Current Affairs JEE Main & Advanced

Type I : An equation of the form   \[(x-a)\,(x-b)\,(x-c)\,(x-d)=A\],   where \[a<b<c<d\], \[b-a=d-c\], can be solved by a change of variable.   i.e.,\[y=\frac{(x-a)+(x-b)+(x-c)+(x-d)}{4}\]   \[y=x-\frac{(a+b+c+d)}{4}\].   Type II : An equation of the form   \[(x-a)\,(x-b)(x-c)(x-d)=A{{x}^{2}}\]   where \[ab=cd\], can be reduced to a collection of two quadratic equations by a change of variable \[y=x+\frac{ab}{x}\].   Type III : An equation of the form \[{{(x-a)}^{4}}+{{(x-b)}^{4}}=A\] can also be solved by a change of variable, i.e., making a substitution \[y=\frac{(x-a)+(x-b)}{2}\].

(1) For the quadratic equation \[a{{x}^{2}}+bx+c=0\].   (i) One root will be reciprocal of the other if \[a=c\].   (ii) One root is zero if \[c=0\].   (iii) Roots are equal in magnitude but opposite in sign if \[b=0\].   (iv) Both roots are zero if \[b=c=0\].   (v) Roots are positive if \[a\] and \[c\] are of the same sign and \[b\] is of the opposite sign.   (vi) Roots are of opposite sign if \[a\] and \[c\] are of opposite sign.   (vii) Roots are negative if \[a,b,c\] are of the same sign.   (2) Let \[f(x)=a{{x}^{2}}+bx+c\], where \[a>0\]. Then   (i) Conditions for both the roots of \[f(x)=0\] to be greater than a given number \[k\] are \[{{b}^{2}}-4ac\ge 0;\,f(k)=0;\,\frac{-b}{2a}>k\].   (ii) Conditions for both the roots of \[f(x)=0\] to be less than a given number \[k\] are \[{{b}^{2}}-4ac\ge 0\], \[f(k)>0,\] \[\frac{-b}{2a}<k\].   (iii) The number \[k\] lies between the roots of \[f(x)=0\], if \[{{b}^{2}}-4ac>0;f(k)<0\].   (iv) Conditions for exactly one root of \[f(x)=0\] to lie between \[{{k}_{1}}\] and \[{{k}_{2}}\] is \[f({{k}_{1}})f({{k}_{2}})<0,\,\,{{b}^{2}}-4ac>0\].   (v) Conditions for both the roots of \[f(x)=0\] are confined between \[{{k}_{1}}\] and \[{{k}_{2}}\] is \[f({{k}_{1}})>0,\,f({{k}_{2}})>0,{{b}^{2}}-4ac\ge 0\] and \[{{k}_{1}}<\frac{-b}{2a}<{{k}_{2}}\], where \[{{k}_{1}}<{{k}_{2}}\].   (vi) Conditions for both the numbers \[{{k}_{1}}\]and \[{{k}_{2}}\] lie between the roots of \[f(x)=0\] is \[{{b}^{2}}-4ac>0;\,f({{k}_{1}})<0;\,f({{k}_{2}})<0\].

Let R and S be two relations from sets A to B and B to C respectively. Then we can define a relation SoR from A to C such that \[(a,\,\,c)\in SoR\Leftrightarrow \exists \,\,b\in B\] such that \[(a,\,\,b)\in R\] and \[(b,\,\,c)\in S\].   This relation is called the composition of R and S.   For example, if \[A=\text{ }\{1,\text{ }2,\text{ }3\},\,\,B=\text{ }\{a,b,c,d\},\,\,C=\{p,q,r,s\}\] be three sets such that \[R=\{(1,\,\,a),\,\,(2,\,\,b),\,\,(1,\,\,c),\,\,(2,\,\,d)\}\] is a relation from A to B and \[S=\{(a,\,\,s),\,\,(b,\,\,r),\,\,(c,\,\,r)\}\] is a relation from B to C. Then SoR is a relation from A to C given by \[SoR=\{(1,\,\,s)\,(2,\,\,r)\,\,(1,\,\,r)\}\]   In this case RoS does not exist.   In general \[RoS\ne SoR\]. Also \[{{(SoR)}^{-1}}={{R}^{-1}}o{{S}^{-1}}\].  

Let R be equivalence relation in \[A(\ne \varphi )\]. Let \[a\in A\]. Then the equivalence class of \[a,\] denoted by \[[a]\] or \[\{\bar{a}\}\] is defined as the set of all those points of A which are related to a under the relation R. Thus \[\left[ a \right]\text{ }=\text{ }\{x\hat{I}A:x\text{ }Ra\}.\]   It is easy to see that   (1) \[b\in [a]\Rightarrow a\in [b]\]          (2) \[b\in [a]\Rightarrow [a]=[b]\]       (3) Two equivalence classes are either disjoint or identical.  

