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Class 12 Maths Notes Chapter 9 Differential Equations

Differential Equation: An equation involving independent variable, dependent variable, derivatives of dependent variable with respect to independent variable and constant is called a differential equation.
e.g.

Ordinary Differential Equation: An equation involving derivatives of the dependent variable with respect to only one independent variable is called an ordinary differential equation.
e.g.

From any given relationship between the dependent and independent variables, a differential equation can be formed by differentiating it with respect to the independent variable and eliminating arbitrary constants involved.

Order of a Differential Equation: Order of a differential equation is defined as the order of the highest order derivative of the dependent variable with respect to the independent variable involved in the given differential equation.
Note: Order of the differential equation, cannot be more than the number of arbitrary constants in the equation.

Degree of a Differential Equation: The highest exponent of the highest order derivative is called the degree of a differential equation provided exponent of each derivative and the unknown variable appearing in the differential equation is a non-negative integer.
Note
(i) Order and degree (if defined) of a differential equation are always positive integers.
(ii) The differential equation is a polynomial equation in derivatives.
(iii) If the given differential equation is not a polynomial equation in its derivatives, then its degree is not defined.

Formation of a Differential Equation: To form a differential equation from a given relation, we use the following steps:
Step I: Write the given equation and see the number of arbitrary constants it has.
Step II: Differentiate the given equation with respect to the dependent variable n times, where n is the number of arbitrary constants in the given equation.
Step III: Eliminate all arbitrary constants from the equations formed after differentiating in step (II) and the given equation.
Step IV: The equation obtained without the arbitrary constants is the required differential equation.

Solution of the Differential Equation
A function of the form y = Φ(x) + C, which satisfies given differential equation, is called the solution of the differential equation.
General solution: The solution which contains as many arbitrary constants as the order of the differential equation, is called the general solution of the differential equation, i.e. if the solution of a differential equation of order n contains n arbitrary constants, then it is the general solution.

Particular solution: A solution obtained by giving particular values to arbitrary constants in the general solution of a differential equation, is called the particular solution.

Methods of Solving First Order and First Degree Differential Equation
Variable separable form: Suppose a differential equation is dydx = F(x, y). Here, we separate the variables and then integrate both sides to get the general solution, i.e. above equation may be written as dydx = h(x) . k(y)
Then, by separating the variables, we get dyk(y) = h(x) dx.
Now, integrate above equation and get the general solution as K(y) = H(x) + C
Here, K(y) and H(x) are the anti-derivatives of 1K(y) and h(x), respectively and C is the arbitrary constant.

Homogeneous differential equation: A differential equation dydx=f(x,y)g(x,y) is said to be homogeneous, if f(x, y) and g(x, y) are homogeneous functions of same degree, i.e. it may be written as

To check that given differential equation is homogeneous or not, we write differential equation as dydx = F(x, y) or dxdy = F(x, y) and replace x by λx, y by λy to write F(x, y) = λ F(x, y).
Here, if power of λ is zero, then differential equation is homogeneous, otherwise not.

Solution of homogeneous differential equation: To solve homogeneous differential equation, we put
y = vx
⇒ dydx = v + x dvdx
in Eq. (i) to reduce it into variable separable form. Then, solve it and lastly put v = yx to get required solution.

Note: If the homogeneous differential equation is in the form of dydx = F(x, y), where F(x, y) is homogeneous function of degree zero, then we make substitution xy = v, i.e. x = vy and we proceed further to find the general solution as mentioned above.

Linear differential equation: General form of linear differential equation is
dydx + Py = Q …(i)
where, P and Q are functions of x or constants.
or dxdy + P’x = Q’ …(ii)
where, P’ and Q’ are functions of y or constants.
Then, solution of Eq. (i) is given by the equation
y × IF = ∫(Q × IF) dx + C
where, IF = Integrating factor and IF = e^∫Pdx
Also, solution of Eq. (ii) is given by the equation
x × IF = ∫ (Q’ × IF) dy + C
where, IF = Integrating factor and IF = e^∫P’dy

Application of Integrals Notes Class 12 Maths Chapter 8

Area under Simple curves:
1. Let us find the area bounded by the curve y = f(x), x-axis, and the ordinates x = a and x – b. Consider the area under the curve as composed of a large number of thin vertical stripes.

Let there be an arbitrary strip of height y and width dx.
Area of elementary strip dA = y dx, where y = f(x).

