The
DSAT
Student Manual - Maths
Index.
01
All you need to
know about SAT Math:
Page no.01
02
Chapter 1Global strategies
Page no.05
03
Chapter 2Linear Equations & Inequalities
Page no.23
04
Chapter 3- Arithmetic
Page no.50
05
Chapter 4Statistics & Graphs
Page no.72
06
Chapter 5Functions and Graphs
Page no.90
07
Chapter 6- Algebra
Page no.105
08
Chapter 7- Geometry
Page no.116
09
Chapter 8- Quadratic
Equation
Page no.130
10
Chapter 9- Circles,
Transformation of graphs
Page no.147
11
Chapter 10- Trigonometry
Page no.161
All you need
to know about
SAT Math:
The SAT Math section is divided into two modules. Your performance in
the first module will define the difficulty of the second module. This
adaptive nature of the math section requires you to be well-prepared
to deal with harder questions in the second module if your accuracy for
the first module is more than 70% to 75%.
The difficulty of the second module will decide the weightage. Hence
you have to ensure that the second module carries more weightage to
have a better total score which demands better accuracy in the first
module.
You will have the access to an on-screen calculator in both modules.
You also have the option to bring your own calculator to the center. Our
recommendation is to carry a physical calculator as it is easy to use
than the on-screen calculator.
Math Section
Module 01
Module 02
Duration
Number of Questions
35 Minutes
22 Questions
(75% MCQ, 25% studentproduced response)
35 Minutes
22 Questions
(75% MCQ, 25% studentproduced response)
01
Areas of Math tested in SAT:
There are four categories into which the SAT Math is divided.
Heart of Algebra:
Heart of Algebra consists of linear equations, systems of linear equations,
and functions. These questions ask you to create equations that represent
a situation, solve equations and systems of equations, and make
connections between different representations of linear relationships.
Problem-Solving and Data Analysis:
Problem-Solving and Data Analysis include using ratios, percentages, and
proportional reasoning to solve problems in real-world situations, including
science, social science, and other contexts.
Passport to Advanced Math:
The Passport to Advanced Math area requires familiarity with more
complex equations or functions like quadratic equations, Polynomials,
Rational Exponents, Radicals, Function Notations, Transformation of Graphs,
etc.
Additional Topics in Math:
The SAT Math Test contains six questions in Additional Topics. They may
include geometry, trigonometry, radian measure, and complex numbers.
SAT Calculator Use:
An on-screen calculator or a physical calculator is allowed for both the
modules of Math section. You must carry your own calculator and ensure e
you’ve got a fresh set of batteries. Don’t carry a new calculator. Practice for
the test using the same calculator you’ll use on test day. Use the calculator
only when it is necessary - get your thoughts down before using the
calculator.
Accepted Calculators:
Most Graphing calculators
All Scientific calculators
All four-function calculators
02
Prohibited Calculators:
Mobile phones, smartwatches, or wearable technology.
Models that use electrical outlets, make noise or have a paper tape.
Models that have a computer-style (QWERTY) keypad, pen input, or
stylus.
Models that can access the Internet, have wireless, Bluetooth, cellular,
audio/video recording and playing, camera, or any other smartphonetype features.
Stylus with Approved calculators.
Use of a calculator is permitted for all questions. A reference sheet,
calculator, and these directions can be accessed throughout the test
Unless otherwise indicated:
All variables and expressions represent real numbers.
Figures provided are drawn to scale All figures lie in a plane.
The domain of a given function f is the set of all real numbers x for
which f(x) is a real number.
For multiple-choice questions, solve each problem and choose the
correct answer from the choices provided. Each multiple-choice question
has a single correct answer.
For student-produced response questions, solve each problem and enter
your answer as described below
If you find more than one correct answer, enter only one answer.
You can enter up to 5 characters for a positive answer and up to 6
characters (including the negative sign) for a negative answer.
If your answer is a fraction that doesn't fit in the provided space, enter
the decimal equivalent.
If your answer is a decimal that doesn't fit in the provided space, enter
it by truncating or rounding at the fourth digit.
1
If your answer is a mixed number (such as 3 ) enter it as an
2
improper fraction (7/2) or its decimal equivalent (3.5)
Don't enter symbols such as a percent sign, comma, or dollar sign
03
Examples:
Answers
Acceptable way to
enter answer
Unacceptable: will NOT
receive credit
3.5
3.5
3.50
7/2
31/2
3 1/2
2/3
2/3
.6666
.6667
0.667
0.666
0.66
.66
0.67
.67
-1/3
-1/3
-.3333
-0.333
-.33
-0.33
04
Chapter 1Global Strategies
Objective of this Chapter:
Global Tips for Time Management and Accuracy:
Work as per your personal order of difficulty
Read and underline the final question before answering
POE (Process Of Elimination)
Solve the Question in Bite Size Pieces
Approach for Solving Questions in SAT Math
Plugging In
Plug In The Answers
Translating Word Problems to Equation
05
1. Work as per your personal order of difficulty
Of the 4 questions which one will you
attempt first? Which one last?
Always do easy
questions first
01
02
03
04
Do easy questions first.
2. Process of Elimination (POE)
Try to eliminate the answers based on the given information before
calculating anything.
01
Try to eliminate the
answers based on the
given information
before calculating
anything.
06
Answer: (B)
Explanation:
Here x<0, it means x is a negative number.
Since question is asking for a x value, final answer will be a negative number.
With this information we can eliminate C & D options.
Now, plug in A or B options in the given equation and check whether it satisfies
or not.
Start with option B (easy to calculate)
Hence, the answer is B.
02
Try to eliminate the
answers based on the
given information
before calculating
anything.
07
03
Try to eliminate the
answers based on the
given information
before calculating
anything.
3. Solve the Question in Bite-Size Pieces
Take one piece at a time and try to eliminate the answers that can not
possibly be correct before moving to the next piece.
Don’t forget to stop between each piece to see which answer can be
eliminated.
Very helpful for word problems in SAT.
08
01
Don't forget to stop
between each piece to
see which answer can
be eliminated
Answer: (D)
Explanation:
Start working with one piece of a question at a time.
2
Let’s add co-efficient of x first.
We can eliminate option ‘C’ because it is 3x .
Now, let’s add co-efficient of x.
2
Before calculating the next part, try to eliminate option/s based on this result.
We can eliminate option ‘A’ because it is -2x.
Now add constants in the equation.
=> -5+1=-4
So, eliminate option ‘B’.
Hence, the answer is D.
09
02
Don't forget to stop
between each piece to
see which answer can
be eliminated
03
Don't forget to stop
between each piece to
see which answer can
be eliminated
10
04
Don't forget to stop
between each piece to
see which answer can
be eliminated
Approaches for solving Questions in SAT Math:
When it comes to solving questions in SAT, the following time-tested
approach is the key to scoring in SAT Math.
Plugging In (PI)
Plug-In The Answers (PITA)
Translating word problems to Equation
Guessing and Pacing
Plugging In (PI)
An effective approach
when it comes to
solving questions in
SAT which can not be
solved easily using
Algebra or framing
equations
This Approach can be
used for the questions
with variables in
question stem and
answer choices
Let's learn to use it!
11
Steps to Approach for Plugging In:
1. Recognise the variables in question stem and in answer choices.
2. Plug-In a good number for the variable(s) that fits the requirements and
makes calculation simple.
3. Work on the problem step by step.
4. Circle the final answer.
5. Check all the options. Select the answer choice that matches the final
answer.
6. Make sure to check all four answer choices before you conclude.
01
What will you prefer?
Algebra or Plugging In
Answer: (B)
Explanation:
All the options are in terms of variables, hence we can use plugging in for
this question.
Let’s plug in for a=2
Then a+6=8
'b' is the average of a & a+6.
So, the average of 2 & 8 will be
If a=2, then a-4=2-4=-2.
'c' is the average of a & a-4.
So, the average of 2 & -2 will be
2-22=0
Sum of b & c=5+0=5
'5' is the final numerical answer.
12
Check all the options and mark which expression gives 5 as the answer.
A. 2a-1 2(2)-1=3
B. 2a+1 2(2)+1=5
C. 2a 2(2)=4
D. 2a+2 2(2)+2=6
Only option ‘B’ is giving answer as ‘5’.
⇒
⇒
⇒
⇒
Hence, the answer is ‘B’.
02
What will you prefer?
Algebra or Plugging In
13
03
What will you prefer?
Algebra or Plugging In
04
What will you prefer?
Algebra or Plugging In
14
Plugging In the Answer
An effective approach
when it comes to
solving questions in
SAT which can not be
solved easily using
Algebra or framing
equations
This Approach can be
used for the questions
with numbers in
answer choices
Let's learn to use it!
