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Algebra formulas for class 11 include quantitative or algebraic rules represented as equations. Complex mathematical disciplines are formed on the basis of algebra formulas. A variable is a statistic whose value changes over time and is commonly denoted by an alphabet. A steady is a quantity with a set value.
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Key terms: Algebra formulas, Expressions, Identities, Equations, Permutations, Combinations, Values, Constant.
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Algebraic Formulas
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An algebraic formula is a quantitative or algebraic rule represented as an equation. It's a two-sided equation with arithmetic operations on both sides. The algebraic technique is a brief, easy-to-remember formula for solving difficult algebraic problems.
These algebraic formulae can be generated for any math topic with an uncertain variable x, and several of the most popular algebraic formulas can be used in any maths topic. The Algebraic Equations of Class 11th are mentioned below:
| 1 | (a + b)2 = a2 + 2ab + b2 |
| 2 | (a – b)2= a2– 2ab + b2 |
| 3 | (a + b)3 = a3 + 3ab(a + b) + b3 |
| 4 | (a – b)3 = a3 – 3ab(a – b) + b3 |
| 5 | a2 – b2= (a + b)(a – b) |
| 6 | a3 – b3 = (a – b)(a2 + ab + b2) |
| 7 | a3 + b3 = (a + b)(a2 – ab + b2) |
| 8 | a4 – b4 = (a2 – b2)(a2 + b2) |
| 9 | a5 – b5 = (a – b)(a4+ a3b + a2b2 + ab3 + b4 ) |
| 10 | a5+ b5 = (a + b)(a4 – a3b + a2b2– ab3 + b4 ) |
| 11 | (a + b + c)2 = a2+ b2 + c2 + 2ab + 2bc + 2ac |
| 12 | (a – b – c)2 = a2 + b2 + c2– 2ab + 2bc – 2ac |
| 13 | a3 + b3+ c3– 3abc = (a + b + c)(a2 + b2 + c2 – ab – bc – ca |
| 14 | If n is even, then an – bn = (a – b)(an-1 + an-2b +an-3b2 + …+ abn-2 + bn-1) |
| 15 | If n is odd, then an + bn = (a + b)(an-1 – an-2b +an-3b2 – …- abn-2 + bn-1) |
| 16 | (am )(an ) = am+n |
| 17 | (ab)m = ambm |
| 18 | (am)n = amn |
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Standard Algebraic Identities
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Algebraic identities are algebraic calculations that are accurate for all configurations of the independent variable in them. Polynomial factoring is also done with them. Algebraic principles are utilized in this way to compute algebraic expressions and solve various polynomials. For any values of the constants, algebraic identity indicates that the left-hand part of the problem is equivalent to the right side of the calculation. Algebraic identities are used to solve problems with unknown variables.
Some of the Standard Algebraic Identities are given below:
- (a + b)2 = a2 + 2ab + b2
- (a – b)2 = a2 – 2ab + b2
- a2 – b2= (a + b)(a – b)
- (x + a)(x + b) = x2 + (a + b) x + ab
- (a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ca
- (a + b)3 = a3 + b3 + 3ab (a + b)
- (a – b)3 = a3 – b3 – 3ab (a – b)
- a3+ b3 + c3 – 3abc = (a + b + c)(a2 + b2 + c2 – ab – bc – ca)
Let's take a glimpse at the algebraic formula (a + b)2 = a2 + 2ab + b2, and see if we can appreciate it in both algebra and geometry. Now let's try to multiply the statement algebraically and obtain the formula as a confirmation of this formula. (a + b)2 = (a + b) × (a + b) = a(a + b) + b(a + b) = a2 + ab + ab + b2 . This statement can be viewed geometrically as the surface of the four sub figures in the square picture below. Furthermore, we can combine (a + b)2= a2 + 2ab + b2 evidence of identity.
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Permutations and Combination
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A permutation is a configuration of a number of entities in a specific order, taken one at a period or all at once. It's all about arranging in a combo. Combinations can be used to calculate the number of possible groups that can be constructed from the items available.
To obtain the permutation and combination of r entities picked from n elements, permutation and combination equations are useful. Permutations and combinations are used to consider alternate arrangements, whereas combinations are used to consider alternate groupings. The permutations are always more than the combos for the specified values of n and r.
Binomial Theorem
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The expanded result of an algebraic expression of the pattern (x + y)n is found using the binomial theorem. Determining the value of (x + y)2, (x + y)3, (a + b + c)2 is simple and may be done simply by multiplying the exponent result by the number of instances.