Total area A of the region between x-axis, ordinates x – a, x = b and the curve y = f(x)
= sum of areas of elementary thin strips across the region PQML.
A = ∫_a^b dA = ∫_a^b ydx = ∫_a^b f(x) dx.

2. The area A of the region bounded by the curve x = g(y), y-axis, and the lines
y = c and y = d is given by
A = ∫_c^d x dy

3. If the curve under consideration lies below x-axis, then f(x) < 0 from x = a to x = b. So, the area bounded by the curve y = f(x) and the ordinates x = a, x = b and x-axis is negative. But the numerical value of the area is to be taken into consideration.
Then, area = |∫_a^b f(x)dx|.

4. It may also happen that some portion of the curve is above the x-axis and some portion is below the x-axis as shown in the figure. Let A₁ be the area below the x-axis and A₂ be the area above the x-axis. Therefore, area A bounded by the curve y = f(x), x-axis and the ordinates x = a and x = b is given by
A = |A₁| + A₂.

Area between two curves:
1. Let the two curves by y = f(x) and y = g(x), as shown in the figure. Suppose these curve intersect at x = a and x = b.
Consider the elementary strip of height y where y = f(x) – g(x), with width dx.
∴ dA = y dx.
⇒ A = ∫_a^b (f(x) – g(x))dx
= ∫_a^b f(x) dx – ∫_a^b g(x) dx.
= Area bounded by the curve y = f(x) – Area bounded by the curve y = g(x), where f(x) > g(x).

2. If the two curves y = f(x) and y = g(x) intersect at x-a,x – c and x = b such that a < c < b, then:

If f(x) > g(x) in [a, c] and f(x) < g(x) in [c, b], then the area of the regions bounded curve
= Area of the region PAQCP + Area of the region QDRBQ
= ∫_a^c f(x) – g(x)) dx + ∫_c^b (g(x) – f(x)) dx.

1. Area Under Simple Curves
(i) Area of the region bounded by the curve y = f (x), x-axis and the linesx = a and x = b(b > a) is given by the formula:
Area = ∫ba ydy = ∫ba f(x) dy.

2. Area of the region bounded by the curve x = g(x),  y-axis and the lines y = c,y = d is given by the formula:
Area = ∫dc xdy = ∫dc g(y) dy.

2. Area Between two Curves
(i) Area of the region enclosed between two curves y = f (x), y = g (x) and the lines x = a, x = b is
∫ba [f(x) -g(x)] dx, where f(x) ≥ g (x) in [a, b],

(ii) Iff (x) ≥ g (x) in [a, c] and f(x) ≤ g (x) in [c, b], a < c < b, then we write the area as:
Area = ∫ca [f(x) – g(x)] dx + ∫bc [g(x) – f(x)] dx.

Class 12 Mathematics Revision Notes Chapter 7 Integrals

Integration is the inverse process of differentiation. In the differential calculus, we are given a function and we have to find the derivative or differential of this function, but in the integral calculus, we are to find a function whose differential is given. Thus, integration is a process which is the inverse of differentiation.
Then, ∫f(x) dx = F(x) + C, these integrals are called indefinite integrals or general integrals. C is an arbitrary constant by varying which one gets different anti-derivatives of the given function.
Note: Derivative of a function is unique but a function can have infinite anti-derivatives or integrals.

Properties of Indefinite Integral
(i) ∫[f(x) + g(x)] dx = ∫f(x) dx + ∫g(x) dx
(ii) For any real number k, ∫k f(x) dx = k∫f(x)dx.
(iii) In general, if f₁, f₂,………, f_n are functions and k₁, k₂,…, k_n are real numbers, then
∫[k₁f₁(x) + k₂ f₂(x)+…+ k_nf_n(x)] dx = k₁ ∫f₁(x) dx + k₂ ∫ f₂(x) dx+…+ k_n ∫f_n(x) dx

Basic Formulae

Integration using Trigonometric Identities
When the integrand involves some trigonometric functions, we use the following identities to find the integral:

• 2 sin A . cos B = sin( A + B) + sin( A – B)
• 2 cos A . sin B = sin( A + B) – sin( A – B)
• 2 cos A . cos B = cos (A + B) + cos(A – B)
• 2 sin A . sin B = cos(A – B) – cos (A + B)
• 2 sin A cos A = sin 2A
• cos² A – sin² A = cos 2A
• sin² A = (1−cos2A2)
• sin² A + cos² A = 1
• sin3A=3sinA−sin3A4
• cos3A=3cosA+cos3A4

Integration by Substitutions
Substitution method is used, when a suitable substitution of variable leads to simplification of integral.
If I = ∫f(x)dx, then by putting x = g(z), we get
I = ∫ f[g(z)] g'(z) dz
Note: Try to substitute the variable whose derivative is present in the original integral and final integral must be written in terms of the original variable of integration.

