1. Question
English: If f(x) = x/(x + 1), find f(f(f(1))).
Hindi: यदि f(x) = x/(x + 1) है, तो f(f(f(1))) ज्ञात कीजिए।
Solution:
Given:
f(x) = x/(x + 1)
First:
f(1)
= 1/(1 + 1)
= 1/2
Now:
f(1/2)
= (1/2)/(1/2 + 1)
= (1/2)/(3/2)
= 1/3
Now:
f(1/3)
= (1/3)/(1/3 + 1)
= (1/3)/(4/3)
= 1/4
Answer: 1/4
2. Question
English: The HCF and LCM of two numbers are 21 and 1260 respectively. If their sum is 399, find the larger number.
Hindi: दो संख्याओं का HCF और LCM क्रमशः 21 और 1260 है। यदि उनका योग 399 है, तो बड़ी संख्या ज्ञात कीजिए।
Solution:
Let numbers be:
21a and 21b
Since HCF = 21, a and b are co-prime.
LCM:
= 21ab
21ab = 1260
ab = 60
Also:
21a + 21b = 399
a + b = 19
Numbers with sum 19 and product 60 are:
15 and 4
Actual numbers:
315 and 84
Larger number:
= 315
Answer: 315
3. Question
English: Simplify: 1/(√7 - √6) + 1/(√7 + √6).
Hindi: सरलीकृत कीजिए: 1/(√7 - √6) + 1/(√7 + √6)।
Solution:
Take LCM:
= [(√7 + √6) + (√7 - √6)] / [(√7 - √6)(√7 + √6)]
Numerator:
= 2√7
Denominator:
= 7 - 6
= 1
Therefore:
= 2√7
Answer: 2√7
4. Question
English: Evaluate: (125^(2/3) × 27^(4/3)) / 81^(3/4).
Hindi: मान ज्ञात कीजिए: (125^(2/3) × 27^(4/3)) / 81^(3/4)।
Solution:
125^(2/3)
= (³√125)²
= 5²
= 25
27^(4/3)
= (³√27)⁴
= 3⁴
= 81
81^(3/4)
= (⁴√81)³
= 3³
= 27
Therefore:
(25 × 81)/27
= 25 × 3
= 75
Answer: 75
5. Question
English: Find the smaller angle between the hour hand and minute hand at 7:20.
Hindi: 7:20 बजे घंटे की सुई और मिनट की सुई के बीच छोटा कोण ज्ञात कीजिए।
Solution:
Minute hand angle:
= 20 × 6
= 120°
Hour hand angle:
= 7 × 30 + 20 × 0.5
= 210 + 10
= 220°
Difference:
= 220 - 120
= 100°
Answer: 100°
6. Question
English: Find the number of odd divisors of 2⁴ × 3⁵ × 5².
Hindi: 2⁴ × 3⁵ × 5² के विषम भाजकों की संख्या ज्ञात कीजिए।
Solution:
Odd divisors cannot contain factor 2.
So power of 2 must be 0.
For 3⁵:
Possible powers:
0,1,2,3,4,5
= 6 choices
For 5²:
Possible powers:
0,1,2
= 3 choices
Total odd divisors:
= 6 × 3
= 18
Answer: 18
7. Question
English: Find the remainder when 9⁸⁷ is divided by 13.
Hindi: 9⁸⁷ को 13 से भाग देने पर शेषफल ज्ञात कीजिए।
Solution:
9² = 81
81 divided by 13 leaves remainder 3.
9³:
= 9 × 3
= 27
Remainder = 1
So:
9³ ≡ 1 (mod 13)
Now:
87 = 3 × 29
Therefore:
9⁸⁷ = (9³)²⁹
Remainder:
= 1²⁹
= 1
Answer: 1
8. Question
English: In an AP, the sum of the 7th and 13th terms is 92, and the 20th term is 86. Find the sum of the first 25 terms.