Basic Approach for Plugging In The Answers (PITA):
1. Recognise the final question and label the answer choices.
2. Start with the middle answer choice (B or C). If the question asks for the
greatest or smallest value, then start there.
3. Work with the question step by step (one piece of a question at a time).
4. Eliminate the answer choices which doesn’t satisfy the test condition. Also,
eliminate the answer choices which are too big or small.
5. Keep Plugging In answers until one of the answer choices satisfy the test
condition.
01
What will you prefer?
Algebra or Plugging In
the Answer
15
02
What will you prefer?
Algebra or Plugging In
the Answer
03
What will you prefer?
Algebra or Plugging In
the Answer
16
04
What will you prefer?
Algebra or Plugging In
the Answer
05
What will you prefer?
Algebra or Plugging In
the Answer
17
Translating Word Problems to Equation
Plugging In and Plug In The Answer
are effective approaches for a wide
variety of questions however some
questions need to be translated to
equations to solve them.
Knowledge to Translate word
problems to equations is very
useful for solving student response
questions and few other simple
questions which are very direct.
ENGLISH
MATH
is, are, were, did, does
=
what
variable (x)
of, product
multiply
out of, quotient
divide
per
divide or multiply
percentage
divided by 100
sum
add
diference
subtract
more than
>
less than
<
at least
≥
no more than
≤
18
01
Which is faster?
Plugging In or
Translating to equation
Answer: (B)
Explanation:
A+R+N=960
Let’s assume total mails received by Ryan & Neil is 'x'.
R+N=x
Given that, Alex received 50% more emails than the sum of the number
of emails received by Ryan and Neil.
A=1.5x
A+R+N=960
1.5x+x=960
2.5x=960
x=384
Number of emails received by Alex is 1.5x.
Therefore, 1.5x=1.5(384)=576
Hence, the answer is ‘B’.
19
02
Which is faster?
Plugging In or
Translating to equation
03
Which is faster?
Plugging In or
Translating to equation
04
Which is faster?
Plugging In or
Translating to equation
20
05
Which is faster?
Plugging In or
Translating to equation
Guessing and Pacing:
When you are stuck or running short on time then it's better to guess and move.
There are two kinds of guesses:
Educated Guess
Blind guess
01
If you are stuck
Blind Guess or
Educated Guess?
21
02
If you are running short
on time and don’t have
sufficient time to use
Plugging In
Blind Guess or
Educated Guess?
Reflect:
1.Global Strategies for Time Management & Accuracy
2.What are the ways to Identify Plug-In ?
3.How do you recognise Plug-In The Answers (PITA)?
22
Chapter 2Linear Equations
and Inequalities
Objective of this Chapter:
Solving Linear Equations
Solving System of Linear Equations
Geometric Meaning of Linear Equation
Equation of line
Properties of lines - parallel, perpendicular.
Case of solutions, no solution, many solutions.
Distance and Mid-point formula
Rules of Reflection
Absolute Value and Inequality
23
Linear Equations
Linear equations are the equations of degree 1. In other words, a linear equation
is an equation of a line on the XY plane.
Example: 2x+3=12, 3x+4y=5
Solving Linear Equations:
Basic rules to remember when you are working with the linear equations.
Whatever mathematical operations you do to one side, you need to do the
same mathematical operations on the other side.
Before cross multiplying, check whether you can simplify it.
Get rid of fractions when you are working with linear equations.
Take variables on one side and numbers on the other side to simplify linear
equations.
Solving Linear Equations with One Variable:
01
Is it required to
completely isolate the
variable?
24
Answer: (D)
Explanation:
Given, 4y-14=36
4y=36+14=50
Here, it is not required to completely isolate the variable because the
question is asking
4y+6.
Directly we can substitute 4y value in the expression.
4y+6=50+6=56.
Hence, the answer is ‘D’.
02
How will solve for x?
03
How will solve for m?
25
Solving Linear equations with Two Variables:
01
What will you prefer?
Solving or Plugging In?
Answer:
Explanation:
Given,
=6
Question is asking 15
Therefore, it is easier to write reciprocal of given equation to answer the
question.
Hence, the answer is
02
What will you prefer?
Solving or Plugging In?
03
What will you prefer?
Solving or Plugging in
the Answer?
26
04
What will you prefer?
Solving or Plugging in
the Answer?
05
What will you prefer?
Solving or Plugging In
the answer?
27
Solving System of Linear
equations:
A system of linear equations is a set of two or more linear equations.
Methods to Solve System of Linear Equation:
1. Elimination Method
2. Substitution Method
The solution to the system of linear equation represents the point of
intersection of the lines
1.Elimination Method:
Steps to follow for Elimination Method:
Example: x+6y=8
2x-y=16
First stack the equations.
Multiply the first equation by 2.
2x+12y=16
2x-y=16
Change the signs of the second equation to the opposite. This way, we can
cancel x terms.
2x+12y=16
2x - y = 16
_(-)__(+)__(-)_________
13y = 0
y=0
Substitute y = 0 in one of the equations to get the value of x.
x+6y=8
x+6(0)=8
x=8
28
2. Substitution Method:
Steps to follow for Substitution Method:
Example: x+6y=8
2x-y=16
We will work with the first equation.
x=8-6y
Substitute this expression into the second equation
2(8-6y)-y=16
16-12y-y=16
-13y=0
y=0
Substitute y = 0 in one of the equations to get the value of x.
x+6y=8
x+6(0)=8
x=8
01
What will you prefer?
Substitution
Elimination
29
02
What will you plug in to
solve easily?
03
What will you prefer?
Plug in the answer
Framing the
equation
04
What will you prefer?
Plug in the answer
Framing the
equation
30
Geometric Meaning of Linear
Equation
Linear equation is the equation of a straight line
The different forms of Linear equation are:
Standard Form: Ax+By=C
Slope Intercept form: y=mx+b
Point Slope form: y-y1=m(x-x1)
Out of all Slope intercept form is very helpful for SAT
Interpretation of Slope Intercept Form: y=mx+b
For y=mx+b,
m is the slope of the line
b is the y intercept of the line
Note:
Slope is the change in y value for 1 unit change in x value​
y intercept is the initial value of the function
At y intercept: x=0​
At x intercept: y=0​
31
01
Recall the definition of
slope
02
Recall the definition of
slope
32
Properties of line:
For Parallel lines: Slopes are equal
m1= m2
For Perpendicular lines: The product of slope of two perpendicular
lines is equal to -1
m1 x m2= -1
Slope of Line parallel to x axis is Zero
Slope of Line Parallel to y axis is Undefined
Direction based on slope of line
Positive Slope
Negative Slope
Zero Slope
Undefined Slope
Equation of Line:
To find the equation of the line you need to find the slope and the y
intercept of the line.
To find the slope you can take any two points on the line with
coordinates (x1,y1) and (x2,y2) and use the formula,
To find the y intercept b, substitute the given point in the equation
y=mx+b
33
01
What do you need to
find the equation of a
line?
02
What do you need to
find the equation of a
line?
34
03
What do you need to
find the equation of a
line?
04
What is true about
parallel lines?
05
What do you know
about perpendicular
lines?
35
06
What is true about a
line perpendicular to
line y=3?
07
Recall the direction of
line based on slope of
line
Interpreting the solution to linear equation:
The solution to the system of linear equation represents the point of
intersection of the lines.
One Solution
No Solution
Infinite many
Solutions
Intersecting Lines: Lines have
different slopes
Parallel Lines: lines have
the same slope &
different y- intercepts
Lines coincide: lines have the
same slopes and same yintercept. The two equation
represents the same line
36
01
Recall the condition for
no solution?
02
Recall the condition for
no solution?
37
03
Recall the condition for
no solution?
Recall the condition for
no infinite solution?
Recall the condition for
no perpendicular lines?
04
Recall the condition for
no solution?
05
What do you mean by
infinite solution
geometrically?
38
Mid-Point Formula:
Rules of Reflection:
39
01
Recall the rule of
reflection about line y=x
02
Recall the rule of
reflection about origin
Absolute Values:
The absolute value of a number is the distance of a number from the origin on
the number line.
40
Understanding
Absolute value
Answer: Absolute value of a number is always non-negative means
it can be zero or any positive number.
01
Mod functions always
gives you positive value
41
02
To solve for a variable
inside the mod, remove
the mod by taking ± on
the other side
42
Linear Inequalities:
Two linear expressions are not equal when compared. These inequalities can be
represented by the following symbols.