Integration by Parts
For a given functions f(x) and q(x), we have
∫[f(x) q(x)] dx = f(x)∫g(x)dx – ∫{f'(x) ∫g(x)dx} dx
Here, we can choose the first function according to its position in ILATE, where
I = Inverse trigonometric function
L = Logarithmic function
A = Algebraic function
T = Trigonometric function
E = Exponential function
[the function which comes first in ILATE should taken as first junction and other as second function]

Note
(i) Keep in mind, ILATE is not a rule as all questions of integration by parts cannot be done by above method.
(ii) It is worth mentioning that integration by parts is not applicable to product of functions in all cases. For instance, the method does not work for ∫√x sinx dx. The reason is that there does not exist any function whose derivative is √x sinx.
(iii) Observe that while finding the integral of the second function, we did not add any constant of integration.

Integration by Partial Fractions
A rational function is ratio of two polynomials of the form p(x)q(x), where p(x) and q(x) are polynomials in x and q(x) ≠ 0. If degree of p(x) > degree of q(x), then we may divide p(x) by q(x) so that p(x)q(x)=t(x)+p1(x)q(x), where t(x) is a polynomial in x which can be integrated easily and degree of p1(x) is less than the degree of q(x) . p1(x)q(x) can be integrated by expressing p1(x)q(x) as the sum of partial fractions of the following type:


where x² + bx + c cannot be factorised further.

Integrals of the types  can be transformed into standard form by expressing 

Integrals of the types  can be transformed into standard form by expressing px + q = A ddx (ax² + bx + c) + B = A(2ax + b) + B, where A and B are determined by comparing coefficients on both sides.

Class 12 Maths Notes Chapter 6 Application of Derivatives

Rate of Change of Quantities: Let y = f(x) be a function of x. Then, dydx represents the rate of change of y with respect to x. Also, [latex s=1]\frac { dy }{ dx }[/latex]x = x0 represents the rate of change of y with respect to x at x = x₀.

If two variables x and y are varying with respect to another variable t, i.e. x = f(t) and y = g(t), then

In other words, the rate of change of y with respect to x can be calculated using the rate of change of y and that of x both with respect to t.
Note: dydx is positive, if y increases as x increases and it is negative, if y decreases as x increases, dx

Marginal Cost: Marginal cost represents the instantaneous rate of change of the total cost at any level of output.
If C(x) represents the cost function for x units produced, then marginal cost (MC) is given by

Marginal Revenue: Marginal revenue represents the rate of change of total revenue with respect to the number of items sold at an instant.
If R(x) is the revenue function for x units sold, then marginal revenue (MR) is given by

Let I be an open interval contained in the domain of a real valued function f. Then, f is said to be

• increasing on I, if x₁ < x₂ in I ⇒ f(x₁) ≤ f(x₂), ∀ x₁, x₂ ∈ I.
• strictly increasing on I, if x₁ < x₂ in I ⇒ f(x₁) < f(x₂), ∀ x₁, x₂ ∈ I.
• decreasing on I, if x₁ < x₂ in I ⇒ f(x₁) ≥ f(x₂), ∀ x₁, x₂ ∈ I.
• strictly decreasing on I, if x₁ < x₂ in f(x₁) > f(x₂), ∀ x₁, x₂ ∈ I.

Let x₀ be a point in the domain of definition of a real-valued function f, then f is said to be increasing, strictly increasing, decreasing or strictly decreasing at x₀, if there exists an open interval I containing x0 such that f is increasing, strictly increasing, decreasing or strictly decreasing, respectively in I.
Note: If for a given interval I ⊆ R, function f increase for some values in I and decrease for other values in I, then we say function is neither increasing nor decreasing.

Let f be continuous on [a, b] and differentiable on the open interval (a, b). Then,

• f is increasing in [a, b] if f'(x) > 0 for each x ∈ (a, b).
• f is decreasing in [a, b] if f'(x) < 0 for each x ∈ (a, b).
• f is a constant function in [a, b], if f'(x) = 0 for each x ∈ (a, b).

Note:
(i) f is strictly increasing in (a, b), if f'(x) > 0 for each x ∈ (a, b).
(ii) f is strictly decreasing in (a, b), if f'(x) < 0 for each x ∈ (a, b).

