1. Question
English: If f(x) = (x + 2)/(x - 2), find f(f(5)).
Hindi: यदि f(x) = (x + 2)/(x - 2) है, तो f(f(5)) ज्ञात कीजिए।
Solution:
First:
f(5)
= (5 + 2)/(5 - 2)
= 7/3
Now:
f(7/3)
= (7/3 + 2)/(7/3 - 2)
= (7/3 + 6/3)/(7/3 - 6/3)
= (13/3)/(1/3)
= 13
Answer: 13
2. Question
English: The HCF and LCM of two numbers are 18 and 1512 respectively. If one number is 216, find the other number.
Hindi: दो संख्याओं का HCF और LCM क्रमशः 18 और 1512 है। यदि एक संख्या 216 है, तो दूसरी संख्या ज्ञात कीजिए।
Solution:
Product of two numbers:
= HCF × LCM
= 18 × 1512
Other number:
= (18 × 1512)/216
= 126
Answer: 126
3. Question
English: Find the LCM of 3/8, 9/16 and 15/32.
Hindi: 3/8, 9/16 और 15/32 का LCM ज्ञात कीजिए।
Solution:
For fractions:
LCM = LCM of numerators / HCF of denominators
Numerators:
3, 9, 15
LCM = 45
Denominators:
8, 16, 32
HCF = 8
Therefore:
LCM = 45/8
Answer: 45/8
4. Question
English: Simplify: (√45 + √80 - √20) / √5.
Hindi: सरलीकृत कीजिए: (√45 + √80 - √20) / √5.
Solution:
√45 = 3√5
√80 = 4√5
√20 = 2√5
Numerator:
= 3√5 + 4√5 - 2√5
= 5√5
Now divide by √5:
= 5
Answer: 5
5. Question
English: Evaluate: (216^(2/3) × 49^(1/2)) / 36^(3/2).
Hindi: मान ज्ञात कीजिए: (216^(2/3) × 49^(1/2)) / 36^(3/2).
Solution:
216^(2/3)
= (³√216)²
= 6²
= 36
49^(1/2)
= 7
36^(3/2)
= (√36)³
= 6³
= 216
Therefore:
(36 × 7)/216
= 7/6
Answer: 7/6
6. Question
English: Find the least positive integer N such that N leaves remainder 4 when divided by 9 and remainder 6 when divided by 11.
Hindi: सबसे छोटी धनात्मक संख्या N ज्ञात कीजिए जो 9 से भाग देने पर 4 शेष और 11 से भाग देने पर 6 शेष छोड़ती है।
Solution:
N = 9k + 4
Also:
9k + 4 leaves remainder 6 when divided by 11.
So:
9k - 2 is divisible by 11
9k ≡ 2 mod 11
Try values:
k = 10
9 × 10 = 90
90 leaves remainder 2 when divided by 11.
Therefore:
N = 9 × 10 + 4
= 94
Answer: 94
7. Question
English: In an AP, the 6th term is 29 and the 14th term is 69. Find the sum of the first 20 terms.
Hindi: एक AP में 6वाँ पद 29 और 14वाँ पद 69 है। पहले 20 पदों का योग ज्ञात कीजिए।
Solution:
6th term:
a + 5d = 29
14th term:
a + 13d = 69
Subtract:
8d = 40
d = 5
Now:
a + 25 = 29
a = 4
Sum of first 20 terms:
S = 20/2 × [2a + 19d]
= 10 × [8 + 95]
= 10 × 103
= 1030
Answer: 1030
8. Question
English: Three numbers are in GP. Their product is 729 and their sum is 39. Find the greatest number.
Hindi: तीन संख्याएँ GP में हैं। उनका गुणनफल 729 और योग 39 है। सबसे बड़ी संख्या ज्ञात कीजिए।
Solution:
Let numbers be:
a/r, a, ar
Their product:
= a³
= 729
a = 9
So numbers:
9/r, 9, 9r
Their sum:
9/r + 9 + 9r = 39
Divide by 3:
3/r + 3 + 3r = 13
Try r = 3
Numbers become:
3, 9, 27
Sum = 39
Greatest number:
= 27
Answer: 27
9. Question
English: How many different arrangements can be made from the letters of the word SUCCESS?
Hindi: SUCCESS शब्द के अक्षरों से कितने अलग-अलग व्यवस्थापन बनाए जा सकते हैं?
Solution:
Total letters = 7
Repeated letters:
S = 3
C = 2
Number of arrangements:
= 7! / (3! × 2!)
= 5040 / 12
= 420
Answer: 420
10. Question
English: A bag contains 4 red, 6 blue and 5 green balls. Three balls are drawn at random. Find the probability that all are blue.
