{ 0 comments }
Property 1 A number having 2, 3, 7 or 8 at unit’s place is never a perfect square. In otherwords, no square number ends in 2, 3, 7 or 8.
Verification: On page 3.1-3.2, the table gives us the squares of first 30 natural numbers.
If we have a glance at the unit’s place of these squares, we observe that they have 0, 1, 4,5,6 or 9 at unit’s place. Thus, the square of a number cannot have 2, 3, 7 or 8 at unit’s place.
Application: Check the digit at unit’s place of the given number, If it is one of the digits 2,3, 7 and 8, then the given number is not a perfect square.
None of the numbers 152, 7693, 14357, 88888, 798328 is a perfect square.
Remark 1 The phrase “number ends in a” means that the unit’s digit of the number is a.
Remark 2 It should be noted that a number having unit’s digit other than the digits 2, 3, 7 and 8 is not necessarily a perfect square. It may or may not be a perfect square. For example, 71, 124, 1500 etc. are not perfect squares.
{ 0 comments }
Sometimes it happens that during rainy season, you carry a raincoat every day and it does not rain for many days. However, by chance, one day you forget to take the raincoat and it rains heavily on that day.
Sometimes it so happens that a student prepares 4 chapters out of 5, very well for a test. But a major question is asked from the chapter that she left unprepared.
Everyone knows that a particular train runs in time but the day you reach
well in time it is late!
You face a lot of situations such as these where you take a chance and it
does not go the way you want it to. Can you give some more examples? These
are examples where the chances of a certain thing happening or not happening are not equal. The chances of the train being in time or being late are not the same. When you buy a ticket which is wait listed, you do take a chance. You hope that it might get confirmed by the time you travel. We however, consider here certain experiments whose results have an equal chance of occurring.
{ 0 comments }
The data regarding choice of subjects showed the occurrence of each of the entries several times. For example, Art is liked by 7 students, Mathematics is liked by 5 students and. This information can be displayed graphically using a pictograph or a bargraph. Sometimes, however, we have to deal with a large data. For example, consider the following marks (out of 50)
If we make a frequency distribution table for each observation, then the table would be too long, so, for convenience, we make groups of observations say, 0-10, 10-20 and so on, and obtain a frequency distribution of the number of observations falling in each group. Thus, the frequency distribution table for the above data can be
Data presented in this manner is said to be grouped and the distribution obtained is called grouped frequency distribution. It helps us to draw meaningful inferences like –
(1) Most of the students have scored between 20 and 40. (2) Eight students have scored more than 40 marks out of 50 and so on. Each of the groups 0-10, 10-20, 20-30, etc., is called a Class Interval (or briefly a class).
Observe that 10 occurs in both the classes, i.e., 0-10 as well as 10-20. Similarly, 20 occurs in classes 10-20 and 20-30. But it is not possible that an observation (say 10 or 20) can belong simultaneously to two classes. To avoid this, we adopt the convention that the common observation will belong to the higher class, i.e., 10 belongs to the class interval 10-20 (and not to 0-10). Similarly, 20 belongs to 20-30 (and not to 10-20). In the class interval, 10-20, 10 is called the lower class limit and 20 is called the upper class limit. Similarly, in the class interval 20-30, 20 is the lower class limit and 30 is the upper class limit. Observe that the difference between the upper class limit and lower class limit for each of the class intervals 0-10, 10-20, 20-30 etc., is equal, (10 in this case). This difference between the upper class limit and lower class limit is called the width or size of the class interval.
This is displayed graphically as in the adjoining graph
{ 0 comments }
Organising Data
data available to us is in an unorganised form called raw data.To draw meaningful inferences, we need to organise the data systematically. For example, a group of students was asked for their favourite subject. The results were as listed below:
Art, Mathematics, Science, English, Mathematics, Art, English, Mathematics, English, Art, Science, Art, Science, Science, Mathematics, Art, English, Art, Science, Mathematics, Science,Art.
Which is the most liked subject and the one least liked?
It is not easy to answer the question looking at the choices written haphazardly.We arrange the data in Table 5.1 using tally marks.
The number of tallies before each subject gives the number of students who like that particular subject.
This is known as the frequency of that subject.
Frequency gives the number of times that a particular entry occurs.