Monotonic Function: A function which is either increasing or decreasing in a given interval I, is called monotonic function.

Approximation: Let y = f(x) be any function of x. Let Δx be the small change in x and Δy be the corresponding change in y.
i.e. Δy = f(x + Δx) – f(x).Then, dy = f'(x) dx or dy = dydx Δx is a good approximation of Δy, when dx = Δx is relatively small and we denote it by dy ~ Δy.
Note:
(i) The differential of the dependent variable is not equal to the increment of the variable whereas the differential of the independent variable is equal to the increment of the variable.
(ii) Absolute Error The change Δx in x is called absolute error in x.

Tangents and Normals
Slope: (i) The slope of a tangent to the curve y = f(x) at the point (x₁, y₁) is given by

(ii) The slope of a normal to the curve y = f(x) at the point (x₁, y₁) is given by

Note: If a tangent line to the curve y = f(x) makes an angle θ with X-axis in the positive direction, then dydx = Slope of the tangent = tan θ. dx

Equations of Tangent and Normal
The equation of tangent to the curve y = f(x) at the point P(x₁, y₁) is given by
y – y₁ = m (x – x₁), where m = dydx at point (x₁, y₁).

The equation of normal to the curve y = f(x) at the point Q(x₁, y₁) is given by
y – y₁ = −1m (x – x₁), where m = dydx at point (x₁, y₁).

If slope of the tangent line is zero, then tanθ = θ, so θ = 0, which means that tangent line is parallel to the X-axis and then equation of tangent at the point (x₁, y₁) is y = y₁.

If θ → π2, then tanθ → ∞ which means that tangent line is perpendicular to the X-axis, i.e. parallel to the Y-axis and then equation of the tangent at the point (x₁, y₁) is x = x₀.

Maximum and Minimum Value: Let f be a function defined on an interval I. Then,
(i) f is said to have a maximum value in I, if there exists a point c in I such that
f(c) > f(x), ∀ x ∈ I. The number f(c) is called the maximum value of f in I and the point c is called a point of a maximum value of f in I.
(ii) f is said to have a minimum value in I, if there exists a point c in I such that f(c) < f(x), ∀ x ∈ I. The number f(c) is called the minimum value of f in I and the point c is called a point of minimum value of f in I.
(iii) f is said to have an extreme value in I, if there exists a point c in I such that f(c) is either a maximum value or a minimum value of f in I. The number f(c) is called an extreme value off in I and the point c is called an extreme point.

Local Maxima and Local Minima
(i) A function f(x) is said to have a local maximum value at point x = a, if there exists a neighbourhood (a – δ, a + δ) of a such that f(x) < f(a), ∀ x ∈ (a – δ, a + δ), x ≠ a. Here, f(a) is called the local maximum value of f(x) at the point x = a. (ii) A function f(x) is said to have a local minimum value at point x = a, if there exists a neighbourhood (a – δ, a + δ) of a such that f(x) > f(a), ∀ x ∈ (a – δ, a + δ), x ≠ a. Here, f(a) is called the local minimum value of f(x) at x = a.

The points at which a function changes its nature from decreasing to increasing or vice-versa are called turning points.
Note:
(i) Through the graphs, we can even find the maximum/minimum value of a function at a point at which it is not even differentiable.
(ii) Every monotonic function assumes its maximum/minimum value at the endpoints of the domain of definition of the function.

Every continuous function on a closed interval has a maximum and a minimum value.

Let f be a function defined on an open interval I. Suppose cel is any point. If f has local maxima or local minima at x = c, then either f'(c) = 0 or f is not differentiable at c.

Critical Point: A point c in the domain of a function f at which either f'(c) = 0 or f is not differentiable, is called a critical point of f.

First Derivative Test: Let f be a function defined on an open interval I and f be continuous of a critical point c in I. Then,

• if f'(x) changes sign from positive to negative as x increases through c, then c is a point of local maxima.
• if f'(x) changes sign from negative to positive as x increases through c, then c is a point of local minima.
• if f'(x) does not change sign as x increases through c, then c is neither a point of local maxima nor a point of local minima. Such a point is called a point of inflection.

Second Derivative Test: Let f(x) be a function defined on an interval I and c ∈ I. Let f be twice differentiable at c. Then,
(i) x = c is a point of local maxima, if f'(c) = 0 and f”(c) < 0. (ii) x = c is a point of local minima, if f'(c) = 0 and f”(c) > 0.
(iii) the test fails, if f'(c) = 0 and f”(c) = 0.

