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1. Question

English: If x + 1/x = 7, find x⁴ + 1/x⁴.

Hindi: यदि x + 1/x = 7 है, तो x⁴ + 1/x⁴ ज्ञात कीजिए।

Solution:

x + 1/x = 7

Square both sides:

x² + 1/x² + 2 = 49

x² + 1/x² = 47

Again square:

x⁴ + 1/x⁴ + 2 = 47²

x⁴ + 1/x⁴ + 2 = 2209

x⁴ + 1/x⁴ = 2207

Answer: 2207

2. Question

English: If a : b = 3 : 4 and b : c = 5 : 6, and a + b + c = 177, find c.

Hindi: यदि a : b = 3 : 4 और b : c = 5 : 6 है, तथा a + b + c = 177 है, तो c ज्ञात कीजिए।

Solution:

a : b = 3 : 4

b : c = 5 : 6

LCM of 4 and 5 = 20

So:

a : b = 15 : 20

b : c = 20 : 24

Therefore:

a : b : c = 15 : 20 : 24

Total ratio:

= 15 + 20 + 24

= 59

59 parts = 177

1 part = 3

c = 24 × 3

= 72

Answer: 72

3. Question

English: In an AP, the sum of the 8th and 12th terms is 80, and the 15th term is 60. Find the sum of the first 20 terms.

Hindi: एक AP में 8वें और 12वें पदों का योग 80 है, और 15वाँ पद 60 है। पहले 20 पदों का योग ज्ञात कीजिए।

Solution:

8th term:

a + 7d

12th term:

a + 11d

Their sum:

2a + 18d = 80

a + 9d = 40

15th term:

a + 14d = 60

Subtract:

5d = 20

d = 4

Now:

a + 9 × 4 = 40

a + 36 = 40

a = 4

Sum of first 20 terms:

S = 20/2 × [2a + 19d]

= 10 × [8 + 76]

= 10 × 84

= 840

Answer: 840

4. Question

English: Three positive numbers are in GP. Their sum is 26 and their product is 216. Find the sum of their squares.

Hindi: तीन धनात्मक संख्याएँ GP में हैं। उनका योग 26 और गुणनफल 216 है। उनके वर्गों का योग ज्ञात कीजिए।

Solution:

Let numbers be:

a/r, a, ar

Product:

= a³ = 216

a = 6

So numbers are:

6/r, 6, 6r

Their sum:

6/r + 6 + 6r = 26

Divide by 2:

3/r + 3 + 3r = 13

r + 1 + 1/r = 13/3

Try r = 2

Numbers become:

3, 6, 12

Sum = 21

Try r = 3/2

Numbers:

4, 6, 9

Sum = 19

Try r = 2 and scaling not working.

Let numbers be:

6/r, 6, 6r

Equation:

1/r + 1 + r = 13/3

Multiply by 3r:

3 + 3r + 3r² = 13r

3r² - 10r + 3 = 0

(3r - 1)(r - 3) = 0

r = 3

Numbers are:

2, 6, 18

Sum of squares:

= 2² + 6² + 18²

= 4 + 36 + 324

= 364

Answer: 364

5. Question

English: Find the least positive integer N such that N ≡ 1 mod 6, N ≡ 2 mod 7, and N ≡ 3 mod 8.

Hindi: सबसे छोटी धनात्मक संख्या N ज्ञात कीजिए जिसके लिए N ≡ 1 mod 6, N ≡ 2 mod 7, और N ≡ 3 mod 8 हो।

Solution:

Check option 163:

163 divided by 6 leaves remainder 1.

163 divided by 7 leaves remainder 2.

163 divided by 8 leaves remainder 3.

Answer: 163

6. Question

English: Find the total number of divisors of 50400.

Hindi: 50400 के कुल भाजकों की संख्या ज्ञात कीजिए।

Solution:

50400 = 504 × 100

504 = 2³ × 3² × 7

100 = 2² × 5²

So:

50400 = 2⁵ × 3² × 5² × 7¹

Number of divisors:

= (5 + 1)(2 + 1)(2 + 1)(1 + 1)

= 6 × 3 × 3 × 2

= 108

Answer: 108

7. Question

English: Find the remainder when 11¹²³ is divided by 17.

Hindi: 11¹²³ को 17 से भाग देने पर शेषफल ज्ञात कीजिए।

Solution:

11² = 121

121 divided by 17 leaves remainder 2.

11⁴ gives remainder 4.