(1) Reflexive relation : A relation R on a set A is said to be reflexive if every element of A is related to itself.   Thus, R is reflexive \[\Leftrightarrow (a,\,\,b)\in R\] for all \[a\in A\].   Example : Let \[A=\{1,\,2,\,3\}\] and \[R=\{(1,\,\,1);\,\,(1,\,\,3)\}\]   Then R is not reflexive since \[3\in A\] but  \[(3,\,\,3)\notin R\]   A reflexive relation on A is not necessarily the identity relation on A.   The universal relation on a non-void set A is reflexive.   (2) Symmetric relation : A relation R on a set A is said to be a symmetric relation \[iff\,(a,b)\,\,\notin R\,\Rightarrow (b,a)\in R\] for all \[a,\,\,b\in A\]   i.e., \[aRb\Rightarrow bRa\] for all \[a,\,b\,\in A\].   it should be noted that R  is symmetric iff \[{{R}^{-1}}=R\]   The identity and the universal relations on a non-void set are symmetric relations.   A reflexive relation on a set A is not necessarily symmetric.   (3) Anti-symmetric relation : Let A be any set. A relation R on set A is said to be an anti-symmetric relation iff \[(a,\,\,b)\in R\]and \[(b,\,\,a)\in R\Rightarrow a=b\] for all \[a,\,\,b\in A\].   Thus, if \[a\ne b\] then a may be related to b or b may be related to a, but never both.   (4) Transitive relation : Let A be any set. A relation R on set A is said to be a transitive relation iff \[(a,\,\,b)\in R\] and \[(b,\,\,c)\in R\Rightarrow (a,\,\,c)\in R\] for all \[a,\,\,b,\,\,c\in A\] i.e.,  \[aRb\] and \[bRC\Rightarrow aRc\] for all \[a,\,\,b,\,\,c\,\in A\].   Transitivity fails only when there exists \[a,\,\,b,\,\,c\] such that a R b, b R c but \[a\,\not{R}\,c\].   Example : Consider the set \[A=\{1,\,\,2,\,\,3\}\] and the relations   \[{{R}_{1}}=\{(1,\,\,2),\,(1,\,3)\};\,\,{{R}_{2}}=\{(1,\,2)\};\,\,{{R}_{3}}=\{(1,\,\,1)\};\]   \[{{R}_{4}}=\{(1,\,\,2),\,\,(2,\,\,1),\,(1,\,\,1)\}\]   Then \[{{R}_{1}}\], \[{{R}_{2}}\], \[{{R}_{3}}\] are transitive while \[{{R}_{4}}\] is not transitive since in \[{{R}_{4}},\,(2,\,\,1)\in {{R}_{4}};\,(1,\,2)\in {{R}_{4}}\] but \[(2,\,2)\notin {{R}_{4}}\].   The identity and the universal relations on a non-void sets are transitive.   (5) Identity relation : Let A be a set. Then the relation \[{{I}_{A}}=\{(a,\,\,a)\,:\,a\in A\}\] on A is called the identity relation on A.   In other words, a relation \[{{I}_{A}}\] on A is called the identity relation if every element of A is related to itself only. Every identity relation will be reflexive, symmetric and transitive.   Example : On the set \[=\{1,\,\,2,\,\,3\},\,\,R=\{(1,\,\,1),\,\,(2,\,\,2),\,\,(3,\,\,3)\}\] is the identity relation on A .   It is interesting to note that every identity relation is reflexive but every reflexive relation need not be an identity relation.   (6) Equivalence relation : A relation R on a set A is said to be an equivalence relation on A iff   (i) It is reflexive i.e. \[(a,\,\,a)\in R\] for all \[a\in A\]   (ii) It is symmetric i.e. \[(a,\,\,b)\,\,\in R\Rightarrow (b,\,\,a)\in R,\] for all \[a,\,\,b\in A\]   (iii) It is transitive i.e. \[(a,\,\,b)\in R\] and \[(b,\,\,c)\in R\Rightarrow (a,\,\,c)\in R\] for all \[a,\,\,b,\,\,c\in A\].   Congruence modulo (m) : Let \[m\] be an arbitrary but fixed integer. Two integers a and b are said to be congruence modulo \[m\] more...