Note
(i) If the test fails, then we go back to the first derivative test and find whether a is a point of local maxima, local minima or a point of inflexion.
(ii) If we say that f is twice differentiable at o, then it means second order derivative exists at a.

Absolute Maximum Value: Let f(x) be a function defined in its domain say Z ⊂ R. Then, f(x) is said to have the maximum value at a point a ∈ Z, if f(x) ≤ f(a), ∀ x ∈ Z.

Absolute Minimum Value: Let f(x) be a function defined in its domain say Z ⊂ R. Then, f(x) is said to have the minimum value at a point a ∈ Z, if f(x) ≥ f(a), ∀ x ∈ Z.

Note: Every continuous function defined in a closed interval has a maximum or a minimum value which lies either at the end points or at the solution of f'(x) = 0 or at the point, where the function is not differentiable.

Let f be a continuous function on an interval I = [a, b]. Then, f has the absolute maximum value and/attains it at least once in I. Also, f has the absolute minimum value and attains it at least once in I.

Class 12 Maths Notes Chapter 5 Continuity and Differentiability

Continuity at a Point: A function f(x) is said to be continuous at a point x = a, if
Left hand limit of f(x) at(x = a) = Right hand limit of f(x) at (x = a) = Value of f(x) at (x = a)
i.e. if at x = a, LHL = RHL = f(a)
where, LHL = limx→a–f(x) and RHL = limx→a+f(x)
Note: To evaluate LHL of a function f(x) at (x = o), put x = a – h and to find RHL, put x = a + h.

Continuity in an Interval: A function y = f(x) is said to be continuous in an interval (a, b), where a < b if and only if f(x) is continuous at every point in that interval.

• Every identity function is continuous.
• Every constant function is continuous.
• Every polynomial function is continuous.
• Every rational function is continuous.
• All trigonometric functions are continuous in their domain.

Standard Results of Limits

Algebra of Continuous Functions
Suppose f and g are two real functions, continuous at real number c. Then,

• f + g is continuous at x = c.
• f – g is continuous at x = c.
• f.g is continuous at x = c.
• cf is continuous, where c is any constant.
• (fg) is continuous at x = c, [provide g(c) ≠ 0]

Suppose f and g are two real valued functions such that (fog) is defined at c. If g is continuous at c and f is continuous at g (c), then (fog) is continuous at c.

If f is continuous, then |f| is also continuous.

Differentiability: A function f(x) is said to be differentiable at a point x = a, if
Left hand derivative at (x = a) = Right hand derivative at (x = a)
i.e. LHD at (x = a) = RHD (at x = a), where Right hand derivative, where

Note: Every differentiable function is continuous but every continuous function is not differentiable.

Differentiation: The process of finding a derivative of a function is called differentiation.

Rules of Differentiation
Sum and Difference Rule: Let y = f(x) ± g(x).Then, by using sum and difference rule, it’s derivative is written as

Product Rule: Let y = f(x) g(x). Then, by using product rule, it’s derivative is written as

Quotient Rule: Let y = f(x)g(x); g(x) ≠ 0, then by using quotient rule, it’s derivative is written as

Chain Rule: Let y = f(u) and u = f(x), then by using chain rule, we may write

Logarithmic Differentiation: Let y = [f(x)]^g(x) ..(i)
So by taking log (to base e) we can write Eq. (i) as log y = g(x) log f(x). Then, by using chain rule

Differentiation of Functions in Parametric Form: A relation expressed between two variables x and y in the form x = f(t), y = g(t) is said to be parametric form with t as a parameter, when

(whenever dxdt≠0)
Note: dy/dx is expressed in terms of parameter only without directly involving the main variables x and y.

Second order Derivative: It is the derivative of the first order derivative.

Some Standard Derivatives

Rolle’s Theorem: Let f : [a, b] → R be continuous on [a, b] and differentiable on (a, b) such that f(a) = f(b), where a and b are some real numbers. Then, there exists at least one number c in (a, b) such that f'(c) = 0.

Mean Value Theorem: Let f : [a, b] → R be continuous function on [a, b]and differentiable on (a, b). Then, there exists at least one number c in (a, b) such that

Note: Mean value theorem is an expansion of Rolle’s theorem.

