#### Chapters

Chapter 2: Inverse Trigonometric Functions

Chapter 3: Matrices

Chapter 4: Determinants

Chapter 5: Continuity and Differentiability

Chapter 6: Application of Derivatives

Chapter 7: Integrals

Chapter 8: Application of Integrals

Chapter 9: Differential Equations

Chapter 10: Vector Algebra

Chapter 11: Three Dimensional Geometry

Chapter 12: Linear Programming

Chapter 13: Probability

## Chapter 1: Relations and Functions

#### NCERT solutions for Class 12 Mathematics Textbook Chapter 1 Relations and Functions [Pages 5 - 7]

Determine whether the following relations are reflective, symmetric and transitive:

Relation R in the set A = {1, 2, 3...13, 14} defined as R = {(x,y):3x - y = 0}

Determine whether each of the following relations are reflexive, symmetric and transitive:

Relation R in the set **N** of natural numbers defined as

R = {(*x*, *y*): *y* = *x* + 5 and *x* < 4}

Determine whether each of the following relations are reflexive, symmetric and transitive:

Relation R in the set *A* = {1, 2, 3, 4, 5, 6} as R = {(*x*, *y*): *y* is divisible by *x*}

Determine whether each of the following relations are reflexive, symmetric and transitive:

Relation R in the set **Z** of all integers defined as

R = {(*x*, *y*): *x* − *y* is as integer}

Relation R in the set *A* of human beings in a town at a particular time given by

(a) R = {(*x*, *y*): *x *and *y* work at the same place}

(b) R = {(*x*, *y*): *x* and *y* live in the same locality}

(c) R = {(*x*, *y*): *x *is exactly 7 cm taller than *y*}

(d) R = {(*x*, *y*): *x* is wife of *y*}

(e) R = {(*x*, *y*): *x* is father of* y*}

Show that the relation R in the set **R **of real numbers, defined as

R = {(*a*, *b*): *a* ≤ *b*^{2}} is neither reflexive nor symmetric nor transitive.

Check whether the relation R defined in the set {1, 2, 3, 4, 5, 6} as R = {(*a*, *b*): *b* = *a* + 1} is reflexive, symmetric or transitive.

Show that the relation R in **R** defined as R = {(*a*, *b*): *a* ≤ *b*}, is reflexive and transitive but not symmetric.

Check whether the relation R on **R** defined as R = {(*a*, *b*): *a* ≤ *b*^{3}} is reflexive, symmetric or transitive.

Show that the relation R in the set {1, 2, 3} given by R = {(1, 2), (2, 1)} is symmetric but neither reflexive nor transitive.

Show that the relation R in the set *A* of all the books in a library of a college, given by R = {(*x*, *y*): *x* and *y* have same number of pages} is an equivalence relation.

Show that the relation R in the set *A* = {1, 2, 3, 4, 5} given by `R = {(a,b) : |a-b| " is even"}` 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}.

Show that each of the relation R in the set A= {x in Z : 0 <= x <= 12} given by R = {(a,b):|a-b| is a multiple of 4}

is an equivalence relation. Find the set of all elements related to 1 in each case.

Show that each of the relation R in the set `A = {x =Z: 0 <= <= 12 }` given by `R = {(a,b) : a= b}

is an equivalence relation. Find the set of all elements related to 1 in each case.

Given an example of a relation. Which is Symmetric but neither reflexive nor transitive.

Given an example of a relation. Which is Transitive but neither reflexive nor symmetric.

Given an example of a relation. Which is Reflexive and symmetric but not transitive.

Given an example of a relation. Which is Reflexive and transitive but not symmetric.

Given an example of a relation. Which is Symmetric and transitive but not reflexive.

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 point related to a point P ≠ (0, 0) is the circle passing through P with origin as centre.

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, *T*_{2} 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?

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?

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 *y* = 2*x* + 4.

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.

Let R be the relation in the set **N **given by R = {(*a*, *b*): *a *= *b* − 2, *b *> 6}. Choose the correct answer.

(A) (2, 4) ∈ R

(B) (3, 8) ∈ R

(C) (6, 8) ∈ R

(D) (8, 7) ∈ R

#### NCERT solutions for Class 12 Mathematics Textbook Chapter 1 Relations and Functions [Pages 10 - 11]

Show that the function *f*: **R**_{*} → **R**_{*} defined by `f(x) = 1/x` is one-one and onto, where **R**_{*} is the set of all non-zero real numbers. Is the result true, if the domain **R**_{*} is replaced by **N** with co-domain being same as **R**?

