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Practice the problems given in the worksheet on addition of matrices. If M and N are the two matrices of the same order, then the matrices are said conformable for addition, and their sum is obtained by adding the corresponding elements of M and N. 1. Find the sum of A and B
Continue reading "Worksheet on Addition of Matrices  Find the Sum of Two Matrices  Ans"
We will discuss about the properties of scalar multiplication of a matrix. If X and Y are two m × n matrices (matrices of the same order) and k, c and 1 are the numbers (scalars). Then the following results are obvious. I. k(A + B) = kA + kB II. (k + c)A = kA + cA III. k(cA)
Continue reading "Properties of Scalar Multiplication of a Matrix Scalar Multiplication"
The operation of multiplying variables by a constant scalar factor may properly be called scalar multiplication and the rule of multiplication of matrix by a scalar is that the product of an m × n matrix A = [aij] by a scalar quantity c is the m × n matrix [bij] where bij
Continue reading "Scalar Multiplication of a Matrix  Examples on Scalar Multiplication"
We proceed to develop the algebra of subtraction of matrices. Two matrices A and B are said to be conformable for subtraction if they have the same order (i.e. same number of rows and columns) and their difference A  B is defined to be the addition of A and (B).
Continue reading "Subtraction of Matrices  Examples on Difference of Two Matrices"
We will discuss about the properties of addition of matrices. 1. Commutative law of addition of matrix: Matrix multiplication is commutative. This says that, if A and B are matrices of the same order such that A + B is defined then A + B = B + A. Proof: Let A = [aij]m × n
Continue reading "Properties of Addition of Matrices  Commutative Law  Associative Law"
We proceed to develop the algebra of matrices. Two matrices A and B are said to be conformable for addition if they have the same order (same number of rows and columns). If A = (aij)m, n and B = (bij)m,n then their sum A + B is the matrix C = (cij)m,n where cij = aij + bij
Continue reading "Addition of Matrices  Example on Sum of Two Matrices"
There are two types of triangular matrices. 1. Upper Triangular Matrix: A square matrix (aij) is said to be an upper triangular matrix if all the elements below the principal diagonal are zero (0). That is, [aij]m × n is an upper triangular matrix if (i) m = n and (ii) aij
Continue reading "Triangular Matrix  Upper Triangular Matrix  Lower Triangular Matrix"
We will solve different types of problems on height and distance with two angles of elevation. Another type of case arises for two angles of elevations. In the given figure, let PQ be the height of pole of ‘y’ units. QR be the one of the distance between the foot of the pole
Continue reading "Height and Distance with Two Angles of Elevation  Solved Problems"
We have already learnt about trigonometry in previous units in detail. Trigonometry has its own applications in mathematics and in physics. One such application of trigonometry in mathematics is “height and distances”. To know about height and distances, we have to start
Continue reading "Angle of Elevation  How to Find out the Angle of Elevation"
A scalar matrix whose diagonal elements are all equal to 1, the identity element of the ground field F, is said to be an identity (or unit) matrix. The identity matrix of order n is denoted by In. A scalar matrix is said to be a unit matrix, if diagonal elements are unity.
Continue reading "Identity Matrix  Unit Matrix If [d] is a scalar matrix then [d] = dI"
Equality of two matrix: Two matrices [aij] and [bij] are said to be equal when they have the same number of rows and columns and aij = bij for all admissible values of i and j. Definition of Equal Matrices: Two matrices A and B are said to be equal if A and B have the same
Continue reading "Definition of Equal Matrices  Examples of Equal Matrices"
If each element of an m × n matrix be 0, the null element of F, the matrix is said to be the null matrix or the zero matrix of order m × n and it is denoted by Om,n. It is also denoted by O, when no confusion regarding its order arises. Null or zero Matrix: Whether A is a
Continue reading "Null Matrix  Null or Zero MatrixZero MatrixProblems on Null Matrix"
Here we will discuss about the column matrix with examples. In an m × n matrix, if n = 1, the matrix is said to be a column matrix. Definition of Column Matrix: If a matrix have only one column then it is called column matrix. Examples of column matrix:
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In an m × n matrix, if m = 1, the matrix is said to be a row matrix. Definition of Row Matrix: If a matrix have only one row then it is called row matrix. Here we will discuss about the row matrix with examples. Examples of row matrix:
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If square matrixes have n rows or columns then the matrix is called the square matrix of order n or an nsquare matrix. Definition of Square Matrix: An n × n matrix is said to be a square matrix of order n. In other words when the number of rows and the number of columns in
