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Class: | 12 |

Subject: | Maths – गणित |

Contents: | NCERT Solutions |

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Table of Contents

- 1 NCERT Solutions for Class 12 Maths
- 1.1 Main points to be recovered
- 1.1.1 Relations and Functions
- 1.1.2 Inverse Trigonometric Functions
- 1.1.3 Matrices
- 1.1.4 Determinants
- 1.1.5 Continuity and Differentiability
- 1.1.6 Applications of Derivatives
- 1.1.7 Integrals
- 1.1.8 Applications of the Integrals
- 1.1.9 Differential Equations
- 1.1.10 Vectors
- 1.1.11 Three – dimensional Geometry
- 1.1.12 Linear Programming
- 1.1.13 Probability

- 1.1 Main points to be recovered
- 2 Related Pages
- 2.0.1 मान लीजिए कि N में *, a * b = a तथा b का H.C.F. द्वारा परिभाषित द्विआधारी संक्रिया है। क्या * क्रमविनिमय है? क्या * साहचर्य है? क्या N में इस द्विआधारी संक्रिया के तत्समक का अस्तित्व है?
- 2.0.2 Show that the function f: R → R given by f(x) = x^3 is injective.
- 2.0.3 निम्नलिखित को सिद्ध कीजिए: 3〖sin〗^(-1) x=〖sin〗^(-1) (3x-4x^3 ), x∈[-1/2,1/2].
- 2.0.4 If A, B are symmetric matrices of same order, then what type of matrix AB – BA is?
- 2.0.5 Which type of matrices have determinants?
- 2.0.6 फलन f(x)=x^2+2x-8,x∈[-4,2] के लिए रोले के प्रमेय को सत्यापित कीजिए।
- 2.0.7 Find two positive numbers whose sum is 16 and the sum of whose cubes is minimum.
- 2.0.8 Find the point on the curve y = x3 – 11x + 5 at which the tangent is y = x – 11.
- 2.0.9 What is integration?
- 2.0.10 Fill in the blank: The area of the region bounded by the curve x = y2, y-axis and the line y = 3 and y = 4 is _______.
- 2.0.11 What is different between a general solutions and a particular solutions.
- 2.0.12 How do you find the dot product of two vectors a and b?
- 2.0.13 Find the Equation of a plane which is at a distance p from the origin with direction cosines of the normal to the plane as l, m, n.
- 2.0.14 What do you understand by LPP?
- 2.0.15 What are independent events?

## NCERT Solutions for Class 12 Maths

**Chapter 1 Relations and Functions****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**

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*is very good to get proper practice. Just like R D Sharma, All in One for Mathematics is also very good book with ample practice questions.*

**Together With Maths***( 15 Sample Question Papers ) is very useful after going through whole syllabus. Buy books according to your need. Download NCERT Solutions class 12 Maths PDF. Take*

**CBSE Mathematics****admission in NIOS**to pass class 12 in easier way.

### Main points to be recovered

**Relations and Functions**

1. *Relations and Functions* – summary: Types of relations: reflexive, symmetric, transitive and equivalence relations. One to one and onto functions, composite functions, inverse of a function. Binary operations.

**Inverse Trigonometric Functions**

2 *Inverse Trigonometric Functions* – summary: Definition, range, domain, principal value branch. Graphs of inverse trigonometric functions. Elementary properties of inverse trigonometric functions.

**Matrices**

3 *Matrices* – summary: Concept, notation, order, equality, types of matrices, zero and identity matrix, transpose of a matrix, symmetric and skew symmetric matrices. Operation on matrices: Addition and multiplication and multiplication with a scalar. Simple properties of addition, multiplication and scalar multiplication. Noncommutativity of multiplication of matrices and existence of non-zero matrices whose product is the zero matrix (restrict to square matrices of order 2).Concept of elementary row and column operations. Invertible matrices and proof of the uniqueness of inverse, if it exists; (Here all matrices will have real entries).

**Determinants**

4 *Determinants* – summary: Determinant of a square matrix (up to 3 x 3 matrices), properties of determinants, minors, co-factors and applications of determinants in finding the area of a triangle. Adjoint and inverse of a square matrix. Consistency, inconsistency and number of solutions of system of linear equations by examples, solving system of linear equations in two or three variables (having unique solution) using inverse of a matrix.

**Continuity and Differentiability**

5 *Continuity and Differentiability* – summary: Continuity and differentiability, derivative of composite functions, chain rule, derivatives of inverse trigonometric functions, derivative of implicit functions. Concept of exponential and logarithmic functions. Derivatives of logarithmic and exponential functions. Logarithmic differentiation, derivative of functions expressed in parametric forms. Second order derivatives. Rolle’s and Lagrange’s Mean Value Theorems (without proof) and their geometric interpretation.

