NCERT Solutions Class 12

NCERT SolutionsMathematics
 Relations and Functions : NCERT Solutions – Class 12 Maths (Ex 1)
 Relations and Functions : NCERT Solutions – Class 12 Maths (Ex 2)
 Relations and Functions : NCERT Solutions – Class 12 Maths (Ex 3)
 Relations and Functions : NCERT Solutions – Class 12 Maths (Ex 4)
 Relations and Functions : NCERT Solutions – Class 12 Maths (Ex 5)
 Inverse Trigonometric Function : NCERT Solutions – Class 12 Maths (Ex 1)
 Inverse Trigonometric Function : NCERT Solutions – Class 12 Maths (Ex 2)
 Inverse Trigonometric Function : NCERT Solutions – Class 12 Maths (Ex 3)
 Matrices : NCERT Solutions – Class 12 Maths (Ex 1)
 Matrices : NCERT Solutions – Class 12 Maths (Ex 2)
 Matrices : NCERT Solutions – Class 12 Maths (Ex 3)
 Matrices : NCERT Solutions – Class 12 Maths (Ex 4)
 Matrices : NCERT Solutions – Class 12 Maths (Ex 5)
 Determinants : NCERT Solutions – Class 12 Maths (Ex 1)
 Determinants : NCERT Solutions – Class 12 Maths (Ex 2)
 Determinants : NCERT Solutions – Class 12 Maths (Ex 3)
 Determinants : NCERT Solutions – Class 12 Maths (Ex 4)
 Determinants : NCERT Solutions – Class 12 Maths (Ex 5)
 Determinants : NCERT Solutions – Class 12 Maths (Ex 6)
 Determinants : NCERT Solutions – Class 12 Maths (Ex 7)
 Continuity and Differentiability : NCERT Solutions – Class 12 Maths (Ex 1)
 Continuity and Differentiability : NCERT Solutions – Class 12 Maths (Ex 2)
 Continuity and Differentiability : NCERT Solutions – Class 12 Maths (Ex 3)
 Continuity and Differentiability : NCERT Solutions – Class 12 Maths (Ex 4)
 Continuity and Differentiability : NCERT Solutions – Class 12 Maths (Ex 5)
 Continuity and Differentiability : NCERT Solutions – Class 12 Maths (Ex 6)
 Continuity and Differentiability : NCERT Solutions – Class 12 Maths (Ex 7)
 Continuity and Differentiability : NCERT Solutions – Class 12 Maths (Ex 8)
 Continuity and Differentiability : NCERT Solutions – Class 12 Maths (Ex 9)

NCERT SolutionsChemistry
 Aldehydes, Ketones and Carboxylic Acids : NCERT Solutions – Class 12 Chemistry
 Alcohols, Phenols and Ethers : NCERT Solutions – Class 12 Chemistry
 Amines : NCERT Solutions – Class 12 Chemistry
 Biomolecules : NCERT Solutions – Class 12 Chemistry
 Chemical Kinetics : NCERT Solutions – Class 12 Chemistry
 Chemistry in Everyday Life : NCERT Solutions – Class 12 Chemistry
 Coordination Compounds : NCERT Solutions – Class 12 Chemistry
 Electrochemistry : NCERT Solutions – Class 12 Chemistry
 General Principles and Processes of Isolation of Elements : NCERT Solutions – Class 12 Chemistry
 Haloalkanes and Haloarenes : NCERT Solutions – Class 12 Chemistry
 Polymers : NCERT Solutions – Class 12 Chemistry
 Surface Chemistry : NCERT Solutions – Class 12 Chemistry
 The dand fBlock Elements : NCERT Solutions – Class 12 Chemistry
 The pBlock Elements : NCERT Solutions – Class 12 Chemistry
 The Solid State : NCERT Solutions – Class 12 Chemistry
 Solutions : NCERT Solutions – Class 12 Chemistry

NCERT SolutionsBiology

NCERT SolutionsPhysics
 Electrostatic Potential And Capacitance : NCERT Solutions – Class 12 Physics
 Electric Charges And Fields : NCERT Solutions – Class 12 Physics
 Semiconductor Electronics: Materials, Devices And Simple Circuits : NCERT Solutions – Class 12 Physics
 Ray Optics And Optical Instruments : NCERT Solutions – Class 12 Physics
 Nuclei : NCERT Solutions – Class 12 Physics
 Moving Charges And Magnetism : NCERT Solutions – Class 12 Physics
 Magnetism And Matter : NCERT Solutions – Class 12 Physics
 Electromagnetic Induction : NCERT Solutions – Class 12 Physics
 Dual Nature Of Radiation And Matter : NCERT Solutions – Class 12 Physics
 Current Electricity : NCERT Solutions – Class 12 Physics
 Communication Systems : NCERT Solutions – Class 12 Physics
 Atoms : NCERT Solutions – Class 12 Physics
 Alternating Current : NCERT Solutions – Class 12 Physics
Relations and Functions : NCERT Solutions – Class 12 Maths (Ex 1)
Exercise 1.1
1. Determine whether each of the following relations are reflexive, symmetric and transitive:
(i) Relation R in the set A = {1, 2, 3, ……….. 13, 14} defined as R = .