Frequency of students who like English is 4
Frequency of students who like Mathematics is 5
The table made is known as frequency distribution table as it gives the number of times an entry occurs.
{ 0 comments }
Take a pair of sticks of equal lengths, say 10 cm. Take another pair of sticks of equal lengths, say, 8 cm. Hinge them up suitably to get a rectangle of length 10 cm and breadth 8 cm.
This rectangle has been created with the 4 available measurements. Now just push along the breadth of the rectangle. Is the new shape obtained, still a rectangle ? Observe that the rectangle has now become a parallelogram. Have you altered the lengths of the sticks? No! The measurements of sides remain the same.
Give another push to the newly obtained shape in a different direction; what do you get? You again get a parallelogram, which is altogether different (Fig 4.3), yet the four measurements remain the same
This shows that 4 measurements of a quadrilateral cannot determine it uniquely. Can 5 measurements determine a quadrilateral uniquely? Let us go back to the activity!
You have constructed a rectangle with two sticks each of length 10 cm and other
two sticks each of length 8 cm. Now introduce another stick of length equal to
BD and tie it along BD. If you push the breadth now, does the shape
change? No! It cannot, without making the figure open. The introduction of the fifth stick has fixed the rectangle uniquely, i.e., there is no other quadrilateral (with the given lengths of sides) possible now.
Thus, we observe that five measurements can determine a quadrilateral uniquely.
But will any five measurements (of sides and angles) be sufficient to draw a unique quadrilateral?
Question
Arshad has five measurements of a quadrilateral ABCD. These are AB = 5 cm,
ÐA = 50°, AC = 4 cm, BD = 5 cm and AD = 6 cm. Can he construct a unique
quadrilateral? Give reasons for your answer.
{ 0 comments }
How to construct a unique quadrilateral given the following measurements:
• When four sides and one diagonal are given.
• When two diagonals and three sides are given.
• When two adjacent sides and three angles are given.
• When three sides and two included angles are given.
• When other special properties are known.
{ 0 comments }
Sum of two numbers is 74. One of the numbers is 10 more than the other. What are the numbers?
We have a puzzle here. We do not know either of the two numbers, and we have to find them. We are given two conditions.
(i) One of the numbers is 10 more than the other.
(ii) Their sum is 74.
We already know from Class VII how to proceed. If the smaller number is taken to be x, the larger number is 10 more than x, i.e., x + 10. The other condition says that the sum of these two numbers x and x + 10 is 74.
This means that x + (x + 10) = 74.
or 2x + 10 = 74
Transposing 10 to RHS, 2x = 74 – 10
or 2x = 64
Dividing both sides by 2, x = 32. This is one number.
The other number is x + 10 = 32 + 10 = 42
The desired numbers are 32 and 42. (Their sum is indeed 74 as given and also one number is 10 more than the other.)
We shall now consider several examples to show how useful this method is.
{ 0 comments }
algebraic expressions and equations.
Some examples of expressions we have so far worked with are:
Some examples of equations are:
Equations use the equality (=) sign; it is missing in expressions
Of these given expressions, many have more than one variable. For example, 2xy + 5
has two variables. We however, restrict to expressions with only one variable when we form equations. Moreover, the expressions we use to form equations are linear. This means that the highest power of the variable appearing in the expression is 1.
These are linear expressions:
These are not linear expressions:
Here we will deal with equations with linear expressions in one variable only. Such
equations are known as linear equations in one variable. The simple equations which
you studied in the earlier classes were all of this type.
Let us briefly revise what we know:
(a) An algebraic equation is an equality involving variables. It has an equality sign.
The expression on the left of the equality sign is the Left Hand Side (LHS). The expression on the right of the equality sign is the Right Hand Side (RHS).
(b) In an equation the values of the expressions on the LHS and RHS are equal. This
happens to be true only for certain values of the variable. These values are the
solutions of the equation.
(c) How to find the solution of an equation? We assume that the two sides of the equation are balanced. We perform the same mathematical operations on both
sides of the equation, so that the balance is not disturbed. A few such steps give the solution.
x = 5 is the solution of the equation
2x – 3 = 7. For x = 5,
LHS = 2 * 5 – 3 = 7 = RHS
On the other hand x = 10 is not a solution of the equation. For x = 10, LHS = 2 × 10 – 3 = 17. This is not equal to the RHS
{ 0 comments }