Symbol
Math Meaning
>
Greater than
<
Lesser than
≥
Greater than or equal to
≤
Lesser than or equal to
≠
Not Equal to
Solving Linear Inequalities:
Solving linear inequalities is almost the same as solving linear equations. But one
important rule is, that if you multiply or divide an inequality by a negative
number, then you must flip the inequality sign.
Let’s take an example:
-5x + 3 > -7
Subtract ‘3’ both sides
-5x + 3 -3 > -7 -3
Since we are subtracting 3, sign won’t change
-5x > -10
Divide both sides by -5
x<2
-------> (Dividing by negative number, sign will flip)
43
01
Solving Inequality is like
solving an equation
with only one
difference?
02
Solving Inequality is like
solving an equation
with only one
difference?
44
03
What is easier than
solving?
04
It is not always required
to completely isolate
the variable
45
05
How can you make it
easier to solve?
06
How can you make it
easier to solve?
46
07
What is easier than
solving?
08
Work in bite size and try
to eliminate before
solving
47
09
Work in bite size and try
to eliminate before
solving
10
Work in bite size and try
to eliminate before
solving
Reflect
1.What kind of curve you get from linear equation?
2.What do you mean by slope of line?
3.Which is the best suited form of linear equation for SAT?
4.What is b in the slope intercept form of linear equation?
5.What is true about the slope of perpendicular lines?
48
6.What do you mean by solution to system of linear equation?
7.What is the condition for no solution to system of equations?
8.What is the condition for infinite solution to system of linear equations?
9.What is the change in coordinates after reflection about line y=x?
10.What is the change in coordinates after reflection about origin?
11.What are the two methods to solve system of linear equations
12.When do you flip the Inequality sign
13.Why is absolute value always positive or non-negative
14.What are the inequality symbol for the following terms:
At least:
No more than:
At most:
49
Chapter 3-
Arithmetic
Objective of this Chapter:
Percentage
Ratio and Proportion
Variation
Rates
Percentage
Percent is a way of describing a portion of a whole expressed in hundredths.
Example: 5%=5/100
50
01
Answer: (A)
Explanation:
Total number of games played=75
Number of games won=55
Number of games lost=20
We know that,
Hence, the answer is ‘A’.
02
51
03
04
52
05
What is easier?
Plug in the Answer or
Algebra?
06
07
53
Ratio and Proportion:
Ratio is a comparison of two quantities. Equating two ratios is a Proportion.
Important factors about Ratio and Proportion:
Make sure the units are the same when you are comparing quantities.
To find out that two ratios are equal and are in proportion, we can use the
cross-multiplication method.
Ratio is the simplest form of the actual quantities.
Example: The number of girls and boys in the class is 12 and 18 respectively.
So, the ratio of girls to boys is 2:3
01
Understanding Ratio
02
Understanding Ratio
54
03
Ratio does not give you
original number
04
The multiplication
factor is constant in the
ratio
55
05
Understanding Ratio
06
The multiplication
factor is constant in the
ratio
Approach for solving Ratio Problems
By Proportion (for easier questions)
Combining Ratio
Ratio Box
Work in Bite Size and frame equation
56
01
Simple Ratio problems
can be easily solved by
setting proportion
Answer: (A)
Explanation:
Given that, the ratio of the money with Adam to Bob is equal to ratio of money
with Casey to David.
So, Adam:Bob=Casey:David
Hence, the answer is ‘A’
02
Try to combine two or
more into a single ratio
to make your job easier
57
Ratio Box
For ratio between two quantities, the ratio box looks like:
Quantity 1
Quantity 2
Total
Ratio
Multiplier
Actual
Note: We have learnt that multiplier remains constant in ratio
Ratio box is effective when one of the actual value is known to you
03
Ratio Box
58
04
How will you set up
the Ratio box here?
05
Do you really need
any ratio concept
here?
How can you make it
simple?
59
06
Can you set up a
ratio box easily?
What is required to
easily set up ratio
box?
07
When time is constant:
Ratio of speed= ratio of
distance
What will you preferAlgebra or PITA?
60
08
Can you easily setup
ratio box here?
Why?
09
What is easier?
Ratio box or Equation
61
Variation:
01
What kind of Variation?
Direct Variation or
Inverse variation?
Which concept is
tested?
Answer: (B)
Explanation:
It is given that, volume of gasoline is varies proportionally to the capacity
of the power engine.
62
Now for driving 300 miles using 600 cc engine the gasoline required is 45 litres
but the question is about the gasoline required for driving 400 miles using 600
cc engine which is given by
Now for driving Hence, the answer is (B).
02
What kind of Variation?
Direct Variation or
Inverse variation?
03
What kind of Variation?
Direct Variation or
Inverse variation?
What is the other way
to solve it?
63
Rate:
Rate: Rate is the ratio of two quantities that have different units.
The SAT uses this knowledge to test on multiple types of problems such as
Unit Cost
Speed, Distance, and Time
Work Rate
Unit Cost Problems ask you to find the cost per item.
Speed, Distance, and Time questions ask you to find
one of the three quantities when you are provided
with the other two.
Rate and Work questions require you to find the
speed of working per unit time.
Formula to be
used in Rate
questions:
𝑇𝑜𝑡𝑎𝑙 𝑐𝑜𝑠𝑡= 𝑈𝑛𝑖𝑡 𝑐𝑜𝑠𝑡×𝑞𝑢𝑎𝑛𝑡𝑖𝑡𝑦
D𝑖𝑠𝑡𝑎𝑛𝑐𝑒=𝑆𝑝𝑒𝑒𝑑×𝑇𝑖𝑚𝑒
𝑊𝑜𝑟𝑘=𝑅𝑎𝑡𝑒×𝑇𝑖𝑚𝑒
64
01
Which formula is
appropriate?
𝑇𝑜𝑡𝑎𝑙 𝑐𝑜𝑠𝑡=
𝑈𝑛𝑖𝑡 𝑐𝑜𝑠𝑡×𝑞𝑢𝑎𝑛𝑡𝑖𝑡𝑦
D𝑖𝑠𝑡𝑎𝑛𝑐𝑒=𝑆𝑝𝑒𝑒𝑑×𝑇𝑖𝑚𝑒
𝑊𝑜𝑟𝑘=𝑅𝑎𝑡𝑒×𝑇𝑖𝑚𝑒
Answer: (C)
Explanation:
We know that,
Total cost=Unit cost× Quantity
5 pens cost $2.
So, 2=Unit cost×5
Unit cost(Per pen cost)=2/5
Cost of 3 pens will be,
Total cost=Unit cost× Quantity
Total cost=2/5×3=1.2
Hence, the answer is ‘c’.
02
Which formula is
appropriate?
𝑇𝑜𝑡𝑎𝑙 𝑐𝑜𝑠𝑡=
𝑈𝑛𝑖𝑡 𝑐𝑜𝑠𝑡×𝑞𝑢𝑎𝑛𝑡𝑖𝑡𝑦
D𝑖𝑠𝑡𝑎𝑛𝑐𝑒=𝑆𝑝𝑒𝑒𝑑×𝑇𝑖𝑚𝑒
𝑊𝑜𝑟𝑘=𝑅𝑎𝑡𝑒×𝑇𝑖𝑚𝑒
65
03
Which formula is
appropriate?
𝑇𝑜𝑡𝑎𝑙 𝑐𝑜𝑠𝑡=
𝑈𝑛𝑖𝑡 𝑐𝑜𝑠𝑡×𝑞𝑢𝑎𝑛𝑡𝑖𝑡𝑦
D𝑖𝑠𝑡𝑎𝑛𝑐𝑒=𝑆𝑝𝑒𝑒𝑑×𝑇𝑖𝑚𝑒
𝑊𝑜𝑟𝑘=𝑅𝑎𝑡𝑒×𝑇𝑖𝑚𝑒
04
Which formula is
appropriate?
𝑇𝑜𝑡𝑎𝑙 𝑐𝑜𝑠𝑡=
𝑈𝑛𝑖𝑡 𝑐𝑜𝑠𝑡×𝑞𝑢𝑎𝑛𝑡𝑖𝑡𝑦
D𝑖𝑠𝑡𝑎𝑛𝑐𝑒=𝑆𝑝𝑒𝑒𝑑×𝑇𝑖𝑚𝑒
𝑊𝑜𝑟𝑘=𝑅𝑎𝑡𝑒×𝑇𝑖𝑚𝑒
05
Which formula is
appropriate?