Check the injectivity and surjectivity of the following functions:

*f*: **N** → **N** given by *f*(*x*) = *x*^{2}

Check the injectivity and surjectivity of the given functions: *f*: **Z** → **Z** given by *f*(*x*) = *x*^{2}

Check the injectivity and surjectivity of the given functions: *f*: **R** → **R** given by *f*(*x*) = *x*^{2}

Check the injectivity and surjectivity of the following functions: *f*: **N **→ **N** given by *f*(*x*) = *x*^{3}

Check the injectivity and surjectivity of the following functions: *f*: **Z** → **Z** given by *f*(*x*) = *x*^{3}

Prove that the Greatest Integer Function *f*: **R **→ **R** given by *f*(*x*) = [*x*], is neither one-once nor onto, where [*x*] denotes the greatest integer less than or equal to *x*.

Show that the Modulus Function *f*: **R **→ **R** given by `f(x) = |x|` is neither one-one nor onto, where `|x|` is *x*, if *x*is positive or 0 and |x| is − *x*, if *x* is negative.

Show that the Signum Function* f*: **R** → **R**, given by

Let *A* = {1, 2, 3}, *B *= {4, 5, 6, 7} and let *f* = {(1, 4), (2, 5), (3, 6)} be a function from *A* to *B*. Show that *f* is one-one.

In each of the following cases, state whether the function is one-one, onto or bijective. Justify your answer.

*f*: **R **→ **R** defined by *f*(*x*) = 3 − 4*x*

In each of the following cases, state whether the function is one-one, onto or bijective. Justify your answer.

*f*: **R **→ **R** defined by *f*(*x*) = 1 + *x*^{2}

Let *A* and *B* be sets. Show that *f*: *A* × *B* → *B *× *A* such that (*a*, *b*) = (*b*, *a*) is bijective function.

Let *f*: **N** → **N** be defined by f(n) = `{((n+1)/2, "if n is odd"),(," for all n ∈ N"), (n/2, if "n is even"):}`

State whether the function f is bijective. Justify your answer.

Let A = **R** − {3} and B = **R** − {1}. Consider the function *f*: A → B defined by `f(x) = ((x- 2)/(x -3))`. Is *f* one-one and onto? Justify your answer.

Let *f*: **R** → **R** be defined as* f*(*x*) = *x*^{4}. Choose the correct answer.

(A) *f* is one-one onto

(B) *f* is many-one onto

(C) *f* is one-one but not onto

(D) *f* is neither one-one nor onto

Let *f*: **R** → **R** be defined as *f*(*x*) = 3*x*. Choose the correct answer.

(A) *f* is one-one onto

(B) *f* is many-one onto

(C) *f* is one-one but not onto

(D) *f* is neither one-one nor onto

#### NCERT solutions for Class 12 Mathematics Textbook Chapter 1 Relations and Functions [Pages 18 - 19]

Let *f*: {1, 3, 4} → {1, 2, 5} and *g*: {1, 2, 5} → {1, 3} be given by *f *= {(1, 2), (3, 5), (4, 1)} and *g *= {(1, 3), (2, 3), (5, 1)}. Write down *g*o*f*.

Let *f*, *g* and *h* be functions from **R **to **R**. Show that

`(f + g)oh = foh + goh`

`(f.g)oh = (foh).(goh)`

Find gof and fog, if f(x) = |x| and g(x) = |5x - 2|

Find *g*o*f *and *f*o*g*, if `f(x) = 8x^3` and `g(x) = x^(1/3)`

if f(x) = `(4x + 3)/(6x - 4), x ≠ 2/3` show that* f*o*f*(*x*) = *x*, for all x ≠ 2/3 . What is the inverse of *f*?

State with reason whether following functions have inverse

*f*: {1, 2, 3, 4} → {10} with

*f* = {(1, 10), (2, 10), (3, 10), (4, 10)}

State with reason whether following functions have inverse *g*: {5, 6, 7, 8} → {1, 2, 3, 4} with *g* = {(5, 4), (6, 3), (7, 4), (8, 2)}

State with reason whether following functions have inverse *h*: {2, 3, 4, 5} → {7, 9, 11, 13} with *h* = {(2, 7), (3, 9), (4, 11), (5, 13)}

Show that *f*: [−1, 1] → **R**, given by f(x) = `x/(x + 2)` is one-one. Find the inverse of the function *f*: [−1, 1] → Range *f*.