(ii) Relation R in the set N of natural numbers defined as R =.
(iii) Relation R in the set A = {1, 2, 3, 4, 5, 6} as R =
(iv) Relation R in the set Z of all integers defined as R =
(v) Relation R in the set A of human beings in a town at a particular time given by:
(a) R =
(b) R =
(c) R =
(d) R =
(e) R =
Ans. (i) R = in A = {1, 2, 3, 4, 5, 6, ……13, 14}
Clearly R = {(1, 3), (2, 6), (3, 9), (4, 12)}
Since, R, R is not reflexive.
Again R but R R is not symmetric.
Also (1, 3) R and (3, 9) R but (1, 9) R, R is not transitive.
Therefore, R is neither reflexive, nor symmetric and nor transitive.
(ii) R = in set N of natural numbers.
Clearly R = {(1, 6), (2, 7), (3, 8)}
Now R, R is not reflexive.
Again R but R R is not symmetric.
Also (1, 6) R and (2, 7) R but (1, 7) R, R is not transitive.
Therefore, R is neither reflexive, nor symmetric and nor transitive.
(iii) R = in A = {1, 2, 3, 4, 5, 6}
Clearly R = (1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6), (2, 2), (2, 4), (2, 6), (3, 3), (3, 6), (4, 4), (5, 5), (6, 6)
Now i.e., (1, 1), (2, 2) and (3, 3) R R is reflexive.
Again i.e., (1, 2) R but R R is not symmetric.
Also (1, 4) R and (4, 4) R and (1, 4) R, R is transitive.
Therefore, R is reflexive and transitive but not symmetric.
(iv) R = in set Z of all integers.
Now i.e., (1, 1) = 1 – 1 = 0 Z R is reflexive.
Again R and R, i.e., and are an integerR is symmetric.
Also Z and Z and
R, R is transitive.
Therefore, R is reflexive, symmetric and transitive.
(v) Relation R in the set A of human being in a town at a particular time.
(a) R =
Since R, because and work at the same place. R is reflexive.
Now, if R and R, since and work at the same place and and work at the same place. R is symmetric.
Now, if R and R R. R is transitive
Therefore, R is reflexive, symmetric and transitive.
(b) R =
Since R, because and live in the same locality. R is reflexive.
Also R R because and live in same locality and and also live in same locality. R is symmetric.
Again R and R R R is transitive.
Therefore, R is reflexive, symmetric and transitive.
(c) R =
is not exactly 7 cm taller than , so R R is not reflexive.
Also is exactly 7 cm taller than but is not 7 cm taller than , so R but R R is not symmetric.
Now is exactly 7 cm taller than and is exactly 7 cm taller than then it does not imply that is exactly 7 cm taller than R is not transitive.
Therefore, R is neither reflexive, nor symmetric and nor transitive.
(d) R =
is not wife of , so R R is not reflexive.
Also is wife of but is not wife of , so R but R R is not symmetric.
Also R and R then R R is not transitive.
Therefore, R is neither reflexive, nor symmetric and nor transitive.
(e) R =
is not father of , so R R is not reflexive.
Also is father of but is not father of ,
so R but R R is not symmetric.
Also R and R then R R is not transitive.
Therefore, R is neither reflexive, nor symmetric and nor transitive.
2. Show that the relation R in the set R of real numbers defined as R = is neither reflexive nor symmetric nor transitive.
Ans. R = , Relation R is defined as the set of real numbers.
(i) Whether R, then which is false. R is not reflexive.
(ii) Whether , then and , it is false. R is not symmetric.
(iii) Now , , which is false. R is not transitive
Therefore, R is neither reflexive, nor symmetric and nor transitive.
3. Check whether the relation R defined in the set {1, 2, 3, 4, 5, 6} as R = is reflexive, symmetric or transitive.
Ans. R = R
Now R = {(1, 2), (2, 3), (3, 4), (4, 5), (5, 6)} and
(i) When which is false, so R, R is not reflexive.
(ii) Whether , then and , false R is not symmetric.
(iii) Now if R, R R
(iv) Then and which is false. R is not transitive.
Therefore, R is neither reflexive, nor symmetric and nor transitive.
4. Show that the relation R in R defined as R = , is reflexive and transitive but not symmetric.
Ans. (i) which is true, so R, R is reflexive.
(ii) but R is not symmetric.
