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1. Question

English: Simplify: (√75 + √48 − √27) / √3

Hindi: सरलीकृत कीजिए: (√75 + √48 − √27) / √3

Solution:

√75 = 5√3

√48 = 4√3

√27 = 3√3

So numerator:

= 5√3 + 4√3 - 3√3

= 6√3

Now:

6√3 / √3 = 6

Answer: 6

2. Question

English: If 2^(x+1) = 32 and 4^(y−1) = 16, find x + y.

Hindi: यदि 2^(x+1) = 32 और 4^(y−1) = 16 है, तो x + y ज्ञात कीजिए।

Solution:

2^(x+1) = 32 = 2⁵

So:

x + 1 = 5

x = 4

Now:

4^(y−1) = 16

16 = 4²

So:

y - 1 = 2

y = 3

Therefore:

x + y = 4 + 3

= 7

Answer: 7

3. Question

English: Evaluate: (27^(2/3) × 16^(3/4)) / (81^(1/2))

Hindi: मान ज्ञात कीजिए: (27^(2/3) × 16^(3/4)) / (81^(1/2))

Solution:

27^(2/3)

= (³√27)²

= 3²

= 9

16^(3/4)

= (⁴√16)³

= 2³

= 8

81^(1/2)

= √81

= 9

Therefore:

(9 × 8)/9

= 8

Answer: 8

4. Question

English: If √(x + 15) = x − 3, find x.

Hindi: यदि √(x + 15) = x − 3 है, तो x ज्ञात कीजिए।

Solution:

√(x + 15) = x - 3

Square both sides:

x + 15 = (x - 3)²

x + 15 = x² - 6x + 9

0 = x² - 7x - 6

Using quadratic formula:

x = [7 ± √(49 + 24)]/2

= [7 ± √73]/2

Answer: (7 + √73)/2

5. Question

English: A and B together can complete a work in 15 days. A alone takes 10 days less than B alone. Find the time taken by A alone.

Hindi: A और B मिलकर किसी कार्य को 15 दिनों में पूरा करते हैं। A अकेले, B से 10 दिन कम समय लेता है। A अकेले कार्य कितने दिनों में करेगा?

Solution:

Let B alone take x days.

Then A alone takes:

x - 10 days

Together:

1/(x - 10) + 1/x = 1/15

x + (x - 10) / x(x - 10) = 1/15

2x - 10 = x(x - 10)/15

30x - 150 = x² - 10x

x² - 40x + 150 = 0

x = [40 ± √(1600 - 600)]/2

= [40 ± √1000]/2

= [40 ± 10√10]/2

= 20 ± 5√10

Taking valid value:

x ≈ 35.81

So A’s time:

≈ 25.81 days

Answer: Approximately 25.8 days

6. Question

English: A train 180 m long crosses another train 220 m long moving in opposite direction in 8 seconds. If the speed of one train is 72 km/h, find speed of the other train.

Hindi: 180 मीटर लंबी ट्रेन 220 मीटर लंबी दूसरी ट्रेन को विपरीत दिशा में 8 सेकंड में पार करती है। यदि एक ट्रेन की गति 72 किमी/घंटा है, तो दूसरी ट्रेन की गति ज्ञात कीजिए।

Solution:

Total distance:

= 180 + 220

= 400 m

Time = 8 seconds

Relative speed:

= 400/8

= 50 m/s

Convert into km/h:

= 50 × 18/5

= 180 km/h

One train speed = 72 km/h

Other train speed:

= 180 - 72

= 108 km/h

Answer: 108 km/h

7. Question

English: The compound interest on a sum for 2 years is ₹2050 and simple interest is ₹2000 at the same rate. Find the principal.