(Hint: For y in Range f, y = `f(x) = x/(x +2)` for some x in [-1, 1] ie x = `2y/(1-y)`

Consider *f*: **R** → **R** given by *f*(*x*) = 4*x* + 3. Show that *f* is invertible. Find the inverse of *f*.

Consider *f*: **R**_{+ }→ [4, ∞) given by *f*(*x*) =* x*^{2} + 4. Show that *f* is invertible with the inverse *f*^{−1} of given *f *by `f^(-1) (y) = sqrt(y - 4)` where **R**_{+} is the set of all non-negative real numbers.

Consider *f*: **R**_{+} → [−5, ∞) given by *f*(*x*) = 9*x*^{2} + 6*x* − 5. Show that *f* is invertible with `f^(-1)(y) = ((sqrt(y +6) - 1)/3)`

Let *f*: *X* → *Y* be an invertible function. Show that* f *has unique inverse. (Hint: suppose *g*_{1} and *g*_{2} are two inverses of *f*. Then for all *y* ∈ *Y*, *f*o*g*_{1}(*y*) = I_{Y}(*y*) = *f*o*g*_{2}(*y*). Use one-one ness of *f*).

Consider *f*: {1, 2, 3} → {*a*, *b*, *c*} given by *f*(1) = *a*, *f*(2) = *b* and *f*(3) = *c*. Find *f*^{−1} and show that (*f*^{−1})^{−1} =* f*.

Let *f*: *X* → *Y* be an invertible function. Show that the inverse of *f*^{−1} is *f*, i.e., (*f*^{−1})^{−1} = *f*.

If* f*: **R **→ **R **be given by `f(x) = (3 - x^3)^(1/3)` , then *f*o*f*(*x*) is

(A) `1/(x^3)`

(B) *x*^{3}

(C) *x*

(D) (3 − *x*^{3})

Let `f:R - {-4/3} -> R` be a function defined as `f(x) = (4x)/(3x + 4)`. The inverse of *f* is map *g* Range `f -> R -{- 4/3}`

(A) `g(y) = (3y)/(3-4y)`

(B) `g(y) = (4y)/(4 - 3y)`

(C) `g(y) = (4y)/(3 - 4y)`

(D) `g(y) = (3y)/(4 - 3y)`

#### NCERT solutions for Class 12 Mathematics Textbook Chapter 1 Relations and Functions [Pages 24 - 26]

Determine whether or not of the definition of ∗ given below gives a binary operation. In the event that ∗ is not a binary operation, give justification for this.

On Z+, define ∗ by a ∗ b = a – b

Determine whether or not of the definition of given below gives a binary operation. In the event that * is not a binary operation, give justification for this.

On **Z**^{+}, define * by *a ** *b *= *ab*

Determine whether or not each of the definition of given below gives a binary operation. In the event that * is not a binary operation, give justification for this.

On **R**, define * by *a ** *b *= *ab*^{2}

Determine whether or not each of the definition of given below gives a binary operation. In the event that * is not a binary operation, give justification for this.

On **Z**^{+}, define * by *a ** *b *= |*a *− *b*|

Determine whether or not each of the definition of given below gives a binary operation. In the event that * is not a binary operation, give justification for this.

On **Z**^{+}, define * by *a ** *b *= *a*

For each binary operation * defined below, determine whether * is commutative or associative.

On **Z**, define *a ** *b *= *a *− *b*

For each binary operation * defined below, determine whether * is commutative or associative.

On **Q**, define *a ** *b *= *ab *+ 1

For each binary operation * defined below, determine whether * is commutative or associative.

On **Q**, define a * b = `(ab)/2`

For each binary operation * defined below, determine whether * is commutative or associative.

On **Z**^{+}, define *a ** *b *= 2^{ab}

For each binary operation * defined below, determine whether * is commutative or associative.

On **Z**^{+}, define *a ** *b *= *a*^{b}

For each binary operation * defined below, determine whether * is commutative or associative.

On **R **− {−1}, define `a*b = a/(b+1)`

Consider the binary operation ∨ on the set {1, 2, 3, 4, 5} defined by *a *∨*b *= min {*a*, *b*}. Write the operation table of the operation∨.

Consider a binary operation * on the set {1, 2, 3, 4, 5} given by the following multiplication table.

(i) Compute (2 * 3) * 4 and 2 * (3 * 4)

(ii) Is * commutative?

(iii) Compute (2 * 3) * (4 * 5).