Let A, B be two sets and let R be a relation from a set A to a set B. Then the inverse of R, denoted by \[{{R}^{-1}},\] is a relation from B to A and is defined by \[{{R}^{-1}}=\{(b,\,a):\,(a,\,b)\,\in R\}\]   Clearly \[(a,\,b)\in R\Leftrightarrow (b,\,a)\in {{R}^{-1}}\]. Also, Dom \[(R)=\,\,\,\text{Range}\,\,\,({{R}^{-1}})\] and Range \[(R)=\,\,\,\text{Dom}\,\,\,({{R}^{-1}})\]   Example :  Let \[A=\{a,\,b,\,c\},\,B=\{1,\,2,\,3\}\] and \[R=\{(a,\,1),\,(a,\,3),\,(b,\,3),\,(c,\,3)\}\].   Then,   (i)  \[{{R}^{-1}}=\{(1,\,a),\,\,(3,\,b),\,(3,\,c)\}\]   (ii)  Dom \[(R)=\{a,\,\,b,\,\,c\}=\] Range \[({{R}^{-1}})\]     (iii)  Range \[(R)=\{1,\,\,3\}=\] Dom \[({{R}^{-1}})\]  

Let A and B be two non-empty sets, then every subset of \[A\times B\] defines a relation from A to B and every relation from A to B is a subset of \[A\times B\].   Let \[R\subseteq A\times B\] and \[(a,\,\,b)\in R\]. Then we say that \[a\] is related to \[b\] by the relation \[R\] and write it as \[a\,R\,b\]. If \[(a,\,b)\in R\],  we write it as \[a\,R\,b\].   (1) Total number of relations : Let A and B be two non-empty finite sets consisting of m and n elements respectively. Then \[A\times B\] consists of mn ordered pairs. So, total number of subset of \[A\times B\] is \[{{2}^{mn}}\]. Since each subset of \[A\times B\] defines relation from A to B, so total number of relations from A to B is \[{{2}^{mn}}\]. Among these \[{{2}^{mn}}\] relations the void relation \[\phi \] and the universal relation \[A\times B\] are trivial relations from A to B.   (2) Domain and range of a relation : Let R be a relation from a set A to a set B. Then the set of all first components or coordinates of the ordered pairs belonging to R is called the domain of R, while the set of all second components or coordinates of the ordered pairs in R is called the range of R.   Thus, Dom \[(R)=\{a\,:(a,\,b)\,\in R\}\] and Range \[(R)=\{b\,:(a,\,\,b)\,\in R\}\].  

Cartesian product of sets : Let A and B be any two non-empty sets. The set of all ordered pairs \[(a,b)\] such that \[a\in A\] and \[b\in B\] is called the cartesian product of the sets A and B and is denoted by \[A\times B\].   Thus, \[A\times B=[(a,\,\,b)\,:\,\,a\in A\] and \[b\in B]\]   If \[A=\phi \] or \[B=\phi ,\] then we define \[A\times B=\phi \].   Example : Let \[A=\{a,\,\,b,\,\,c\}\] and \[B=\{p,\,q\}\].   Then \[A\times B=\{(a,\,p),\,(a,\,q),\,(b,\,p),\,(b,\,q),\,(c,\,p),\,(c,\,q)\}\]   Also \[B\times A=\{(p,\,a),\,(p,\,b),\,(p,\,c),\,\,(q,\,\,a),\,\,(q,\,\,b),\,\,(q,\,\,c)\}\]   Important theorems on cartesian product of sets :   Theorem 1 : For any three sets A, B, C   (i)  \[A\times (B\cup C)=(A\times B)\cup (A\times C)\]   (ii) \[A\times (B\cap C)=(A\times B)\cap (A\times C)\]   Theorem 2 :  For any three sets A, B, C    \[A\times (B-C)=(A\times B)-(A\times C)\]   Theorem 3 :  If A and B are any two non-empty sets, then   \[A\times B=B\times A\Leftrightarrow A=B\]   Theorem 4 :  If \[A\subseteq B,\] then \[A\times A\subseteq (A\times B)\cap (B\times A)\]   Theorem 5 :  If \[A\subseteq B,\] then \[A\times C\subseteq B\times C\] for any set C.   Theorem 6 :  If \[A\subseteq B\] and \[C\subseteq D,\] then \[A\times C\subseteq B\times D\]   Theorem 7 :  For any sets A, B, C, D   \[(A\times B)\cap (C\cup D)=(A\cap C)\times (B\cap D)\]   Theorem 8 :  For any three sets A, B, C   (i)  \[A\times (B'\times C')'=(A\times B)\cap (A\times C)\]   (ii) \[A\times (B'\cap C')'=(A\times B)\cup (A\times C)\]  