Hindi: किसी राशि पर 2 वर्षों के लिए चक्रवृद्धि ब्याज ₹2050 और उसी दर पर साधारण ब्याज ₹2000 है। मूलधन ज्ञात कीजिए।

Solution:

Difference between CI and SI:

= 2050 - 2000

= 50

For 2 years:

Difference = P × R² / 10000

Also:

SI = PRT/100

2000 = P × R × 2 /100

PR = 100000

Now:

P × R² /10000 = 50

Using PR = 100000:

100000 × R /10000 = 50

10R = 50

R = 5%

Now:

PR = 100000

P × 5 = 100000

P = 20000

Answer: ₹20000

8. Question

English: A shopkeeper marks an article 40% above cost price and gives successive discounts of 10% and 20%. Find profit percentage.

Hindi: एक दुकानदार वस्तु का अंकित मूल्य क्रय मूल्य से 40% अधिक रखता है और 10% तथा 20% की क्रमिक छूट देता है। लाभ प्रतिशत ज्ञात कीजिए।

Solution:

Let cost price = 100

Marked price = 140

After 10% discount:

= 140 × 90/100

= 126

After 20% discount:

= 126 × 80/100

= 100.8

Profit:

= 100.8 - 100

= 0.8

Profit percentage:

= 0.8%

Answer: 0.8%

9. Question

English: If the ratio of curved surface area and total surface area of a cylinder is 2 : 3, find the ratio of radius to height.

Hindi: यदि किसी बेलन के वक्र पृष्ठीय क्षेत्रफल और कुल पृष्ठीय क्षेत्रफल का अनुपात 2 : 3 है, तो त्रिज्या और ऊँचाई का अनुपात ज्ञात कीजिए।

Solution:

CSA = 2πrh

TSA = 2πr(h + r)

Given:

2πrh / 2πr(h + r) = 2/3

h/(h + r) = 2/3

3h = 2h + 2r

h = 2r

Therefore:

r : h = 1 : 2

Answer: 1 : 2

10. Question

English: If sec θ + tan θ = 3, find sec θ − tan θ.

Hindi: यदि sec θ + tan θ = 3 है, तो sec θ − tan θ ज्ञात कीजिए।

Solution:

(sec θ + tan θ)(sec θ - tan θ) = 1

3(sec θ - tan θ) = 1

sec θ - tan θ = 1/3

Answer: 1/3

11. Question

English: The sum of two numbers is 27 and their HCF is 3. If their LCM is 54, find the larger number.

Hindi: दो संख्याओं का योग 27 और उनका HCF 3 है। यदि उनका LCM 54 है, तो बड़ी संख्या ज्ञात कीजिए।

Solution:

Let numbers be:

3a and 3b

Since HCF = 3, a and b are co-prime.

LCM:

= 3ab

3ab = 54

ab = 18

Also:

3a + 3b = 27

a + b = 9

Numbers with sum 9 and product 18 are:

3 and 6

Actual numbers:

9 and 18

Larger number = 18

Answer: 18

12. Question

English: Find the value of: (0.04)^(-3/2)

Hindi: मान ज्ञात कीजिए: (0.04)^(-3/2)

Solution:

0.04 = 1/25

So:

(1/25)^(-3/2)

= 25^(3/2)

= (√25)³

= 5³

= 125

Answer: 125

13. Question

English: If x² − 12x + 1 = 0, find x³ + 1/x³.

Hindi: यदि x² − 12x + 1 = 0 है, तो x³ + 1/x³ ज्ञात कीजिए।

Solution:

x² − 12x + 1 = 0

Divide by x:

x - 12 + 1/x = 0

x + 1/x = 12

Now:

x³ + 1/x³

= (x + 1/x)³ - 3(x + 1/x)

= 12³ - 3 × 12

= 1728 - 36

= 1692

Answer: 1692

14. Question

English: Two pipes fill a tank in 12 hours and 15 hours respectively. A third pipe empties 8 litres per hour. If all pipes are opened together, the tank fills in 20 hours. Find capacity of tank.

