CLASS XII
MATHEMATICS
Units Weightage (Marks)
(i) Relations and Functions 10
(ii) Algebra (Matrices and Determinants) 13
(iii) Calculus 44
(iv) Vector and Three dimensional Geometry. 17
(v) Linear Programming 06
(vi) Probability 10
Total : 100
Design
Type of Questions Weightage of Number of Total Marks
each question questions
(i) Very short answer (VSA) 01 10 10
(ii) Short Answer (SA) 04 12 48
(iii) Long Answer (LA) 06 07 42
Internal Choice
There will be internal choice in 4 questions of short answer type and in 2 questions of Long answer
type.
NOTE
Questions requiring Higher Order thinking skills (HOTS) have been added in every chapter. Such
questions are marked with a star, and to help the students, hints to their solutions are given along with
the answers.
116 XII – Maths
, CHAPTER 1
RELATIONS AND FUNCTIONS
POINTS TO REMEMBER
1. Empty relation is the relation R in X given by R = φ ⊂ X × X.
2. Universal relation is the relation R in X given by R = X × X.
3. Reflexive relation R in X is a relation with (a, a) ∈ R, ∀ a ∈ X.
4. Symmetric relation R in X is a relation satisfying (a, b) ∈ R ⇒ (b, a) ∈ R.
5. Transitive relation R in X is a relation satisfying
(a, b) ∈ R and (b, c) ∈ R ⇒ (a, c) ∈ R.
6. Equivalence relation R in X is a relation which is reflexive, symmetric and transitive.
7. A function f = X → Y is one-one (or injective) if
f(x1) = f(x2) ⇒ x1 = x2, ∀ x1, x2 ∈ X
8. A function f : X → Y is onto (or surjective) if given any y ∈ Y, ∃ x ∈ X such that f(x) = y.
9. A function f : X → Y is called bijective if it is one-one and onto.
10. For f : A → B and g : B → C, the function gof : A → C is given by (gof) (x) = g[f(x)] ∀ x ∈ A.
11. A function f : X → Y is invertible if ∃ g : Y → X such that go f = Ix and fog = Iy.
12. A function f : X → Y is invertible if and only if f is one-one and onto.
13. A binary operation * on a set A is a function * : A × A → A.
14. An operation * on A is commutative if a * b = b * a, ∀ a, b ∈ A.
15. An operation * on A is associative if (a * b) * c = a * (b * c) ∀ a, b, c ∈ A.
16. An element e ∈ A, is the identity element for * : A × A → A if
a * e = a = e * a, ∀ a, ∈ A.
17. An element a ∈ A is invertible for * : A × A → A if there exists b ∈ A such that a * b = e = b
–1
* a, where e is the identity for *. The element b is called inverse of a and is denoted by a .
117 XII – Maths
, VERY SHORT ANSWER TYPE QUESTIONS
1. If A is the set of students of some boys school then write, which types of following relations are.
(Universal, Empty or neither of the two).
R1 = {(a, b) : a, b are ages of students and |a – b| ≥ 0}
R2 = {(a, b) : a, b are weights of students, and |a – b| < 0}
R3 = {(a, b) : a, b are weights of students and |a – b| > 0}
R4 = {(a, b) : a, b are students studying in same class}
R5 = {(a, b) : age of a is greater than age of b}
2. If A = {2, 3, 4, 5} then write whether each of the following relations on set A is a function or not?
Give reasons also.
(i) {(2, 3), (3, 4), (4, 5), (5, 2)}
(ii) {(2, 4), (3, 4), (5, 4), (4, 4)}
(iii) {(2, 3), (2, 4), (5, 4)}
(iv) {(2, 3), (3, 5), (4, 5)}
(v) {(2, 2), (2, 3), (4, 4), (4, 5)}
*3. If f : R → R, g : R → R defined by
3x – 7 8x + 7
f (x) = , g (x) = then
8 3
find (i) (fog) (7) =
(ii) (gof) (7) =
4. If f, g are the functions,
given by f = {(1, 2), (2, 3), (3, 7), (4, 6)}
g = {(0, 4), (1, 2), (2, 1)}
find fog.
x
5. If f (x ) = ∀ x ≠ –1
x +1
write (fof) (x)
6. If f : R → R defined by
2x − 1
f (x ) = (x ) = ?