Continue reading "Square Matrix  Definition of Square Matrix Diagonal of Square Matrix"
A rectangular array of mn elements aij into m rows and n columns, where the elements aij belongs to field F, is said to be a matrix of order m × n (or an m × n matrix) over the field F. Definition of a Matrix: A matrix is a rectangular arrangement or array of numbers
Continue reading "Matrix  Definition of a Matrix  Examples of a Matrix  Elements"
One variable quantity is said to vary jointly as a number of other variable quantities, when it varies directly as their product. If the variable A varies directly as the product of the variables B, C and D, i.e., if.A ∝ BCD or A = kBCD (k = constant ), then A varies jointly
Continue reading "Joint Variation  Solving Joint Variation Problems and Application"
When two variables change in inverse proportion it is called as indirect variation. In indirect variation one variable is constant times inverse of other. If one variable increases other will decrease, if one decrease other will also increase. This means that the variables
Continue reading "Indirect Variation  Inverse Variation  Inverse or Indirect Variation"
When two variables change in proportion it is called as direct variation. In direct variation one variable is constant times of other. If one variable increases other will increase, if one decrease other will also decease. This means that the variables change in a same ratio
Continue reading "Direct Variation  Solving Direct Variation Word Problem"
There are different methods for solving simultaneous linear Equations: 1. Elimination of a variable 2. Substitution 3. Crossmultiplication 4. Evaluation of proportional value of variables This topic is purely based upon numerical examples. So, let us solve some examples
Continue reading "Methods of Solving Simultaneous Linear Equations  Solved Examples"
The next method of solving linear equations in two variables that we are going to learn about is method of cross multiplication. Let us see the steps followed while soling the linear equation by method of cross multiplication: Assume two linear equation be A1 x + B1y + C1=
Continue reading "Method of Cross MultiplicationSolve by Method of Cross Multiplication"
We will discuss about some of the properties of angles of a triangle. 1. The three angles of a triangle are together equal to two right angles. ABC is a triangle. Then ∠ZXY + ∠XYZ + ∠YZX = 180° Using this property, let us solve some of the examples. Solved examples
Continue reading "Properties of Angles of a Triangle Sum of Three Angles of a Triangle "
The three altitudes of triangle are concurrent. The point at which they intersect is known as the orthocentre of the triangle. In the adjoining figure, the three altitudes XP, YQ and ZR intersect at the orthocentre O.
Continue reading "Geometrical Property of AltitudesAltitudes of Triangle are Concurrent"
Here we will discuss about Medians and Altitudes of a Triangle. Median: The straight line joining a vertex of a triangle to the midpoint of the opposite side is called a median. A triangle has three medians. Here XL, YM and ZN are medians. A geometrical property of medians
Continue reading "Medians and Altitudes of a Triangle Three Altitudes and Three Medians"
Here we will discuss about classification of triangles on the basis of their sides and angles Equilateral triangle: An equilateral triangle is a triangle whose all three sides are equal. Here, XYZ is an equilateral triangle as XY = YZ = ZX. Isosceles triangle: An isosceles
Continue reading "Classification of Triangles on the Basis of Their Sides and Angles"
A triangle is a plane figure bounded by three straight lines. A triangle has three sides and three angles, and each one of them is called an element of the triangle. Here, PQR is a triangle, its three sides are line segments PQ, QR and RP; ; ∠PQR, ∠QRP and ∠RPQ are its
Continue reading "Triangle  Exterior Opposite AnglesInterior Opposite AnglesPerimeter"
We will solve different types of problems on change the subject of a formula. The subject of a formula is a variable whose relation with other variables of the context is sought and the formula is written in such a way that subject is expressed in terms of the other
Continue reading "Problem on Change the Subject of a Formula  Changing the Subject"
Practice the questions given in the worksheet on change of subject When a formula involving certain variables is known, we can change the subject of the formula. What is the subject in each of the following questions? Change the subject as indicated.
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Practice the questions given in the worksheet on framing a formula. I. Frame a formula for each of the following statements: 1. The side ‛s’ of a square is equal to the square root of its area A.
Continue reading "Worksheet on Framing a Formula  Framing Formulas  Frame an Equation"
We will discuss here about establishing an equation. In a given context, the relation between variables expressed by equality (or inequality) is called a formula. When a formula is expressed by an equality, the algebraic expression is called an equation.
