Algebra Quick Refresh, Exercises of Mathematics

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Shortcuts, Formulas & Tips

Vol. 2: Algebra & Modern Math

present

Quadratic and Other Equations

For a quadratic equation, ax 2 + bx + c = 0, its roots

 𝛼 𝑜𝑜𝑜𝑜 𝛽 = −𝑏𝑏^ ±^ √𝑏𝑏^

(^2) −4𝑎𝑎𝑐𝑐 2𝑎𝑎  Sum of roots = 𝛼 + 𝛽 = −𝑏𝑏 𝑎𝑎  Product of roots = 𝛼𝛽 = 𝑐𝑐 𝑎𝑎

Discriminant ∆ = 𝑏𝑏 2 − 4 𝑎𝑎𝑐

Condition Nature of Roots

∆ < 0 Complex Conjugate

∆ = 0 Real and equal

∆ > 0 and a perfect square Rational and unequal

∆ > 0 and not a perfect square Irrational and unequal

Cubic equation ax 3 +bx 2 +cx+d = 0

 Sum of the roots = - b/a  Sum of the product of the roots taken two at a time = c/a  Product of the roots = -d/a

Biquadratic equation ax 4 +bx 3 +cx 2 +dx+e = 0

 Sum of the roots = - b/a  Sum of the product of the roots taken two at a time = c/a  Sum of the product of the roots taken three at a time = -d/a  Product of the roots = e/a

Tip: If c = a, then roots are reciprocal of each other Tip: If b =0, then roots are equal in magnitude but opposite in sign. Tip: Provided a, b and c are rational  If one root is p + iq, other root will be p – iq  If one root is p + (^) �𝑞𝑞, other root will be p – (^) �𝑞𝑞

If a,b,c …. k are n positive quantities and m is a natural number, then

𝑎𝑎 𝑚𝑚^ +𝑏𝑏 𝑚𝑚^ +𝑐𝑐 𝑚𝑚….+𝑘 𝑚𝑚

𝑛𝑛 >^ �^

𝑎𝑎+𝑏𝑏+𝑐𝑐…+𝑘

𝑚𝑚

𝑎𝑎+𝑏𝑏+𝑐𝑐+⋯+𝑘

𝑛𝑛

Tip:

𝑎𝑎𝑚𝑚+𝑏𝑏𝑚𝑚

𝑎𝑎+𝑏𝑏

𝑚𝑚

[𝑚𝑚 ≤ 0 𝑜𝑜𝑜𝑜 𝑚𝑚 ≥ 1 ]

𝑎𝑎 𝑚𝑚^ +𝑏𝑏 𝑚𝑚

2 <^ �^

𝑎𝑎+𝑏𝑏

𝑚𝑚

[0 < 𝑚𝑚 < 1]

Tip: For any positive integer n, 2≤ �1 + (^) 𝑛𝑛^1 �

𝑛𝑛 ≤ 3

Tip: a m^ bnc p……..will be greatest when (^) 𝑚𝑚𝑎𝑎= 𝑏𝑏𝑛𝑛=𝑐𝑐𝑝𝑝

Tip: If a > b and both are natural numbers, then

 𝑎𝑎 𝑏𝑏^ < 𝑏𝑏 𝑎𝑎^ {Except 32 > 2^3 & 4^2 = 2^4 }

Tip: (n!)^2 ≥ nn

Tip: If the sum of two or more positive quantities is constant, their product is greatest when they are equal and if their product is constant then their sum is the least when the numbers are equal.

 If x + y = k, then xy is greatest when x = y  If xy = k, then x + y is least when x = y

Continued >>

Logarithm

𝑎𝑎 𝑏𝑏 �^ =^ 𝐿𝑜𝑜𝑔(𝑎𝑎)^ −^ 𝐿𝑜𝑜𝑔(𝑏𝑏) 𝐿𝑜𝑜𝑔(𝑎𝑎 𝑛𝑛) = 𝑛𝑛 𝐿𝑜𝑜𝑔(𝑎𝑎)

𝐿𝑜𝑜𝑔𝑏𝑏(𝑎𝑎) = 𝑙𝑜𝑔 𝑙𝑜𝑔𝑐(𝑎𝑎) 𝑐(𝑏𝑏)

𝐿𝑜𝑜𝑔𝑏𝑏 𝑏𝑏 = 1

𝐿𝑜𝑜𝑔𝑏𝑏 1 = 0

𝐿𝑜𝑜𝑔𝑏𝑏 𝑏𝑏 𝑥^ = 𝑥𝑥

Ln x means log𝑒 𝑥𝑥

𝑥𝑥 = 𝑏𝑏 log𝑏^ 𝑥

Functions

f(x) = |x|

If we consider –f(x), it gets mirrored in the X-Axis.

If we consider f(x+2), it shifts left by 2 units

If we consider f(x-2), it shifts right by 2 units.

If we consider f(x) + 2, it shifts up by 2 units.

If we consider f(x) – 2, it shifts down by 2 units.