(1) Idempotent laws : For any set A, we have   (i)   \[A\cup A=A\]   (ii)  \[A\cap A=A\]   (2) Identity laws : For any set A, we have   (i) \[A\cup \phi =A\]                (ii) \[A\cap U=A\]   i.e., \[\phi \] and U are identity elements for union and intersection respectively.   (3) Commutative laws : For any two sets A and B, we have   (i) \[A\cup B=B\cup A\]   (ii) \[A\cap B=B\cap A\]   (iii) \[A\Delta B=B\Delta A\]   i.e., union, intersection and symmetric difference of two sets are commutative.   (iv) \[A-B\ne B-A\]   (v) \[A\times B\ne B\times A\]   i.e., difference and cartesian product of two sets are not commutative   (4) Associative laws : If A, B and C are any three sets, then   (i) \[(A\cup B)\cup C=A\cup (B\cup C)\]   (ii) \[A\cap (B\cap C)=(A\cap B)\cap C\]   (iii) \[(A\Delta B)\Delta C=A\Delta (B\Delta C)\]   i.e., union, intersection and symmetric difference of two sets are associative.   (iv) \[(A-B)-C\ne A-(B-C)\]      (v) \[(A\times B)\times C\ne A\times (B\times C)\]   i.e., difference and cartesian product of two sets are not associative.   (5) Distributive law : If A, B and C are any three sets, then   (i) \[A\cup (B\cap C)=(A\cup B)\cap (A\cup C)\]   (ii) \[A\cap (B\cup C)=(A\cap B)\cup (A\cap C)\]   i.e., union and intersection are distributive over intersection and union respectively.   (iii) \[A\times (B\cap C)=(A\times B)\cap (A\times C)\]   (iv) \[A\times (B\cup C)=(A\times B)\cup (A\times C)\]   (v) \[A\times (B-C)=(A\times B)-(A\times C)\]   (6) De-Morgan’s law : If A, B and C are any three sets, then   (i) \[(A\cup B)'=A'\cap B'\]   (ii) \[(A\cap B)'=A'\cup B'\]   (iii) \[A-(B\cap C)=(A-B)\cup (A-C)\]   (iv) \[A-(B\cup C)=(A-B)\cap (A-C)\]   (7) If A and B are any two sets, then   (i) \[A-B=A\cap B'\]   (ii)  \[B-A=B\cap A'\]   (iii) \[AB=A\Leftrightarrow A\cap B=\phi \]   (iv) \[(AB)\cup B=A\cup B\]   (v) \[(AB)\cap B=\phi \]   (vi) \[A\subseteq B\Leftrightarrow B'\subseteq A'\]   (vii) \[(AB)\cup (BA)=(A\cup B)(A\cap B)\]   (8) If A, B and C are any three sets, then   (i) \[A\cap (BC)=(A\cap B)(A\cap C)\]   (ii) \[A\cap (B\Delta C)=(A\cap B)\Delta (A\cap C)\]  

If A, B and C are finite sets and U be the finite universal set, then   (1) \[n(A\cup B)=n(A)+n(B)-n(A\cap B)\]   (2) \[n(A\cup B)=n(A)+n(B)\Leftrightarrow A,\,\,B\]  are disjoint non-void sets.   (3) \[n(A-B)=n(A)-n(A\cap B)\] i.e., \[n(A-B)+n(A\cap B)=n(A)\]     (4) \[n(A\Delta B)=\] Number of elements which belong to exactly one of A or B  \[=n((A-B)\cup (B-A))=n(A-B)+n(B-A)\]     \[[\because \,\,(A-B)\] and \[(B-A)\] are disjoint]   \[=n(A)n(A\cap B)+n(B)n(A\cap B)=n(A)+n(B)2n(A\cap B)\]   (5) \[n(A\cup B\cup C)=n(A)+n(B)+n(C)n(A\cap B)n(B\cap C)n(A\cap C)+n(A\cap B\cap C)\]   (6) n (Number of elements in exactly two of the sets A, B, C) \[=n(A\cap B)+n(B\cap C)+n(C\cap A)3n(A\cap B\cap C)\]   (7) n(Number of elements in exactly one of the sets A, B, C) \[=n(A)+n(B)+n(C)\]   \[2n(A\cap B)2n(B\cap C)2n(A\cap C)+3n(A\cap B\cap C)\]   (8) \[n(A'\cap B')\text{ }=n(A\cap B)'=n(U)n(A\cap B)\]   (9) \[n(A'\cap B')\text{ }=n(A\cap B)'=n(U)n(A\cup B)\]