Hindi: एक AP में 7वें और 13वें पदों का योग 92 है, और 20वाँ पद 86 है। पहले 25 पदों का योग ज्ञात कीजिए।
Solution:
7th term:
a + 6d
13th term:
a + 12d
Their sum:
2a + 18d = 92
a + 9d = 46
20th term:
a + 19d = 86
Subtract:
10d = 40
d = 4
Now:
a + 9 × 4 = 46
a + 36 = 46
a = 10
Sum of first 25 terms:
S = 25/2 × [2a + 24d]
= 25/2 × [20 + 96]
= 25/2 × 116
= 25 × 58
= 1450
Answer: 1450
9. Question
English: Three positive numbers are in GP. Their product is 1000 and their sum is 35. Find the sum of their squares.
Hindi: तीन धनात्मक संख्याएँ GP में हैं। उनका गुणनफल 1000 और योग 35 है। उनके वर्गों का योग ज्ञात कीजिए।
Solution:
Let numbers be:
a/r, a, ar
Their product:
= a³
= 1000
a = 10
So numbers are:
10/r, 10, 10r
Their sum:
10/r + 10 + 10r = 35
Divide by 5:
2/r + 2 + 2r = 7
Try r = 2
Numbers:
5, 10, 20
Sum = 35
Now sum of squares:
= 5² + 10² + 20²
= 25 + 100 + 400
= 525
Answer: 525
10. Question
English: In how many ways can the letters of the word EDUCATION be arranged if all vowels are together?
Hindi: EDUCATION शब्द के अक्षरों को कितने तरीकों से व्यवस्थित किया जा सकता है यदि सभी स्वर साथ हों?
Solution:
Word: EDUCATION
Total letters = 9
Vowels:
E, U, A, I, O
= 5 vowels
Consonants:
D, C, T, N
= 4 consonants
Treat all vowels as one group.
Now total groups:
= 5
These can be arranged in:
= 5!
Vowels among themselves:
= 5!
Total arrangements:
= 5! × 5!
= 120 × 120
= 14400
Answer: 14400
11. Question
English: Three cards are drawn from a standard deck. Find the probability of getting exactly one ace.
Hindi: एक सामान्य ताश की गड्डी से 3 पत्ते निकाले जाते हैं। ठीक 1 इक्का आने की प्रायिकता ज्ञात कीजिए।
Solution:
Total cards = 52
Aces = 4
Non-aces = 48
Exactly 1 ace means:
Choose 1 ace from 4:
= 4C1
= 4
Choose 2 non-aces from 48:
= 48C2
= 1128
Favourable ways:
= 4 × 1128
= 4512
Total ways:
= 52C3
= 22100
Probability:
= 4512/22100
= 1128/5525
Answer: 1128/5525
12. Question
English: A and B together can complete a work in 10 days, B and C together in 12 days, and A and C together in 15 days. Find the time taken by A, B and C together.
Hindi: A और B मिलकर कार्य 10 दिनों में, B और C मिलकर 12 दिनों में तथा A और C मिलकर 15 दिनों में करते हैं। A, B और C मिलकर कार्य कितने दिनों में करेंगे?
Solution:
A + B = 1/10
B + C = 1/12
A + C = 1/15
Add all:
2(A + B + C)
= 1/10 + 1/12 + 1/15
LCM = 60
= 6/60 + 5/60 + 4/60
= 15/60
= 1/4
So:
A + B + C = 1/8
Therefore, together they complete work in:
= 8 days
Answer: 8 days
13. Question
English: Pipe A fills a tank in 10 hours, pipe B fills it in 15 hours and pipe C empties it in 20 hours. If all are opened together, find the time to fill the tank.
Hindi: पाइप A टंकी को 10 घंटे में भरता है, पाइप B 15 घंटे में भरता है और पाइप C 20 घंटे में खाली करता है। तीनों को साथ खोलने पर टंकी कितने समय में भरेगी?
Solution:
A’s 1-hour work:
= 1/10
B’s 1-hour work:
= 1/15
C’s 1-hour work:
= -1/20
Net work:
= 1/10 + 1/15 - 1/20
LCM = 60
= 6/60 + 4/60 - 3/60
= 7/60
Time taken:
= 60/7 hours
Answer: 60/7 hours
14. Question
English: In a 400 m race, A beats B by 40 m. In a 360 m race, B beats C by 30 m. By how many metres will A beat C in a 600 m race?