𝑇𝑜𝑡𝑎𝑙 𝑐𝑜𝑠𝑡=
𝑈𝑛𝑖𝑡 𝑐𝑜𝑠𝑡×𝑞𝑢𝑎𝑛𝑡𝑖𝑡𝑦
D𝑖𝑠𝑡𝑎𝑛𝑐𝑒=𝑆𝑝𝑒𝑒𝑑×𝑇𝑖𝑚𝑒
𝑊𝑜𝑟𝑘=𝑅𝑎𝑡𝑒×𝑇𝑖𝑚𝑒
66
For questions on work done
01
Answer: (B)
Explanation:
Use the formula,
(Number of hours not given in the questions. So, you can ignore h in the formula)
Let’s plug in work as ‘1’ on both sides.
Hence, the answer is ‘B’.
67
02
03
04
Which formula is
appropriate?
𝑇𝑜𝑡𝑎𝑙 𝑐𝑜𝑠𝑡=
𝑈𝑛𝑖𝑡 𝑐𝑜𝑠𝑡×𝑞𝑢𝑎𝑛𝑡𝑖𝑡𝑦
D𝑖𝑠𝑡𝑎𝑛𝑐𝑒=𝑆𝑝𝑒𝑒𝑑×𝑇𝑖𝑚𝑒
𝑊𝑜𝑟𝑘=𝑅𝑎𝑡𝑒×𝑇𝑖𝑚𝑒
68
05
Which two concepts
are tested?
06
When people and
machine work together,
the rate is added up
Timed Drill:
01
69
02
03
70
Reflect
1.The percent change formula is:
2. The ratio of students in classes A and B is 4:3 and there are 35 students in
total. Draw the Ratio-Box for this.
3.Formula for direct variation is:
4.Formula for inverse variation is:
5.Work = ______ x ______
6.Distance = ______ x ______
71
Chapter 4-
Statistics &
Graphs
Objective Of Today’s Class
Statistics
Graphs
Standard deviation
Scatterplot
Box-Whisker Plots
72
Mean, Median & Mode:
Combined Mean or Weighted Average:
Combined Mean can be calculated by multiplying the mean or average of
each group by the number of values in the group and adding these products.
Then divide the sum of products by the total number of the values in all of the
groups.
Example: The average age of 15 students in class A is 20. The average age of
10 students in class B is 22. Find the average age of the students in both the
classes combined?
Sum of ages of students in class A = 15 x 20 = 300
Sum of ages of students in class B = 10 x 22 = 220
Median:
Median is the middle most number in the given set. Before finding median
make sure the numbers are arranged in either ascending or descending order.
Finding Median:
A ={0,2,3,5,7,10}
Set A has 6 numbers. So, 6/2=3rd & 4th numbers will be in the middle.
To find the median, take the average of 3rd & 4th numbers in the set.
Therefore, Median
B={4,5,8,12,17}
Set B has 5 numbers. So,
3rd, number will be in the middle.
Therefore, Median=8
Note:
If the set has even number of terms, then Median will be average of
terms. Where ‘n’ is number of terms in the given set.
If the set has odd number of terms, then Median will be
Where ‘n’ is number of terms in the given set.
73
01
What is important to
ensure before finding
the median of list of
data?
Mode:
Mode is the most frequently occurring number in the set.
02
Is it required to sort the
data in an order to find
the mode?
74
03
Is it required to sort the
data in an order to find
the mode?
Range:
Range is the difference between maximum and minimum values of the set. In
other words, it is the maximum difference between any two numbers in the
given set.
04
Range= maximum
value- minimum value
05
75
06
07
What can you plug in for to
make it easier to solve?
08
Which is easier to check
first?
Do you really need to work
for all?
09
What does "no unique mode"
hints you on?
76
10
What are the two ways to
solve this question?
Standard Deviation:
Standard deviation is all about how well the data points spread out in the
set. The more spread out of the data, the higher the standard deviation
and vice versa.
Standard Deviation
Standard deviation is the deviation of data from mean value.
It measures how far or how close the data as a whole is to the mean value
If a data set has a small standard deviation, it means that data as a whole
is closer to the mean.
Data with a small standard deviation has little variation, or differences, in the
data.
If a data set has a large standard deviation, it means that the data is
further away from the mean.
Data with a large standard deviation has more variation in the data.
Note:
In SAT, they never ask you to calculate the standard deviation.
Questions in SAT mainly will ask you to compare the standard deviation of
the different sets.
77
01
Answer: (A)
Explanation:
Standard deviation is all about the spread of the data. The more spread out of
the data, the higher the standard deviation and vice versa.
In school A, The spread of the data is less compared to school B.
Hence, the answer is ‘A’.
02
03
78
Bar Graphs & Tables
Graphs are used to visually summarise data in a way that is concise and
easy to read.
Before attempting to answer questions based on a graph, you should
quickly scan the graph to get an overview of what data and information are
being represented.
Pay close attention to the type of graph and its title. take note of any
descriptive labels and units of measurement of the graph.
Circle Graphs
A circle graph or pie chart shows how an
entire quantity has been divided. The circle
represents 100 percent of the quantity.
The information is usually presented on the
face of each section, telling you exactly
what the section stands for and the value
of that section in comparison to the other
parts of the graph.
Each slice of a pie chart is typically labelled
with percent that part of the whole.
The sum of the slices of the pie chart add
up to 100%
01
What is complete angle at
the center of the circle?
What total value
corresponds to complete
angle?
79
Answer: (C)
Explanation:
First read the title of pie chart before reading the question.
Total score from singles=8
Total score=8+24+9+28+54=123
Total angle in circle is 360.
Therefore, angle made by the sector representing singles
Hence, the answer is ‘c’.
Bar Graphs
The bar graph shows how the information
is compared by using broad lines, called
bars, of varying lengths.
It compare similar categories of items
using rectangular bars. The base of each
bar has a non numerical label that
describes a category.
The height or length of each bar
represents a numerical amount
associated with that category.
01
How do you find percentage
change?
80
Histograms
The bar graph shows how the information
is compared by using broad lines, called
bars, of varying lengths.
It compare similar categories of items
using rectangular bars. The base of each
bar has a non numerical label that
describes a category.
The height or length of each bar
represents a numerical amount
associated with that category.
01
How do you find percentage
change?
Line Graphs
It is used to represent how one
data variable changes with
another particularly when one
of the variables is time.
If a line segment slants up in a
time interval, the amount is
increasing for that time
interval.
If a line segment falls in a time
interval, the amount is
decreasing in that interval.
A horizontal line segment
indicates no change.
81
01
What will you read first?
Is it really needed to read
the wordy information?
Which concept is tested in
the question?
01
What do you mean by
frequency?
02
Recall:
82
Scatterplot
A scatterplot is a type of data
display that shows the
relationship between two
numerical variables. It can
indicate visually the type of
relationship, if any, that exists
between two sets of data.
Line of Best Fit
It is the line that best represents
the relationship between the two
sets of data graphed in a
scatterplot.
Typically, the line of best fit passes
through some, of the plotted points
with approximately the same
number of data points falling on
either side of it.
A line of best fit is useful for
predicting the y-value for any
particular x-value that was not
included in the original data.
83
01
Answer:
Explanation:
Fuel consumption when
emission is 75g=2.4
Fuel consumption when
emission is 55g=1.6
Hence, the answer is ‘B’.
Recognising the type of association
from a Scatterplot
A scatterplot may suggest different types of relationships, or no relationship, between
two sets of measurements.
84
02
03
04
85
05
Box-Whisker Plot
Box and Whisker plot displays the five-number summary of a set of data.
Minimum Value: Least number in
the given set
Maximum Value: Greatest
number in the given set.
Median (Q2): Middlemost value
of the given set when the
numbers are arranged in order.
Lower Quartile (Q1): Median of
the lower half of the data.
Upper Quartile (Q3): Median of
the upper half of the data.
Let’s Learn how to find out parts of the Box-Whisker Plot:
Calculate the quartiles of the data set: 20, 2, 1, 12, 4, 8, 9, 6, and draw the box plot.
Let us arrange them in ascending order.
1, 2, 4, 6, 8, 9, 12, 20
Minimum Value: 1
Maximum Value: 20
Median (Q2): 7
Lower Quartile (Q1): 3
Lower half of the data: 1, 2, 4, 6
Median= 3
Upper Quartile (Q3): 10.5
Upper half of the data: 8, 9, 12, 20
Median= 10.5
86
01
Which of these is not a correct statement for the Box plot shown?
A.The first 25% of data is closely packed
B.Interquartile Range is 40
C.The average of data is 105
D.The box plot divides the data in 4 equal parts
Stem-Leaf Plots:
Stem-leaf Plots are another way of representing the data in the given set.
What is Median?