(Hint: use the following table)

* | 1 | 2 | 3 | 4 | 5 |

1 | 1 | 1 | 1 | 1 | 1 |

2 | 1 | 2 | 1 | 2 | 1 |

3 | 1 | 1 | 3 | 1 | 1 |

4 | 1 | 2 | 1 | 4 | 1 |

5 | 1 | 1 | 1 | 1 | 5 |

Let*′ be the binary operation on the set {1, 2, 3, 4, 5} defined by *a **′ *b *= H.C.F. of *a *and *b*. Is the operation *′ same as the operation * defined in Exercise 4 above? Justify your answer.

Let * be the binary operation on **N **given by *a* * *b *= L.C.M. of *a *and *b*. Find

(i) 5 * 7, 20 * 16

(ii) Is * commutative?

(iii) Is * associative?

(iv) Find the identity of * in **N**

(v) Which elements of **N **are invertible for the operation *?

Is * defined on the set {1, 2, 3, 4, 5} by *a ** *b *= L.C.M. of *a *and *b *a binary operation? Justify your answer.

Let * be a binary operation on the set **Q **of rational numbers as follows:

(i) *a ** *b *= *a *− *b *

(ii) *a ** *b *= *a*^{2} + *b*^{2}

(iii) *a ** *b *= *a *+ *ab *

(iv) *a ** *b *= (*a *− *b*)^{2}

(v) a * b = ab/4

(vi) *a ** *b *= *ab*^{2}

Find which of the binary operations are commutative and which are associative.

Find which of the operations given above has identity.

Let A = **N **× **N **and * be the binary operation on A defined by (*a*, *b*) * (*c*, *d*) = (*a *+ *c*, *b *+ *d*)

Show that * is commutative and associative. Find the identity element for * on A, if any.

State whether the following statements are true or false. Justify.

For an arbitrary binary operation * on a set **N**, *a ** *a *= ∀ *a * *a ** **N**.

State whether the following statements are true or false. Justify.

If * is a commutative binary operation on **N**, then *a ** (*b ** *c*) = (*c ** *b*) * *a*

Consider a binary operation * on **N **defined as *a ** *b *= *a*^{3} + *b*^{3}. Choose the correct answer.

(A) Is * both associative and commutative?

(B) Is * commutative but not associative?

(C) Is * associative but not commutative?

(D) Is * neither commutative nor associative?

#### NCERT solutions for Class 12 Mathematics Textbook Chapter 1 Relations and Functions [Pages 29 - 31]

Let *f*: **R** → **R **be defined as *f*(*x*) = 10*x* + 7. Find the function *g*: **R** → **R** such that *g* o *f* = *f* o *g *= 1_{R}.

Let *f*: W → W be defined as *f*(*n*) = *n* − 1, if is odd and *f*(*n*) = *n* + 1, if *n* is even. Show that* f* is invertible. Find the inverse of *f*. Here, W is the set of all whole numbers.

If *f*: **R **→ **R** is defined by *f*(*x*) = *x*^{2} − 3*x *+ 2, find *f*(*f*(*x*)).

Show that function *f*: **R** → {*x* ∈ **R**: −1 < *x* < 1} defined by *f*(*x*) =`x/(1 + |x|)` , *x *∈**R** is one-one and onto function.

Show that the function *f*: **R** → **R** given by *f*(*x*) = *x*^{3} is injective.

Give examples of two functions *f*: **N** → **Z** and *g*: **Z** → **Z** such that *g* o *f* is injective but *g*is not injective.

(Hint: Consider *f*(*x*) = *x* and *g*(*x*) =|x|)

Given examples of two functions *f*: **N** → **N** and *g*: **N** → **N** such that *g*o*f* is onto but *f *is not onto.

(Hint: Consider *f*(*x*) = *x* + 1 and `g(x) = {(x-1, ifx >1),(1, if x = 1):}`

Given a non empty set *X*, consider P(*X*) which is the set of all subsets of *X*.

Define the relation R in P(*X*) as follows:

For subsets *A*, *B* in P(*X*), *A*R*B* if and only if *A* ⊂ *B*. Is R an equivalence relation on P(*X*)? Justify you answer:

Given a non-empty set *X*, consider the binary operation *: P(*X*) × P(*X*) → P(*X*) given by *A* * *B* = *A* ∩ *B* &mnForE; *A*, *B* in P(*X*) is the power set of *X*. Show that *X *is the identity element for this operation and *X *is the only invertible element in P(*X*) with respect to the operation*.

Find the number of all onto functions from the set {1, 2, 3, … , *n*) to itself.