Continue reading "Establishing an Equation  Framing a Formula  Framing Linear Equation"
What is Rectangular Cartesian Coordinates? Let O be a fixed point on the plane of this page; draw mutually perpendicular straight line XOX’ and YOY’ through O. Clearly, these lines divide the plane
Continue reading "Rectangular Cartesian Coordinates  Abscissa  Ordinate  Oblique Coordinate "
What is coordinate geometry? The subject coordinate geometry is that particular branch of mathematics in which geometry is studied with the help of algebra. This branch of mathematics was fir
Continue reading "What is Coordinate Geometry?  Analytical Geometry Cartesian Coordinate"
We will discuss here the method of using the table of tangents and cotangents. This table shown below is also known as the table of natural tangents and natural cotangents. Using the table we can
Continue reading "Table of Tangents and Cotangents  Natural Tangents and Natural Cotangents "
We will discuss here the method of using the table of sines and cosines: The above table is also known as the table of natural sines and natural cosines. Using the table we can find the values
Continue reading "Table of Sines and Cosines Trigonometric TableTable of Natural sines & cosines"
We will solve different types of problems on properties of triangle. 1. If in any triangle the angles be to one another as 1 : 2 : 3, prove that the corresponding sides are 1 : √3 : 2.
Continue reading "Problems on Properties of Triangle  Angle Properties of Triangles"
We will discuss the list of properties of triangle formulae which will help us to solve different types of problems on triangle.
Continue reading "Properties of Triangle Formulae  Triangle Formulae  Properties of Triangle"
We will discuss here about the law of tangents or the tangent rule which is required for solving the problems on triangle. In any triangle ABC,
Continue reading "Law of Tangents The Tangent RuleProof of the Law of TangentsAlternative Proof"
If ∆ be the area of a triangle ABC, Proved that, ∆ = ½ bc sin A = ½ ca sin B = ½ ab sin C That is, (i) ∆ = ½ bc sin A (ii) ∆ = ½ ca sin B (iii) ∆ = ½ ab sin C
Continue reading "Area of a Triangle  ∆ = ½ bc sin A  ∆ = ½ ca sin B  ∆ = ½ ab sin C"
We will discuss here about the law of cosines or the cosine rule which is required for solving the problems on triangle. In any triangle ABC, Prove that, (i) b\(^{2}\)
Continue reading "The Law of Cosines  The Cosine Rule  Cosine Rule Formula  Cosine Law Proof"
The geometrical interpretation of the proof of projection formulae is the length of any side of a triangle is equal to the algebraic sum of the projections of other sides upon it. In Any Triangle
Continue reading "Proof of Projection Formulae  Projection Formulae  Geometrical Interpretation "
Projection formulae is the length of any side of a triangle is equal to the sum of the projections of other two sides on it. In Any Triangle ABC, (i) a = b cos C + c cos B
Continue reading "Projection Formulae  a = b cos C + c cos B  b = c cos A + a cos C"
Proof the theorem on properties of triangle p/sin P = q/sin Q = r/sin R = 2K. Proof: Let O be the circumcentre and R the circumradius of any triangle PQR. Let O be the circumcentre and R
Continue reading "Theorem on Properties of Triangle  p/sin P = q/sin Q = r/sin R = 2K"
We will discuss here about the law of sines or the sine rule which is required for solving the problems on triangle. In any triangle the sides of a triangle are proportional to the sines
Continue reading "The Law of Sines  The Sine Rule  The Sine Rule Formula  Law of sines Proof"
In trigonometry we will discuss about the different properties of triangles. We know any triangle has six parts, the three sides and the three angles are generally called the elements of the triangle.
Continue reading "Properties of Triangles  Semiperimeter CircumcircleCircumradiusInradius "
We will solve different types of problems on inverse trigonometric function. 1. Find the values of sin (cos\(^{1}\) 3/5)
Continue reading "Problems on Inverse Trigonometric Function  Inverse Circular Function Problems"
We will learn how to find the principal values of inverse trigonometric functions in different types of problems. The principal value of sin\(^{1}\) x for x > 0, is the length of the arc of a unit
Continue reading "Principal Values of Inverse Trigonometric Functions Different types of Problems"
We will discuss the list of inverse trigonometric function formula which will help us to solve different types of inverse circular or inverse trigonometric function.
Continue reading "Inverse Trigonometric Function Formula  Inverse Circular Function Formula"
We will learn how to prove the property of the inverse trigonometric function 3 arctan(x) = arctan(\(\frac{3x  x^{3}}{1  3 x^{2}}\)) or, 3 tan\(^{1}\) x = tan\(^{1}\)
Continue reading "3 arctan(x)  3 tan\(^{1}\) x 3 tan inverse x  Inverse Trigonometric Function"
We will learn how to prove the property of the inverse trigonometric function 3 arccos(x) = arccos(4x\(^{3}\)  3x) or, 3 cos\(^{1}\) x = cos\(^{1}\) (4x\(^{3}\)  3x)
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