87
01
02
03
Stem
Leaf
0
4
1
0,7,8
2
3,3,4,7,8
3
2,2,2,3,5,7,7
4
0,0,1,1,3
5
6,7
Stem
Leaf
0
4
1
0,7,8
2
3,3,4,7,8
3
2,2,2,3,5,7,7
4
0,0,1,1,3
5
6,7
Stem
Leaf
0
4
1
0,7,8
2
3,3,4,7,8
3
2,2,2,3,5,7,7
4
0,0,1,1,3
5
6,7
The mode of the data is:
A.23
B.32
C.37
D.40
The median of the data is:
A.23
B.32
C.37
D.40
The range of the data is:
A.23
B.32
C.37
D.40
88
Reflect
1.Define the following terms.
Mean _________________________________________
Median _________________________________________
Mode _________________________________________
Range _________________________________________
Standard Deviation___________________________________
2. Good habits to follow when you are answering graph questions:
________________________________________________
________________________________________________
________________________________________________
3. What is the line of best fit?
________________________________________________
4. Things to remember when you are working with Drawing Conclusions
________________________________________________
________________________________________________
5. Define the following terms:
Inter-Quartile Range ________________________________
Upper Quartile ____________________________________
Lower Quartile ____________________________________
6. How many digits numbers can be used in Stem-Leaf plots?
________________________________________________
7. Two effective methods to solve ‘Identifying the best representation of graphs’:
________________________________________________
________________________________________________
89
Chapter 5-
Functions &
Graphs
Objective Of Today’s Class
Functions
Polynomials
Graphs of Higher degree polynomial
90
Understanding Functions:
01
Function Definition:
Answer: (C)
Explanation:
Function in math relates to input to output. In other words, it defines the
relationship between two variables.
Hence, the answer is ‘c’.
Remember:
If ‘x’ is the input, then ‘y’ will be the output. So, f(x)=y.
In function word problems, analyse the equation given in the question
and read the keywords carefully.
02
What do you need
to find f(2) or
f(-2) ?
91
03
What will go in
place of x ?
Compound Functions:
Compound function is a combination of multiple functions. Always work from
the innermost part on compound functions.
Note: a(b(x))= aob(x) [Read it as a of b of x]
04
For compound
functions always
start from inner
function to outer
function
92
05
What will be the
value of x in
function h(x)?
How can you make
it easy?
06
What will go in
place of x ?
93
07
What is needed to
find particular value
of h(x) from given
table?
08
What remains
constant for a linear
function?
09
How will you find
f(1), f(4) from the
graph of a function?
What do you need
to find f(k)?
94
10
To find the x values
where f(x)=b
Draw a horizontal
line from y=b and
observe the
intersecting points
11
To find the x values
where f(x)=b
Draw horizontal line
from y=b and
observe the
intersecting points
95
12
To find the x values
where f(x)=b
Draw horizontal line
from y=b and
observe the
intersecting points
Inverse Function:
In mathematics, an inverse function is a function that reverses another
function.
If the function f applied to an input a gives a result of b, then applying its
inverse function g to b gives the result a, and vice versa.
f(a)=b if and only if 'g(b) = a'.
Likewise, we could also say that f(x) is the inverse of g(x) and denote it by
So, applying a function f and then its inverse f¹ gives us the original value back
again:
96
01
What other way can
be used to answer
this question
without finding an
inverse?
97
02
What other way can
be used to answer
this question
without finding an
inverse?
03
98
04
Polynomials:
A polynomial is any mathematical expression that contains variables ,
constants, coefficients, and /or non-negative integer exponents.
Degree of a Polynomial
99
01
What is the y value
at the x intercept?
Answer: (A)
Explanation:
X-intercept means the point at which the line cuts the x-axis. At that point value
of y will be zero.
We know that, f(x)=y
Substitute f(x)=y=0 in the given function to find out the x-intercept.
f(x)=3x+5
0=3x+5
02
What is the x value
at the y intercept?
100
03
Remainder
Theorem:
If a polynomial P(x)
is divided by (x−a),
then the remainder
= P(a).
04
What are the two
ways to solve it?
05
Solution is the value
of x that satisfies
the equation
06
How can you solve it
easily? Algebra?
Use Coefficient
Comparison when
two expression are
equated to each
other
101
07
What is easier than
algebra?
08
Plugging in or
Coefficient
Comparison Method?
09
Root is the value of x at
which y value is zero.
Hence root are also called
x intercept
To find the root , factorize
the function and equate
each factor to zero
But to find the sum of roots
for cubic equation use:
Sum of roots = -b/a
Product of roots = -d/a
102
Understanding the graph of
higher-order functions:
1.Graph of Quadratic Function:
The graph of a quadratic function is called a Parabola which has a curved
shape. A quadratic function is a polynomial function of degree 2.
The general form of a quadratic function is:
Where a, b, & c represents real numbers and a≠0
2.Graph of Cubic Function:
A Cubic function is a polynomial function of degree 3.
The general form of cubic function is:
103
3. Graph of Bi-Quadratic Function:
A Bi-Quadratic function is a polynomial function of degree 4.
The general form of a Bi-quadratic function is:
01
Plugging in or
Coefficient
Comparison Method?
Reflect
1.If ‘x’ is the input and ‘y’ is the output, then f(x)=_____
2.
= _____
3.Quadratic function is a function of degree ____
4.Cubic function is a function of degree ____
5.Bi-Quadratic function is a function of degree ____
104
Chapter 6-
Algebra
Objective Of Today’s Class
Exponents and Roots
Growth and Decay
Imaginary Numbers/Complex Numbers
Your turn
Exponents and Roots:
The questions on exponents and roots can be easily solved by using the rules of
exponents. There are different rules defined for exponents. The important rules
of exponents are given below:
Now let us discuss all the laws one by one with examples here.
105
01
Answer: (A)
Explanation:
The rule of exponent can be applied on same base, so the first step is to make
the base same
2
We know that, 9=3
2
Substitute 9=3 in the expression
02
03
What is the square
root of a negative
number?
106
04
What are two ways to
solve this question?
05
How can you make it
simpler?
06
What is easier?
Solving for x and y or
Plugging in.
07
When the base is
same the exponents
can be equated
08
107
09
When can you
compare the
exponent on both
sides of an equation?
10
11
What options can be
eliminated before
solving?
12
Isolate y and z in
terms of x
108
13
What approach can
be used to solve it
easily?
14
Try to reduce the
square root in
simplest term
15
Which identity will
you use to solve it?
Exponential Functions:
x
An exponential function is a function of the form f(x)= a , where
“x” is the variable and
“a” is known as a constant which is also known as the base of the function and it
should always be greater than the value zero.
Some examples of exponential functions:
x
f(x)=2
x+3
f(x) = 5
Exponential functions are of two types:
Growth Function
Decay Function
When something increases or decreases by some percent or by
some multiple over a period of time, use growth and decay
function to solve it.
109
When growth or decay is in terms of %, use
Number of changes
Final value= Initial value(1±rate)
When growth or decay is in terms of multiplier, use
Number of changes
Final value= Initial value(multiplier)
01
Use Growth or decay
formula
Answer: (B)
Explanation:
Since the population is increasing therefore it’s a case of exponential growth.
Now recall the growth formula to be used, since increase is in terms of
percentage therefore the growth formula to be used here is:
Final value= Initial value(1+rate)Number of changes
Notice here,
Initial Value = 90,000
Since the population is increasing once in every year hence the
Number of changes = 2015-2010 = 5 changes
Now, substituting each value in the formula gives:
02
Use Growth or decay
formula
110
03
Use Growth or decay
formula
04
Use Growth or decay
formula
Imaginary Numbers:
Imaginary Numbers are the numbers which are not real numbers.
Imaginary number is basically the square root of a negative number.
Example:
-1
-2
-10
Complex Numbers:
Complex numbers are the numbers that are expressed in the form of a+ib
where, a and b are real numbers and ‘i’ is an imaginary number called “iota”.
The value of i = (√-1)
Examples:
2+√(-1) = 2+i
3+√(-4) = 3+2i
2-√(-25) = 2-5i
111
Dealing
with 'iota"
01
-9
i
02
9
(1+2i) x (3-4i)=
03
2
i = -1
112
04
Your Turn:
Use the below mentioned global tips learnt in first class for better
accuracy and time management.
Do the easy question first
Read and underline the relevant pieces of information
Underline the final question before answering
Work in Bite Size Pieces and do POE after every piece of information
When it comes to solving, use the Time-Tested Approach and
Techniques learnt in first class for better accuracy and time
management.
Plugging In
Plug in the answer
Use Algebra and make equation only when you are dead sure.