Let *S* = {*a*, *b*, *c*} and *T* = {1, 2, 3}. Find F^{−1} of the following functions F from *S* to *T*, if it exists.

*F* = {(*a*, 3), (*b*, 2), (*c*, 1)}

Let *S* = {*a*, *b*, *c*} and *T* = {1, 2, 3}. Find F^{−1} of the following functions F from *S* to *T*, if it exists.

*F* = {(*a*, 2), (*b*, 1), (*c*, 1)}

Consider the binary operations*: **R **×**R **→ and o: **R** × **R** → **R **defined as a * b = |a - b| and *a*o *b* = *a*, &mnForE;*a*, *b* ∈ **R**. Show that * is commutative but not associative, o is associative but not commutative. Further, show that &mnForE;*a*, *b*, *c* ∈ **R**, *a**(*b* o *c*) = (*a** *b*) o (*a* * *c*). [If it is so, we say that the operation * distributes over the operation o]. Does o distribute over *? Justify your answer.

Given a non-empty set *X*, let *: P(*X*) × P(*X*) → P(*X*) be defined as *A* * *B* = (*A* − *B*) ∪ (*B* −*A*), &mnForE; *A*, *B* ∈ P(*X*). Show that the empty set *Φ* is the identity for the operation * and all the elements *A* of P(*X*) are invertible with *A*^{−1} = *A*. (Hint: (*A* − *Φ*) ∪ (*Φ* − *A*) = *A*and (*A* − *A*) ∪ (*A* − *A*) = *A* * *A* = *Φ*).

Define a binary operation *on the set {0, 1, 2, 3, 4, 5} as

a * b = `{(a+b, "if a+b < 6"), (a + b - 6, if a +b >= 6):}`

Show that zero is the identity for this operation and each element *a* ≠ 0 of the set is invertible with 6 − *a* being the inverse of *a*.

Let *A* = {−1, 0, 1, 2}, *B* = {−4, −2, 0, 2} and *f*, *g*: *A* → *B* be functions defined by *f*(*x*) = *x*^{2} − *x*, *x* ∈ A and g(x) = `2|x - 1/2|- 1, x in A`. Are *f* and *g* equal?

Justify your answer. (Hint: One may note that two function *f*: *A* → *B* and g: *A* → B such that *f*(*a*) = g(*a*) &mn For E;*a* ∈*A*, are called equal functions).

Let *A* = {1, 2, 3}. Then number of relations containing (1, 2) and (1, 3) which are reflexive and symmetric but not transitive is

(A) 1 (B) 2 (C) 3 (D) 4

Let *A* = {1, 2, 3}. Then number of equivalence relations containing (1, 2) is

(A) 1

(B) 2

(C) 3

(D) 4

Let *f*: **R** → **R** be the Signum Function defined as

f(x) = `{(1,x>0), (0, x =0),(-1, x< 0):}`

and *g*: **R** → **R **be the Greatest Integer Function given by *g*(*x*) = [*x*], where [*x*] is greatest integer less than or equal to *x*. Then does *f*o*g* and *g*o*f* coincide in (0, 1]?

Number of binary operations on the set {*a*, *b*} are

(A) 10

(B) 16

(C) 20

(D) 8

## Chapter 1: Relations and Functions

## NCERT solutions for Class 12 Mathematics Textbook chapter 1 - Relations and Functions

NCERT solutions for Class 12 Mathematics Textbook chapter 1 (Relations and Functions) include all questions with solution and detail explanation. This will clear students doubts about any question and improve application skills while preparing for board exams. The detailed, step-by-step solutions will help you understand the concepts better and clear your confusions, if any. Shaalaa.com has the CBSE Class 12 Mathematics Textbook solutions in a manner that help students grasp basic concepts better and faster.

Further, we at Shaalaa.com provide such solutions so that students can prepare for written exams. NCERT textbook solutions can be a core help for self-study and acts as a perfect self-help guidance for students.

Concepts covered in Class 12 Mathematics Textbook chapter 1 Relations and Functions are Composition of Functions and Invertible Function, Types of Functions, Types of Relations, Introduction of Relations and Functions, Concept of Binary Operations, Inverse of a Function.

Using NCERT Class 12 solutions Relations and Functions exercise by students are an easy way to prepare for the exams, as they involve solutions arranged chapter-wise also page wise. The questions involved in NCERT Solutions are important questions that can be asked in the final exam. Maximum students of CBSE Class 12 prefer NCERT Textbook Solutions to score more in exam.

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