox:$|x|=x,$ if $x\geq 0$ and $|x|=-x,$ if $x$<0.$f(x)=6^x+6^{|x|}$ $=2.6^x,$ if $x\geq0$...

Given $f(f(x))=\large\frac{k.f(x)}{1+f(x)}$ $=x$$\Rightarrow\:\large\frac{k.\large\frac{kx}{1+x}}{1...

$f(-1)=(-1)^2=1,\:\:f(0)=0\:\:and\:\:f(1)=1$since $f(-1)=f(1)$ $f$ is not 1-1.Also since $-1\in A$ ...

Toolbox:For $f^{-1}$ to exist $f$ should be bijective function.( i.e., both 1-1 and onto.)$f(-2)=f(2...

$y=f(x)=3x-5$$\Rightarrow\:x=\large\frac{y+5}{3}$$=f^{-1}(y)$$\therefore\:f^{-1}(x)=\large\frac{x+5}...

Toolbox:$sin(-\theta)=-sin\theta$$sin(\large\frac{\pi}{2}$$+\theta)=cos\theta$$sin^{-1}x=cos^{-1}\sq...

Toolbox: Distance of any point $ (x_1,y_1,z_1)$ from the plane $ ax+by+cz+d=0$ is given by...

Toolbox: Distance of any point $ (x_1,y_1,z_1)$ from the plane $ ax+by+cz+d=0$ is given by...

$f(x)=0$ if $ x=[x]$ that is when $x\in Z$ When $x$ is not an integer then $0<2(x-[x...

Toolbox:$|x|=x,\:ifx\geq 0$ and $|x|=-x$ if $x<0$$|x|-x>0$$=0$ if $x>0\:x\:and\:=-2x\:\: if...

$goh(x)=\large\frac{\sqrt [3] {x}}{1-\sqrt [3] {x}}$$fogoh(x)=\large\frac{\large\frac{\sqrt [3] {x}}...

if $f(x)=sin^2x$ and $g(x)=\sqrt x$ then $f(g(x))=sin^2(\sqrt x)=(sin\sqrt x)^2$  

$y=\sqrt x$ is not a function in R because negative numbersdo not have squareroot in R

Ans: $f(x)=x+1$Because if if $f(x)=2x$ then 3 has no image in BIf $f(x)=\sqrt x$ then none of the el...

Toolbox:$(A\cap B)'=A'\cup B'$$A\cap (A\cap B)'=A\cap(A'\cup B')$$=(A\cap A')\cup (A\cap B')$$=\phi\...

Ans: $[4,5)$Because, if $\geq\:\:or\:\:\leq$ comes then use closed bracket [ ].If < or > come...

$B\in A$ is correct answer. Because $ 4\notin A$ and  $ \{B\}\subset A$ not B. ...

Toolbox:$A\cap (B\cup C)=(A\cap B)\cup (A\cap C)$$A\cap (A\cup B)=(A\cap A)\cup (A\cap B)$$=A\cup (A...

$f(x)=\large\frac{4^x}{4^x+2}$$f(1-x)=\large\frac{4^{1-x}}{4^{1-x}+2}$$=\large\frac{4}{4+2\times 4^x...