Hindi: 400 मीटर की दौड़ में A, B को 40 मीटर से हराता है। 360 मीटर की दौड़ में B, C को 30 मीटर से हराता है। 600 मीटर की दौड़ में A, C को कितने मीटर से हराएगा?
Solution:
When A runs 400 m, B runs 360 m.
So:
A : B = 400 : 360
= 10 : 9
When B runs 360 m, C runs 330 m.
So:
B : C = 360 : 330
= 12 : 11
Now combine:
A : B = 10 : 9
B : C = 12 : 11
Make B same.
LCM of 9 and 12 = 36
A : B = 40 : 36
B : C = 36 : 33
So:
A : C = 40 : 33
In a 600 m race, when A runs 600 m, C runs:
= 600 × 33/40
= 15 × 33
= 495 m
A beats C by:
= 600 - 495
= 105 m
Answer: 105 m
15. Question
English: A 50-litre mixture contains 30% acid. How much pure acid should be added to make the acid concentration 50%?
Hindi: 50 लीटर मिश्रण में 30% अम्ल है। अम्ल की मात्रा 50% करने के लिए कितना शुद्ध अम्ल मिलाया जाना चाहिए?
Solution:
Initial acid:
= 30% of 50
= 15 litres
Let pure acid added = x litres.
New acid:
= 15 + x
New mixture:
= 50 + x
Required concentration = 50%
So:
(15 + x)/(50 + x) = 1/2
2(15 + x) = 50 + x
30 + 2x = 50 + x
x = 20
Answer: 20 litres
16. Question
English: The sides of a triangle are 13 cm, 20 cm and 21 cm. Find its inradius.
Hindi: एक त्रिभुज की भुजाएँ 13 सेमी, 20 सेमी और 21 सेमी हैं। उसकी अंतःत्रिज्या ज्ञात कीजिए।
Solution:
Semi-perimeter:
= (13 + 20 + 21)/2
= 54/2
= 27
Area:
= √[27 × (27 - 13) × (27 - 20) × (27 - 21)]
= √[27 × 14 × 7 × 6]
= √15876
= 126
Inradius:
= Area / semi-perimeter
= 126/27
= 14/3 cm
Answer: 14/3 cm
17. Question
English: Find the distance between the parallel lines 3x - 4y + 7 = 0 and 6x - 8y - 9 = 0.
Hindi: समानांतर रेखाओं 3x - 4y + 7 = 0 और 6x - 8y - 9 = 0 के बीच की दूरी ज्ञात कीजिए।
Solution:
First make coefficients of x and y same.
Second line:
6x - 8y - 9 = 0
Divide by 2:
3x - 4y - 9/2 = 0
Now lines are:
3x - 4y + 7 = 0
3x - 4y - 9/2 = 0
Distance between parallel lines:
= |c1 - c2| / √(a² + b²)
= |7 - (-9/2)| / √(3² + (-4)²)
= |7 + 9/2| / 5
= |14/2 + 9/2| / 5
= 23/2 ÷ 5
= 23/10
Answer: 23/10
18. Question
English: If f(n) = 3f(n - 1) + 2 and f(1) = 1, find f(4).
Hindi: यदि f(n) = 3f(n - 1) + 2 और f(1) = 1 है, तो f(4) ज्ञात कीजिए।
Solution:
f(1) = 1
f(2) = 3f(1) + 2
= 3 × 1 + 2
= 5
f(3) = 3f(2) + 2
= 3 × 5 + 2
= 17
f(4) = 3f(3) + 2
= 3 × 17 + 2
= 53
Answer: 53
19. Question
English: If the roots of x² - px + 45 = 0 are positive and differ by 4, find p.
Hindi: यदि x² - px + 45 = 0 के मूल धनात्मक हैं और उनका अंतर 4 है, तो p ज्ञात कीजिए।
Solution:
Let roots be a and b.
ab = 45
a - b = 4
Now:
(a + b)² = (a - b)² + 4ab
= 4² + 4 × 45
= 16 + 180
= 196
a + b = 14
In equation:
x² - px + 45 = 0
Sum of roots = p
So:
p = 14
Answer: 14
20. Question
English: If the price of an item increases by 20% and its consumption decreases by 10%, find the percentage change in expenditure.