**Applications of Derivatives**

6 *Applications of Derivatives* – summary: rate of change of bodies, increasing/decreasing functions, tangents and normals, use of derivatives in approximation, maxima and minima (first derivative test motivated geometrically and second derivative test given as a provable tool). Simple problems (that illustrate basic principles and understanding of the subject as well as real-life situations).

**Integrals**

7 *Integrals* – summary: Integration as inverse process of differentiation.Integration of a variety of functions by substitution, by partial fractions and by parts, Evaluation of simple integrals of the types given in the syllabus and problems based on them. Definite integrals as a limit of a sum, Fundamental Theorem of Calculus (without proof).Basic properties of definite integrals and evaluation of definite integrals.

**Applications of the Integrals**

8 *Applications of the Integrals* – summary: Applications in finding the area under simple curves, especially lines, circles/parabolas/ellipses (in standard form only), Area between any of the two above said curves (the region should be clearly identifiable).

**Differential Equations**

9 *Differential Equations* – summary: Definition, order and degree, general and particular solutions of a differential equation.Formation of differential equation whose general solution is given.Solution of differential equations by method of separation of variables solutions of homogeneous differential equations of first order and first degree. Solutions of linear differential equation of the type given in the syllabus.

**Vectors**

10 *Vectors* – summary: Vectors and scalars, magnitude and direction of a vector.Direction cosines and direction ratios of a vector. Types of vectors (equal, unit, zero, parallel and collinear vectors), position vector of a point, negative of a vector, components of a vector, addition of vectors, multiplication of a vector by a scalar, position vector of a point dividing a line segment in a given ratio. Definition, Geometrical Interpretation, properties and application of scalar (dot) product of vectors, vector (cross) product of vectors, scalar triple product of vectors.

**Three – dimensional Geometry**

11 *Three – Dimensional Geometry* – summary: Direction cosines and direction ratios of a line joining two points.Cartesian equation and vector equation of a line, coplanar and skew lines, shortest distance between two lines.Cartesian and vector equation of a plane.Angle between (i) two lines, (ii) two planes, (iii) a line and a plane.Distance of a point from a plane.

**Linear Programming**

12 *Linear Programming* – summary: Introduction, related terminology such as constraints, objective function, optimization, different types of linear programming (L.P.) problems, mathematical formulation of L.P. problems, graphical method of solution for problems in two variables, feasible and infeasible regions(bounded and unbounded), feasible and infeasible solutions, optimal feasible solutions (up to three non-trivial constraints).

**Probability**

13 *Probability* – summary: Conditional probability, multiplication theorem on probability, independent events, total probability, Bayes’ theorem, Random variable and its probability distribution, mean and variance of random variable. Repeated independent (Bernoulli) trials and Binomial distribution.

Also see the complete solutions for NCERT (All Classes and Subjects).

- NCERT Solutions for class 12 Biology
- NCERT Solutions for class 12 Physics
- NCERT Solutions for class 12 Chemistry

Table of Contents

- 1 मान लीजिए कि N में *, a * b = a तथा b का H.C.F. द्वारा परिभाषित द्विआधारी संक्रिया है। क्या * क्रमविनिमय है? क्या * साहचर्य है? क्या N में इस द्विआधारी संक्रिया के तत्समक का अस्तित्व है?
- 2 Show that the function f: R → R given by f(x) = x^3 is injective.
- 3 निम्नलिखित को सिद्ध कीजिए: 3〖sin〗^(-1) x=〖sin〗^(-1) (3x-4x^3 ), x∈[-1/2,1/2].
- 4 If A, B are symmetric matrices of same order, then what type of matrix AB – BA is?
- 5 Which type of matrices have determinants?
- 6 फलन f(x)=x^2+2x-8,x∈[-4,2] के लिए रोले के प्रमेय को सत्यापित कीजिए।
- 7 Find two positive numbers whose sum is 16 and the sum of whose cubes is minimum.
- 8 Find the point on the curve y = x3 – 11x + 5 at which the tangent is y = x – 11.
- 9 What is integration?
- 10 Fill in the blank: The area of the region bounded by the curve x = y2, y-axis and the line y = 3 and y = 4 is _______.
- 11 What is different between a general solutions and a particular solutions.
- 12 How do you find the dot product of two vectors a and b?
- 13 Find the Equation of a plane which is at a distance p from the origin with direction cosines of the normal to the plane as l, m, n.
- 14 What do you understand by LPP?
- 15 What are independent events?

