Summary
A rational number is a number of the form where p and q are integers and . Problems involving rational numbers are simplified using 'BODMAS' rule. A rational number can be represented in the decimal form. When a rational number is repr..
A rational number is a number of the form where p and q are integers and . Problems involving rational numbers are simplified using 'BODMAS' rule. A rational number can be represented in the decimal form. When a rational number is repr..Homogeneous Equations (Constant = 0)
Consider the homogeneous equations a 1 x + b 1 y + c 1 z = 0 a 2 x + b 2 y + c 2 z = 0 a 3 x + b 3 y + c 3 z = 0 The homogenous system of equations is always consistent because x = 0, y = 0, z = 0 satisfies all the equations in the system. This solution is called the trivial so..
Consider the homogeneous equations a 1 x + b 1 y + c 1 z = 0 a 2 x + b 2 y + c 2 z = 0 a 3 x + b 3 y + c 3 z = 0 The homogenous system of equations is always consistent because x = 0, y = 0, z = 0 satisfies all the equations in the system. This solution is called the trivial so..Question 2
Question: List the elements of the following sets and specify whether they are finite or infinite: Answer: (i) Since there is no natural number lying between 2 and 3, the set is a null set i.e. f . This is a finite set. (i) Since there are infinite number of..
Question: List the elements of the following sets and specify whether they are finite or infinite: Answer: (i) Since there is no natural number lying between 2 and 3, the set is a null set i.e. f . This is a finite set. (i) Since there are infinite number of..Remark:
The sum and product of two conjugate numbers is always real. 2a + i(0) 2a ..
The sum and product of two conjugate numbers is always real. 2a + i(0) 2a ..Some important results
Given a, b, c and d are non-zero real numbers, we can deduce other proportions by simple Algebra. These results are often referred by the names mentioned along each of the properties obtained. (1) If then bc = ad This property is known as INVERTENDO. (2) If , then ad = bc This p..
Given a, b, c and d are non-zero real numbers, we can deduce other proportions by simple Algebra. These results are often referred by the names mentioned along each of the properties obtained. (1) If then bc = ad This property is known as INVERTENDO. (2) If , then ad = bc This p..Suggested answer:
Note that we can also evaluate the determinant D 1 , D 2 and D 3 directly without using the properties of determinant. The solution of the system is given by It is important to mention here the consistency and inconsistency of a system of linear equations with three unknown..
Note that we can also evaluate the determinant D 1 , D 2 and D 3 directly without using the properties of determinant. The solution of the system is given by It is important to mention here the consistency and inconsistency of a system of linear equations with three unknown..Proof:
By property (1), we have Also we have = f(x) + g(x) (2) From (1) and (2), we have (4) For any real number k, By property (1) Also = kf(x) (ii) From (i) and (ii..
By property (1), we have Also we have = f(x) + g(x) (2) From (1) and (2), we have (4) For any real number k, By property (1) Also = kf(x) (ii) From (i) and (ii..Proof:
Draw AL, BM and CN perpendicular to x-axis. LM = x 2 -x 1 MN = x 3 -x 2 LN = x 3 -x 1 Area of D ABC = area of trap {ALMB + BMNC - ALNC} This can be expressed in the form of a determinant Another form which is convenient to use for the area of triangles but which is very much useful wh..
Draw AL, BM and CN perpendicular to x-axis. LM = x 2 -x 1 MN = x 3 -x 2 LN = x 3 -x 1 Area of D ABC = area of trap {ALMB + BMNC - ALNC} This can be expressed in the form of a determinant Another form which is convenient to use for the area of triangles but which is very much useful wh..Proof
Let A (x 1 , y 1 ) and B (x 2 , y 2 ) be two points in the plane. Let d = distance between the points A and B. Draw AL and BM perpendicular to x-axis (parallel to y-axis). Draw AC perpendicular to BM to cut BM at C. In the figure, OL = x 1 , OM = x 2 [AC = LM = OM - OL = x 2 - x 1 ] MB = y 2 , MC =..
Let A (x 1 , y 1 ) and B (x 2 , y 2 ) be two points in the plane. Let d = distance between the points A and B. Draw AL and BM perpendicular to x-axis (parallel to y-axis). Draw AC perpendicular to BM to cut BM at C. In the figure, OL = x 1 , OM = x 2 [AC = LM = OM - OL = x 2 - x 1 ] MB = y 2 , MC =..Proof:
Each circular permutation corresponds to n linear permutations depending on where we start. Since there are exactly n! linear permutations, there are exactly permutations. Hence, the number of circular permutations is the same as (n-1)..
Each circular permutation corresponds to n linear permutations depending on where we start. Since there are exactly n! linear permutations, there are exactly permutations. Hence, the number of circular permutations is the same as (n-1)..See what our Users say :
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