Hindi: यदि किसी वस्तु का मूल्य 20% बढ़ता है और उसकी खपत 10% घटती है, तो खर्च में प्रतिशत परिवर्तन ज्ञात कीजिए।
Solution:
Let original price = 100
Let original consumption = 100
Original expenditure:
= 100 × 100
New price after 20% increase:
= 120
New consumption after 10% decrease:
= 90
New expenditure:
= 120 × 90
= 10800
Original expenditure:
= 10000
Change:
= 800 increase
Percentage change:
= 800/10000 × 100
= 8% increase
Answer: 8% increase
21. Question
English: The average of 20 numbers is 56. The average of the first 12 numbers is 52 and that of the last 9 numbers is 60. Find the 12th number.
Hindi: 20 संख्याओं का औसत 56 है। पहली 12 संख्याओं का औसत 52 और अंतिम 9 संख्याओं का औसत 60 है। 12वीं संख्या ज्ञात कीजिए।
Solution:
Total sum of 20 numbers:
= 20 × 56
= 1120
Sum of first 12 numbers:
= 12 × 52
= 624
Sum of last 9 numbers:
= 9 × 60
= 540
Now first 12 numbers and last 9 numbers overlap at the 12th number.
So:
624 + 540 = total sum + 12th number
1164 = 1120 + 12th number
12th number:
= 1164 - 1120
= 44
Answer: 44
22. Question
English: Find the HCF of 3/4, 9/10 and 15/16.
Hindi: 3/4, 9/10 और 15/16 का HCF ज्ञात कीजिए।
Solution:
For fractions:
HCF = HCF of numerators / LCM of denominators
Numerators:
3, 9, 15
HCF = 3
Denominators:
4, 10, 16
LCM = 80
So:
HCF = 3/80
Answer: 3/80
23. Question
English: A cube is painted on all faces and cut into 216 equal smaller cubes. How many smaller cubes have exactly two faces painted?
Hindi: एक घन की सभी सतहों को रंगा गया और उसे 216 बराबर छोटे घनों में काटा गया। कितने छोटे घनों की ठीक दो सतहें रंगी होंगी?
Solution:
216 = 6³
So cube is cut into 6 parts along each edge.
Cubes with exactly two faces painted are on edges, excluding corners.
Formula:
= 12 × (n - 2)
Here n = 6
= 12 × (6 - 2)
= 12 × 4
= 48
Answer: 48
24. Question
English: If f(x) = x² and g(x) = 2x + 1, find (g ∘ f)(3) - (f ∘ g)(2).
Hindi: यदि f(x) = x² और g(x) = 2x + 1 है, तो (g ∘ f)(3) - (f ∘ g)(2) ज्ञात कीजिए।
Solution:
First:
(g ∘ f)(3) = g(f(3))
f(3) = 3² = 9
g(9) = 2 × 9 + 1
= 19
Now:
(f ∘ g)(2) = f(g(2))
g(2) = 2 × 2 + 1
= 5
f(5) = 5²
= 25
Required value:
= 19 - 25
= -6
Answer: -6
25. Question
English: Find the least positive integer N such that N leaves remainder 5 when divided by 17 and remainder 7 when divided by 19.
Hindi: सबसे छोटी धनात्मक संख्या N ज्ञात कीजिए जो 17 से भाग देने पर 5 शेष और 19 से भाग देने पर 7 शेष छोड़ती है।
Solution:
N leaves remainder 5 when divided by 17.
So:
N = 17k + 5
It also leaves remainder 7 when divided by 19.
So:
17k + 5 leaves remainder 7 when divided by 19
17k - 2 should be divisible by 19
17k = 2 mod 19
Now try k values.
k = 18 gives:
17 × 18 = 306
306 divided by 19 leaves remainder 2.
So:
N = 17 × 18 + 5
= 306 + 5
= 311
Answer: 311