If we consider f(2x) or 2f(x) ,the slope doubles and the rise and fall become much sharper than earlier

If we consider f(x/2) or ½ f(x), the slope halves and the rise and fall become much flatter than earlier.

Binomial Theorem

For some basic values:

(𝑎𝑎 + 𝑏𝑏)^0 = 1 (𝑎𝑎 + 𝑏𝑏)^1 = 𝑎𝑎 + 𝑏𝑏 (𝑎𝑎 + 𝑏𝑏)^2 = 𝑎𝑎 2 + 2𝑎𝑎𝑏𝑏 + 𝑏𝑏 2 (𝑎𝑎 + 𝑏𝑏)^3 = 𝑎𝑎 3 + 3𝑎𝑎 2 𝑏𝑏 + 3𝑎𝑎𝑏𝑏 2 + 𝑏𝑏 3 (𝑎𝑎 + 𝑏𝑏)^4 = 𝑎𝑎 4 + 4𝑎𝑎 3 𝑏𝑏 + 6𝑎𝑎 2 𝑏𝑏 2 + 4𝑎𝑎𝑏𝑏 3 + 𝑏𝑏 4 (𝑎𝑎 + 𝑏𝑏)^5 = 𝑎𝑎 5 + 5𝑎𝑎 4 𝑏𝑏 + 10𝑎𝑎 3 𝑏𝑏 2 + 10𝑎𝑎 2 𝑏𝑏 3 + 5𝑎𝑎𝑏𝑏 4 + 𝑏𝑏 5

Theorem (𝒂 + 𝒃)𝒏𝒏^ = (^) 𝟎𝒏𝒏𝑪^ 𝒂 𝒏𝒏^ 𝒃𝟎^ + (^) 𝟏𝒏𝒏𝑪^ 𝒂 𝒏𝒏−𝟏^ 𝒃𝟏^ + (^) 𝟐𝒏𝒏𝑪^ 𝒂 𝒏𝒏−𝟐^ 𝒃𝟐^ … +

𝒏𝒏𝒏𝒏^ 𝑪^ 𝒂^ 𝟎^ 𝒃𝒏𝒏

(𝑥𝑥 + 1)𝑛𝑛^ = 𝑥𝑥 𝑛𝑛^ + 𝑛𝑛𝑥𝑥 𝑛𝑛−1^ + 𝑛𝑛 2 𝐶𝑥𝑥 𝑛𝑛−2^ … + 𝑛𝑛𝑥𝑥 + 1 𝑛𝑛 0 𝐶 (^) + 𝑛𝑛 1 𝐶 (^) + 𝑛𝑛 2 𝐶 (^) … + (^) 𝑛𝑛𝑛𝑛𝐶 (^) = 2𝑛𝑛

𝑛𝑛 0 𝐶 (^) + 𝑛𝑛 2 𝐶 (^) + 𝑛𝑛 4 𝐶 (^) … = 𝑛𝑛 1 𝐶+ 𝑛𝑛 3 𝐶+ 𝑛𝑛 5 𝐶… =

Some basic properties

Tip: There is one more term than the power of the exponent, n. That is, there are terms in the expansion of (a + b) n.

Tip: In each term, the sum of the exponents is n, the power to which the binomial is raised.

Tip: The exponents of a start with n, the power of the binomial, and decrease to 0. The last term has no factor of a. The first term has no factor of b, so powers of b start with 0 and increase to n.

Tip: The coefficients start at 1 and increase through certain values about “half”-way and then decrease through these same values back to 1.

Tip: To find the remainder when (x + y) n^ is divided by x, find the remainder when y n^ is divided by x.

Tip: (1+x)n^ ≅ 1 + nx, when x<<

Permutation & Combination

When two tasks are performed in succession, i.e., they

are connected by an 'AND' , to find the total number of

ways of performing the two tasks, you have to MULTIPLY

the individual number of ways. When only one of the two

tasks is performed, i.e. the tasks are connected by an

' OR ', to find the total number of ways of performing the

two tasks you have to ADD the individual number of

ways.

Eg: In a shop there are‘d’ doors and ‘w’ windows.

Case1: If a thief wants to enter via a door or window, he

can do it in – (d+w) ways.

Case2: If a thief enters via a door and leaves via a

window, he can do it in – (d x w) ways.

Linear arrangement of ‘r’ out of 'n' distinct items (nPr ):

The first item in the line can be selected in 'n' ways AND the second in (n — 1) ways AND the third in (n — 2) ways AND so on. So, the total number of ways of arranging 'r' items out of 'n' is

(n)(n - 1)(n — 2)...(n - r + 1) =

𝒏𝒏! (𝒏𝒏−𝒓𝒓)!

Circular arrangement of 'n' distinct items: Fix the first item and then arrange all the other items linearly with respect to the first item. This can be done in (n — 1)! ways.

Tip: In a necklace, it can be done in

(𝒏𝒏−𝟏)! 𝟐

ways.

Selection of r items out of 'n' distinct items (nCr): Arrange of r items out of n = Select r items out of n and then arrange those r items on r linear positions.

n

Pr =

n

C r x r! →

n

C r =

nrP 𝑟!

