Toolbox:Eqn. of a line is $\large\frac{x-a_1}{b_1}=\frac{y-a_2}{b_2}=\frac{z-a_3}{b_3}=\lambda$ wher...

$log_x1$ to any base $x$=0$\therefore x=5^{64}$

Toolbox: If $f(-x)=-f(x)$ then $f(x)$ is said to be odd function. $cot(-x)=-cotx$ ...

Toolbox: $(A\times B)\cap (A\times C)=A\times (B\cap C)$ $n(A\times B)=n(A).n(B)$$n(S_1)=20,\:...

Toolbox:$logxy=logx+logy$$log\large\frac{x}{y}$=$logx-logy$$cos(A+B)=cosAcosB-sinAsinB$$cos(A-B)=cos...

Toolbox:Roster form of representing a set is writing the elements of the set individually in proper ...

Toolbox: In set builder form the set is to be explained. Here $x\neq x$ is false f...

If $A\cap B=B$ then B is smaller set and is within A$B\subset A$

If $A\cup B=A$ then A is the bigger set and B is within A.$\Rightarrow B\subset A$

Toolbox:$|x| =x$ if $x\geq 0$ and $|x|=-x$ if $x<0$.$\large\frac{|x|}{x}$ $=\large\frac{x}{x}=1$ ...

Toolbox:$ logx$ is defined for x$ > 0$$1+x > 0$$\Rightarrow x > -1$Domain is $(-1,\infty)$

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

Toolbox:If a set has n elements then no. of subsets of the set = $2^n$Null set and the same set are ...

Only when $x=-2, \:x+2=0,\:x^2+2x=0\:and\:x^2+x-2=0$$\therefore$ at $x=-2$ V=R=S

$f(1)=5,\:\: g(5)=8,\:\Rightarrow\:g(f(1))=8$$Similarly \:g(f(2))=9\:\:and\:\:g(f(3))=7$$\Rightarrow...

$f(1)=0$$f(2)=-1$$f(3)=1$$f(4)=-2$ ........Each natural number is related with only one integer$\Rig...

Two or more different matrices can have same determinent. $\therefore $ $f$ is many to one but not 1...

Toolbox:$x-[x]$ is fractional function =a, where $a\in [0,1)$Let,$g(x)=b$ where $b\in [1,2)$$fog(x)...

$fog(x)=\big(2^x\big)^2=2^{2x}$$gof(x)=2^{x^2}$$2^{2x}=2^{x^2}\:\Rightarrow\:x^2=2x$$\Rightarrow\:x=...

$fof(x)=\large\frac{1}{1-\big(\frac{1}{1-x}\big)}=\large-\frac{1-x}{x}$$=\large\frac{x-1}{x}$$fofof(...

$2*3=6-3=3$$(2*3)*4=3*4=9-4=5$

Toolbox: If $e\in Q^+$ is the identity element , then $a*e=a \:\:\forall\:a\in\: Q^+$ ...

Toolbox:If $e\in Q^+$ is the identity element , then $a*e=a \:\:\forall\:a\in\: Q^+$If $a*b=b*a=e$...

Toolbox:$If[x]=k,\: \:then\:\: k\leq x< k+1$given: $[2x-3]=5$$\Rightarrow \:5\leq 2x-3 <\:6$$...

Toolbox:$log(ab)=loga+logb$Ans- (C)given: $f(x)=log\bigg(\large\frac{1+x}{1-x}\bigg)$$f(x)+f(y)=log\...

Toolbox:Any point on the line $\large\frac{x-x_1}{l}=\frac{y-y_1}{m}=\frac{z-z_1}{n}=\lambda$ is giv...

Ans- (A)$gof(x)=sin(x^2),$ $gogof(x)=sin(sin(x^2))$$fogogof(x)=\big[sin(sin(x^2))]^2$Given $fogogof...

Ans- (A)Given $f(x+1)+f(x-1)=2f(x)\:\forall x\in R$Put $x=1$ $\Rightarrow\:f(2)+f(0)=2f(1)$$f(0)=0\R...

Toolbox:$f(x)$ is said to be a periodic function with period a if $f(x)=f(x+a)$Ans (C)Given $f(2x+3)...

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

Toolbox:Domain of $y=sin^{-1}x\:\:is\:\:[-1,1]$Ans- (C)$-1\leq (1+3x+2x^2)\leq 1$$\Rightarrow\:0\le...

Ans- (B)We know that $(A-B)\cup B=A\cup B$But given that $(A-B)\cup B=A$$\Rightarrow\:A\cup B=A$$\Ri...

Toolbox:$-1\leq \sin x \leq 1$Ans- (C)$f(x)=\sin x-\sqrt3 \cos x+1$$=2(\large\frac{1}{2}$$\sin x-\fr...

Ans- (D) Given $f(1)=a$ and $f(x+y)=f(x)f(y)$ $\Rightarrow\:f(2)=f(1+1)=f(1)f(1)=a^2$$\Rightarro...

Toolbox:If f(g(x))=x then g is inverse of x and f is inverse of gGiven f is inverse of itself.$\Righ...

Toolbox:For inverse of a function to be defined it should be bijective function.$f(x)=x-n$ when ...

Toolbox:Express x in terms of y to get the inverse.$y=\large\frac{e^x-e^{-x}}{e^x+e^{-x}}$$\Rightarr...

For f(x) to be defined $[x]-1\neq0$$\Rightarrow\:[x]\neq 1$$\Rightarrow\:x$ is any Real number and $...

Toolbox: No. of of subsets of a set having n elements =$2^n$ Let n(A)=n and n(B)=m $2^n=56+2^...

Ans: (A) 1 $R=\{(1,1),(2,2),(3,3),(1,2),(2,1),(1,3),(3,1)\}$Here (3,1),(1,2)$\in\:$ R but (3,2) $\no...

Toolbox: If a*b=b*a then * is commutative If (a*b)*c=a*(b*c) then * is associative ...

Toolbox:If n(A)=m, n(B)=n then no. of functions from A to B=$n^m$No. of one-one functions = $^nP_m$...

Toolbox: Angle between two lines having d.r as $ b_1(l_1, m_1,n_1) \: and \: b_2 (l_2,m_2,...

Toolbox: Angle betwen two lines $\overrightarrow r=\overrightarrow {a_1}+\lambda\overright...

Toolbox:$log\frac{a}{b}=loga-l$ogb$log_b\large a=\frac{loga}{logb} $ (with any base)$log_a\large a=1...

Toolbox: If $n(A)=n,\:then\:n(A\times A)=n^2$ Ans: (A) 36 $A=\{3,6,9,............

Toolbox:A relation in any set S is a sub set of $S\times\:S$.If no. of elements in $S= n(S)=n$, then...

  Ans  (D)  $[1,\infty)$ Both numerator and denominator of $f(x)$ is pos...

Toolbox:In Onto function all the elements of B should be related.Ans: (B) =14case (i)Any 3 elements...

Toolbox:Median is the segment joining mid point of any side with its opposite vertex.Section formula...