Some Useful Substitutions for Finding Derivatives Expression

Class 12 Maths Notes Chapter 4 Determinants

Determinant: Determinant is the numerical value of the square matrix. So, to every square matrix A = [a_ij] of order n, we can associate a number (real or complex) called determinant of the square matrix A. It is denoted by det A or |A|.
Note
(i) Read |A| as determinant A not absolute value of A.
(ii) Determinant gives numerical value but matrix do not give numerical value.
(iii) A determinant always has an equal number of rows and columns, i.e. only square matrix have determinants.

Value of a Determinant
Value of determinant of a matrix of order 2, A = \(\begin{bmatrix} { a }_{ 11 } & { a }_{ 12 } \\ { a }_{ 21 } & { a }_{ 22 } \end{bmatrix}\) is

Value of determinant of a matrix of order 3, A = \(\left[ \begin{matrix} { a }_{ 11 } & { a }_{ 12 } & { a }_{ 13 } \\ { a }_{ 21 } & { a }_{ 22 } & { a }_{ 23 } \\ { a }_{ 31 } & { a }_{ 32 } & { a }_{ 33 } \end{matrix} \right]\) is given by expressing it in terms of second order determinant. This is known as expansion of a determinant along a row (or column).

Note
(i) For easier calculations of determinant, we shall expand the determinant along that row or column which contains the maximum number of zeroes.
(ii) While expanding, instead of multiplying by (-1)^i+j, we can multiply by +1 or -1 according to as (i + j) is even or odd.

Let A be a matrix of order n and let |A| = x. Then, |kA| = kⁿ |A| = kⁿ x, where n = 1, 2, 3,…

Minor: Minor of an element ay of a determinant, is a determinant obtained by deleting the ith row and jth column in which element ay lies. Minor of an element a_ij is denoted by M_ij.
Note: Minor of an element of a determinant of order n(n ≥ 2) is a determinant of order (n – 1).

Cofactor: Cofactor of an element a_ij of a determinant, denoted by A_ij or C_ij is defined as A_ij = (-1)^i+j M_ij, where M_ij is a minor of an element a_ij.
Note
(i) For expanding the determinant, we can use minors and cofactors as

(ii) If elements of a row (or column) are multiplied with cofactors of any other row (or column), then their sum is zero.

Singular and non-singular Matrix: If the value of determinant corresponding to a square matrix is zero, then the matrix is said to be a singular matrix, otherwise it is non-singular matrix, i.e. for a square matrix A, if |A| ≠ 0, then it is said to be a non-singular matrix and of |A| = 0, then it is said to be a singular matrix.
Theorems
(i) If A and B are non-singular matrices of the same order, then AB and BA are also non-singular matrices of the same order.
(ii) The determinant of the product of matrices is equal to the product of their respective determinants, i.e. |AB| = |A||B|, where A and B are a square matrix of the same order.

Adjoint of a Matrix: The adjoint of a square matrix ‘A’ is the transpose of the matrix which obtained by cofactors of each element of a determinant corresponding to that given matrix. It is denoted by adj(A).
In general, adjoint of a matrix A = [a_ij]_n×n is a matrix [A_ji]_n×n, where A_ji is a cofactor of element a_ji.

Properties of Adjoint of a Matrix
If A is a square matrix of order n × n, then

• A(adj A) = (adj A)A = |A| I_n
• |adj A| = |A|^n-1
• adj (A^T) = (adj A)^T

The area of a triangle whose vertices are (x₁, y₁), (x₂, y₂) and (x₃, y₃) is given by

NOTE: Since the area is a positive quantity we always take the absolute value of the determinant.

Properties of Determinants
To find the value of the determinant, we try to make the maximum possible zero in a row (or a column) by using properties given below and then expand the determinant corresponding that row (or column).
Following are the various properties of determinants:
1. If all the elements of any row or column of a determinant are zero, then the value of a determinant is zero.

2. If each element of any one row or one column of a determinant is a multiple of scalar k, then the value of the determinant is a multiple of k. then the value of the determinant is a multiple of k. i.e.

3. If in a determinant any two rows or columns are interchanged, then the value of the determinant obtained is negative of the value of the given determinant. If we make n such changes of rows (columns) indeterminant ∆ and obtain determinant ∆ , then ∆₁ = (-1)ⁿ ∆.

4. If all corresponding elements of any two rows or columns of a determinant are identical or proportional, then the value of the determinant is zero.

[∴ R₁ and R₃ are identical.]

5. The value of a determinant remains unchanged on changing rows into columns and columns into rows. It follows that, if A is a square matrix, then |A’| = |A|.

Note: det(A) = det(A’), where A’ = transpose of A.