11⁸ gives remainder 16 = -1

11¹⁶ gives remainder 1

123 divided by 16 leaves remainder 11.

So:

11¹²³ has same remainder as 11¹¹

Now:

11⁸ remainder = -1

11² remainder = 2

11 remainder = 11

So:

11¹¹ remainder

= -1 × 2 × 11

= -22

-22 divided by 17 leaves remainder 12.

Answer: 12

8. Question

English: How many 5-digit numbers greater than 30000 can be formed using digits 0 to 7 without repetition, if the number must be divisible by 5?

Hindi: 0 से 7 तक के अंकों का बिना पुनरावृत्ति उपयोग करके 30000 से बड़ी कितनी 5-अंकीय संख्याएँ बनाई जा सकती हैं, यदि संख्या 5 से विभाज्य होनी चाहिए?

Solution:

Number divisible by 5 means last digit must be 0 or 5.

Case 1: Last digit = 0

First digit must be greater than or equal to 3.

Choices for first digit:

3, 4, 5, 6, 7

= 5 choices

Middle 3 places:

6 × 5 × 4

= 120

Total:

= 5 × 120

= 600

Case 2: Last digit = 5

First digit cannot be 0 or 5.

Choices:

3, 4, 6, 7

= 4 choices

Middle places:

6 × 5 × 4

= 120

Total:

= 4 × 120

= 480

Total numbers:

= 600 + 480

= 1080

Answer: 1080

9. Question

English: A bag contains 5 red and 4 blue balls. Three balls are drawn at random. Find the probability that exactly 2 balls are red.

Hindi: एक थैले में 5 लाल और 4 नीली गेंदें हैं। यादृच्छया 3 गेंदें निकाली जाती हैं। ठीक 2 गेंदों के लाल होने की प्रायिकता ज्ञात कीजिए।

Solution:

Total ways:

9C3 = 84

Favourable ways:

Choose 2 red from 5:

5C2 = 10

Choose 1 blue from 4:

4C1 = 4

Favourable ways:

= 10 × 4

= 40

Probability:

= 40/84

= 10/21

Answer: 10/21

10. Question

English: A can finish a work in 24 days, B in 36 days and C in 48 days. All three work together for 4 days, then A leaves. In how many more days will B and C finish the remaining work?

Hindi: A किसी कार्य को 24 दिनों में, B 36 दिनों में और C 48 दिनों में पूरा करता है। तीनों 4 दिन साथ काम करते हैं, फिर A चला जाता है। शेष कार्य B और C कितने और दिनों में पूरा करेंगे?

Solution:

A’s 1-day work = 1/24

B’s 1-day work = 1/36

C’s 1-day work = 1/48

Together:

= 1/24 + 1/36 + 1/48

LCM = 144

= 6 + 4 + 3 /144

= 13/144

Work done in 4 days:

= 52/144

= 13/36

Remaining work:

= 1 - 13/36

= 23/36

Now B and C work together:

= 1/36 + 1/48

= 4 + 3 /144

= 7/144

Time required:

= 23/36 ÷ 7/144

= 23/36 × 144/7

= 92/7 days

Answer: 92/7 days

11. Question

English: Pipe A fills a tank in 20 hours, pipe B fills it in 30 hours, and pipe C empties it in 60 hours. If all three are opened together, in how many hours will the tank be filled?

Hindi: पाइप A टंकी को 20 घंटे में भरता है, पाइप B 30 घंटे में भरता है, और पाइप C 60 घंटे में खाली करता है। तीनों को साथ खोलने पर टंकी कितने घंटे में भरेगी?

Solution:

A’s 1-hour work = 1/20

B’s 1-hour work = 1/30

C’s 1-hour work = -1/60

Total work:

= 1/20 + 1/30 - 1/60

LCM = 60

= 3 + 2 - 1 /60

= 4/60

= 1/15

So tank fills in:

15 hours

Answer: 15 hours

12. Question

English: A invests ₹12000 for 8 months, B invests ₹15000 for 6 months, and C invests for 10 months. If their profit ratio is 16 : 15 : 12, find C's investment.

Hindi: A ₹12000 को 8 महीने के लिए, B ₹15000 को 6 महीने के लिए निवेश करता है, और C 10 महीने के लिए निवेश करता है। यदि उनका लाभ अनुपात 16 : 15 : 12 है, तो C का निवेश ज्ञात कीजिए।

Solution:

Profit ratio proportional to:

Investment × time

A:

12000 × 8 = 96000

B:

15000 × 6 = 90000

Ratio:

96000 : 90000 = 16 : 15

So C’s contribution must correspond to 12.

