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1. Question
English: If f(x) = (3x - 1)/(x + 2), find f⁻¹(2).
Hindi: यदि f(x) = (3x - 1)/(x + 2) है, तो f⁻¹(2) ज्ञात कीजिए।
Solution:
f⁻¹(2) means value of x for which:
(3x - 1)/(x + 2) = 2
3x - 1 = 2x + 4
x = 5
Answer: 5
2. Question
English: The HCF and LCM of two numbers are 24 and 1440. If the sum of the numbers is 408, find their difference.
Hindi: दो संख्याओं का HCF और LCM क्रमशः 24 और 1440 है। यदि उनका योग 408 है, तो उनका अंतर ज्ञात कीजिए।
Solution:
Let numbers be:
24a and 24b
Since HCF = 24, a and b are co-prime.
LCM:
= 24ab
24ab = 1440
ab = 60
Also:
24a + 24b = 408
a + b = 17
Numbers whose sum is 17 and product is 60:
12 and 5
Actual numbers:
288 and 120
Difference:
= 288 - 120
= 168
Answer: 168
3. Question
English: Simplify: (√7 + √5)² + (√7 - √5)².
Hindi: सरलीकृत कीजिए: (√7 + √5)² + (√7 - √5)².
Solution:
Using identity:
(a + b)² + (a - b)²
= 2(a² + b²)
Here:
a = √7
b = √5
So:
= 2(7 + 5)
= 2 × 12
= 24
Answer: 24
4. Question
English: Evaluate: (243^(2/5) × 32^(3/5)) / 27^(1/3).
Hindi: मान ज्ञात कीजिए: (243^(2/5) × 32^(3/5)) / 27^(1/3).
Solution:
243^(2/5)
= (⁵√243)²
= 3²
= 9
32^(3/5)
= (⁵√32)³
= 2³
= 8
27^(1/3)
= 3
Therefore:
(9 × 8)/3
= 72/3
= 24
Answer: 24
5. Question
English: Find the least positive number which leaves remainder 5 when divided by 8, 12 and 15, and is exactly divisible by 7.
Hindi: वह सबसे छोटी धनात्मक संख्या ज्ञात कीजिए जो 8, 12 और 15 से भाग देने पर 5 शेष छोड़े तथा 7 से पूर्णतः विभाजित हो।
Solution:
Required number:
= LCM(8,12,15) × k + 5
LCM of 8,12,15:
8 = 2³
12 = 2² × 3
15 = 3 × 5
LCM = 120
So:
Number = 120k + 5
Now check divisibility by 7.
120 mod 7 = 1
So:
k + 5 divisible by 7
k = 2 gives:
120 × 2 + 5
= 245
Answer: 245
6. Question
English: In an AP, the sum of the 7th and 13th terms is 100, and the 20th term is 84. Find the sum of the first 25 terms.
Hindi: एक AP में 7वें और 13वें पदों का योग 100 है, और 20वाँ पद 84 है। पहले 25 पदों का योग ज्ञात कीजिए।
Solution:
7th term:
a + 6d
13th term:
a + 12d
Their sum:
2a + 18d = 100
a + 9d = 50
20th term:
a + 19d = 84
Subtract:
10d = 34
d = 3.4
Now:
a + 9 × 3.4 = 50
a + 30.6 = 50
a = 19.4
Sum of first 25 terms:
S = 25/2 × [2a + 24d]
= 25/2 × [38.8 + 81.6]
= 25/2 × 120.4
= 1505
Answer: 1505
7. Question
English: Three positive numbers are in GP. Their product is 512 and their sum is 28. Find the sum of their squares.
Hindi: तीन धनात्मक संख्याएँ GP में हैं। उनका गुणनफल 512 और योग 28 है। उनके वर्गों का योग ज्ञात कीजिए।
Solution:
Let numbers be:
a/r, a, ar
Their product:
= a³
= 512
a = 8
So numbers are:
8/r, 8, 8r
Their sum:
8/r + 8 + 8r = 28
Divide by 4:
2/r + 2 + 2r = 7
r + 1 + 1/r = 7/2
Try r = 2
Numbers become:
4, 8, 16
Sum = 28
Now sum of squares:
= 4² + 8² + 16²
= 16 + 64 + 256
= 336
Answer: 336
8. Question
English: How many different arrangements can be made from the letters of the word MISSISSIPPI?
