# Simplification Questions with solution

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1. Simplify||सरलीकरण कीजिए $$\frac{1}{3+\frac{1}{2-\frac{1}{\frac{7}{9}}}}+\frac{17}{22}$$

1. $$\frac{12}{22}$$

2. $$\frac{22}{5}$$

3. $$\frac{5}{22}$$

4. 1

= $$\frac{1}{3+\frac{1}{2-\frac{9}{7}}}+\frac{17}{22}$$
= $$\frac{1}{3+\frac{1}{\frac{14-9}{7}}}+\frac{17}{22}$$
= $$\frac{1}{3+\frac{7}{5}}+\frac{17}{22}$$
= $$\frac{1}{\frac{15+7}{5}}+\frac{17}{22}$$
= $$\frac{5}{22}+\frac{17}{22}$$
= $$\frac{22}{22}$$
= 1

2. If $$x = 1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{2}}}}$$, then $$2x+\frac{7}{4}$$ =?

यदि $$x = 1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{2}}}}$$ हो, तो $$2x+\frac{7}{4}$$ = ?

1. 3

2. 4

3. 5

4. 6

$$x = 1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{2}}}}$$
= $$1+\frac{1}{1+\frac{1}{1+\frac{1}{\frac{3}{2}}}}$$
= $$1+\frac{1}{1+\frac{1}{1+\frac{2}{3}}}$$
= $$1+\frac{1}{1+\frac{1}{\frac{3+2}{3}}}$$
= $$1+\frac{1}{1+\frac{3}{5}}$$
= $$1+\frac{1}{\frac{5+3}{5}}$$
= $$1+\frac{5}{8}$$
= $$\frac{13}{8}$$
∴$$2x+\frac{7}{4}$$
= $$2✕\frac{13}{8}+\frac{7}{4}$$
= $$\frac{13}{4}+\frac{7}{4}$$
= $$\frac{20}{4}$$
= 5

3. Simplify||सरलीकरण कीजिए $$\sqrt{\frac{4\tfrac{1}{7}-2\tfrac{1}{4}}{3\tfrac{1}{2}+1\tfrac{1}{7}}\div \frac{2}{2+\frac{1}{2+\frac{1}{5-\frac{1}{5}}}}}$$

1. 1

2. 4

3. 3

4. 2

$$\sqrt{\frac{4\tfrac{1}{7}-2\tfrac{1}{4}}{3\tfrac{1}{2}+1\tfrac{1}{7}}\div \frac{2}{2+\frac{1}{2+\frac{1}{5-\frac{1}{5}}}}}$$
= $$\sqrt{\frac{\frac{29}{7}-\frac{9}{4}}{\frac{7}{2}+\frac{8}{7}}\div \frac{2}{2+\frac{1}{2+\frac{1}{\frac{25-1}{5}}}}}$$
= $$\sqrt{\frac{\frac{116-63}{28}}{\frac{49+16}{14}}\div \frac{2}{2+\frac{1}{2+\frac{5}{24}}}}$$
= $$\sqrt{\frac{53}{28}\times\frac{14}{65}\div \frac{2}{2+\frac{1}{\frac{48+5}{24}}}}$$
= $$\sqrt{\frac{53}{130}\div \frac{2}{2+\frac{24}{53}}}$$
= $$\sqrt{\frac{53}{130}\div \frac{2}{\frac{106+24}{53}}}$$
= $$\sqrt{\frac{53}{130}\div \frac{53}{130}}$$
= $$\sqrt{\frac{53}{130}\times \frac{130}{53}}$$
= $$\sqrt{1}$$
= 1

4. $$\frac{\frac{1}{4}\times \frac{3}{4}\div 1\tfrac{1}{4}of\frac{2}{5}+\frac{1}{8}}{\frac{2}{5}\div 1\tfrac{2}{3}of\frac{3}{25}+1\tfrac{2}{3}of\frac{3}{25}+1\tfrac{3}{4}-\frac{1}{2}}\div \frac{\frac{1}{3}-\frac{1}{3}of\frac{3}{5}}{\frac{1}{5}-\frac{1}{5}of\frac{5}{7}}$$