Guess and Go on killer questions
01
113
02
03
04
05
114
06
Reflect
1.What must be same to apply the rules of exponents?
________________________________________________
2. When do you use growth and decay formula?
_______________________________________________
3. What is the difference between growth rate and growth multiplier?
________________________________________________
4. What is the growth factor for 10% growth rate?
________________________________________________
5. What is an imaginary number?
________________________________________________
6. What is the value of iota?
________________________________________________
_
115
Chapter 7-
Geometry
Objective Of Today’s Class
Homework Discussion
Lines and Angles
Triangles and its properties
Quadrilaterals and Circles
Volume
116
Formula sheet:
SAT exam booklet will have the geometry formula sheet as shown below, so you
do not need to learn these formulas.
You can use the information in this sheet as and when required for solving SAT
questions.
The same
formulae will be
available in your
test booklet.​​
Types of Angles:
Knowing the different types of angle will make it easier to understand the
question. The different types of angles are:
117
01
What is a
complete angle?
Parallel line Angles
The two kinds of angles- Big and
Small angles are created by a
line which intersects two parallel
lines.
Big + Small = 180 degrees
Small angles = Small angles
Big angles = Big angles
02
Small angles are
always _____?
In the figure below, m and n are parallel lines. What is
the value of r?
A) 90
B) 85
C) 75
D) 70
118
Types of Triangles and
their Properties:
Types of Triangles
Properties of Triangles
119
01
Do you see any
exterior angles?
Similar Triangles and its Properties:
Similar TrianglesSame shape, but not necessarily the same size.
Corresponding angles are equal.
Corresponding sides are in the same ratio.
a
p
=
b
q
=
c
r
For Similar figures, following properties holds true:
120
02
PQR:XYZ=?
Which triangle is
smaller?
03
Look for ratio of
the sides
Commonly used
Pythagorean
Triplets are(3,4,5), (6,4,10),
(5,12,13),(8,15,17)
Triangle PQR is similar to triangle XYZ. If the perimeter
of triangle PQR is thrice the length of the perimeter of
triangle XYZ, what is the ratio of the area of triangle
PQR to the area of triangle XYZ?
A) 3:1
B) 1:3
C) 9:1
D) 2:1
If in the figure below, CD=3, AB=6, and AD=12, then
what is the length of BC?
A) 5
B) 8
C) 10
D) 15
04
MO:NO, then
MQ: ?
121
05
Commonly used
Pythagorean
Triplets are:
(3,4,5), (6,4,10),
(5,12,13),(8,15,17)
Special Right Triangles:
Use the information of special right triangles from the formula sheet to solve
questions based on special right triangles.
06
Identify the
Isosceles right
triangle
∠
In the figure below, S and
What is the length of QR?
∠Q are right angles.
A) 2√2
B) 2√6
C) 4√2
D) 8
122
07
Which side is
common for
both triangles?
Quadrilaterals and Circles
Quadrilateral is a closed figure with four sides.
Some examples of quadrilaterals tested in SAT with their area
formula is given below:
01
The sum of
Adjacent angles
will be?
123
02
Will the tiles cover
the area of the
kitchen floor?
Polygons:
Polygon is a closed figure with n sides.
Some examples of polygons are:
Triangles − closed figure with 3 sides
Square − closed figure with 4 sides
Parallelogram − closed figure with 4 sides
Rectangle − closed figure with 4 sides
Pentagon − closed figure with 5 sides
Hexagon − closed figure with 6 sides
Note:
Sum of all interior angles in a polygon is (n-2) ×180°, where n is the number of
sides of the polygon
Sum of all exterior angles in a polygon is 360°
Regular Polygon:
All the sides and interior angles of
a Regular Polygon are all equal.
The bisectors of the interior
angles of a Regular Polygon meet
at its center.
The perpendiculars drawn from
the Centre of a Regular Polygon
to its sides are all equal.
The lines joining the center of the
Polygon to its vertices are all
equal Polygon is a closed figure
with n sides.
124
03
Is there anything
common between
Hexagon and
Square in the given
figure?
Circles:
For solving questions on circles easily, the following properties of circle are
must to remember.
125
In the figure below, if the radius length of circle O is 5,
OY AB, and AB = 8, what is the length of segment XY?
01
What will happen if
radius
to the
chord ?
⊥
02
What is the radius
of the larger circle?
⊥
A) 2
B) 3
C) 4
D) 5
In the figure below, point P is the center of each circle.
The circumference of the larger circle exceeds the
circumference of the smaller circle by 14π. What is the
width, w, of the region between the two circles?
A) 4
B) 6
C) 7
D) 8
03
P is the mid-point
of AO, it means?
The circle shown below has center O and a radius
length of 6. If P is the midpoint of OA and AB is tangent
to circle O at B, what is the area of the shaded region?
A) 5π
B) 12π
C) 30π
D) 36π
126
04
Angle tangent to
the circle will be?
In the figure below, PA is tangent to circle O at
point A, PB is tangent to circle O at point B.
Angle AOB measures 120° and OP =18/π. What
is the length of minor arc AB?
Volume:
Volume is the space occupied by an object in its surroundings.
The formula for volume of solid objects (cone, pyramid, cylinder, sphere and
cuboid) is given in the formula sheet so you do not need to learn it.
Note:
When one solid object is melted and re-casted to another solid object, the
volume remains same.
To find the greatest distance between two vertices in a cuboid, use the Super
Pythagorean
127
01
Remember
Pythagorean
theorem?
02
Equal volumes?
03
What are the
possible length and
width values of
rectangle having
area 18?
128
Reflect
1.What are the two types of angles created when two parallel lines are
intersected by a third line?
_______________________________________________
2. 1 BIG Angle + 1 SMALL Angle =
_______________________________________________
3. What is the condition for two triangles to be similar?
_______________________________________________
4. How the ratio of area is related to ratio of corresponding sides in similar
triangles?
_______________________________________________
5. What is a Polygon?
_______________________________________________
6. What is the formula for sum of all angles in a polygon?
_______________________________________________
7. What do you mean by regular polygon?
_______________________________________________
8. What is the formula for the area of parallelogram?
_______________________________________________
9. If the angle at the circumference of circle is 80 degrees, what is the angle at
the centre of circle?
_______________________________________________
10. What is the formula for finding the area of sector?
_______________________________________________
11. What is the formula for finding the arc length?
_______________________________________________
12. The angle subtended by the diameter of circle at the circumference is
_______________________________________________
129
Chapter 8-
Quadratic
Equations
Objective Of Today’s Class
Quadratic Equations
Solving Quadratic equations by factoring
Quadratic Formula,Discriminant & Nature of Roots
Classic Quadratics
Quadratic word Problems
Probability
130
Quadratic Equations
Standard Form of a Quadratic Equation:
The standard form of a quadratic equation is: y=ax2 +bx+c
Here,
a is the coefficient of x 2
b is the coefficient of x
c is the constant term
Quadratic equation gives a U-shaped Curve which is called
Parabola as shown below:
Important Note:
The x intercepts of parabola are called solutions or roots of the
quadratic equation.
The y value at the x intercept is zero therefore to find the roots of
quadratic equation, put y=0 in the standard form.
2
ax +bx+c=0 is the standard form of the quadratic equation to find
roots of the equation.
131
Solutions of the Quadratic
Equation:
Roots are called x intercepts of parabola which is also known as zeros or
solutions of the quadratic equation.
There are two ways to find the roots or solutions of the quadratic equation:
Factorization method (Splitting the Middle Term)
Quadratic formula
Factorization method (Splitting the Middle Term):
Follow the steps to learn solving Quadratic equations by factorization method:
2
1. Compare the given polynomial to standard form ax +bx+c=0 and
identify ‘a’, ‘b’ and ‘c’.
2. Multiply ‘a’ and ‘c’.
3. Find two numbers whose addition = ‘b’ and multiplication = a×c
4. Use the two numbers we get in step 3 to split the middle term ‘bx’.
Let’s look at an example and understand splitting the middle term this
better:
2
Find the solution of the quadratic equation y=6x +17x+5
1.Since solutions are roots of equation or x intercepts of the parabola hence put
2
y=0 in the equation y=6x +17x+5 and compare the equation with standard
form
ax 2+bx+c=0 and identify ‘a’, ‘b’ and ‘c’
2
6x +17x+5=0
Here,
a = 6, b = 17 and c = 5
2. Next, multiply ‘a’ and ‘c’
6 × 5 = 30
3. Now our goal is to find two numbers whose:
addition = b = (17) and
multiplication = a×c=(30)
4. Now, to get these two numbers, think of all the possible factor pairs of 30:
30=1×30
30=2×15
30=3×10
132
30=5×6
Notice that out of the four options we have here, only 2+15 gives 17.