Hindi: एक थैले में 4 लाल, 6 नीली और 5 हरी गेंदें हैं। यादृच्छया 3 गेंदें निकाली जाती हैं। सभी के नीली होने की प्रायिकता ज्ञात कीजिए।
Solution:
Total balls:
= 4 + 6 + 5
= 15
Total ways:
= 15C3
= 455
Ways to choose 3 blue balls:
= 6C3
= 20
Probability:
= 20/455
= 4/91
Answer: 4/91
11. Question
English: A and B together complete a work in 16 days. B and C together complete it in 24 days. A and C together complete it in 48 days. Find the time taken by A alone.
Hindi: A और B मिलकर कार्य 16 दिनों में, B और C मिलकर 24 दिनों में तथा A और C मिलकर 48 दिनों में करते हैं। A अकेला कार्य कितने दिनों में करेगा?
Solution:
A + B = 1/16
B + C = 1/24
A + C = 1/48
Now add first and third:
(A + B) + (A + C)
= 1/16 + 1/48
= 3/48 + 1/48
= 4/48
= 1/12
So:
2A + B + C = 1/12
But:
B + C = 1/24
Therefore:
2A + 1/24 = 1/12
2A = 1/12 - 1/24
2A = 1/24
A = 1/48
So A alone takes:
48 days
Answer: 48 days
12. Question
English: Pipe A fills a tank in 12 hours, B fills it in 18 hours and C empties it in 24 hours. If all are opened together, find the time taken to fill the tank.
Hindi: पाइप A टंकी को 12 घंटे में भरता है, B 18 घंटे में भरता है और C 24 घंटे में खाली करता है। तीनों को साथ खोलने पर टंकी कितने समय में भरेगी?
Solution:
A’s 1-hour work:
= 1/12
B’s 1-hour work:
= 1/18
C’s 1-hour work:
= -1/24
Total 1-hour work:
= 1/12 + 1/18 - 1/24
LCM = 72
= 6/72 + 4/72 - 3/72
= 7/72
Time taken:
= 72/7 hours
Answer: 72/7 hours
13. Question
English: Find the smaller angle between the hour hand and minute hand at 2:50.
Hindi: 2:50 बजे घंटे की सुई और मिनट की सुई के बीच छोटा कोण ज्ञात कीजिए।
Solution:
Minute hand angle:
= 50 × 6
= 300°
Hour hand angle:
= 2 × 30 + 50 × 0.5
= 60 + 25
= 85°
Difference:
= 300 - 85
= 215°
Smaller angle:
= 360 - 215
= 145°
Answer: 145°
14. Question
English: In a 500 m race, A beats B by 50 m. In a 450 m race, B beats C by 45 m. By how many metres will A beat C in a 1000 m race?
Hindi: 500 मीटर की दौड़ में A, B को 50 मीटर से हराता है। 450 मीटर की दौड़ में B, C को 45 मीटर से हराता है। 1000 मीटर की दौड़ में A, C को कितने मीटर से हराएगा?
Solution:
When A runs 500 m, B runs 450 m.
So:
A : B = 500 : 450
= 10 : 9
When B runs 450 m, C runs 405 m.
So:
B : C = 450 : 405
= 10 : 9
Therefore:
A : C = 10 × 10 : 9 × 9
= 100 : 81
In 1000 m race, when A runs 1000 m, C runs:
= 1000 × 81/100
= 810 m
A beats C by:
= 1000 - 810
= 190 m
Answer: 190 m
15. Question
English: A 72-litre mixture contains milk and water in the ratio 7 : 2. How much water should be added to make the ratio 7 : 4?
Hindi: 72 लीटर मिश्रण में दूध और पानी का अनुपात 7 : 2 है। अनुपात 7 : 4 करने के लिए कितना पानी मिलाया जाना चाहिए?
Solution:
Milk : Water = 7 : 2
Total parts = 9
Milk:
= 72 × 7/9
= 56 litres
Water:
= 72 × 2/9
= 16 litres
New ratio should be 7 : 4.
Milk remains 56 litres.
If 7 parts = 56
Then 1 part = 8
Required water:
= 4 × 8
= 32 litres
Water to be added:
= 32 - 16
= 16 litres
Answer: 16 litres
16. Question
English: The sides of a triangle are 25 cm, 39 cm and 56 cm. Find its area.
Hindi: एक त्रिभुज की भुजाएँ 25 सेमी, 39 सेमी और 56 सेमी हैं। उसका क्षेत्रफल ज्ञात कीजिए।
Solution:
Semi-perimeter:
= (25 + 39 + 56)/2
= 120/2
= 60
Area:
= √[60 × (60 - 25) × (60 - 39) × (60 - 56)]
= √[60 × 35 × 21 × 4]
= √176400
= 420 cm²
Answer: 420 cm²
17. Question
English: Find the reflection of point (5, -2) in the line y = x.
Hindi: बिंदु (5, -2) का रेखा y = x में प्रतिबिंब ज्ञात कीजिए।
Solution:
Reflection in line y = x swaps x-coordinate and y-coordinate.
So:
(5, -2) becomes (-2, 5)
Answer: (-2, 5)
18. Question
English: If tan θ = 5/12, find sin θ + cos θ.