#### मान लीजिए कि N में *, a * b = a तथा b का H.C.F. द्वारा परिभाषित द्विआधारी संक्रिया है। क्या * क्रमविनिमय है? क्या * साहचर्य है? क्या N में इस द्विआधारी संक्रिया के तत्समक का अस्तित्व है?

हम जानते हैं कि a और b का H.C.F. = b और a का H.C.F., सभी a, b ∈ N के लिए

∴ a * b = b * a, अतः, संक्रिया * क्रमविनिमय है।

सभी a, b, c ∈ N के लिए, (a * b)* c = (a और b का H.C.F.) * c = a, b और c का H.C.F.

तथा a *(b * c) = a *(b और c का H.C.F.) = a, b और c का H.C.F.

∴ (a * b) * c = a * (b * c), अतः, संक्रिया * साहचर्य है।

अब, कोई अवयव e ∈ N, संक्रिया * में तत्समक होगा यदि a * e = a = e* a, सभी a ∈ N के लिए

लेकिन ये संबंध किसी भी a ∈ N के लिए सत्य नहीं है।

अतः, N में इस द्विआधारी संक्रिया * के तत्समक का अस्तित्व नहीं है।

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

For one – one

Suppose f(x) = f(y), where x, y ∈ R.

⇒ x^3 = y^3 … (1)

Now, we need to show that x = y.

Suppose x ≠ y, their cubes will also not be equal.

⇒ x^3 ≠ y^3

However, this will be a contradiction to (1).

∴ x = y

Hence, f is injective.

#### निम्नलिखित को सिद्ध कीजिए: 3〖sin〗^(-1) x=〖sin〗^(-1) (3x-4x^3 ), x∈[-1/2,1/2].

RHS = 〖sin〗^(-1) (3x-4x^3 )

=〖sin〗^(-1) (3 sinθ-4〖sin〗^3 θ)

=〖sin〗^(-1) (sin3θ )

= 3θ

= 3〖sin〗^(-1) x

= LHS

#### If A, B are symmetric matrices of same order, then what type of matrix AB – BA is?

⇒(𝐴𝐵 − 𝐵𝐴)′

=−(𝐴𝐵 − 𝐵𝐴),

Therefore,the matrix (AB − BA) is a skew symmetric matrix.

#### Which type of matrices have determinants?

#### फलन f(x)=x^2+2x-8,x∈[-4,2] के लिए रोले के प्रमेय को सत्यापित कीजिए।

(ii) f'(x)=2x+2

अतः, फलन f विवृत अंतराल (-4,2) में अवकलनीय है।

(iii) f(-4)=(-4)^2+2(-4)-8=16-8-8=0

तथा f(2)=(2)^2+2(2)-8=4+4-8=0

⇒f(-4)=f(2)

यहाँ, रोले की तीनों परिस्थितियाँ सत्य हैं। इसलिए, विवृत अंतराल (-4,2) में किसी ऐसे c का अस्तित्व है कि f'(c)=0 है।

⇒f^’ (c)=2c+2=0

⇒c=-1∈(-4,2)

अतः, फलन f(x)=x^2+2x-8,x∈[-4,2] के लिए रोले की प्रमेय सत्यापित हो जाती है।

#### Find two positive numbers whose sum is 16 and the sum of whose cubes is minimum.

Let the sum of the cubes of these numbers be denoted by S(x). Then,

⇒S(x)=x^3+(16-x)^3

∴〖 S〗^’ (x)=3x^2-3(16-x)^2, S^” (x)=6x+6(16-x)

Now, S^’ (x)=0

⇒ 3x^2-3(16-x)^2 = 0

⇒ x^2-(16-x)^2=0

⇒ x^2-256-x^2+32x = 0

⇒ x = 256/32 = 8

Now S^” (8)=6(8)+6(16-8)=48+48=96>0

∴ By second derivative test, x = 8 is the point of local minima of S.

Hence, the sum of the cubes of the numbers is the minimum when the numbers are 8 and 16 − 8 = 8.

#### Find the point on the curve y = x3 – 11x + 5 at which the tangent is y = x – 11.

Slope of the line y=x-11 is 1.

Given that:

The slope of tangent to the curve y=x^3-11x+5 = the slope of the line y=x-11

⇒2x^2-11=1

⇒3x^2=12

⇒x^2=4

⇒x=±2

If x=2, then y=(2)^3-11(2)+5=-9, therefore, the point = (2,-9)

If x=-2, then y=(-2)^3-11(-2)+5=18, therefore, the point = (-2,19)

Out of the points (2,-9) and (-2,19), only (2,-9) satisfy the equation of line y=x-11.

Hence, (2,-9) is the point on y=x^3-11x+5 at which tangent is y=x-11.

#### What is integration?

process of finding anti derivatives is called integration.

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