(iii) and which is true. R is transitive.
Therefore, R is reflexive and transitive but not symmetric.
5. Check whether the relation R in R defined by R = is reflexive, symmetric or transitive.
Ans. (i) For which is false. R is not reflexive.
(ii) For and which is false. R is not symmetric.
(iii) For and , which is false. R is not transitive.
Therefore, R is neither reflexive, nor symmetric and nor transitive.
6. Show that the relation in the set A = {1, 2, 3} given by R = {(1, 2), (2, 1)} is symmetric but neither reflexive nor transitive.
Ans. R = {(1, 2), (2, 1)}, so for , (1, 1) R. R is not reflexive.
Also if then RR is symmetric.
Now R and then does not imply R R is not transitive..
Therefore, R is symmetric but neither reflexive nor transitive.
7. Show that the relation R in the set A of all the books in a library of a college, given by R = is an equivalence relation.
Ans. Books and have same number of pages R R is reflexive.
If R R, so R is symmetric.
Now if R, R R R is transitive.
Therefore, R is an equivalence relation.
8. Show that the relation R in the set A = {1, 2, 3, 4, 5} given by R = is an equivalence relation. Show that all the elements of {1, 3, 5} are related to each other and all the elements of {2, 4} are related to each other. But no element of {1, 3, 5} is related to any element of {2, 4}.
Ans. A = {1, 2, 3, 4, 5} and R = , then R = {(1, 3), (1, 5), (3, 5), (2, 4)}
(a) For which is even. R is reflexive.
If is even, then is also even. R is symmetric.
Now, if and is even then is also even. R is transitive.
Therefore, R is an equivalence relation.
(b) Elements of {1, 3, 5} are related to each other.
Since all are even numbers
Elements of {1, 3, 5} are related to each other.
Similarly elements of (2, 4) are related to each other.
Since an even number, then no element of the set {1, 3, 5} is related to any element of (2, 4).
Hence no element of {1, 3, 5} is related to any element of {2, 4}.
9. Show that each of the relation R in the set A = given by:
(i) R =
(ii) R = is an equivalence relation. Find the set of all elements related to 1 in each case.
Ans. (a) (i) A = A = {0, 1, 2, 3, ……….., 12}
Now R =
R = {(4, 0), (0, 4), (5, 1), (1, 5), (6, 2), (2, 6), ….. (12, 9), (9, 12), …. (8, 0), (0, 8), ….. (8, 4), (4, 8), …… (12, 12)}
Here, is a multiple of 4.
is a multiple of 4. R is reflexive.
Also we observe that R is symmetric.
And is the multiple of 4. R is transitive.
Hence R is an equivalence relation.
(ii) R = and A = {0, 1, 2, 3, ……….., 12}
R = {(0, 0), (1, 1), (2, 2), …….. (12, 12)}
For R is reflexive.
As then R is symmetric.
Also then R is transitive.
Hence R is an equivalence relation.
(b) Now set of all elements related to 1 in each case.
(i) Required set = {1, 5, 9} (ii) Required set = {1}
10. Give an example of a relation, which is:
(i) Symmetric but neither reflexive nor transitive.
(ii) Transitive but neither reflexive nor symmetric.
(iii) Reflexive and symmetric but not transitive.
(iv) Reflexive and transitive but not symmetric.
(v) Symmetric and transitive but not reflexive.
Ans. (i) The relation “is perpendicular to” is not perpendicular to
If then , however if and then is not perpendicular to
So it is clear that R “is perpendicular to” is a symmetric but neither reflexive nor transitive.
(ii) Relation R =
We know that is false. Also but is false and if , this implies
Therefore, R is transitive, but neither reflexive nor symmetric.
(iii) “is friend of” R =
It is clear that is friend of R is reflexive.
Also is friend of and is friend of R is symmetric.
Also if is friend of and is friend of then
cannot be friend of R is not transitive.
Therefore, R is reflexive and symmetric but not transitive.
(iv) “is greater or equal to” R =
It is clear that R is reflexive.
And does not imply R is not symmetric.
But , R is transitive.
Therefore, R is reflexive and transitive but not symmetric.
(v) “is brother of” R =
It is clear that is not the brother of R is not reflexive.
Also is brother of and is brother of R is symmetric.
Also if is brother of and is brother of then
can be brother of R is transitive.
Therefore, R is symmetric and transitive but not reflexive.
11. Show that the relation R in the set A of points in a plane given by R = {(P. Q) : distance of the point P from the origin is same as the distance of the point Q from the origin}, is an equivalence relation. Further, show that the set of all points related to a point P (0, 0) is the circle passing through P with origin as centre.
Ans. Part I: R = {(P, Q): distance of the point P from the origin is the same as the distance of the point Q from the origin}
Let P and Q and O (0, 0).
