# Algebra Exam Preparation

### Algebra important Concepts – Chapter wise (Maharashtra Board):

#### Chapter 1 Linear Equations in Two Variables:

An equation of the form ax + by + c = 0 where a ¹ 0 and b ¹ 0 is the standard form of linear equation in two variables x and y where a, b, c = R. When we consider two linear equations in two variables at the same time & get a unique common solution, such set of equations is known as simultaneous equations. Further, solving equations through Cramer’s rule is the most important topic to study. Steps for solving simultaneous equations using Cramer’s rule are as follows:

•  Step I: Write the given equations in the form of ax + by = c
• Step II: Find the value of D, Dx and Dy
• Step III : Now apply Cramer’s rule to get the value of x & y.
• x = Dx, D and y = Dy

The equation ax² + bx + c = 0, where a, b, c are real numbers & a ¹ 0 refers to quadratic equation in variable x. The equation ax²+ bx + c = 0 refers to the standard form of quadratic equation. The highest index of variable is 2. So, it is a second degree equation in one variable. Factorization method, Completing square method & Formula method are the methods to solve Quadratic Equations.

• If b² – 4ac = 0, the roots of the equation are real & equal.
• If b²– 4ac > 0, the roots of the equation are real & unequal.
• If b²– 4ac < 0, the roots of the equation are not real.

#### Chapter 3 Arithmetic Progression:

If a set of numbers are arranged in a definite order according to some precise rule, then it refers to a sequence. Each number in the sequence refers to ‘a term of a sequence’. The first term of the sequence is denoted by t1 or a, the second term by t2, the third term is denoted by t3 and so on. In general, the nth term is denoted by tn. A sequence is usually denoted by {tn}n. A sequence in which the difference between any two consecutive terms (tn+1 – tn) is a fixed number (constant) it refers to ‘Arithmetic Progression’ (A.P.) The fixed number is called the common difference of the A.P. and is denoted by ‘d’.The value of ‘d’ may be positive, negative or zero. The nth term of an A.P. with the first term as ‘a’ and the common difference ‘d’ is given as  tn = a + (n – 1) d

#### Chapter 4 Financial Planning:

In this Chapter you will learn the various aspects of GST (Goods and Service Tax) CGST, i.e. Central Goods and Service Tax is to be paid to the Central Government whereas SGST, i.e. State Goods and Service Tax is to be  paid to the State Government. It is compulsory to get GSTIN, if the turnover of a dealer in the previous financial year exceeds 20 lacs. GSTIN includes 15 alpha numerals. For example, 27 A B C D E 1 2 3 4 H 1 Z 5 10 digit PAN of the firm 1 represents registration Uniform for all(By Default). Check Sum digit 2 digit for (Confirms validity of GSTIN State Code(Confirms validity of GSTIN when entered on GST portal. Additionally, ‘27’ is Maharashtra State Code. Form 27, one can understand that a person or a firm is registered in Maharashtra.

Furthermore, different tax rates for products as well as services are explained. When a trader pays GST at the time of purchase, it is called ‘Input tax’ and he collects GST at the time of sale which is called ‘Output tax’. At the time of paying GST to the government a trader deducts the input tax from the output tax and pays the remaining tax. This deduction of input tax refers to Input Tax Credit. GST payable = Output tax – ITC. Towards the end involvement of various stakeholders in the business is explained along with their Roles and Use of GST w.r.t. Investments.

#### Chapter 5 Probability:

The set of all the possible outcomes of a random experiment refers to the sample space. It is denoted by ‘S’ or ‘W’ (omega). Each element of sample space is known as ‘sample point’. Number of elements of the set S is denoted by n(s).  If n(S) is finite, then the sample space is called finite sample space. Moreover, the various types of events like

• Certain/Sure event where S = { 1, 2, 3, 4, 5, 6 } and A = { 1, 2, 3, 4, 5, 6 }
• Impossible event Here, S = { 1, 2, 3, 4, 5, 6 } and A = { }
• Complementary event where, A  B = f and A È B = S
• Simple/Elementary event
• Mutually Exclusive event where, A Ç B =  (Empty set)
• Exhaustive events where, A È B = S

The probability of an event ‘A’ in finite sample space S is written as P(A) and is defined as, P (A) = Number of elements in A/Number of elements in S  = n(A) / n(S).

#### Chapter 6 Statistics:

In this chapter you will learn the various measures to determine the Central tendency of Grouped Data such as Pie-Diagram, Histogram, Frequency Polygon or Frequency Curve, Mode. Mean, Assume mean and mean derivation method are the three types of method to find out the mean of a data. Median is the middle value of a data where Classes have to be continuous & the Cumulative frequency less than type has to be found for every sum. The observation having maximum frequency is mode.