1. $$\frac{3}{4}$$

2. $$\frac{6}{91}$$

3. $$\frac{31}{4}$$

4. 1

V - Vinculum
B - Bracket
O - Of
D - Division
M - Multiplication
S - Subtraction
= $$\frac{\frac{1}{4}\times \frac{3}{4}\div \frac{5}{4}of\frac{2}{5}+\frac{1}{8}}{\frac{2}{5}\div \frac{5}{3}of\frac{3}{25}+\frac{5}{3}of\frac{3}{25}+\frac{7}{4}-\frac{1}{2}}\div \frac{\frac{1}{3}-\frac{1}{3}of\frac{3}{5}}{\frac{1}{5}-\frac{1}{5}of\frac{5}{7}}$$
= $$\frac{\frac{1}{4}\times \frac{3}{4}\div \frac{1}{2}+\frac{1}{8}}{\frac{2}{5}\div \frac{1}{5}+\frac{7}{4}-\frac{1}{2}}\div \frac{\frac{1}{3}-\frac{1}{5}}{\frac{1}{5}-\frac{1}{7}}$$
= $$\frac{\frac{1}{4}\times \frac{3}{4}\times \frac{2}{1}+\frac{1}{8}}{\frac{2}{5}\times \frac{5}{1}+\frac{7}{4}-\frac{1}{2}}\div \frac{\frac{5-3}{15}}{\frac{7-5}{35}}$$
= $$\frac{\frac{3}{8}+\frac{1}{8}}{2+\frac{7}{4}-\frac{1}{2}}\div \frac{\frac{2}{15}}{\frac{2}{35}}$$
= $$\frac{\frac{3}{8}+\frac{1}{8}}{2+\frac{7}{4}-\frac{1}{2}}\div \left ( \frac{2}{15}\times\frac{35}{2} \right )$$
= $$\frac{\frac{4}{8}}{2+\frac{7-2}{4}}\div \left ( \frac{7}{3} \right )$$
= $$\frac{\frac{4}{8}}{\frac{13}{4}}\times \left ( \frac{3}{7} \right )$$
= $$\frac{1}{2}\times\frac{4}{13}\times \frac{3}{7}$$
= $$\frac{2}{13}\times \frac{3}{7}$$
= $$\frac{6}{91}$$

5. Find the value of $$\frac{1}{5}+999\tfrac{494}{495}\times 99$$

$$\frac{1}{5}+999\tfrac{494}{495}\times 99$$ का मान ज्ञात कीजिए।

1. 90000

2. 99000

3. 90900

4. 99990

$$\frac{1}{5}+999\tfrac{494}{495}\times 99$$
= $$\frac{1}{5}+\frac{999\times 495 + 494}{495}\times 99$$
= $$\frac{1}{5}+\frac{999\times 495 + 494}{5}$$
= $$\frac{1 + (1000-1)\times 495 + 494}{5}$$
= $$\frac{1 + 495000 - 495 + 494}{5}$$
= $$\frac{1 + 495000 - 1}{5}$$
= $$\frac{495000}{5}$$
= 99000

6. Simplify||सरलीकरण कीजिए $$8.7-\left [ 7.6-\left\{6.5 -\left ( 5.4-\overline{4.3-2} \right )\right\} \right ]$$

1. 2.5

2. 3.5

3. 4.5

4. 5.5

$$8.7-\left [ 7.6-\left\{6.5 -\left ( 5.4-\overline{4.3-2} \right )\right\} \right ]$$
= $$8.7-\left [ 7.6-\left\{6.5 -\left ( 5.4-2.3 \right )\right\} \right ]$$
= $$8.7-\left [ 7.6-\left\{6.5 -3.1\right\} \right ]$$
= $$8.7-\left [ 7.6-3.4 \right ]$$
= $$8.7-4.2$$
= 4.5

7. Simplify||सरलीकरण कीजिए $$\frac{0.05\times0.05\times0.05-0.04\times0.04\times0.04}{0.05\times0.05+0.002+0.04\times0.04}$$

1. 1

2. 0.1

3. 0.01

4. 0.001

$$\frac{0.05\times0.05\times0.05-0.04\times0.04\times0.04}{0.05\times0.05+0.002+0.04\times0.04}$$
∵ a3 - b3 = (a - b)(a2 + ab + b2)
= $$\frac{0.05^{3}-0.04^{3}}{0.05^{2}+0.002+0.04^{2}}$$
= $$\frac{\left ( 0.05 - 0.04 \right )\left ( 0.05^{2}+0.05\times0.04+0.04^{2} \right )}{0.05^{2}+0.002+0.04^{2}}$$
= 0.05 - 0.04
= 0.01

8. If '+' means '÷', '✕' means '-', '÷' means '✕' and '-' means '+', then what will be the value of the following expression?

यदि '+' का अर्थ है '÷', '✕' का अर्थ है '-', '÷' का अर्थ है '✕' और '-' का अर्थ है '+' तो निम्नलिखित व्यंजक का मान क्या होगा?

9 + 3 ÷ 4 - 8 ✕ 2 = ?