6. If some or all elements of a row or column of a determinant are expressed as a sum of two or more terms, then the determinant can be expressed as the sum of two or more determinants, i.e.

7. In the elements of any row or column of a determinant, if we add or subtract the multiples of corresponding elements of any other row or column, then the value of determinant remains unchanged, i.e.

In other words, the value of determinants remains the same, if we apply the operation R_i → R_i + kE_j or C_i → C_j → kC_j.

Inverse of a Matrix and Applications of Determinants and Matrix
1. Inverse of a Square Matrix: If A is a non-singular matrix (i.e. |A| ≠ 0), then

Note: Inverse of a matrix, if exists, is unique.

Properties of a Inverse Matrix

• (A^-1)^-1 = A
• (A^T)^-1=(A^-1)^T
• (AB)^-1 = B^-1A^-1
• (ABC)^-1 =C^-1B^-1A^-1
• adj (A^-1) = (adj A)^-1

2. Solution of system of linear equations using inverse of a matrix.
Let the given system of equations be a₁x + b₁y + c₁z = d₁; a₂x + b₂y + c₂z = d₂ and a₃x + b₃y + c₃z = d₃.
We write the following system of linear equations in matrix form as AX = B, where

Case I: If |A| ≠ 0, then the system is consistent and has a unique solution which is given by X = A^-1B.
Case II: If |A| = 0 and (adj A) B ≠ 0, then system is inconsistent and has no solution.
Case III: If |A| = 0 and (adj A) B = 0, then system may be either consistent or inconsistent according to as the system have either infinitely many solutions or no solutions

Class 12 Maths Notes Chapter 2 Inverse Trigonometric Functions

Inverse Trigonometric Functions: Trigonometric functions are many-one functions but we know that inverse of function exists if the function is bijective. If we restrict the domain of trigonometric functions, then these functions become bijective and the inverse of trigonometric functions are defined within the restricted domain. The inverse of f is denoted by ‘f^-1‘.
Let y = f(x) = sin x, then its inverse is x = sin^-1 y.

Domain and Range of Inverse Trigonometric Functions

sin^-1(sinθ) = θ; ∀ θ ∈ [−π2,π2]

cos^-1(cosθ) = θ; ∀ θ ∈ [0, π]

tan^-1(tanθ) = θ; ∀ θ [−π2,π2]

cosec^-1(cosecθ) = 0; ∀ θ ∈ [−π2,π2] , θ ≠ 0

sec^-1(secθ) = θ; ∀ θ ∈ [0, π], θ ≠ π2

cot^-1(cotθ) = θ; ∀ θ ∈ (0, π)

sin(sin^-1 x) = x, ∀ x ∈ [-1, 1]

cos(cos^-1 x) = x; ∀ x ∈ [-1, 1]

tan(tan^-1x) = x, ∀ x ∈ R

cosec(cosec^-1x) = x, ∀ x ∈ (-∞, -1] ∪ [1, ∞)

sec(sec^-1 x) = x, ∀ x ∈ (-∞, -1] ∪ [1, ∞)

cot(cot^-1 x) = x, ∀ x ∈ R

Note: sin^-1(sinθ) = θ ; sin^-1 x should not be confused with (sinx)^-1 = 1sinx or sin^-1 x = sin^-1(1x) land similarly for other trigonometric functions.

The value of an inverse trigonometric function, which lies in the range of principal value branch, is called the principal value of the inverse trigonometric function.
Note: Whenever no branch of an inverse trigonometric function is mentioned, it means we have to consider the principal value branch of that function.

Properties of Inverse Trigonometric Functions

Following substitutions are used to write inverse trigonometric functions in simplest form:

Remember Points
(i) Sometimes, it may happen, that some of the values of x that we find out does not satisfy the given equation.
(ii) While solving an equation, do not cancel the common factors from both sides.

Class 12 Maths Notes Chapter 1 Relations and Functions

Relation: A relation R from set X to a set Y is defined as a subset of the cartesian product X × Y. We can also write it as R ⊆ {(x, y) ∈ X × Y : xRy}.

Note: If n(A) = p and n(B) = q from set A to set B, then n(A × B) = pq and number of relations = 2^pq.