Hindi: दो पाइप किसी टंकी को क्रमशः 12 घंटे और 15 घंटे में भरते हैं। तीसरा पाइप 8 लीटर प्रति घंटा पानी निकालता है। तीनों को साथ खोलने पर टंकी 20 घंटे में भरती है। टंकी की क्षमता ज्ञात कीजिए।

Solution:

Let capacity = x litres.

First pipe fills:

x/12 litres/hour

Second pipe fills:

x/15 litres/hour

Third pipe empties:

8 litres/hour

Net filling rate:

x/20 litres/hour

So:

x/12 + x/15 - 8 = x/20

LCM = 60

5x + 4x - 480 = 3x

6x = 480

x = 80 litres

Answer: 80 litres

15. Question

English: In a class, 60% students passed in Mathematics, 70% passed in English and 20% failed in both. Find percentage of students who passed in both subjects.

Hindi: एक कक्षा में 60% विद्यार्थी गणित में उत्तीर्ण हुए, 70% अंग्रेजी में उत्तीर्ण हुए और 20% दोनों में अनुत्तीर्ण हुए। दोनों विषयों में उत्तीर्ण विद्यार्थियों का प्रतिशत ज्ञात कीजिए।

Solution:

Students passing at least one subject:

= 100 - 20

= 80%

Now:

Math + English - Both = At least one

60 + 70 - Both = 80

130 - Both = 80

Both = 50%

Answer: 50%

16. Question

English: A man covers half the distance at 60 km/h, one-third at 30 km/h and remaining at 20 km/h. Find average speed.

Hindi: एक व्यक्ति आधी दूरी 60 किमी/घंटा, एक-तिहाई दूरी 30 किमी/घंटा तथा शेष दूरी 20 किमी/घंटा की गति से तय करता है। औसत गति ज्ञात कीजिए।

Solution:

Let total distance = 6 km

Half distance:

= 3 km

One-third distance:

= 2 km

Remaining distance:

= 1 km

Time for 3 km at 60 km/h:

= 3/60

Time for 2 km at 30 km/h:

= 2/30

Time for 1 km at 20 km/h:

= 1/20

Total time:

= 3/60 + 2/30 + 1/20

= 1/20 + 1/15 + 1/20

= 1/10 + 1/15

= 5/30

= 1/6 hour

Average speed:

= total distance / total time

= 6 ÷ 1/6

= 36 km/h

Answer: 36 km/h

17. Question

English: A cone and a hemisphere have equal radii. If their volumes are equal, find ratio of height of cone to radius.

Hindi: एक शंकु और एक अर्धगोले की त्रिज्या समान है। यदि उनके आयतन समान हैं, तो शंकु की ऊँचाई और त्रिज्या का अनुपात ज्ञात कीजिए।

Solution:

Volume of cone:

= 1/3 × π × r² × h

Volume of hemisphere:

= 2/3 × π × r³

Given both volumes are equal:

1/3 × π × r² × h = 2/3 × π × r³

Cancel common terms:

h = 2r

So:

h : r = 2 : 1

Answer: 2 : 1

18. Question

English: Find the number of diagonals in a polygon having 20 sides.

Hindi: 20 भुजाओं वाले बहुभुज में विकर्णों की संख्या ज्ञात कीजिए।

Solution:

Number of diagonals:

= n(n - 3)/2

Here n = 20

= 20 × 17 / 2

= 10 × 17

= 170

Answer: 170

19. Question

English: If 5 men or 8 women can complete a work in 24 days, then in how many days can 10 men and 8 women complete the work?

Hindi: यदि 5 पुरुष या 8 महिलाएँ किसी कार्य को 24 दिनों में पूरा कर सकती हैं, तो 10 पुरुष और 8 महिलाएँ मिलकर कार्य कितने दिनों में पूरा करेंगे?

Solution:

5 men complete work in 24 days.

So 1 man’s 1-day work:

= 1 / (5 × 24)

= 1/120

8 women complete work in 24 days.