Hindi: MISSISSIPPI शब्द के अक्षरों से कितने अलग-अलग व्यवस्थापन बनाए जा सकते हैं?
Solution:
Total letters = 11
Repeated letters:
M = 1
I = 4
S = 4
P = 2
Number of arrangements:
= 11! / (4! × 4! × 2!)
= 39916800 / 1152
= 34650
Answer: 34650
9. Question
English: A bag contains 3 red, 4 blue and 5 green balls. Three balls are drawn at random. Find the probability that all three are of different colours.
Hindi: एक थैले में 3 लाल, 4 नीली और 5 हरी गेंदें हैं। यादृच्छया 3 गेंदें निकाली जाती हैं। तीनों के अलग-अलग रंग की होने की प्रायिकता ज्ञात कीजिए।
Solution:
Total balls:
= 3 + 4 + 5
= 12
Total ways:
= 12C3
= 220
Favourable ways:
Choose 1 red from 3
= 3C1 = 3
Choose 1 blue from 4
= 4C1 = 4
Choose 1 green from 5
= 5C1 = 5
Total favourable ways:
= 3 × 4 × 5
= 60
Probability:
= 60/220
= 3/11
Answer: 3/11
10. Question
English: Pipe A fills a tank in 15 hours, pipe B fills it in 20 hours and pipe C empties it in 30 hours. If all three are opened together, find the time taken to fill the tank.
Hindi: पाइप A टंकी को 15 घंटे में भरता है, पाइप B 20 घंटे में भरता है और पाइप C 30 घंटे में खाली करता है। तीनों को साथ खोलने पर टंकी कितने समय में भरेगी?
Solution:
A’s 1-hour work:
= 1/15
B’s 1-hour work:
= 1/20
C’s 1-hour work:
= -1/30
Net work:
= 1/15 + 1/20 - 1/30
LCM = 60
= 4 + 3 - 2 /60
= 5/60
= 1/12
Therefore tank fills in:
12 hours
Answer: 12 hours
11. Question
English: A invests ₹15000 for 8 months, B invests ₹12000 for 10 months, and C invests for 6 months. If their profit ratio is 10 : 10 : 9, find C's investment.
Hindi: A ₹15000 को 8 महीने के लिए, B ₹12000 को 10 महीने के लिए और C 6 महीने के लिए निवेश करता है। यदि उनका लाभ अनुपात 10 : 10 : 9 है, तो C का निवेश ज्ञात कीजिए।
Solution:
Profit depends on investment × time.
A’s capital-time:
= 15000 × 8
= 120000
B’s capital-time:
= 12000 × 10
= 120000
So A : B = 10 : 10
Now C’s part = 9
If 10 parts = 120000
1 part = 12000
9 parts = 108000
Let C’s investment = x
x × 6 = 108000
x = 18000
Answer: ₹18000
12. Question
English: Find the smaller angle between the hour hand and minute hand at 4:40.
Hindi: 4:40 बजे घंटे की सुई और मिनट की सुई के बीच छोटा कोण ज्ञात कीजिए।
Solution:
At 4:40:
Minute hand angle:
= 40 × 6
= 240°
Hour hand angle:
= 4 × 30 + 40 × 0.5
= 120 + 20
= 140°
Difference:
= 240 - 140
= 100°
Answer: 100°
13. Question
English: In a 200 m race, A beats B by 20 m. In a 180 m race, B beats C by 30 m. By how many metres will A beat C in a 300 m race?
Hindi: 200 मीटर की दौड़ में A, B को 20 मीटर से हराता है। 180 मीटर की दौड़ में B, C को 30 मीटर से हराता है। 300 मीटर की दौड़ में A, C को कितने मीटर से हराएगा?
Solution:
When A runs 200 m, B runs 180 m.
So:
A : B = 200 : 180
= 10 : 9
When B runs 180 m, C runs 150 m.
So:
B : C = 180 : 150
= 6 : 5
Now combine:
A : B = 10 : 9
B : C = 6 : 5
Make B same.