Types of Relation
Empty Relation: A relation R in a set X, is called an empty relation, if no element of X is related to any element of X,
i.e. R = Φ ⊂ X × X

Universal Relation: A relation R in a set X, is called universal relation, if each element of X is related to every element of X,
i.e. R = X × X

Reflexive Relation: A relation R defined on a set A is said to be reflexive, if
(x, x) ∈ R, ∀ x ∈ A or
xRx, ∀ x ∈ R

Symmetric Relation: A relation R defined on a set A is said to be symmetric, if
(x, y) ∈ R ⇒ (y, x) ∈ R, ∀ x, y ∈ A or
xRy ⇒ yRx, ∀ x, y ∈ R.

Transitive Relation: A relation R defined on a set A is said to be transitive, if
(x, y) ∈ R and (y, z) ∈ R ⇒ (x, z) ∈ R, ∀ x, y, z ∈ A
or xRy, yRz ⇒ xRz, ∀ x, y,z ∈ R.

Equivalence Relation: A relation R defined on a set A is said to be an equivalence relation if R is reflexive, symmetric and transitive.

Equivalence Classes: Given an arbitrary equivalence relation R in an arbitrary set X, R divides X into mutually disjoint subsets A, called partitions or sub-divisions of X satisfying

• all elements of A_i are related to each other, for all i.
• no element of A_i is related to any element of A_j, i ≠ j
• A_i ∪ A_j = X and A_i ∩ A_j = 0, i ≠ j. The subsets A_i and A_j are called equivalence classes.

Function: Let X and Y be two non-empty sets. A function or mapping f from X into Y written as f : X → Y is a rule by which each element x ∈ X is associated to a unique element y ∈ Y. Then, f is said to be a function from X to Y.
The elements of X are called the domain of f and the elements of Y are called the codomain of f. The image of the element of X is called the range of X which is a subset of Y.
Note: Every function is a relation but every relation is not a function.

Types of Functions
One-one Function or Injective Function: A function f : X → Y is said to be a one-one function, if the images of distinct elements of x under f are distinct, i.e. f(x₁) = f(x₂ ) ⇔ x₁ = x₂, ∀ x₁, x₂ ∈ X
A function which is not one-one, is known as many-one function.

Onto Function or Surjective Function: A function f : X → Y is said to be onto function or a surjective function, if every element of Y is image of some element of set X under f, i.e. for every y ∈ y, there exists an element X in x such that f(x) = y.
In other words, a function is called an onto function, if its range is equal to the codomain.

Bijective or One-one and Onto Function: A function f : X → Y is said to be a bijective function if it is both one-one and onto.

Composition of Functions: Let f : X → Y and g : Y → Z be two functions. Then, composition of functions f and g is a function from X to Z and is denoted by fog and given by (fog) (x) = f[g(x)], ∀ x ∈ X.
Note
(i) In general, fog(x) ≠ gof(x).
(ii) In general, gof is one-one implies that f is one-one and gof is onto implies that g is onto.
(iii) If f : X → Y, g : Y → Z and h : Z → S are functions, then ho(gof) = (hog)of.

Invertible Function: A function f : X → Y is said to be invertible, if there exists a function g : Y → X such that gof = I_x and fog = I_y. The function g is called inverse of function f and is denoted by f^-1.
Note
(i) To prove a function invertible, one should prove that, it is both one-one or onto, i.e. bijective.
(ii) If f : X → V and g : Y → Z are two invertible functions, then gof is also invertible with (gof)^-1 = f^-1og^-1

Domain and Range of Some Useful Functions

Binary Operation: A binary operation * on set X is a function * : X × X → X. It is denoted by a * b.

Commutative Binary Operation: A binary operation * on set X is said to be commutative, if a * b = b * a, ∀ a, b ∈ X.

Associative Binary Operation: A binary operation * on set X is said to be associative, if a * (b * c) = (a * b) * c, ∀ a, b, c ∈ X.
Note: For a binary operation, we can neglect the bracket in an associative property. But in the absence of associative property, we cannot neglect the bracket.

Identity Element: An element e ∈ X is said to be the identity element of a binary operation * on set X, if a * e = e * a = a, ∀ a ∈ X. Identity element is unique.
Note: Zero is an identity for the addition operation on R and one is an identity for the multiplication operation on R.

Invertible Element or Inverse: Let * : X × X → X be a binary operation and let e ∈ X be its identity element. An element a ∈ X is said to be invertible with respect to the operation *, if there exists an element b ∈ X such that a * b = b * a = e, ∀ b ∈ X. Element b is called inverse of element a and is denoted by a^-1.
Note: Inverse of an element, if it exists, is unique.

Operation Table: When the number of elements in a set is small, then we can express a binary operation on the set through a table, called the operation table.