So 1 woman’s 1-day work:

= 1 / (8 × 24)

= 1/192

10 men’s 1-day work:

= 10/120

= 1/12

8 women’s 1-day work:

= 8/192

= 1/24

Together 1-day work:

= 1/12 + 1/24

= 2/24 + 1/24

= 3/24

= 1/8

So they complete the work in 8 days.

Answer: 8 days

20. Question

English: A vessel contains 80 litres milk. 20 litres are removed and replaced with water. This process is repeated once more. Find quantity of milk left.

Hindi: एक बर्तन में 80 लीटर दूध है। 20 लीटर निकालकर पानी मिलाया जाता है। यह प्रक्रिया एक बार और दोहराई जाती है। शेष दूध की मात्रा ज्ञात कीजिए।

Solution:

Initial milk = 80 litres

Each time removed fraction:

= 20/80

= 1/4

Remaining fraction each time:

= 3/4

After 2 operations:

Milk left = 80 × 3/4 × 3/4

= 80 × 9/16

= 45 litres

Answer: 45 litres

21. Question

English: If x = 2 + √3, find value of x² + 1/x².

Hindi: यदि x = 2 + √3 है, तो x² + 1/x² का मान ज्ञात कीजिए।

Solution:

x = 2 + √3

So:

1/x = 2 - √3

Therefore:

x + 1/x = 2 + √3 + 2 - √3

= 4

Now:

x² + 1/x²

= (x + 1/x)² - 2

= 4² - 2

= 16 - 2

= 14

Answer: 14

22. Question

English: The difference between two numbers is 20% of the larger number. If the smaller number is 80, find the larger number.

Hindi: दो संख्याओं का अंतर बड़ी संख्या का 20% है। यदि छोटी संख्या 80 है, तो बड़ी संख्या ज्ञात कीजिए।

Solution:

Let larger number = x

Smaller number = 80

Difference:

= x - 80

Given:

x - 80 = 20% of x

x - 80 = x/5

5x - 400 = x

4x = 400

x = 100

Answer: 100

23. Question

English: A trader mixes tea costing ₹72/kg with tea costing ₹48/kg. In what ratio should they be mixed so that profit after selling at ₹66/kg is 10%?

Hindi: एक व्यापारी ₹72/किग्रा वाली चाय को ₹48/किग्रा वाली चाय में मिलाता है। यदि मिश्रण को ₹66/किग्रा पर बेचने पर 10% लाभ हो, तो उन्हें किस अनुपात में मिलाना चाहिए?

Solution:

Selling price = ₹66/kg

Profit = 10%

So cost price of mixture:

= 66 × 100/110

= ₹60/kg

Now use allegation:

Dearer tea = ₹72

Cheaper tea = ₹48

Mean price = ₹60

Ratio of ₹72 tea : ₹48 tea

= (60 - 48) : (72 - 60)

= 12 : 12

= 1 : 1

Answer: 1 : 1

24. Question

English: The area of a sector of angle 72° in a circle is 154 cm². Find radius of the circle.

Hindi: एक वृत्त में 72° कोण वाले सेक्टर का क्षेत्रफल 154 सेमी² है। वृत्त की त्रिज्या ज्ञात कीजिए।

Solution:

Sector area:

= angle/360 × πr²

154 = 72/360 × 22/7 × r²

72/360 = 1/5

So:

154 = 1/5 × 22/7 × r²

154 × 5 = 22/7 × r²

770 = 22/7 × r²

r² = 770 × 7 / 22

r² = 245

r = 7√5

Answer: 7√5 cm

25. Question

English: If tan θ + sec θ = 5, find tan θ.

Hindi: यदि tan θ + sec θ = 5 है, तो tan θ ज्ञात कीजिए।

Solution:

tan θ + sec θ = 5

We know:

sec θ + tan θ = 5

Also:

sec θ - tan θ = 1/5

Subtract:

(sec θ + tan θ) - (sec θ - tan θ)

= 5 - 1/5

2tan θ = 24/5

tan θ = 12/5

Answer: 12/5