1. $$6\tfrac{1}{4}$$

2. $$6\tfrac{3}{4}$$

3. $$-1\tfrac{3}{4}$$

4. 18

= 9 + 3 ÷ 4 - 8 ✕ 2
= 9 ÷ 3 ✕ 4 + 8 - 2
= $$= \frac{9}{3}\times 4+8-2$$
= 3 ✕ 4+6
= 18

9. Simplify||सरलीकरण कीजिए

$$\left ( 113^{2}+115^{2}+117^{2}-113\times 115-115\times 117-117\times 113 \right)$$

1. 0

2. 4

3. 8

4. 12

$$a^{2}+b^{2}+c^{2}-ab-bc-ac$$
$$= \frac{1}{2}\left ( 2a^{2}+2b^{2}+2c^{2}-2ab-2bc-2ac \right )$$
∵$$= \frac{1}{2}\left [ (a-b)^{2}+(b-c)^{2}+(c-a)^{2} \right ]$$
$$= \frac{1}{2}\left [ (113-115)^{2}+(115-117)^{2}+(117-113)^{2} \right ]$$
where|जहाँ a = 113, b = 115, c = 117
$$= \frac{1}{2}\left [ (-2)^{2}+(-2)^{2}+(4)^{2} \right ]$$
$$= \frac{1}{2}\left ( 4+4+16 \right )$$
$$= \frac{1}{2}\left ( 24 \right )$$
= 12

10.
$$\left ( 999\tfrac{1}{7}+999\tfrac{2}{7}+999\tfrac{3}{7}+999\tfrac{4}{7}+999\tfrac{5}{7}+999\tfrac{6}{7} \right )=?$$

1. 2997

2. 5979

3. 5997

4. 5994

∵=$$\left ( 6\times999 \right )+\frac{1}{7}+\frac{2}{7}+\frac{3}{7}+\frac{4}{7}+\frac{5}{7}+\frac{6}{7}$$
$$= 6\times \left ( 1000-1 \right )+\frac{21}{7}$$
= (6000-6+3)
= 5997

11. $$\frac{0.342\times 0.684}{0.000342\times 0.000171}$$ का वर्गमूल है-

$$\frac{0.342\times 0.684}{0.000342\times 0.000171}$$ is the square root of-

1. 250

2. 2500

3. 2000

4. 4000

= $$\frac{0.342\times 0.684}{0.000342\times 0.000171}$$
= $$\frac{342\times 684\times 1000000}{342\times 171}$$
= $$\frac{684\times 1000000}{171}$$
= 4✕1000000
square root|वर्गमूल = $$\sqrt{4\times 1000000}$$
= 2000

12. If the expression $$\frac{x^{2}}{y^{2}}+tx+\frac{y^{2}}{4}$$ is a perfect square, then what will be the value of t?

यदि व्यंजक $$\frac{x^{2}}{y^{2}}+tx+\frac{y^{2}}{4}$$ एक पूर्ण वर्ग हो, तो t का मान कितना होगा?

1. ±1

2. ±2

3. 0

4. ±3

= If|यदि $$a=\frac{x}{y}, b=\frac{y}{2}$$
= So|तो,$$\frac{342\times 684\times 1000000}{342\times 171}$$
= $$\pm2ab=\pm 2\times \frac{x}{y}\times \frac{y}{2}$$
= ±x
∴tx = ±x
t = ±1

13. If (x + 7954✕7956) is a square number, then what will be the value of 'x'?

यदि (x + 7954✕7956) एक वर्ग संख्या हो तो 'x' का मान क्या होगा?

1. 1

2. 16

3. 9

4. 4

= (x + 7954✕7956)
= x + 7954✕(7954 + 2)
= (7954)2 + 2✕7954✕1 + x
For the above to be a whole number|उपर्युक्त को पूर्ण संख्या होने के लिए,
x = 12 = 1

14. The smallest number which when added to 680621 makes the sum a perfect square number?

वह सबसे छोटी संख्या, जिसे 680621 में जोड़ने पर योगफल एक पूर्ण वर्ग संख्या बन जाए?

1. 4

2. 5

3. 6

4. 8

∴(824)2 < 680621 < (825)2
= (825)2 - 680621
= 680625 - 680621
= 4
Short trick
भाजक + 1 - शेषफल | divisor + 1 - remainder
= 1648+1 - 1645
= 1649 - 1645
= 4

15. For the number 1276* to be a perfect square, the value of the missing digit (*) will be?

संख्या 1276* को एक पूर्ण वर्ग संख्या होने के लिए, लुप्त अंक (*) का मान होगा?