OP = OQ = =
Now, For (P, P), OP = OP R is reflexive.
Also OP = OQ and OQ = OP (P, Q) = (Q, P) R R is symmetric.
Also OP = OQ and OQ = OR OP = OQ R is transitive.
Therefore, R is an equivalent relation.
Part II: As = = (let) which represents a circle with centre (0, 0) and radius
12. Show that the relation R defined in the set A of all triangles as R = {(T_{1}, T_{2}) : T_{1} is similar to T_{2}}, is equivalence relation. Consider three right angle triangles T_{1} with sides 3, 4, 5, T2 with sides 5, 12, 13 and T_{3} with sides 6, 8, 10. Which triangles among T_{1}, T_{2} and T_{3} are related?
Ans. Part I: R = {(T_{1}, T_{2}): T_{1} is similar to T_{2}} and T_{1}, T_{2} are triangle.
We know that each triangle similar to itself and thus (T_{1}, T_{2}) R R is reflexive.
Also two triangles are similar, then T_{1} T_{2} T_{1} T_{2} R is symmetric.
Again, if then T_{1} T_{2} and then T_{2} T_{3} then T_{1} T_{3} R is transitive.
Therefore, R is an equivalent relation.
Part II: It is given that T_{1}, T_{2} and T_{3} are right angled triangles.
T_{1} with sides 3, 4, 5 T_{2} with sides 5, 12, 13 and
T_{3} with sides 6, 8, 10
Since, two triangles are similar if corresponding sides are proportional.
Therefore,
Therefore, T_{1} and T_{3} are related.
13. Show that the relation R defined in the set A of all polygons as R = {(P_{1}, P_{2}) : P_{1} and P_{2} have same number of sides}, is an equivalence relation. What is the set of all elements in A related to the right angle triangle T with sides 3, 4, and 5?
Ans. Part I: R = {(P_{1}, P_{2}): P_{1} and P_{2} have same number of sides}
(i) Consider the element (P_{1}, P_{2}), it shows P_{1} and P_{2} have same number of sides. Therefore, R is reflexive.
(ii) If (P_{1}, P_{2}) R then also (P_{2}, P_{1}) R
(P_{1}, P_{2}) = (P_{2}, P_{1}) as P_{1} and P_{2} have same number of sides, therefore, R is symmetric.
(iii) If (P_{1}, P_{2}) R and (P_{2}, P_{3}) R then also (P_{1}, P_{3}) R as P_{1}, P_{2} and P_{3} have same number of sides, therefore, R is transitive.
Therefore, R is an equivalent relation.
Part II: we know that if 3, 4, 5 are the sides of a triangle, then the triangle is right angled triangle. Therefore, the set A is the set of right angled triangle.
14. Let L be the set of all lines in XY plane and R be the relation in L defined as R = {(L_{1}, L_{2}) : L_{1} is parallel to L_{2}}. Show that R is an equivalence relation. Find the set of all lines related to the line
Ans. Part I: R = {(L_{1}, L_{2}): L_{1} is parallel to L_{2}}
(i) It is clear that L_{1} L_{1} i.e., (L_{1}, L_{1}) R R is reflexive.
(ii) If L_{1} L_{2} and L_{2} L_{1} then (L_{1}, L_{2}) R R is symmetric.
(iii) If L_{1} L_{2} and L_{2} L_{3} L_{1} L_{3} R is transitive.
Therefore, R is an equivalent relation.
Part II: All the lines related to the line and where is a real number.
15. Let R be the relation in the set {1, 2, 3, 4} given by R = {(1, 2), (2, 2), (1, 1), (4, 4), (1, 3), (3, 3), (3, 2)}. Choose the correct answer:
(A) R is reflexive and symmetric but not transitive.
(B) R is reflexive and transitive but not symmetric.
(C) R is symmetric and transitive but not reflexive.
(D) R is an equivalence relation.
Ans. Let R be the relation in the set {1, 2, 3, 4} is given by
R = {(1, 2), (2, 2), (1, 1), (4, 4), (1, 3), (3, 3), (3, 2)}
(a) (1, 1), (2, 2), (3, 3), (4, 4) R R is reflexive.
(b) (1, 2) R but (2, 1) R R is not symmetric.
(c) If (1, 3) R and (3, 2) R then (1, 2) R R is transitive.
Therefore, option (B) is correct.
16. Let R be the relation in the set N given by R = Choose the correct answer:
(A) (2, 4) R
(B) (3, 8) R
(C) (6, 8) R
(D) (8, 7) R
Ans. Given:
(A) , Here is not true, therefore, this option is incorrect.
(B) and 3 = 6, which is false.
Therefore, this option is incorrect.
(C) and 6 = 6, which is true.
Therefore, option (C) is correct.
(D) and 8 = 5, which is false.