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Part II: Geometry and statistics simplified!
The countdown for Class X board examination has already begun for most students. If any of you have still not started this backward counting, please pull up your socks and get going.
This step is not to add to the stress you are probably already experiencing, but to calm you down and make you feel better before you actually take on the first major academic challenge in your life.
In mathematics, if you are fast, you are likely to make lesser number of errors. Therefore, it is easier to improve your speed and accuracy. At the end of the day you will save valuable time which you can use for rechecking your answers and ensuring perfection.
Experts from TCYonline.com have discussed below a few facts that will help you solve questions fast. We are presenting the number system, algebra and geometry units here in the first part of the feature.
Number system
~ LCM of two coprime numbers is always equal to the product of the two numbers. We can use this fact to check whether the two given numbers are coprime or not.
~ Remember that HCF of two numbers is always less than or equal to the smaller number and LCM of two numbers is always greater than or equal to the larger number.
This fact is very important in solving word problems. Consider the following problem from the NCERT textbook.
There is a circular path around a sports field. Sonia takes 18 minutes to drive one round of the field, while Ravi takes 12 minutes for the same. Suppose they both start at the same time and go in the same direction. After how many minutes will they meet again at the starting point?
One thing is clear that your answer will be either HCF or LCM of 12 and 18. Now if you take HCF of 12 and 18, you will get a number less than or equal to 12. But it is not possible for Ravi to reach the starting point again in less than 18 minutes. Hence, your answer will be the LCM of 12 and 18.
Algebra
~ While dividing a polynomial with another polynomial, make sure that the two polynomials are in their standard forms.
For example, when you are dividing x^{3} + 3x + 1 with x + 2, you should write the dividend as x^{3 }+ 0x^{2} + 3x + 1. This will avoid making mistakes (that are later on termed as 'careless mistakes' by students).
~ To get rid of denominators in linear equations and quadratic equations multiply the total equation with the LCM of the denominators. This will give you a very simple equation that you would prefer to solve.
Geometry
~ Remember, BPT will apply only when the line intersecting two sides of triangle is parallel to the third side.
~ Understand the meaning of similarity of two triangles. If triangle ABC is similar to triangle DEF, then remember the following:
AB/DE = BC/EF = AC/DF = Altitude/Altitude = Median/Median = Angle Bisector segment/Angle Bisector segment
~ Plus, the ratio of their areas will be equal to the square of any of the above ratios.
~ While applying Pythagoras theorem take care of the following relations.
~ The mid point of hypotenuse of all the right angled triangles is the circumcenter of the triangle and hence equidistant from all the three vertices of the triangle.
In circles, remember the following facts:
~ Cyclic trapeziums are isosceles and viceversa.
~ Cyclic parallelograms are rectangles and viceversa.
~ The line segment joining the points of contact of two parallel tangents to a circle will be a diameter of the circle.
~ All triangles are cyclic and viceversa.
~ The two tangents from an external point and the chord whose end points are points of contact will make an isosceles triangle.
~ The two tangents from an external point and the chord whose end points are points of contact will make an isosceles triangle.
Remember these facts to have an instant grip over the situation.
Part II: Geometry and statistics simplified!
Top Careers & You has been preparing national and international candidates for highend tests viz. GRE, GMAT, SAT, NTSE and Class X Boards since 1998.
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