The binomial theorem, on the other hand, can be used to find the enhanced version of (x + y)17 or other expressions with greater exponential values. This binomial formula expansion's factorial value could be a fraction or a negative integer.
Let’s do expansion of (x+3)5 by taking the binomial theorem. Here a = 3 and n = 5. By Substitution and expansion, we will get:
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Quadratic Equation
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An algebraic statement of the second degree in x is called a quadratic equation. ax2 + bx + c = 0 is the usual form of the quadratic equation, in which a, b are the coefficients, x is the independent, and c is the positive constant. The component of x2 is a non-zero term(a 0), which is the first criterion for an expression to be a quadratic equation.
The x2 term is written first, then the x term, and lastly the constant term when expressing a quadratic equation in proper format. The numerical values of a, b, and c are usually expressed as integral values rather than fraction or decimals.
The list of significant formulas below can be used to solve quadratic problems.
- ax2 + bx + c = 0 is the usual form of the quadratic equation.
- D = b2 - 4ac is the quadratic equation's exponent.
- The origins are real and distinguishable for D > 0.
- The roots of D = 0 are real and equal.
- The roots for D<0 either don't exist or are fictitious.
Principle Of Mathematical Induction
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P(n), wherein n is a natural number, is an example of a statement. Then apply the following method to assess the correctness of P(n) for each n:
Step 1: Verify that the given point is valid when n = 1.
Step 2: Consider that the above assertion P(n) holds for n = k, with k being any positive integer.
Step 3: For just any positive integer k, demonstrate that the conclusion is true for P(k+1).
The concept begins with a factual statement and ends with a conditional assertion. As shown in this, if the given assertion is true only for certain positive integer k, then the claim P(n) is valid for n = k + 1. This is also referred as the inductive phase, and the induction hypothesis is the supposition that P(n) is true for n=k.
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Sequence and Series
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The sequence is a collection of numbers arranged in a specific order or according to a set of criteria. The numbers of a sequence are added together to form a series. An one phrase can appear multiple times in a sequence. There are two sorts of sequences: endless terms and limiting terms. The succession and series will next be formed by combining the sequence's components. In some instances, the sum of infinite elements in a series is also conceivable.
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Things to Remember
- Numbers and letters are both included in algebra. The value of digits is known since they are fixed.
- The unknown values in the algebra formula are represented by letters or alphabets.
- An expression or formula is made up of numbers, characters, factorials, matrices, and other variables. This is basically the algebra technique.
- Acknowledging how algebraic equations are derived can help you memorise them.
- y=f is the notation of an algebraic function (x). This method's input and output are x and y, respectively.
Read Also: Elimination Method of Solving a Pair of Linear Equations
Sample Questions
Ques. What are Linear Inequalities? (2 Marks)
Ans. In math, an inequality is a mathematical representation that has two sides that are not equal. In mathematics, inequality occurs when a connection performs a non-equal balance between different formulations or two numbers. In this scenario, the inequality signs larger than symbol (>), less than symbol (<), equal to or higher to symbol (>), less than or equivalent to symbol (<), or not equivalent to symbol () substitute the equal sign "=" in the statement. Polynomial disparity, rational inequality, and definite integral inequality are examples of inequalities in mathematics.
Ques. What are geometric Progressions? (2 Marks)
Ans. A geometric progression is a sort of progression in which each term has a fixed ratio known as the uniform probability. GP stands for geometric progression. The geometric succession is usually written as a, ar, ar2,..., where an is the first concept and r is the progression's commonly used metric. Both negative and positive numbers are possible for the similar response. We simply need the initial term and the steady ratio to find the components of a geometric sequence.
Ques. Determine the roots of x2+5x+6=0, a quadratic expression. Quadratic problems are solved using algebra formulas. (3 Marks)
Ans. The equation presented is x2 + 5x + 6 = 0. By Comparison to ax2+bx+c=0, we will get: a=1; b=5; c=6. Substitution of these values in the quadratic formula:
Ques. Determine the constant k so that the line with expression y = kx is tangent to the circle with formula (x - 3)2 + (y - 5)2 = 4. (3 Marks)
Ans. The Given Expression is (x - 3)2 + (y - 5)2 = 4
By Substitution ‘Y’ by ‘X’: (x - 3)2+ (kx - 5)2 = 4
By Expansion & Writing of equation in Standard way: x2(1 + k2) - x(6 + 10k) + 21 = 0
Now, the circle and the line y = kx to be tangent, the discriminant of the abovementioned quadratic Expression must be equivalent to zero. (6 + 10k)2 - 4(1 + k2)(21) = 0
By Expansion of aforementioned expression: 16k2 + 120k - 48 = 0
Solved, the above mentioned formula: k = (-15 + √(273)/4 , k = (-15 - √(273)/4
Ques. What are linear Equations? What is the Standard formula? (3 Marks)
Ans. A linear equation is expressed using the linear equation algorithm. A linear formula can be expressed in simple form, slope-intercept form, or point-slope form, for example. Let's look at the normal state of a linear equation and see how it's written. We see now that it fluctuates depending on the number of variables, and it's important to understand that all parameters in the equation should have the maximum (and only) degree of 1.