Let investment of C = x

x × 10 corresponds to 12.

Using ratio:

96000/16 = x × 10 /12

6000 = 10x/12

72000 = 10x

x = 7200

Answer: ₹7200

13. Question

English: Two runners run on a circular track of length 600 m with speeds 18 km/h and 12 km/h in the same direction. After how much time will they meet again at the same point?

Hindi: दो धावक 600 मीटर लंबी वृत्ताकार पथ पर 18 किमी/घंटा और 12 किमी/घंटा की गति से एक ही दिशा में दौड़ते हैं। वे फिर से कितने समय बाद उसी बिंदु पर मिलेंगे?

Solution:

Relative speed:

= 18 - 12

= 6 km/h

Convert into m/s:

= 6 × 5/18

= 5/3 m/s

Time:

= distance / relative speed

= 600 ÷ 5/3

= 600 × 3/5

= 360 seconds

= 6 minutes

Answer: 6 minutes

14. Question

English: A train crosses a pole in 8 seconds and a 240 m long platform in 20 seconds. Find the speed of the train.

Hindi: एक ट्रेन एक खंभे को 8 सेकंड में और 240 मीटर लंबे प्लेटफॉर्म को 20 सेकंड में पार करती है। ट्रेन की गति ज्ञात कीजिए।

Solution:

Let length of train = L

Crossing pole:

Speed = L/8

Crossing platform:

Speed = (L + 240)/20

So:

L/8 = (L + 240)/20

20L = 8L + 1920

12L = 1920

L = 160 m

Speed:

= 160/8

= 20 m/s

Convert into km/h:

= 20 × 18/5

= 72 km/h

Answer: 72 km/h

15. Question

English: The sides of a triangle are 13 cm, 14 cm and 15 cm. Find its inradius.

Hindi: एक त्रिभुज की भुजाएँ 13 सेमी, 14 सेमी और 15 सेमी हैं। उसकी अंतःत्रिज्या ज्ञात कीजिए।

Solution:

Semi-perimeter:

= (13 + 14 + 15)/2

= 42/2

= 21

Area by Heron formula:

= √[21 × 8 × 7 × 6]

= √7056

= 84

Inradius:

= Area / semi-perimeter

= 84/21

= 4 cm

Answer: 4 cm

16. Question

English: Find the reflection of point (3,4) in the line x + y = 0.

Hindi: बिंदु (3,4) का रेखा x + y = 0 में प्रतिबिंब ज्ञात कीजिए।

Solution:

Line x + y = 0 means y = -x.

Reflection rule in line y = -x:

(x, y) becomes (-y, -x)

So:

(3,4) becomes (-4,-3)

Answer: (-4,-3)

17. Question

English: A product is sold at 20% loss. If the selling price is increased by ₹96, there is a profit of 12%. Find the cost price.

Hindi: एक वस्तु 20% हानि पर बेची जाती है। यदि विक्रय मूल्य ₹96 बढ़ा दिया जाए, तो 12% लाभ होता है। क्रय मूल्य ज्ञात कीजिए।

Solution:

Let cost price = 100x

Selling price at 20% loss:

= 80x

Selling price at 12% profit:

= 112x

Difference:

= 112x - 80x

= 32x

Given difference = ₹96

32x = 96

x = 3

Cost price:

= 100x

= ₹300

Answer: ₹300

18. Question

English: A 40-litre mixture contains milk and water in the ratio 5 : 3. How much water should be added to make the ratio 1 : 1?

Hindi: 40 लीटर मिश्रण में दूध और पानी का अनुपात 5 : 3 है। अनुपात 1 : 1 करने के लिए कितना पानी मिलाया जाना चाहिए?

Solution:

Total mixture = 40 litres

Milk : Water = 5 : 3

Total parts = 8

Milk:

= 40 × 5/8

= 25 litres

Water:

= 40 × 3/8

= 15 litres

To make ratio 1 : 1, water should also be 25 litres.

Water to be added:

= 25 - 15

= 10 litres

Answer: 10 litres

19. Question

English: The difference between compound interest and simple interest for 2 years at 12% per annum is ₹72. Find the principal.