1. 1

2. 4

3. 8

4. 9

Obviously 9 will come in place of *.|स्पष्टतः * के स्थान पर 9 आएगा।

16. What is the smallest number subtracted from 4750 to get a perfect square number?

4750 में से किस सबसे छोटी संख्या को घटाने पर एक पूर्ण वर्ग संख्या प्राप्त होगी?

1. 126

2. 162

3. 216

4. 612

So on subtracting 126, the remaining number will become a perfect square of 68.|अतः 126 घटाने पर शेष संख्या 68 का पूर्ण वर्ग बन जाएगा।

17. What is the least positive integer to be subtracted from 3011 ✕ 3012 so that the remainder is a perfect square?

3011 ✕ 3012 से कौन सा न्यूनतम धन पूर्णांक घटाया जाए कि शेषफल पूर्ण वर्ग हो?

1. 3009

2. 3010

3. 3011

4. 3012

3011 ✕ 3012
= 3011 ✕ (3011 + 1)
= (3011)2 + 3011
Obviously subtracting 3011 will make the number a perfect square.|स्पष्टतः 3011 घटाने पर संख्या पूर्ण वर्ग बन जाएगी।

18. By what smallest number should 5808 be multiplied so that it becomes a perfect square?

5808 को किस छोटी से छोटी संख्या से गुणा किया जाए कि वह पूर्ण वर्ग बन जाए?

1. 2

2. 7

3. 11

4. 3

= 2 ✕ 2 ✕ 2 ✕ 2 ✕ 3 ✕ 11 ✕ 11
Obviously multiplying by 3 will make the number a perfect square.|स्पष्टतः 3 से गुणा करने पर संख्या पूर्ण वर्ग बन जाएगी।

19. What is the least number by which 20184 will be divided to get a perfect square?

वह छोटी से छोटी संख्या जिससे 20184 को भाग देने पर एक पूर्ण वर्ग प्राप्त होगा?

1. 2

2. 3

3. 5

4. 6

∴20184 = 2 ✕ 2 ✕ 2 ✕ 3 ✕ 29 ✕ 29
= 22 ✕ 2 ✕ 3 ✕ 292
= 22 ✕ 6 ✕ 292
Obviously dividing by 6 will make the number a perfect square.|स्पष्टतः 6 से भाग करने पर संख्या पूर्ण वर्ग बन जाएगी।

20. The average of the two numbers is 7.5 and the square root of their product is 6. Those are numbers.

दो संख्याओं का औसत 7.5 है और उनके गुणनफल का वर्गमूल 6 है। वे संख्याएँ हैं।

1. 13, 2

2. 9, 6

3. 10, 5

4. 12, 3

Let the numbers|माना संख्याएँ = x, y
$$\therefore \frac{x+y}{2}=7.5$$
x + y = 7.5✕2
x + y = 15....(1)
xy = 62 = 36.....(2)
from eq. 1 and 2|समीकरण 1 और 2 से,
x(15 -x) = 36
15x - x2 = 36
x2 - 15x + 36 = 0
x2 - 12x - 3x + 36 = 0
x(x - 12) -3(x - 12) = 0
(x - 12)(x - 3) =0
x = 12, 3
Shortcut- Do it through mental action.इसे मानसिक क्रिया द्वारा करें।
x + y = 15, xy = 36
x + y = 12 + 3, xy = 12 ✕ 3
x and y = 12, 3

21. The general of the army wants to make a square array of 36562 soldiers, after making the array, some soldiers were left. How many soldiers were left?

सेना का जनरल 36562 सैनिकों का वर्गाकार व्यूह बनाना चाहता है व्यूह बनाने के बाद कुछ सैनिक बच गए। कुल कितने सैनिक बचे?

1. 81

2. 36

3. 97

4. 65

Total soldiers left|कुल सैनिक बचे = 81

22. What will be the square root of $$\frac{2+\sqrt{3}}{2}$$?

$$\frac{2+\sqrt{3}}{2}$$ का वर्गमूल होगा?

