Wednesday, 30 March 2016

[Q] Algebra Tricks - 3 [Approximation]

This is another useful article for you all and requires no prior knowledge of any sort.

Approximation is a very important tool that can help you solve some complex and time taking questions. I will solve the below questions from SSC CGL with approximation technique to give you an idea of how it works. But before that, some basic rules of approximation:

1. Establish a limit within which the variable is falling.
2. Neglect the smaller terms of the expression (fractions with Denominator>Numerator)
3. Please use this technique only when the options have a significant difference between them. E.g. If in a question the 4 options are A. 4, B. 5, C. 6, D. 7, you can't use the approximation technique because the options are fairly close.

These rules will make sense once you go through the below CGL questions-

Q. 1

Now how will you approach this question if you dont know how to solve it?
Given, x^4 + 1/x^4 = 119
We can safely assume that 3<x<4 because 3^4 = 81 and 4^4 = 256  (119 lies between 81 and 256). Moreover x will be closer to 3 as 119 is more close to 81 than 256
We have established the limit of the variable.
Let us take our first value. Go with x = 3.2
3.2^4 = 104 (approx), which is still a little away from 119
Hence let us take x = 3.3 as our second value
3.3^4 = 118 (approx) [PERFECT]
Now we have to find x^3 - 1/x^3
Note that 1/x^3 is negligible and hence we can neglect it
So just find the value of 3.3^3
Answer : (C)
Note : You won't take much time in calculating 3.2^4 or 3.3^4 if you know a fast method to calculate squares. I have written an article about it. You can check it here. 




Q. 2

Here again no need to figure out how to solve the question
√3 = 1.73, √5 = 2.23
Hence √x = 1.73 - 2.23 or x = 0.25
Put the value of x
(0.25)^2 - 16*0.25 + 6
= 0.0625 - 4 + 6
= 2 (approx)
Answer : (C)

Q. 3

x = √5 + 2 = 4.23
(x^4 - 1)x^2 = x^2 - 1/x^2
Neglect 1/x^2
x^2 = 4.23^2 = 17 (approx)
Answer : (A)


Q. 4

Again if you dont know how to solve the above question, then observe the above equation
If a=1, then LHS = 5.33 (which is little more than RHS, i.e., 5). We need to decrease the value of 'a'.
Hence let's take a = 0.9
5a + 1/3a = 4.8
Now LHS is more than RHS. We need to increase the value of 'a' slightly
So lets lock the final value a = 0.95 (Now no need to check the value of LHS for a=0.95)
9a^2 + 1/25a^2
Neglect 1/25a^2
9a^2 = 8 (approx)
Answer : (D)


Q. 5

x = 2 √3 = 2 + 1.73 = 3.73
Now you have to find the value of √x + 1/√x
√x = √3.73
You know that 19^2 = 361
Hence √3.73 = 1.9 (approx)
1/√x = 1/1.9 = 0.5
√x + 1/√x = 1.9 + 0.5 = 2.4 (which is close to √6)
Answer : (B)


Q. 6

3x - 1/4y = 6
Put x=1 and solve the equation for y
y = -1/12
Put x = 1 and y=-1/12 in the expression (4x - 1/3y)
You will get 8
Answer : (D)


Q. 7

Here again we can say that the approx value of x is 8, because 8^2 = 64
Put x = 8 in the expression
 = (64 - 1 + 16)/8
 = 10 (approx)
Answer : (A)

Q. 8)



Put x=2, the LHS becomes 6 and it is little more than RHS. So we need to decrease its value slightly
Let us take x = 1.8
LHS = (1.8)^2 + 1.8 = 5(approx)
LHS is almost equal to RHS, hence x=1.8 is a perfect value
Neglect 1/(x + 3)^3.
Now we only need to find the value of (x + 3)^3
(x + 3)^3 = (1.8 + 3)^3 = 110 (approx)
Answer : (A)


Below are some important formulas for Algebra, see if you can memorize them :)



If you have read CI-SI Tricks Part-1, I would request you to read the "annual instalments" section again. I have updated that topic...

Keep Reading :)

To buy the super-hit SSC Hack Book, follow the below link-

34 comments:

  1. Bhai English b dalde error waghrh

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  2. bhai english ki kuchh tricks do please

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  3. Thankyou sir awesome tricks. Kudos

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  4. Thanks sir..just few days back I came to know your website.. Nice tricks..very useful..I am taking time in solving maths problem s.minimum I will take 2 to 3 min for a single problem which should not be taken. People like me find these shortcuts very useful. You said 3.2 ^4 ,is there any shortcut for multiplication? Generally I will calculate 32^4 and keep the points to get answer.. But these calculations are time taking sir.

    ReplyDelete
    Replies
    1. 3.2^4 = 3.2^2 * 3.2^2
      you can find square of a number quickly by going thru my article
      http://sschacks.blogspot.in/2016/03/q-fast-calculation-tricks.html
      (Copy paste above link in your browser)
      So 3.2^4 = 10.24*10.24 (Now neglect the second decimal place)
      3.2^4 = 10.2*10.2 = 104 (Approx)

      Delete
    2. Sir Jo apne last question me 110 approx nikala he wo direct kaise Nikala means answer could be 120 also na.
      4.8 ^3 which is close to 5^3

      Delete
    3. 4.8^3 calculate karna padega, maine usme assume kuch nai kiya hai.
      4.8*4.8*4.8 = 110.59
      Hence the answer is 110
      and 4.8^3 is not at all close to 5^3
      Itna bada approximation mat karo :)

      Delete
  5. Sir Plz Do write abt Number system questions ...
    It is really helpful sir..