Ax + B = 0 is the conventional or basic form of linear equations in one component, in Which A and B are real values and x is the new element. Ax + By = C is the conventional form of a two-variable linear model, where A, B, and C are any actual values, and x and y are the constants.
Ax + B = 0, Ax + By = C
Ques. What is Arithmetic Progression? Determine the general term of the arithmetic progression -3, -(1/2), 2…! (4 Marks)
Ans. A linear equation is expressed using the linear equation algorithm. This can be accomplished in a variety of ways. A linear formula can be expressed in simple form, slope-intercept form, or point-slope form, for example. Let's look at the normal state of a linear equation and see how it's written. We see now that it fluctuates depending on the range of variables, and it's important to understand that all parameters in the equation should have the maximum (and only) degree of 1.
The Sequence Presented is -3, -(1/2),2…
Therefore, the first term is a=-3, and the common distinction is, d = -(1/2) -(-3) = -(1/2)+3 = 5/2
By Arithmetic Progression formulas, the generic term of an AP is determined by the formula:
an = a+(n-1)d
an = -3 +(n-1) 5/2
= -3+ (5/2)n - 5/2
= 5n/2 - 11/2
Hence, the generic term of the presented AP is: an = 5n/2 - 11/2.
Ques. A three-person committee will be constituted, comprising two male members and one female member. Determine the myriad of options this panel can be made up of 5 male and 4 female individuals. (4 Marks)
Ans. The goal is to assemble a three-person committee, with two men and one woman.
The number of male participants is five.
The total proportion of female members is four.
We can make this committee by selecting two male members from among the five male members and one female person from among the four female members.
To come up with a solution, we use the combinations equation.
There are 5C2 × 4C15C2 × 4C1 different ways to form this panel.
5C2 × 4C15C2 × 4C1
=5 !/ [2! (3)!] × 4 !/ [1! (3)!]
= [120/12] × [24/6]
= 10 × 4 = 40
Hence, The panel can be made in 40 ways.
Ques. Patricia must select 5 marbles from a total of 12 marbles. How many different options does she have? (3 Marks)
Ans. Patricia must select five marbles from a total of twelve. It makes no difference what sequence things are done in. As a result, the combinations used here. She has a total of 12C5 options.
Therefore, there are 792 ways.
Ques. A class consists of ten individuals. Jack and Daniel are 2 of such students that do not get along. How many various contexts can the professor arrange the kids in a row to keep Jack and Daniel apart? (3 Marks)
Ans. The 10 pupils' overall number of configurations is 10 P 10 = 10! Let us now record the number of times Jack and Daniel are in the same arrangement. We get a total of 9 things that we can permute if we treat Jack and Daniel as a single entity (let's name it JD). There are 9P9= 9! methods to permute these 9 entities. J and D, on the other hand, can be discretized amongst themselves about 2! or 2 ways for each of the 9! iterations: JD or DJ. As a result, there are 2 9 combinations in which Jack and Daniel get paired!
We come to the conclusion that there are 10 variations in which Jack and Daniel are not paired! - (2 9!)
Ques. C(x) = 20 x + 4000 and R(x) = 60x + 2000, accordingly, are the cost and profit functions for a product, where x is the number of pieces manufactured and sold. To make a profit, how many things must be sold? (4 Marks)
Ans. As a result,
Cost, C(x) = 20 x + 4000
Revenue, R(x) = 60x + 2000
Profit = Revenue – Cost, as we all know.
Substitute the given data into the formula above.
R(x) − C = Profit (x)
(60x + 2000)-(20 x + 4000) = Profit
Now let's break it down:
60x + 2000 -20x -4000 = Profit
40x – 2000 = Profit
40x – 2000 > 0 to make a profit
40x is more than 2000.
⇒ x>2000/40
x is greater than 50.
To make a profit, the maker needs to sell more than 50 goods.
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