𝑛𝑛! 𝑟!(𝑛𝑛−𝑟)! Continued >>

Probability

P(A) =

𝑁𝑢𝑚𝑚𝑏𝑏𝑒𝑟 𝑜𝑓 𝑓𝑎𝑎𝑣𝑜𝑟𝑎𝑎𝑏𝑏𝑙𝑒 𝑜𝑢𝑡𝑐𝑐𝑜𝑚𝑚𝑒𝑠 𝑇𝑜𝑡𝑎𝑎𝑙 𝑛𝑛𝑢𝑚𝑚𝑏𝑏𝑒𝑟 𝑜𝑓 𝑜𝑢𝑡𝑐𝑐𝑜𝑚𝑚𝑒𝑠

For Complimentary Events: P(A) + P(A’) = 1

For Exhaustive Events: P(A) + P(B) +P(C)… = 1

Addition Rule:

P (A U B) = P(A) + P(B) – P(A ∩ B)

For Mutually Exclusive Events P(A ∩ B) = 0

 P (A U B) = P(A) + P(B)

Multiplication Rule :

P(A ∩ B) = P(A) P(B/A) = P(B) P(A/B)

For Independent Events P(A/B) = P(B) and P(B/A) = P(B)

 P(A ∩ B) = P(A).P(B)  P (A U B) = P(A) + P(B) – P(A).P(B )

Odds

Odds in favor =

𝑁𝑢𝑚𝑚𝑏𝑏𝑒𝑟 𝑜𝑓 𝑓𝑎𝑎𝑣𝑜𝑟𝑎𝑎𝑏𝑏𝑙𝑒 𝑜𝑢𝑡𝑐𝑐𝑜𝑚𝑚𝑒𝑠 𝑁𝑢𝑚𝑚𝑏𝑏𝑒𝑟 𝑜𝑓 𝑢𝑛𝑛𝑓𝑎𝑎𝑣𝑜𝑟𝑎𝑎𝑏𝑏𝑙𝑒 𝑜𝑢𝑡𝑐𝑐𝑜𝑚𝑚𝑒𝑠

Odds against =

𝑁𝑢𝑚𝑚𝑏𝑏𝑒𝑟 𝑜𝑓 𝑢𝑛𝑛𝑓𝑎𝑎𝑣𝑜𝑟𝑎𝑎𝑏𝑏𝑙𝑒 𝑜𝑢𝑡𝑐𝑐𝑜𝑚𝑚𝑒𝑠 𝑁𝑢𝑚𝑚𝑏𝑏𝑒𝑟 𝑜𝑓 𝑓𝑎𝑎𝑣𝑜𝑟𝑎𝑎𝑏𝑏𝑙𝑒 𝑜𝑢𝑡𝑐𝑐𝑜𝑚𝑚𝑒𝑠

Tip: If the probability of an event occurring is P, then the probability of that event occurring ‘r’ times in ‘n’ trials is = n C r x P r^ x (1-P) n-r

Sequence, Series & Progression

Arithmetic Progression

𝑎𝑎𝑛𝑛 = 𝑎𝑎 1 + (𝑛𝑛 − 1)𝑑

𝑆𝑛𝑛 = 𝑛𝑛 2 (𝑎𝑎 1 + 𝑎𝑎𝑛𝑛) = 𝑛𝑛 2 [2𝑎𝑎 1 + (𝑛𝑛 − 1)𝑑]

Geometric Progression

𝑎𝑎𝑛𝑛 = 𝑎𝑎𝑜𝑜 𝑛𝑛−

𝑆𝑛𝑛 =

𝑎𝑎(1−𝑟 𝑛𝑛) 1−𝑟

Sum till infinite terms = (^) 1−𝑟𝑎𝑎 (Valid only when r<1)

Sum of first n natural numbers

 1 + 2 + 3 … + 𝑛𝑛 = 𝒏𝒏(𝒏𝒏+𝟏 𝟐 )

Sum of squares of first n natural numbers

 12 + 2^2 + 3^2 … + 𝑛𝑛 2 =

𝒏𝒏(𝒏𝒏+𝟏)(𝟐𝒏𝒏+𝟏) 𝟔 Sum of cubes of first n natural numbers

 13 + 2^3 + 3^3 … + 𝑛𝑛 3 = � 𝒏𝒏

(𝒏𝒏+𝟏) 𝟐 �^

𝟐

Tip: Number of terms = 𝑎𝑎𝑛𝑛− 𝑑𝑑𝑎𝑎 1 + 1 Tip: Sum of first n odd numbers  1 + 3 + 5 … + ( 2 𝑛𝑛 − 1 )^ = 𝑛𝑛^2  Tip: Sum of first n even numbers  2 + 4 + 6 … 2𝑛𝑛 = 𝑛𝑛(𝑛𝑛 + 1)

Tip: If you have to consider 3 terms in an AP, consider {a-d,a,a+d}. If you have to consider 4 terms, consider {a-3d,a-d,a+d,a+3d}

Tip: If all terms of an AP are multiplied with k or divided with k, the resultant series will also be an AP with the common difference dk or d/k respectively.