So, the two numbers we finalize are: 2 and 15, because
2×15=30 and
2+15=17
5. Now we are ready to split the middle term 17x as 2x and 15x
Therefore,
2
2
6x +17x+5=0 can be written as 6x +2x+15x+5=0
6. We can now factorize by taking 2x common from the first two terms
and 5 common from the last two terms as shown
2x(3x+1)+5(3x+1)=0
7. Now, from the two terms 2x(3x+1) and 5(3x+1) we can take (3x+1)
common as shown
(3x+1)(2x+5)=0
8. So, we have factorized the given polynomial by splitting the middle
term:
2
6x +17x+5= (3x+1)(2x+5)
01
Recall the standard
form of Quadratic
Equation.
02
Divides evenly
means?
133
03
Convert into
Quadratic form.
04
Is there anything
common between
numerator and
denominator?
05
Identify the
quadratic equation
in it
06
Don't like fractions?
Then get rid of it.
134
Quadratic formula:
The roots of the quadratic equation that cannot be factorized easily can
be found using the quadratic formula.
The Discriminant and Nature of Roots:
The Discriminant in the quadratic formula is used to find the number of
solutions of the quadratic equation.
01
Is it easy to solve
this by factoring?
135
02
What is the
constant term
here?
03
b^2- 4ac < 0 ?
04
“Solutions” means
where they
intersect.
136
Sum and Product of the Roots:
To find the sum of the solutions or the product of the solutions of any
quadratic equation, use the formula:
Note: Roots are called x intercepts of parabola which is also known as
zeros or solutions of the quadratic equation.
01
Which options can
be eliminated
directly?
02
What is the easiest
way to solve this?
137
03
Can you see a
quadratic equation
in it?
04
Recall the standard
form of a quadratic
equation
Quadratic Word Problems:
SAT will test you on word problems that may require you to create and solve a
quadratic equation. Since we have learnt much about quadratic equation so
let’s move and solve some word problems.
01
Stuck in figuring
out numbers. Use
quadratic concept
here.
138
02
Geometry or
Algebra?
03
Can you see a
quadratic equation
in it?
Classic Quadratics:
Remember these classic quadratics for solving the questions easily.
139
01
Which algebraic
formula?
02
Simplify the left
part
03
Trial and error?
04
Which algebraic
formula?
140
Timed Drill (5 mins):
Try a timed drill of 5 minutes to check your pace and understanding of this
chapter. Remember to do the easy questions first.
01
02
03
04
Probability
Important Note:
Addition Principle: Either of the events is happening (When ‘or’ comes
between the events)
P(A or B) = P(A) + P(B) - P(A & B)
Multiplication Principle: Both the events are happening (When ‘and’ comes
between the events)
P(A and B)= P(A) x P(B)
P(Event happening) + P (Same Event not happening) = 1
141
01
Total number of
outcomes?
Number of
favorable
outcomes?
Answer: (C)
Explanation:
Hence, the answer is ‘c’.
02
P(A)= 1-P(Not A)
142
03
04
143
05
06
07
In how many ways
you can pick one
red and one black
ball?
144
08
To find the
probability of
consecutive events:
Multiply the
probability of each
event
Reflect
1.What is the standard form of quadratic equation?
________________________________________________
2. What do you mean by roots of quadratic equation?
________________________________________________
3. What is the y value when you have to find the roots of quadratic equation?
________________________________________________
4. What are the other names of Roots?
________________________________________________
5. What is the condition for 1 real root?
________________________________________________
6. How may x intercepts of parabola be there for no real root?
________________________________________________
7. What are the two ways to find roots of the quadratic equation?
________________________________________________
8. How will you identify the nature of roots and number of solutions of quadratic
equation?
___________________________________________
145
9. What is the formula for finding sum and product of roots?
________________________________________________
10. What is (a-b)(a+b)?
________________________________________________
11. Formula for Probability is:
________________________________________________
12. Probability of event A and B happening can be written as:
________________________________________________
146
Chapter 9Objective Of Today’s Class
Standard and vertex form of Quadratic Equation
Circles
Transformation of Graphs
Standard Form of a Quadratic
Equation:
In the last chapter, we learnt the standard and the factored form of the
quadratic equation as described here,
Standard form: y=ax 2+bx+c
Here,
2
a is the coefficient of x
b is the coefficient of x
c is the constant term
147
Factored form: y=a(x-α)(x-β),
where α and β are the roots of the equation
Now we are very much aware that Quadratic equation gives a U-shaped Curve
which is called Parabola as shown below:
Notice the vertex here is the point where the parabola takes a turn. The
coordinates of the vertex as shown in figure is (h,k) where,
Understanding more about Quadratic function and parabola:
The sign of a will determine the orientation of the Parabola- w
upwards or downwards.
If a is positive then it will curve upwards
If a is negative then parabola curve downwards
He we will illustrate graph of parabola y=x 2 and y=-x2
148
Vertex Form of Quadratic Equation:
Knowing the standard and the factored form, lets look at the vertex form of
quadratic equation. The vertex form of quadratic equation or the vertex form of
parabola helps in identifying the coordinates of the vertex of the parabola.
01
How do you find the
x coordinate of the
vertex of parabola?
149
02
Where do you see the
maximum or minimum
value of a quadratic
function
How do you find the x
coordinate and y
coordinate of the vertex
03
What does the
maximum time
denotes in this
quadratic equation?
04
What does the
maximum time
denotes in this
quadratic equation?
150
05
What are the two ways
to find maximum value
of quadratic function?
06
What is the y value at x
intercept?
07
What is the x value at y
intercept?
151
08
What is the y value at x
intercept?
09
What is c representing?
152
10
Recall axis of symmetry
11
Determine the axis of
symmetry
How will the axis of
symmetry help solve
this?
153
Equation of Circle:
The equation of a circle in the standard form is:
Where,
(h,k) are the corordinates of the center of the circle
r is the radius of the circle
In order to convert the general form to standard form of the equation, use
completing the square method for both x and y as described.
Step 1:
Coefficient of x 2 and y 2 should be one: If anything is multiplied on the squared
terms (and it'll be the same value on each of the squared-variable terms),
then divide through on both sides of the equation by that value.
Step 2:
Collect Like Terms: Collect the x-containing terms together, and collect the ycontaining terms together.
Step 3:
Take all constants on to the other side of the equation: Move any stand-alone
numbers to the other side of the "equals" sign.
Step 4:
Take half of the coefficient of linear terms and Square it: Find half of the
coefficients of each of the linear x and y terms (not the squared terms!).Square
each of these new values and add the squared value on both side of the
equation
Step 5:
Complete the x and y square: Convert the left-hand side to completed-square
2
2
form by using the identity (a+b) and (a-b) as applicable.
Step 6:
Find the value of r 2 term on the other side of equation: Convert the numerical
expression on the right-hand side to radius form by simplifying which is the
value of r 2 in the standard form. You can find the radius by taking the square
root of this simplified value.
154
2
Example:
Find the coordinates and radius of the circle having the following equation :
Step 1:
Step 2:
Step 3:
Step 4:
Coefficient of x is 2
Coefficient of y is 3
Half of co-effiecient of x is 1 and square of this is 1
Half of coefficient of y is 3/2 and square of this is 9/4
Now add this square value on both side of the equation
Step 5:
Step 6:
155
01
Convert to standard
form to find center
coordinates and radius
02
Which options can be
eliminated?
03
Find the equation of the
circle in standard form
to test if a point lies
outside, inside or on the
circumference of the
circle
Transformation of functions:
In this part of chapter we will observe the changes in the graph of the parent
function when some constant is added, subtracted or multiplied in to the
function.
Following rules will help to learn the changes in the graph of the parent fuction:
156
Use online desmos graphing calculator and apply the changes as specified in
the table above to observe the changes for better learning.
Example:
Observe the change in the parabola when constant is multiplied to the parent
function:
157
01
Recall Transformation
Rules!
158
03
Recall Transformation
Rules!
04
05
Apply rules of
transformation in steps
159
Reflect
1.What factor determines the direction of the parabola?
______________________________________________
2. What do you mean by vertex of parabola?
______________________________________________
3. How do you find the x and y coordinate of the vertex?
______________________________________________
4. What is vertex form of quadratic equation?
______________________________________________
5. Which form is best suited for finding the roots of quadratic equation?
______________________________________________
6. Which form is best suited for finding the maximum or minimum of quadratic
equation?