LCM of 9 and 6 = 18
A : B = 20 : 18
B : C = 18 : 15
So:
A : C = 20 : 15
= 4 : 3
In a 300 m race, when A runs 300 m, C runs:
= 300 × 3/4
= 225 m
A beats C by:
= 300 - 225
= 75 m
Answer: 75 m
14. Question
English: A vessel contains 64 litres of pure milk. One-fourth is removed and replaced with water. This process is repeated 3 times. Find the quantity of milk left.
Hindi: एक बर्तन में 64 लीटर शुद्ध दूध है। उसका 1/4 भाग निकालकर पानी से भर दिया जाता है। यह प्रक्रिया 3 बार दोहराई जाती है। शेष दूध की मात्रा ज्ञात कीजिए।
Solution:
Each time removed fraction = 1/4
Remaining fraction each time:
= 3/4
After 3 times:
Milk left:
= 64 × 3/4 × 3/4 × 3/4
= 64 × 27/64
= 27 litres
Answer: 27 litres
15. Question
English: The sides of a triangle are 10 cm, 24 cm and 26 cm. Find its circumradius.
Hindi: एक त्रिभुज की भुजाएँ 10 सेमी, 24 सेमी और 26 सेमी हैं। उसकी परित्रिज्या ज्ञात कीजिए।
Solution:
Check:
10² + 24² = 100 + 576
= 676
26² = 676
So it is a right triangle.
For a right triangle, circumradius = hypotenuse/2
= 26/2
= 13 cm
Answer: 13 cm
16. Question
English: Find the area of the triangle formed by the points (1,2), (5,6) and (7,2).
Hindi: बिंदुओं (1,2), (5,6) और (7,2) से बने त्रिभुज का क्षेत्रफल ज्ञात कीजिए।
Solution:
Points are:
(1,2), (5,6), (7,2)
Base from (1,2) to (7,2):
= 6 units
Height from (5,6) to line y = 2:
= 4 units
Area:
= 1/2 × base × height
= 1/2 × 6 × 4
= 12 sq units
Answer: 12 sq units
17. Question
English: If tan θ + cot θ = 10/3, find sin 2θ.
Hindi: यदि tan θ + cot θ = 10/3 है, तो sin 2θ ज्ञात कीजिए।
Solution:
tan θ + cot θ
= sin θ/cos θ + cos θ/sin θ
= (sin² θ + cos² θ) / sin θ cos θ
= 1 / sin θ cos θ
Given:
tan θ + cot θ = 10/3
So:
1 / sin θ cos θ = 10/3
sin θ cos θ = 3/10
Now:
sin 2θ = 2sin θ cos θ
= 2 × 3/10
= 3/5
Answer: 3/5
18. Question
English: A sum becomes double in 5 years at compound interest. In how many years will it become 8 times?
Hindi: एक राशि चक्रवृद्धि ब्याज पर 5 वर्षों में दोगुनी हो जाती है। वह कितने वर्षों में 8 गुनी होगी?
Solution:
Amount doubles in 5 years.
8 times means:
8 = 2³
So it needs 3 doubling periods.
Time:
= 5 × 3
= 15 years
Answer: 15 years
19. Question
English: A man sells two articles for ₹990 each. On one he gains 10% and on the other he loses 10%. Find the overall result.
Hindi: एक व्यक्ति दो वस्तुओं को प्रत्येक ₹990 में बेचता है। एक पर 10% लाभ और दूसरी पर 10% हानि होती है। कुल परिणाम ज्ञात कीजिए।
Solution:
Selling price of each article = ₹990
For first article, gain = 10%
Cost price:
= 990 × 100/110
= 900
For second article, loss = 10%
Cost price:
= 990 × 100/90
= 1100
Total cost price:
= 900 + 1100
= 2000
Total selling price:
= 990 + 990
= 1980
Loss:
= 2000 - 1980
= 20
Loss percentage:
= 20/2000 × 100
= 1%
Answer: 1% loss
20. Question
English: If the radius and height of a cylinder are in the ratio 3 : 7, find the ratio of its total surface area to curved surface area.
