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# PS-2

When two quantities are in inverse proportion, what remains constant? Their product remains constant.For example, speed and time are in inverse, hence S*T remains constant. So when you see a product of two numbers, they are in inverse proportion.
When two quantities are in direct proportion, what remains constant? Their ratio remains constant. For example, distance and time are in direct proportion, hence S/T remains constant.
How to calculate the amount after certain percentage increase? For 25% increase, we do 100+25=125. So original*1.25=Final. For 150% increase, 100+150=250. Thus we have to multiply by 2.5
How to calculate percentage decrease? If percentage decrease is 20%, then 100-20=80. So Final=0.8*original. A natural number cannot decrease by more than 100%; at 100% it becomes 0. But if its real number, then it can.
NCK = ? NC(N-K)
When is the multiplication principle applied? In counting number of "ways". Remember example of ordering food from a menu with 5 entrees and 3 desserts.
If R is the rate of compound interest paid annually, in how many years will the amount double? In nearly 70/R years.
When to use a Venn diagram and when to use a matrix? When the same population is divided differently on 2 (or more) attributes, use a matrix. When there is only one attribute for segregation, use Venn diagram.
What is the sum of the interior angles of a polygon? (N-2)*180. Also remember that each interior angle is less than 180
What is the relation between the sum of any two sides of a triangle & the third side? Sum of any two sides > Third side
If two angles of a triangle are equal, what can you say about the opposite sides? The sides opposite to the equal angles are also equal. It is infact, an isosceles triangle
The sides of a 30-60-90 triangle are in what ratio? 1:Root(3):2, where 1 is the side opposite to 30 and root(3) is side opposite to 60
What can be said of the altitude in an equilateral and isosceles triangle? The altitude bisects the base
Formula for distance between two points in the coordinate plane? Root( (y2-y1)^2 + (x2-x1)^2 ). It is derived by forming a right angled triangle with the points and using Pythagorus theorem
It is possible to form the equation of a line given two points on it? Yes. Use the two points to get the slope using y2-y1/x2-x1. Then use formula y-y1=m(x-x1) to get the equation of the line
What does the polarity of the slope of a line convey? If the slope is negative, the line slants downwards from left to right. If it is positive, it slants upwards from left to right
What is the slope of a line parallel to y axis and a line parallel to x axis? Parallel to y axis = 0Parallel to x axis = undefinedJust use the slope formula!
How do two linear equations with no common solution appear in coordinate plane? They will be parallel to each other. 2x+3y=4 and 6x+9y=12 represent the same line and thus have infinite common points
What does the graph of a quadratic equation look like? It is a parabola. The x intercepts are the two solutions of the equation. The y intercept is the value of f(0)
x^2+4=0 has real roots? No
What are the roots of quadratic equation ax^2+bx+c=0? x= (-b + root(b^2-4ac)/2a and x= (-b - root(b^2-4ac)/2a
How should you imagine x^r? Imagine it as x*x*x*x...r times. For example, x^3 = x*x*x. If you think of it this way, all exponent rules will flow automatically
Does multiplying or dividing both sides of an inequality by a positive number alter the inequality? No. Only multiplying or dividing both sides of an inequality by a negative number reverses the inequality
How do you solve |x+2|= 7? +(x+2)=7 and -(x+2)=7. Now solve
Of 25 students in a class, 20 study Math and 18 study History. How many study both? 25-20=5 18-5=1313 is the answer
If a cube ends in 9, its root ends in? 9
The avg of 30 students is 65, and avg of 40 students is 85. What is the combined average? 1) The ratio is 30:40=3:4. Inverse it i.e. 4:32) Divide the difference in the inverse ratio20 * 4/7 = 11.423) The average is 65+11.42 = 76.42
How to divide 5 chocolates between 3 children such that each gets at least 1? Theory of partitions 1) Write 5 as 1 1 1 1 1. There are 4 gaps in between.2) x+y+z=5 There are 2 + signs in the equation3) Answer is 4C2=6
r+s is divisible by 3 if BOTH r & s are divisible by 3 r & s could BOTH not be divisible by 3, but r+s could still be divisible.
What should be your strategy when they ask you to count ways in a word problem? Setup an equation, simplify it and then get your count from the equation instead of doing the counting manually or by logic. This reduces chances of error
What can you say about r^x when 0<r<1? The value of r^x decreases as x increases. Hence r^2<r<r^1/2. From this it can be inferred that x^2 > x only when x>1
Created by: deepak.bhavi