Hindi: 12% वार्षिक दर से 2 वर्षों के लिए चक्रवृद्धि ब्याज और साधारण ब्याज का अंतर ₹72 है। मूलधन ज्ञात कीजिए।

Solution:

Difference between CI and SI for 2 years:

= P × R² / 10000

Given:

Difference = ₹72

Rate = 12%

So:

72 = P × 12² / 10000

72 = P × 144 / 10000

P = 72 × 10000 / 144

P = 5000

Answer: ₹5000

20. Question

English: The present ages of A and B are in the ratio 4 : 5. Six years ago, their ages were in the ratio 7 : 9. Find the sum of their present ages.

Hindi: A और B की वर्तमान आयु का अनुपात 4 : 5 है। 6 वर्ष पहले उनकी आयु का अनुपात 7 : 9 था। उनकी वर्तमान आयु का योग ज्ञात कीजिए।

Solution:

Let present ages be 4x and 5x.

Six years ago:

A = 4x - 6

B = 5x - 6

Given:

(4x - 6) / (5x - 6) = 7/9

9(4x - 6) = 7(5x - 6)

36x - 54 = 35x - 42

x = 12

Present ages:

A = 4 × 12 = 48

B = 5 × 12 = 60

Sum:

= 48 + 60

= 108 years

Answer: 108 years

21. Question

English: If x + y = 13 and x² + y² = 89, find x³ + y³.

Hindi: यदि x + y = 13 और x² + y² = 89 है, तो x³ + y³ ज्ञात कीजिए।

Solution:

Given:

x + y = 13

x² + y² = 89

First find xy:

(x + y)² = x² + y² + 2xy

13² = 89 + 2xy

169 = 89 + 2xy

2xy = 80

xy = 40

Now:

x³ + y³ = (x + y)³ - 3xy(x + y)

= 13³ - 3 × 40 × 13

= 2197 - 1560

= 637

Answer: 637

22. Question

English: If tan θ = 2, find (sin θ + cos θ)².

Hindi: यदि tan θ = 2 है, तो (sin θ + cos θ)² ज्ञात कीजिए।

Solution:

tan θ = 2

Take:

Perpendicular = 2

Base = 1

Hypotenuse = √5

So:

sin θ = 2/√5

cos θ = 1/√5

Now:

sin θ + cos θ = 3/√5

Therefore:

(sin θ + cos θ)²

= 9/5

Answer: 9/5

23. Question

English: The length, breadth and height of a cuboid are in the ratio 3 : 2 : 1. If its volume is 384 cm³, find its total surface area.

Hindi: एक घनाभ की लंबाई, चौड़ाई और ऊँचाई का अनुपात 3 : 2 : 1 है। यदि उसका आयतन 384 सेमी³ है, तो उसका कुल पृष्ठीय क्षेत्रफल ज्ञात कीजिए।

Solution:

Let length = 3x

Breadth = 2x

Height = x

Volume:

= 3x × 2x × x

= 6x³

Given:

6x³ = 384

x³ = 64

x = 4

So:

Length = 12 cm

Breadth = 8 cm

Height = 4 cm

Total surface area:

= 2(lb + bh + hl)

= 2(12 × 8 + 8 × 4 + 4 × 12)

= 2(96 + 32 + 48)

= 2 × 176

= 352 cm²

Answer: 352 cm²

24. Question

English: A cylinder of radius 6 cm and height 28 cm is melted into small spheres of radius 3 cm. Find the number of spheres formed.

Hindi: 6 सेमी त्रिज्या और 28 सेमी ऊँचाई वाले बेलन को पिघलाकर 3 सेमी त्रिज्या वाले छोटे गोलों में बदला जाता है। बने हुए गोलों की संख्या ज्ञात कीजिए।

Solution:

Volume of cylinder:

= π × 6² × 28

= 1008π

Volume of one sphere:

= 4/3 × π × 3³

= 36π

Number of spheres:

= 1008π / 36π

= 28

Answer: 28

25. Question

English: In a class of 100 students, 60 study Mathematics, 55 study English, and 10 study neither. Find the number of students who study exactly one of the two subjects.

Hindi: 100 विद्यार्थियों की कक्षा में 60 गणित पढ़ते हैं, 55 अंग्रेजी पढ़ते हैं और 10 कोई भी विषय नहीं पढ़ते। ठीक एक विषय पढ़ने वाले विद्यार्थियों की संख्या ज्ञात कीजिए।

Solution:

Total students = 100

Students studying neither = 10

Students studying at least one subject:

= 100 - 10

= 90

Now:

Math students = 60

English students = 55

Using formula:

Math + English = At least one + Both

60 + 55 = 90 + Both

115 = 90 + Both

Both = 25

Students studying exactly one subject:

= At least one - Both

= 90 - 25

= 65

Answer: 65