We know that parallelogram is also a quadrilateral. Let us split such a quadrilateral into two triangles. Find their areas and subsequently that of the parallelogram. Does this process in tune with the formula that you already know?

What would happen if the quadrilateral is not convex? Consider quadrilateral ABCD. Split it into two triangles and find the sum of the interior angles. What is the sum of interior angles of a concave quadrilateral?

Represent the following situations with suitable mathematical equations. The hypotenuse of a right triangle is 25 cm. We know that the difference in lengths of the other two sides is 5 cm. We would like to find out the length of the two sides?

Draw circles having different radius on à graph paper. Find the area by counting the number of squares, Also find the area by using formula. Compare the two answers.

observe the below diagram showing variation in beetle population and its impact. Let us consider a group of eweleve beetles. They live in bushes on green leaves. Their population will grow by sexual reproduction. So they were able to generate variations in population. Let us assume crows eat these red beetles. If the crows eat more Red beeles, their population is slowly reduced. let us disuss the above 3 different situations in detail. A. Situation-1 : in this situation a colour variation arises during reproduction. So that there appears one beetle that is green n colour instead of red moreover this green coloured beetle passes it's colur to it's offspring (Progeny ). So that all its progeny are green. Crows cannot see the green coloured beetles on green leaves of the bushes and therefore crows cannot eat them. But crows can see the red beetles and eat them As a result there are more and more gren beetles than red ones which decrease in their number . the variation of colour in beetle green gave a survival advantage to green beetles' than red beetles. in other word it was naturaly selected. We can see that the natural selection was exerted by the crows. The more crows there are, the more red beetles would be eaten and the more number of green beetles in the population would be. thus the natural selection is directing evolution in the beetle population. it results in adaptation in the beetle population to fit in their envirnment better. Let us think of another situation. Situation-2: In this situation a colour variation occurs again in its progeny during reproducation but now it results in blue colour beetles instead of red colour beetle. this blue colour beetle can pass its colour to its progeny. So that all its progeny are blue. crows can see blue coloured beetles on the green leaves of the bushes and the red ones as well. And therefore crows can eat both red and blue coloured beetles. In this case there is no survival advantage for blue coloured beetles as we have seen in case of green coloured beetles What happens initially in the population, there are a few blue beetles, butmost are red. Imagine at this point an elephant comes by and stamps on the bushes where the beetles live. this kills most of the beetles. By chance the few beetles survived are mostly blue. Again the beetle population slowly increases. But in the beetle population most of them are in blue colour. Thus sometimes accidents may also result in changes in certain characters of the population. Characters as we know are governed by genes. Thus there is change in the frequency of genes in small populations. this is known as genetic drift, which provides diversity in the population. Situation-3: In this case beetles population is increasing, but suddently bushes were affected by a lant disease in which leaf material were destroyed or in which leaves are affected by this beetles got les food material. So beetles are poorly nourished. So the weight of beetles decrease but no changes take place in their genetic material (DNA). After a few years the plant diseases are eliminated. Bushes are healthy with plenty of leaves. (Q) What do you think will be conditin of the beetles?

The two adjacent sides of a parallelogram are 2hati-4hatj+5kandhati-2hatj-3hatk . Find the unit vector parallel to its diagonal Also , find its area.

Think of this puzzle What do you need to find a chosen number from this square? Four of the clues below are true but do nothing to help in finding the number. Four of the clues are necessary for finding it. Here are eight clues to use: a. The number is greater than 9. b. The number is not a multiple of 10. c. The number is a multiple of 7. d. The number is odd. e. The number is not a multiple of 11. f. The number is less than 200. g. Its ones digit is larger than its tens digit. h. Its tens digit is odd. What is the number? Can you sort out the four clues that help and the four clues that do not help in finding it? First follow the clues and strike off the number which comes out from it. Like - from the first clue we come to know that the number is not from 1 to 9. (strike off numbers from 1 to 9). After completing the puzzle, see which clue is important and which is not?

Classically, an electron can be in any orbit around the nucleus of an atom. Then what determines the typical atomic size? Why is an atom not, say, thousand times bigger than its typical size? The question had greatly puzzled Bohr before he arrived at his famous model of the atom that you have learnt in the text. To simulate what he might well have done before his discovery, let us play as follows with the basic constants of nature and see if we can get a quantity with the dimensions of length that is roughly equal to the known size of an atom (~ 10^(-10)m) . (a) Construct a quantity with the dimensions of length from the fundamental constants e, m_e, and c . Determine its numerical value. (b) You will find that the length obtained in (a) is many orders of magnitude smaller than the atomic dimensions. Further, it involves c. But energies of atoms are mostly in non-relativistic domain where c is not expected to play any role. This is what may have suggested Bohr to discard c and look for ‘something else’ to get the right atomic size. Now, the Planck’s constant h had already made its appearance elsewhere. Bohr’s great insight lay in recognising that h, m_e, and e will yield the right atomic size. Construct a quantity with the dimension of length from h, me, and e and confirm that its numerical value has indeed the correct order of magnitude.

Do you know we also use some particulate pollutants? Could you list some of them?

Represent the following situations mathematically The hypotenuse of a right triangle is 25cm. We know that the difference in lengths of the other two sides is 5 cm. We would like to find out the length of the two sides.