______________________________________________
7. How to find the x coordinate of vertex from roots of equation?
______________________________________________
8. What is the standard form of equation of circle?
______________________________________________
9. How to identify if a point lies outside, inside or on the circle?
______________________________________________
10. Which factor accounts for the left and right shift of the curve?
______________________________________________
160
Chapter 10-
Trigonometry
Objective Of Today’s Class
Trigonometric Ratios
Sine and Cosine rule in a triangle
Degree and Radians
Identifying Best Representing Graphs
Drawing Conclusion and Meaning in Context
Trigonometric Ratio:
Trigonometric ratios are the ratios of the length of sides of a triangle.
These ratios in trigonometry relate the ratio of sides of a right triangle to
the respective angle.
The basic trigonometric ratios are sin, cos, and tan, namely sine, cosine,
and tangent ratios.
161
01
Draw a right angle
triangle and label it
02
162
03
04
05
163
06
07
Trigonometric ratio in all
Four Quadrants:
Knowing the trigonometric ratio in all four quadrants help us to
identify if the trigonometric ratio in that quadrant is either positive or
negative.
164
01
02
Degree and Radians:
Radian and Degree are the units of angle measurement.
One complete anticlockwise revolution can be represented by 2π (in
radians) or 360° (in degrees). Therefore, degree and radian can be
equated as:
2π = 360°
And
π = 180°
Hence, from the above equation, we can say, 180 degrees is equal to π
radian.
165
Remember the conversion method as shown below:
01
To convert
From degrees to
radians
x π/180°
The circle shown below has the centre at point 0. The
area of the sector OCB is 12.5 Square units. The radius
of the circle is 5 units. What is the radian measure of α
shown in the figure?
02
To convert
From degrees to
radians
x π/180°
166
03
To convert
From degrees to
radians
x π/180°
Identifying the bestrepresenting graphs:
We have to understand the given information and identify the best
representation graph of the given data.
In these types of questions, don’t forget to underline the important
information while reading.
Work with one piece of a question at a time and then stop & check whether
any option/s can be eliminated.
Bite-sized pieces and Process of Elimination (POE) are effective methods to
solve these types of questions.
167
01
Michel is fishing 10 miles from his
home. At 4:00 pm he decided to
drive back to home. At halfway to
home he suddenly realized that he
has forgot his fishing net. And
decided to return back to the fishing
point and spent some time talking
to his friend who was fishing there.
After about 20 minutes talking to his
friend he drove straight home.
Which of the following graphs best
represents Michel’s activity?
Answer: (A)
Explanation:
In the graph, the horizontal axis represents time & vertical axis represents the
distance from home.
Given that, ‘Michel is fishing 10 miles from his home.’
Based on this information we can eliminate options ‘B’ & ‘D’ because these
options say that he started from home (distance from home is 0) which is not
correct.
The next statement in the question says that ‘at 4 pm he decided to drive back
home. At halfway home he realised that he forgot his fishing net and returned
back to the fishing point.’
With this information, you can eliminate option ‘C’, because this option says that
‘at 4 pm he drove back home and then realised that he forgot his fishing net’
which is not correct.
Hence, the answer is (A).
02
Marshal has stopped at a food
point and then accelerated
gradually until he reached a
constant speed on the highway.
After driving few miles, he
encountered some problem in the
car and slowed down significantly.
After few minutes the car stopped
completely due to unknown failure.
Which of the following graph could
represent Marshal's journey, as
described above?
168
03
A reaction between two chemical
reactants is endothermic, which
means it consumes heats from its
environment.As the reactants are
consumed, the rate of reaction
slows and the amount of heat being
consumed decreases. Which of the
following graphs, showing
temperature in degree celsius as a
function of time in seconds, best
illustrates the temperature graph of
the reaction?
Drawing conclusions:
Drawing conclusions are word problems in SAT. They will ask for the best
conclusion drawn from an experiment or a survey.
Remember:
Observe the sample size and representativeness of the sample given
in the question.
Don’t make any assumptions.
Check all the options before you mark the answer.
01
Alex invested his savings in two
business A and B. The revenue
earned from business A is denoted
by function g(x) and from business
B is denoted by function h(x).Which
of the following statement is true?
A. Business A consistently earned
more revenue that business B
B. Business B initially earned more
revenue than business A but was
eventually overtaken
C. Business B consistently earned
more revenue that business A
D. Business A initially earned more
revenue than business B but was
eventually overtaken
169
Answer: (D)
Explanation:
Given that,
The revenue earned from business A is denoted by function g(x).
The revenue earned from business B is denoted by function h(x).
In the graph, x-axis represents time and y-axis represents revenue.
If you observethe graph, initially business A (staright line) was more than
business B (curved line) but after some time curved line g(x) (business B)
overtakes the straight line h(x) (business A).
Hence, the answer is ‘d’.
02 A survey was done to record number helpless people taking shelter
in orphanage and it is found that 60% of the helpless people take
shelter in orphanage and are getting all the facilities to sustain their
livelihood. Which of the following conclusions is most valid based on
this observation?
A. Approximately 40% of helpless people are deprived of facilities.
B. Approximately 40% of helpless people are living roadside.
C. Approximately 60% of helpless people are in Orphanage and
getting facilities.
D. Approximately 60% of the helpless people are living happily.
03 In a study conducted by environmentalist it was observed that the
use of Air conditioners has increased 1000% between 2000 and 2020
in the United States, and the ozone layer depleted significantly in the
united states and worldwide over the same time period. Which of the
following is most reasonable conclusion that can be drawn from
data?
A. As Air conditioners increased worldwide, the ozone depleted
significantly.
B. Increased usage of Air conditioners increased ozone depletion.
C. If Air conditioners continues to increase, there will be continue
increase in ozone depletion.
D. Increased use of Air conditioner in United States occur at the same
time when ozone layer depleted significantly.
170
04 A recent Survey on 200 people in the society revealed that 143
people prefer metro and 57 cars .If there are 45376 people in the
society, which of the following statement is most likely to be correct?
A. Approximately 13000 people in the society prefer metro.
B. Approximately 18000 people in the society prefer metro.
C. Approximately 29000 people in the society prefer metro.
D. Approximately 32000 people in the society prefer metro
05 Mike is conducting a survey in her society to identify how many
people in the society prefers evening walk. He visited the local park in
the society for few days and counted the number of people visiting
the park for evening walk. He found that 35% of the people in his
society visit that park regularly for evening walk , which of the
following statements must be true?
A. This method of sampling is flawed and may generate a biased
estimate of the number of people in the society who prefers evening
walk
B. This method of sampling is not flawed and likely generated an
unbiased estimate of the number of people in the society who
prefers evening walk
C. Approximately 35% of people in the society prefers evening walk.
D. Approximately 65% of people in the society does not prefer
evening walk.
06 ABC corporation, hoping to determine how to encourage more
people to buy their products, surveyed its employees on what
motivates them to buy their products. A total of 500 people
responded to the survey, while the remaining members did not
respond. Which of the following factors most calls into question the
organization’s ability to make a reliable conclusion about how to
encourage more people to buy their products?
A. The number of members who did not respond to the survey
B. The size of the survey sample.
C. The method of distributing the survey
D. The group affiliation of the respondents
171
Meaning in Context:
In some word problems, you will have to find out what a number or expression
means in the context. Don’t forget to read the final question carefully. Use POE
and Bite-Sized pieces to answer the questions.
01
What does y intercept represent in
the graph?
A. The speed of ball after 9
seconds
B. The decrease in speed 1 unit
increase in time
C. The time until the speed
becomes zero
D. The initial speed of the ball
The graph displays the speed of ball in meter per
second with time in seconds.
Answer: (D)
Explanation:
In the graph, the x-axis represents the time and the y-axis represents the
speed.
Y-intercept means the point at which the line cuts the y-axis.
At the y-intercept, the value of x will be zero. It means that the time will be zero.
So, it is the initial speed of the ball.
Hence, the answer is ‘d’.
02
What does the slope of line in the
graph represents?
A. The speed of ball after t
seconds
B. The initial speed of the ball
C. The time at which the speed
becomes zero
D.The change in speed for one
unit change in time
The graph displays the speed of ball in meter per
second with time in seconds.
172
03
The graph displays the speed of ball in meter per
second with time in seconds.
Reflect
1.What is cos(90-x) equal to?
______________________________________________
2. In which quadrant tan is positive?
______________________________________________
3. In which quadrant sin is positive?
______________________________________________
4. How to convert radian to a degree?
______________________________________________
5. How to convert degree to radian?
______________________________________________
6. Two effective methods to solve ‘Identifying the best representation of graphs’:
______________________________________________
173