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51.

Using the sutra Shunyam Samyaschahye, Solve the equation:1. 1/(x + 4) + 1/(x - 6) = 02. 5/(3x + 2) + 5/(2x + 8) = 03. (2x + 4)/(2x + 1) = (2x + 1)/(2x + 4)4. (3x + 2)/(5x + 7) = (x + 1)/(3x - 1)

Answer»

1. 1/(x + 4) + 1/(x - 6) = 0

Here Numerator of two fractions are same = 1, 

So According to formula : 

x + 4 + x- 6 = 0 

⇒ 2x – 2 = 0 

⇒ 2x = 

⇒ x = 1

2. 5/(3x + 2) + 5/(2x + 8) = 0

Here, Numerator of two fractions are same = 5, 

So according to formula : 

3x + 2 + 2x + 8 = 0 

⇒ 3x + 2x + 2 + 8 = 0 

⇒ 5x + 10 = 0 

⇒ 5x = -10 

⇒ x = -2

3. (2x + 4)/(2x + 1) = (2x + 1)/(2x + 4)

Sum of numerators of both sides

= 2x + 4 + 2x + 1 = 4x + 5

Sum of denominators in both sides

= 2x + 1 + 2x + 4 = 4x + 5 

Two sums are equal, so by the formula 4x + 5 = 0

⇒ 4x = -5

⇒ x = -5/4

4. (3x + 2)/(5x + 7) = (x + 1)/(3x - 1)

Sum of numerators of two sides 

= 3x + 2 + x + 1 = 4x + 3 …..(i) 

Sum of denominators of two sides 

= 5x + 7 + 3x – 1 = 8x + 6 …(ii) 

Ratio of (i) and (ii) is 1 : 2.

So, by formula, equation any sum equal to zero, 

4x + 3 = 0 

⇒ 4x = – 3 

⇒ x = - 3/4
Discussion
52.

Find the square of the following numbers1. 1032. 953. 2044. 225

Answer»

1. 103

1032= 103 + 03 / (03)2

= 106 / 09 = 10609

2. 95

952 = 95 – 05 / (-05)2

Here deviation = (- 5)

= 90/25 = 9025

3. 204

2042 = 2 (204 + 4) / (04)2

= 2 (208)/16

= 416/16

= 41616

4. 225

2252 = 2 (225 + 25) / 252

= 2(250) /625

= 500 /625

= 50625
Discussion
53.

Find the cube of 103.

Answer»

Base = 100, Deviation = + 03 

(103)3 = 103 + 2 x 03 / 3 x (03)2 / (03)3 

= 103 + 6 / 27 / 27

= 1092727
Discussion
54.

Find the cube of 15.

Answer»

Base = 10, Deviation = + 5 

Then, (15)3 = 15 + 2 x 5 / 3 x 52 / 53

= 15 + 10 / 75 / 125 

= 25 / 75 / 125 

= 25 /8 7 / 5 

= 33/7/5 

= 3375
Discussion
55.

Find the cube of 96.

Answer»

Base = 100, Deviation = (- 04) 

(96)3 = 96 + 2 x (- 04)/3 x (- 04)2 /(- 04)3 

= 96 – 08 / 3 x 16 / – 64 

= 88 / 48 / – 64

= 88 / 48 /1 – 64 

= 88 / 47 / 100 – 64 

= 88 / 47 / 36 

= 884736
Discussion
56.

Sub-base digit is calculated as (a) Base ÷ Sub-base (b) Sub-base – Base (c) Sub-base ÷ Base (d) Base – Sub-base

Answer»

(c) Sub-base ÷ Base
Discussion
57.

If the number is 22 and the base is 20, then the deviation is (a) 10 (b) 20 (c) 2 (d) – 2

Answer»

If the number is 22 and the base is 20, then the deviation is 2.
Discussion
58.

If base is 10 and sub-base is 30 then what sub-base digit will be (a) 3 (b) 2 (c) 1 (d) 43

Answer»

If base is 10 and sub-base is 30 then what sub-base digit will be 3.
Discussion
59.

How many groups makes when three digit number is multiplied by three digit number (a) 3 (b) 5 (c) 6 (d) 9

Answer»

5 groups makes when three digit number is multiplied by three digit number.
Discussion
60.