1. $$\pm\frac{1}{\sqrt{2}}\left ( \sqrt{3}+1 \right )$$

2. $$\pm\frac{1}{2}\left ( \sqrt{3}- 2 \right )$$

3. $$\pm\frac{1}{2}\left ( \sqrt{3}- 1 \right )$$

4. None of the above||इनमें से कोई नहीं

$$\frac{2+\sqrt{3}}{2}$$
= $$\frac{2\left ( 2+\sqrt{3} \right )}{4}$$
= $$\frac{\left ( 4+2\sqrt{3} \right )}{4}$$
= $$\frac{\left ( 3 + 1 +2\sqrt{3} \right )}{4}$$
= $$\frac{\left (\sqrt{3} \right )^{2}+1^{2}+2\times \sqrt{3}\times 1}{4}$$
= $$\left ( \frac{\sqrt{3}+1}{4} \right )^{2}$$
= $$\therefore \sqrt{\frac{2+\sqrt{3}}{2}}=\pm \frac{\sqrt{3}+1}{2}$$
= $$\pm\frac{1}{2}\left ( \sqrt{3}+ 1 \right )$$

23. If p = 999, then the value of $$\sqrt[3]{p(p^{2}+3p+3)+1}$$will be?

यदि p = 999 हो, तो $$\sqrt[3]{p(p^{2}+3p+3)+1}$$ का मान होगा?

1. 1000

2. 999

3. 998

4. 1002

$$\sqrt[3]{p(p^{2}+3p+3)+1}$$
= $$\sqrt[3]{p^{3}+3p^{2}+3p+1}$$
= $$\sqrt[3]{p^{3}+3p^{2}+3p+1}$$
= $$\sqrt[3]{(p+1)^{3}}$$
p + 1 = 999 + 1 = 1000

24. What will be the value of (1001)3?

(1001)3 का मान क्या होगा?

1. 1003003001

2. 100303001

3. 100300301

4. 103003001

(1001)2
= 1002001
See the pattern|पैटर्न को देखें
(1001)3
1003003001

25. Find the least number by which the quotient obtained by dividing 625 is a perfect cube root.

वह लघुत्तम संख्या ज्ञात कीजिए जिसके द्वारा 625 में भाग दी जाने पर भागफल पूर्ण घनफल हो।

1. 25

2. 5

3. 2

4. 3

625 = 5 ✕ 5 ✕ 5 ✕ 5
= 53 ✕ 5
Divide by 5 to make a perfect cube|पूर्ण घन बनाने के लिए 5 से भाग देना होगा
625 ÷ 5 = 125 = 53

26. The sum of the cubes of two numbers is 793. If the sum of the numbers is 13, then find the difference between the two numbers.

दो संख्याओं के घनों का योग 793 है। संख्याओं का योग 13 है, तो दोनों संख्याओं का अंतर बताइए।

1. 7

2. 6

3. 5

4. 8

Let the numbers|माना संख्याएँ = x, y
where x > y
x3 + y3 = 793
and x + y = 13
∴(a + b)3 = a3 + b3 + 3ab(a + b)
(13)3 = 793 + 3xy✕13
2197 = 793 + 39xy
$$xy = \frac{1404}{39}=36$$
∴(a - b)2 = (a + b)2 - 4ab
= (13)2 - 4 ✕ 36
= 169 - 144 = 25
$$(a - b) = \sqrt{25}=5$$

27. Find the cube root of (-13824).

(-13824) का घनमूल ज्ञात कीजिए।

1. 38

2. -38

3. 24

4. -24

∴13824 = 23 ✕ 23 ✕ 23 ✕ 33
∴$$\sqrt[3]{-13824}=\sqrt[3]{(-1)^{3}(2)^{3}\times2^{3}\times2^{3}\times3^{3} }$$
= -1 ✕ 2 ✕ 2 ✕ 2 ✕ 3 = -24

28. If the square root of x is the cube root of y, then the relation between x and y is.

यदि x का वर्गमूल y का घनमूल है तो x और y के बीच सम्बन्ध है।

1. x3 = y2

2. x2 = y3

3. x = y

4. x6 = y5

$$\sqrt{x}=\sqrt[3]{y}$$
$$x^{\frac{1}{2}}=y^{\frac{1}{3}}$$
$$\left ( x^{\frac{1}{2}} \right )^{6}=\left ( y^{\frac{1}{3}} \right )^{6}$$
$$x^{3}=y^{2}$$

29. The difference between $$\frac{3}{4}$$ and $$\frac{4}{7}$$ of a number is 100, what will that number be?

किसी संख्या के $$\frac{3}{4}$$ तथा $$\frac{4}{7}$$ का अंतर 100 है वह संख्या क्या होगी?

1. 520

2. 540

3. 500

4. 560

Let the number|माना संख्या = x
$$\frac{3}{4}x -\frac{4}{7}x$$ = 100
21x - 16x = 2800
5x = 2800
x = 560

30. 103x = 125, then what will be the value of 10-2x?

यदि 103x = 125 हो, तो 10-2x का मान क्या होगा?

1. $$\frac{1}{5}$$

2. $$\frac{1}{25}$$

3. 25

4. $$\frac{-1}{25}$$

∴$$10^{-2x}=\frac{1}{10^{2x}} = \frac{1}{25}$$