Hindi: यदि किसी बेलन की त्रिज्या और ऊँचाई का अनुपात 3 : 7 है, तो उसके कुल पृष्ठीय क्षेत्रफल और वक्र पृष्ठीय क्षेत्रफल का अनुपात ज्ञात कीजिए।
Solution:
Radius : Height = 3 : 7
Let r = 3 and h = 7
Total surface area:
= 2πr(r + h)
Curved surface area:
= 2πrh
Ratio:
= 2πr(r + h) : 2πrh
= (r + h) : h
= (3 + 7) : 7
= 10 : 7
Answer: 10 : 7
21. Question
English: If f(n) = f(n - 1) + 2n for n ≥ 2 and f(1) = 3, find f(10).
Hindi: यदि n ≥ 2 के लिए f(n) = f(n - 1) + 2n और f(1) = 3 है, तो f(10) ज्ञात कीजिए।
Solution:
Given:
f(1) = 3
Now:
f(2) = f(1) + 2 × 2
= 3 + 4
= 7
f(3) = 7 + 6
= 13
Instead of writing all terms, add from 2 to 10:
f(10) = f(1) + 2(2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10)
Sum from 2 to 10:
= 54
So:
f(10) = 3 + 2 × 54
= 3 + 108
= 111
Answer: 111
22. Question
English: Find the number of even divisors of 2⁵ × 3³ × 5².
Hindi: 2⁵ × 3³ × 5² के सम भाजकों की संख्या ज्ञात कीजिए।
Solution:
Number is:
2⁵ × 3³ × 5²
For an even divisor, power of 2 must be at least 1.
Power of 2 can be:
1, 2, 3, 4, 5
So 5 choices.
Power of 3 can be:
0, 1, 2, 3
So 4 choices.
Power of 5 can be:
0, 1, 2
So 3 choices.
Total even divisors:
= 5 × 4 × 3
= 60
Answer: 60
23. Question
English: A boat takes 6 hours upstream and 4 hours downstream to cover the same distance. Find the ratio of speed of boat in still water to speed of stream.
Hindi: एक नाव समान दूरी धारा के विपरीत 6 घंटे में और धारा के साथ 4 घंटे में तय करती है। स्थिर जल में नाव की गति और धारा की गति का अनुपात ज्ञात कीजिए।
Solution:
Let same distance = 24 km
Upstream time = 6 hours
Upstream speed:
= 24/6
= 4 km/h
Downstream time = 4 hours
Downstream speed:
= 24/4
= 6 km/h
Speed of boat in still water:
= (6 + 4)/2
= 5 km/h
Speed of stream:
= (6 - 4)/2
= 1 km/h
Required ratio:
= 5 : 1
Answer: 5 : 1
24. Question
English: In a class of 100 students, 70 study Mathematics, 65 study Reasoning, 50 study English and 10 study none. If 30 study all three subjects, find the number of students who study exactly two subjects.
Hindi: 100 विद्यार्थियों की कक्षा में 70 गणित, 65 रीजनिंग, 50 अंग्रेजी पढ़ते हैं और 10 कोई विषय नहीं पढ़ते। यदि 30 विद्यार्थी तीनों विषय पढ़ते हैं, तो ठीक दो विषय पढ़ने वाले विद्यार्थियों की संख्या ज्ञात कीजिए।
Solution:
Total students = 100
Students studying none = 10
Students studying at least one subject:
= 100 - 10
= 90
Let exactly two subjects = x
Students studying all three = 30
Now count total subject memberships:
Mathematics + Reasoning + English
= 70 + 65 + 50
= 185
This includes:
Exactly one subjects counted 1 time.
Exactly two subjects counted 2 times.
All three subjects counted 3 times.
Let exactly one subject = y.
So:
x + y + 30 = 90
x + y = 60
Also:
2x + y + 3 × 30 = 185
2x + y + 90 = 185
2x + y = 95
Subtract:
2x + y = 95
x + y = 60
x = 35
Answer: 35
25. Question
English: If the roots of x² - px + 36 = 0 are positive and differ by 5, find p.
Hindi: यदि x² - px + 36 = 0 के मूल धनात्मक हैं और उनका अंतर 5 है, तो p ज्ञात कीजिए।
Solution:
Let roots be a and b.
Given:
ab = 36
a - b = 5
Now:
(a + b)² = (a - b)² + 4ab
= 5² + 4 × 36
= 25 + 144
= 169
a + b = 13
In equation:
x² - px + 36 = 0
Sum of roots = p
So:
p = 13
Answer: 13