Multiply with the help of Sutra Ekanyunena Purvena(i) 54 x 99(ii) 214 x 999(iii) 47 x 999(iv)  342 x 99999(v) 73 x 9(vi) 467 x 99

Answer»

(i) 54 x 99

L.H.S = 54 – 1 = 53

R.H.S = 99 – 53 = 46

∴54 x 99 = 54 – 1/99 – 53

= 5346

(ii) 214 x 999

L.H.S = 214 – 1 = 213

R.H.S = 9 99 – 213 = 786

∴ 214 x 999 = 214 – 1/999 – 213

= 213786

(iii) 47 x 999

L.H.S = 047 – 1 = 046

R.H.S = 999 – 046 = 953

∴ 47 x 999 = 047 – 1/999 – 046

= 046953

= 46953

(iv)  342 x 99999

L.H.S = 00342 – 1 = 00341

R.H.S = 99999 – 00341 = 99658

∴00342 x 99999

= 00342 – 1 /99999 – 00341

= 0034199658

(v) 73 x 9

L.H.S = 73 – 1 = 72

R.H.S = 9 – 72

∴ 73 x 9 = 73 – 1/9 – 72

= 72/9 – 72

= 729 – 72

= 657

(vi) 467 x 99

L.H.S = 467 – 1 = 466

R.H.S = 99 – 466

∴ 467 x 99 = 467 – 1/99 – 466

= 466/99 – 466

= 46699 – 466

= 46233
Discussion
61.

Find the cube of the following numbers1. 152. 913. 324. 208

Answer»

1. 15

153 = 15 + 2 x (5) / 3 x (5)2 / 53

[Base = 10, deviation = 5] 

= 25/75/125 = 25/75/125

= 25/87/5 = 25/87/5 = 33 / 7 / 5

2. 91

913

= 91 + 2 x (-09) / 3 x (-09)2 / (-09)3

[Base = 100, deviation = (-09)]

= 73 / 243 / (-729)

= 73 /2 43 / (-729)

= 75 / 43 / -729

= 754300 – 729

= 753571

3. 32

323 = 32 (32 + 2 x 2) / 3 x 3 x (2)2 / 23

[Base =10, sub-base = 3, deviation = 2]

= 9 x 36/9 x 4/8

= 324 / 36 / 8

= 324 /3 6 / 8

= 327 / 6 / 8

= 32768

4. 208

2083 = 22 (208 + 2 x 8)/2 x 3 x (8)2 / (8)3

[Base = 100, sub-base = 2, deviation = 8]

= 4 (224) / 384 / 512

= 896 /384 / 512

= 899 / 84 /512

= 899 / 89 / 12

= 8998912
Discussion
62.

In multiplication process by Urdhva Tiryagbhyam, the number of groups is known by (a) (2n – 1) (b) (2n + 2) (c) (2n + 1) (d) (2n – 2)

Answer»

In multiplication process by Urdhva Tiryagbhyam, the number of groups is known by (2n – 1).
Discussion
63.

Multiply with the help of Sutra Ekanyunena Purvena(a) 8567 x 9999(b) 512 x 99

Answer»

(a) 8567 x 9999

L.H.S = 8567 – 1 = 8566

R.H.S = 9999 – 8566 = 1433

∴ 8567 x 9999

= 8567 – 1 / 9999 – 8566

= 85661433

(b) 512 x 99

L.H.S = 512 – 1 = 511

R.H.S = 99 – 511

∴ 512 x 99

= 511 / 99 – 511

= 51199 – 511

= 50688
Discussion
64.

Find the square of the following numbers by Dvanda yoga method or Duplex process:(a) 53214(b) 31321

Answer»

(a) 532142

= Dvanda of 5 / Dvanda of 53 / Dvanda of 532 / Dvanda of 5321 / Dvanda of 53214 / Dvanda of 3214 / Dvanda of 214 / Dvanda of 14 / Dvanda of 4

= 52 / 2 x 5 x 3 / 2 x (2 x 5) + 32 / 2 x (5 x 1) + 2 (3 x 2) / 2 (5 x 4) + 2 (3 x 1) + 22 / 2 (3 x 4) + 2 (2 x 1) / 2 x (2 x 4) + 12 / 2 x 1 x 4 / 42

= 25 / 30 / 29 / 22 / 50 / 28 / 17 / 8 / 16

= 2831729796

(b) 313212

= Dvanda of 3 / Dvanda of 13 / Dvanda of 132 / Dvanda of 3132 / Dvanda of 31321 / Dvanda of 1321 / Dvanda of 321 / Dvanda of 21 / Dvanda of 1

= 32 / 2 x 1 x 3 / 2 (1 x 2) + 32 / 2 x (3 x 2) + 2 (1 x 3) / 2 (3 x 1) + 2 (1 x 2) + 32 / 2 (1 x 1) + 2 (3 x 2) / 2 x (3 x 1) + 22 / 2 x 2 x 1 / 12

= 9/6/13/18/19/14/10/4/1

= 981005041
Discussion