–1
, find f
5
118 XII – Maths
, 7. Check the following functions for one-one. Also, give the reason for your answer.
(i) f : R → R s.t f(x) = x2 + 1 ∀ x ∈ R
(ii) f : R – {0} → R – {0} such that x · f(x) = 1
(iii) f : R → R such that f(x) = |x|.
(iv) f : R → R such that f(x) = x3.
(v) f : R → R such that f(x) = (x – 1) (x – 2) (x – 3)
(vi) f : R → R such that f(x) = [x] ∀ x ∈ R
where [ . ] denotes the greatest integer function.
(vii) f : R → R, f(x) = sin x ∀ x ∈ R
(viii) f : [0, π] → [–1, 1], f(x) = cos x ∀ x ∈ [ 0, π ]
π π
(ix) f : – , → R, f ( x ) = tan x .
2 2
8. Check whether the following functions are onto or not. Give one reason for your Answer.
−π π −1
(i) f : [ − 1, 1] → , , f ( x ) = sin x.
2 2
π π −1
(ii) f : R → − , , f ( x ) = tan x.
2 2
(iii) f : R → R, f (x) = x3
1
(iv) f : R – { 0} → R, f (x ) = .
x
(v) f : N → N, f(x) = x 3.
(vi) f : (0 ∞) → R, f(x) = x2.
x
(vii) f : R → { –1} → R , f (x ) = .
x +1
sin π [ x ]
(viii) f :R → [ – 1, 1] , f (x ) = 2
where
x +1
[ . ] denotes the greatest integer function.
9. If ‘*’ is a Binary operation defined on R then if
(i) a * b = a2 – b2, write 8 * (3 * 1)
ab
(ii) a*b = write ( 4 * 2 ) * 6
2
119 XII – Maths
MATHEMATICS
Units Weightage (Marks)
(i) Relations and Functions 10
(ii) Algebra (Matrices and Determinants) 13
(iii) Calculus 44
(iv) Vector and Three dimensional Geometry. 17
(v) Linear Programming 06
(vi) Probability 10
Total : 100
Design
Type of Questions Weightage of Number of Total Marks
each question questions
(i) Very short answer (VSA) 01 10 10
(ii) Short Answer (SA) 04 12 48
(iii) Long Answer (LA) 06 07 42
Internal Choice
There will be internal choice in 4 questions of short answer type and in 2 questions of Long answer
type.
NOTE
Questions requiring Higher Order thinking skills (HOTS) have been added in every chapter. Such
questions are marked with a star, and to help the students, hints to their solutions are given along with
the answers.
116 XII – Maths
, CHAPTER 1
RELATIONS AND FUNCTIONS
POINTS TO REMEMBER
1. Empty relation is the relation R in X given by R = φ ⊂ X × X.
2. Universal relation is the relation R in X given by R = X × X.
3. Reflexive relation R in X is a relation with (a, a) ∈ R, ∀ a ∈ X.
4. Symmetric relation R in X is a relation satisfying (a, b) ∈ R ⇒ (b, a) ∈ R.
5. Transitive relation R in X is a relation satisfying
(a, b) ∈ R and (b, c) ∈ R ⇒ (a, c) ∈ R.
6. Equivalence relation R in X is a relation which is reflexive, symmetric and transitive.
7. A function f = X → Y is one-one (or injective) if
f(x1) = f(x2) ⇒ x1 = x2, ∀ x1, x2 ∈ X
8. A function f : X → Y is onto (or surjective) if given any y ∈ Y, ∃ x ∈ X such that f(x) = y.
9. A function f : X → Y is called bijective if it is one-one and onto.
10. For f : A → B and g : B → C, the function gof : A → C is given by (gof) (x) = g[f(x)] ∀ x ∈ A.
11. A function f : X → Y is invertible if ∃ g : Y → X such that go f = Ix and fog = Iy.
12. A function f : X → Y is invertible if and only if f is one-one and onto.
13. A binary operation * on a set A is a function * : A × A → A.
14. An operation * on A is commutative if a * b = b * a, ∀ a, b ∈ A.
15. An operation * on A is associative if (a * b) * c = a * (b * c) ∀ a, b, c ∈ A.
16. An element e ∈ A, is the identity element for * : A × A → A if
a * e = a = e * a, ∀ a, ∈ A.