To find mode for the given data, Classes have to be continuous & If maximum frequency occurs twice in the data, then data will have two Modal classes. Hence, we find Mode for both the modal classes. Moving ahead, the difference in the gap between the origin & lower limit of the first class and class width, then we draw a krink mark near the origin on the X-axis. We use this mark on the Y-axis to draw a graph of the optimum site for Histogram. When it Comes to Frequency Polygon, there are two methods to draw i.e.

• (i) Using histogram
• (ii) Plotting the points.

The main terms related to pie diagrams are sector and central angle.

### Do’s and Don’ts for Algebra Board Exams:

• For CBSE students, Solve each and every question of NCERT : 70% to 80% questions asked are directly from NCERT.
• If your target is 100 % in algebra then you should focus on Statistics and Probability
• Be ready for both short and long questions, overall you need to solve 30 Questions from 4 sections.
• Board will ask you tricky questions, so focus more  on practicing the sums.
• Make a Separate Note while revising your topics.
• Most important, solve the Easier Questions First.
• Don’t forget to use the Units and Decimal Points wherever necessary, one single mistake can cost you your marks.
• Provide Diagram Wherever Necessary
• Practice case study questions majorly from all the chapters but pay special attention to the chapters like Arithmetic Progression, Surface Areas & Volumes.
• Keep your Exam paper neat as well as clean, avoid overwriting and more cuttings.
• Make sure your rough work is neat and clean.
• Your basics should be clear and you should continuously practice as well brush up the concepts.
• Try to do Group studies you will practice more as well as get a more detailed solution along with fun.
• Note down all the formulas in a book.

### Class 10 Algebra Important Formulas & Concepts:

#### Formula for Algebraic Identities

1. (a + b)2 = a2 + 2ab + b2
2. (a + b)(a – b) = a2 – b2
3. (a − b)2 = a2 − 2ab + b2
4. (x + a)(x + b) = x2 + (a + b)x + ab
5. (x + a)(x – b) = x2 + (a – b)x – ab
6. (x – a)(x + b) = x2 + (b – a)x – ab
7. (x – a)(x – b) = x2 – (a + b)x + ab
8. (a + b)3 = a3 + b3 + 3ab(a + b)
9. (a – b)3 = a3 – b3 – 3ab(a – b)
10. (x + y + z)2 = x2 + y2 + z2 + 2xy + 2yz + 2xz
11. (x – y – z)2 = x2 + y2 + z2 – 2xy + 2yz – 2x
12. (x + y – z)2 = x2 + y2 + z2 + 2xy – 2yz – 2xz
13. (x – y + z)2 = x2 + y2 + z2 – 2xy – 2yz + 2xz
14. (x + a)(x + b)(x + c) = x3 + (a + b + c)x2 + (ab + bc + ca)x + abc
15. x3 + y3 + z3 – 3xyz = (x + y + z)(x2 + y2 + z2 – xy – yz − xz)
16. x2 + y2 + z2 − xy – yz – zx = 12[(x − y)2 + (y − z)2 + (z − x)2]
17. x2 + y2 = 1/2[(x + y)2 + (x – y)2]
18. x3 + y3 = (x + y)(x2 – xy + y2)
19. x3 – y3 =(x – y)(x2 + xy + y2)

For ax2 + bx + c = 0, (α, β) =[−b±(b2−4ac)/√2ac, when α and β are the roots of the equation.

•  If b2 − 4ac < 0, then the quadratic equation has two imaginary roots.
• If b2 − 4ac > 0, then the quadratic equation has two distinct real roots.
• If b2 − 4ac = 0, then the quadratic equation has two equal real roots.

#### Formula for Arithmetic Progression:

• nth term of an arithmetic sequence: an = a + (n – 1)d
• Sum of n terms of an arithmetic sequence: Sn = n/2 2a+(n–1)d
• nth term of a geometric sequence: an = a.rn-1
• Sum of n terms of a geometric sequence: Sn = a (1 – rn)/ (1 – r), r≠1
• Sum of infinite terms of a geometric sequence: S = a/ (1 – r)

#### Formula for Linear Equations in two variables:

• a1x + b1 + c1 = 0 and a2x + b2 + c2= 0
• When a1, b1, c1, & a2, b2, c2 are real numbers where a12+b12 ≠ 0 & a22 + b22 ≠ 0

To conclude, Class 10 students should follow a schedule so that they can give equal time for all the chapters for their subjects.

### Last Minute Algebra Exam Tips to Score Good marks.

• Get up early and use that time to study important topics
• Share problems you are facing and look for solutions
• Review the summaries you have rather than the entire chapters
• Reduce your Screen Time & avoid using Technology
•  Consider taking breaks, refresh yourself and start again.
• Get everything ready and in place for the next day
• Give more focus on your weak points
• Solve Mock papers
• Have a clear overview of your Exam