Hindi: यदि tan θ = 5/12 है, तो sin θ + cos θ ज्ञात कीजिए।
Solution:
tan θ = 5/12
So take:
Perpendicular = 5
Base = 12
Hypotenuse = 13
sin θ = 5/13
cos θ = 12/13
So:
sin θ + cos θ
= 5/13 + 12/13
= 17/13
Answer: 17/13
19. Question
English: A sum becomes ₹14520 in 2 years and ₹15972 in 3 years at compound interest. Find the annual rate.
Hindi: एक राशि चक्रवृद्धि ब्याज पर 2 वर्षों में ₹14520 और 3 वर्षों में ₹15972 हो जाती है। वार्षिक दर ज्ञात कीजिए।
Solution:
Amount after 2 years = ₹14520
Amount after 3 years = ₹15972
One-year multiplier:
= 15972/14520
= 1.10
So annual rate:
= 10%
Answer: 10%
20. Question
English: An article is sold at 15% loss. If it is sold for ₹96 more, there is a 9% gain. Find the cost price.
Hindi: एक वस्तु 15% हानि पर बेची जाती है। यदि उसे ₹96 अधिक में बेचा जाए, तो 9% लाभ होता है। क्रय मूल्य ज्ञात कीजिए।
Solution:
Let cost price = 100x
Selling price at 15% loss:
= 85x
Selling price at 9% gain:
= 109x
Difference:
= 109x - 85x
= 24x
Given difference = ₹96
24x = 96
x = 4
Cost price:
= 100x
= ₹400
Answer: ₹400
21. Question
English: If f(n) = f(n - 1) + 3n - 1 and f(1) = 2, find f(8).
Hindi: यदि f(n) = f(n - 1) + 3n - 1 और f(1) = 2 है, तो f(8) ज्ञात कीजिए।
Solution:
Given:
f(1) = 2
Now:
f(2) = f(1) + 3 × 2 - 1
= 2 + 6 - 1
= 7
f(3) = 7 + 9 - 1
= 15
f(4) = 15 + 12 - 1
= 26
f(5) = 26 + 15 - 1
= 40
f(6) = 40 + 18 - 1
= 57
f(7) = 57 + 21 - 1
= 77
f(8) = 77 + 24 - 1
= 100
Answer: 100
22. Question
English: If the roots of x² - px + 64 = 0 are positive and differ by 12, find p.
Hindi: यदि x² - px + 64 = 0 के मूल धनात्मक हैं और उनका अंतर 12 है, तो p ज्ञात कीजिए।
Solution:
Let roots be a and b.
Given:
ab = 64
a - b = 12
Now:
(a + b)² = (a - b)² + 4ab
= 12² + 4 × 64
= 144 + 256
= 400
a + b = 20
In equation:
x² - px + 64 = 0
Sum of roots = p
So:
p = 20
Answer: 20
23. Question
English: The average of 15 numbers is 36. If each of the first 5 numbers is increased by 4 and each of the last 5 numbers is decreased by 2, find the new average.
Hindi: 15 संख्याओं का औसत 36 है। यदि पहली 5 संख्याओं में प्रत्येक को 4 बढ़ाया जाए और अंतिम 5 संख्याओं में प्रत्येक को 2 घटाया जाए, तो नया औसत ज्ञात कीजिए।
Solution:
Original average = 36
Total numbers = 15
Original total:
= 15 × 36
= 540
Increase in first 5 numbers:
= 5 × 4
= 20
Decrease in last 5 numbers:
= 5 × 2
= 10
Net increase:
= 20 - 10
= 10
New total:
= 540 + 10
= 550
New average:
= 550/15
= 110/3
= 36⅔
Answer: 36⅔
24. Question
English: A cube is painted on all faces and cut into 343 equal smaller cubes. How many smaller cubes have exactly one face painted?
Hindi: एक घन की सभी सतहों को रंगा गया और उसे 343 बराबर छोटे घनों में काटा गया। कितने छोटे घनों की ठीक एक सतह रंगी होगी?
Solution:
343 = 7³
So cube is cut into 7 parts along each edge.
Small cubes with exactly one face painted are on face interiors.
Formula:
= 6 × (n - 2)²
Here n = 7
= 6 × (7 - 2)²
= 6 × 5²
= 6 × 25
= 150
Answer: 150
25. Question
English: If x + y = 10 and xy = 21, find x⁴ + y⁴.
Hindi: यदि x + y = 10 और xy = 21 है, तो x⁴ + y⁴ ज्ञात कीजिए।
Solution:
Given:
x + y = 10
xy = 21
First:
x² + y² = (x + y)² - 2xy
= 10² - 2 × 21
= 100 - 42
= 58
Now:
x⁴ + y⁴ = (x² + y²)² - 2x²y²
= 58² - 2 × 21²
= 3364 - 2 × 441
= 3364 - 882
= 2482
Answer: 2482