17. An element a ∈ A is invertible for * : A × A → A if there exists b ∈ A such that a * b = e = b
–1
* a, where e is the identity for *. The element b is called inverse of a and is denoted by a .
117 XII – Maths
, VERY SHORT ANSWER TYPE QUESTIONS
1. If A is the set of students of some boys school then write, which types of following relations are.
(Universal, Empty or neither of the two).
R1 = {(a, b) : a, b are ages of students and |a – b| ≥ 0}
R2 = {(a, b) : a, b are weights of students, and |a – b| < 0}
R3 = {(a, b) : a, b are weights of students and |a – b| > 0}
R4 = {(a, b) : a, b are students studying in same class}
R5 = {(a, b) : age of a is greater than age of b}
2. If A = {2, 3, 4, 5} then write whether each of the following relations on set A is a function or not?
Give reasons also.
(i) {(2, 3), (3, 4), (4, 5), (5, 2)}
(ii) {(2, 4), (3, 4), (5, 4), (4, 4)}
(iii) {(2, 3), (2, 4), (5, 4)}
(iv) {(2, 3), (3, 5), (4, 5)}
(v) {(2, 2), (2, 3), (4, 4), (4, 5)}
*3. If f : R → R, g : R → R defined by
3x – 7 8x + 7
f (x) = , g (x) = then
8 3
find (i) (fog) (7) =
(ii) (gof) (7) =
4. If f, g are the functions,
given by f = {(1, 2), (2, 3), (3, 7), (4, 6)}
g = {(0, 4), (1, 2), (2, 1)}
find fog.
x
5. If f (x ) = ∀ x ≠ –1
x +1
write (fof) (x)
6. If f : R → R defined by
2x − 1
f (x ) = (x ) = ?
–1
, find f
5
118 XII – Maths
, 7. Check the following functions for one-one. Also, give the reason for your answer.
(i) f : R → R s.t f(x) = x2 + 1 ∀ x ∈ R
(ii) f : R – {0} → R – {0} such that x · f(x) = 1
(iii) f : R → R such that f(x) = |x|.
(iv) f : R → R such that f(x) = x3.
(v) f : R → R such that f(x) = (x – 1) (x – 2) (x – 3)
(vi) f : R → R such that f(x) = [x] ∀ x ∈ R
where [ . ] denotes the greatest integer function.
(vii) f : R → R, f(x) = sin x ∀ x ∈ R
(viii) f : [0, π] → [–1, 1], f(x) = cos x ∀ x ∈ [ 0, π ]
π π
(ix) f : – , → R, f ( x ) = tan x .
2 2
8. Check whether the following functions are onto or not. Give one reason for your Answer.
−π π −1
(i) f : [ − 1, 1] → , , f ( x ) = sin x.
2 2
π π −1
(ii) f : R → − , , f ( x ) = tan x.
2 2
(iii) f : R → R, f (x) = x3
1
(iv) f : R – { 0} → R, f (x ) = .
x
(v) f : N → N, f(x) = x 3.
(vi) f : (0 ∞) → R, f(x) = x2.
x
(vii) f : R → { –1} → R , f (x ) = .
x +1
sin π [ x ]
(viii) f :R → [ – 1, 1] , f (x ) = 2
where
x +1
[ . ] denotes the greatest integer function.
9. If ‘*’ is a Binary operation defined on R then if
(i) a * b = a2 – b2, write 8 * (3 * 1)
ab
(ii) a*b = write ( 4 * 2 ) * 6
2
119 XII – Maths