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  6. Ok sir but mene ye assume kiya ki since it should be less than 125 and only difference is of .2 so may be answer from option is 120

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  7. Q No. 3 me jab answer 17 aaya to option "A" answer kaise ho gaya,Please explain dis ??????

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  8. thanks a lot sir....please guide reasoning sec..for..aa_ bda_ _ ca these type series

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  9. C.I of 2 years is rs 156 and 3 years is rs 254. What is rate? (Any short trick)

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  10. C.I of 2 years is rs 156 and 3 years is rs 254. What is rate? (Any short trick)

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  11. Please give some tricks on solving number series.

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  12. This comment has been removed by the author.

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  13. Sir,your tricks are awesome sir it helps a lot.
    Sir I have a doubt about the application, I have submitted the application form for that I even got a mail of successfully submitted but when i checked my application status it is showing that my application is pending.I am worrying about this sir please reply

    ReplyDelete
  14. Prashant sir,
    tried to solve the following questions with the tricks you have provided, but could not solve them. Need your guidance sir.

    1. Find the value of (3+2√2)^-3 + (3-2√2)^-3
    2. If x(x-3)=-1, then x^3(x^3-18) =?
    3. if x=332, y=333, z=335 then x^3+y^3+z^3-3xyz =?
    4. if {√(a+2b) + √(a-2b)} / {√(a+2b) - √(a-2b)}= √3 then find a:b

    apart from this sir, there is a question which comes many times,

    Q. x^51 + 51 is divided by (x+1) , the remainder will be ?
    any short tricks to solve this type of question ?

    ReplyDelete
    Replies
    1. 1. Let a = 3+2√2, b = 3-2√2. Then you have to find
      a^3 + b^3/a^3.b^3
      a^3 + b^3 = (a + b)^3 - 3ab(a + b)
      = 6^3 - 3.1.6 [because a + b = 6 and a.b = 1]
      = 216 - 18 = 198
      Now denominator = (ab)^3 = 1
      Answer : 198

      2. Given x + 1/x = 3, then x^3 + 1/x^3 = 3^3 - 3*3 - 18
      Hence x^3(x^3-18) = 0

      3. x^3+y^3+z^3-3xyz = 1/2(a + b + c)((a - b)^2 + (b - c)^2 + (c - a)^2)

      4. Rationalize and solve.

      5. Put x = 1
      Answer : 0

      Delete
    2. sir 2nd question me x+1/x = 3
      whole cube - x^3+1/x^3+3(x+1/x)=3*3*3
      x^3+1/x^3=27-3*3 =18 =>x^3=18-1/x^3

      now x^3(x^3-18)=x^3[(18-1/x^3)-18]
      x^3 [-1/x^3] = -1

      sir aapka zero kaise aa rha h?
      wo cube wali equation glt lg rhi h

      Delete
    3. Yes...sorry it will be -1 only :)

      Delete
  15. Thank you so much sir .. But didnt get the solution to Q.5 If (x^51 + 51) is divided by (x+1) , the remainder will be ? kindly thora explain kar dijiye .

    ReplyDelete
    Replies
    1. Put any value for x
      If you put x = 1
      (x^51 + 51) will become 52
      (x + 1) will become 2
      52 when divided by 2 gives zero remainder
      Hence Answer is 0.

      Delete
  16. sir, pls. solve this ques.
    [IMG]http://i67.tinypic.com/2qbu3yq.png[/IMG]

    ReplyDelete
    Replies
    1. Given (x - 2) = root(y)
      (x - 2)^3 = x^3 - 6x^2 - 12x - 8
      So you have to find the value of-
      [(x - 2)^3/root(y)] - y
      = 0 [Put (x - 2) = root(y)
      Answer : 0

      Delete
    2. sir,[(x-2)^3/root(y)]-y kese kiya 2nd line k bad??

      Delete
  17. sir, please post some tricks on number systems too!!

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  18. sir post more algebra hacks

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  19. Here again no need to figure out how to solve the question
    √3 = 1.73, √5 = 2.23
    Hence √x = 1.73 - 2.23 or x = 0.25
    Put the value of x
    (0.25)^2 - 16*0.25 + 6
    = 0.0625 - 4 + 6
    = 2 (approx)

    ===================
    Sir, how is x = 0.25?
    √x = 1.73 - 2.23 = 0.5 ... how can x= 0.25? Please explain

    ReplyDelete
  20. Sir,how to solve this problem, xy/x+y=a,xz/x+z=b,yz/y+z=c then x=? Option=1)2abc/ab+bc-ca 2)2abc/ac+bc-ab sir please tell me how to solve this

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  21. Sir,how to solve this problem, xy/x+y=a,xz/x+z=b,yz/y+z=c then x=? Option=1)2abc/ab+bc-ca 2)2abc/ac+bc-ab sir please tell me how to solve this

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  22. sir, please tell the just above question's answer
    same doubt here

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  23. In Que 4 , how to caculate 5a+1/3a = 4.8 orally?

    ReplyDelete