1. Question
English: If a + b + c = 10, ab + bc + ca = 31 and abc = 30, find a³ + b³ + c³.
Hindi: यदि a + b + c = 10, ab + bc + ca = 31 और abc = 30 है, तो a³ + b³ + c³ ज्ञात कीजिए।
Solution:
a³ + b³ + c³ - 3abc = (a + b + c)[(a + b + c)² - 3(ab + bc + ca)]
= 10 × [10² - 3 × 31]
= 10 × [100 - 93]
= 10 × 7
= 70
Now:
a³ + b³ + c³ - 3abc = 70
3abc = 3 × 30 = 90
a³ + b³ + c³ = 70 + 90
= 160
Answer: 160
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2. Question
English: If α and β are roots of x² - 7x + 10 = 0, find the equation whose roots are α² and β².
Hindi: यदि x² - 7x + 10 = 0 के मूल α और β हैं, तो α² और β² मूलों वाला समीकरण ज्ञात कीजिए।
Solution:
α + β = 7
αβ = 10
α² + β² = (α + β)² - 2αβ
= 7² - 2 × 10
= 49 - 20
= 29
α²β² = (αβ)²
= 10²
= 100
Required equation:
x² - 29x + 100 = 0
Answer: x² - 29x + 100 = 0
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3. Question
English: Find the least positive integer N such that N ≡ 2 mod 5, N ≡ 4 mod 7, and N ≡ 6 mod 11.
Hindi: सबसे छोटी धनात्मक संख्या N ज्ञात कीजिए जिसके लिए N ≡ 2 mod 5, N ≡ 4 mod 7, और N ≡ 6 mod 11 हो।
Solution:
Check option 347:
347 divided by 5 leaves remainder 2.
347 divided by 7 leaves remainder 4.
347 divided by 11 leaves remainder 6.
So least positive integer is 347.
Answer: 347
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4. Question
English: Find the equation of the circle whose diameter endpoints are (4,-5) and (-2,7).
Hindi: उस वृत्त का समीकरण ज्ञात कीजिए जिसके व्यास के अंतिम बिंदु (4,-5) और (-2,7) हैं।
Solution:
Centre is midpoint:
x-coordinate = (4 - 2) / 2 = 1
y-coordinate = (-5 + 7) / 2 = 1
Centre = (1,1)
Radius square:
= (4 - 1)² + (-5 - 1)²
= 3² + (-6)²
= 9 + 36
= 45
Equation:
(x - 1)² + (y - 1)² = 45
Expand:
x² + y² - 2x - 2y + 2 = 45
x² + y² - 2x - 2y - 43 = 0
Answer: x² + y² - 2x - 2y - 43 = 0
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5. Question
English: Four dice are thrown. Find the probability of getting exactly two sixes.
Hindi: चार पासे फेंके जाते हैं। ठीक दो पासों पर 6 आने की प्रायिकता ज्ञात कीजिए।
Solution:
For exactly two sixes:
Choose 2 dice out of 4:
4C2 = 6
Probability of getting six on selected dice:
= 1/6 × 1/6
Probability of not getting six on remaining 2 dice:
= 5/6 × 5/6
Required probability:
= 6 × 1/36 × 25/36
= 150/1296
= 25/216
Answer: 25/216
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6. Question
English: If sin θ + cos θ = 1/3, find sin 2θ.
Hindi: यदि sin θ + cos θ = 1/3 है, तो sin 2θ ज्ञात कीजिए।
Solution:
sin θ + cos θ = 1/3
Square both sides:
sin² θ + cos² θ + 2sin θ cos θ = 1/9
1 + sin 2θ = 1/9
sin 2θ = 1/9 - 1
= -8/9
Answer: -8/9
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7. Question
English: A cone of radius 14 cm and height 21 cm is cut by a plane parallel to its base at 9 cm from the vertex. Find the volume of the frustum.
Hindi: 14 सेमी त्रिज्या और 21 सेमी ऊँचाई वाले शंकु को शीर्ष से 9 सेमी दूरी पर आधार के समानांतर काटा जाता है। छिन्नक का आयतन ज्ञात कीजिए।
Solution:
Original cone radius = 14 cm
Original cone height = 21 cm
Small cone height = 9 cm
By similarity:
Small cone radius = 14 × 9 / 21
= 6 cm
Volume of original cone:
= 1/3 × π × 14² × 21
= 1372π
Volume of small cone:
= 1/3 × π × 6² × 9
= 108π
Volume of frustum:
= 1372π - 108π
= 1264π cm³
Answer: 1264π cm³
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8. Question
English: Two trains of lengths 180 m and 300 m cross each other in 12 seconds in opposite directions and in 48 seconds in the same direction. Find the difference of their speeds.
Hindi: 180 मीटर और 300 मीटर लंबाई वाली दो ट्रेनें विपरीत दिशा में 12 सेकंड में और समान दिशा में 48 सेकंड में एक-दूसरे को पार करती हैं। उनकी गतियों का अंतर ज्ञात कीजिए।
Solution:
Total length = 180 + 300
= 480 m
Same direction crossing time = 48 seconds
Difference of speeds:
= 480/48
= 10 m/s
Convert into km/h:
= 10 × 18/5
= 36 km/h
Answer: 36 km/h
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9. Question
English: Pipe A fills a tank in 12 hours, pipe B fills it in 18 hours, and pipe C empties it in 36 hours. If all are opened together, find the time to fill the tank.
Hindi: पाइप A टंकी को 12 घंटे में भरता है, पाइप B 18 घंटे में भरता है और पाइप C 36 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 = -1/36
Together work:
= 1/12 + 1/18 - 1/36
LCM = 36
= 3/36 + 2/36 - 1/36
= 4/36
= 1/9
So tank will be filled in 9 hours.
Answer: 9 hours
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10. Question
English: In how many ways can the letters of the word BALLOON be arranged if all vowels are together?
Hindi: BALLOON शब्द के अक्षरों को कितने तरीकों से व्यवस्थित किया जा सकता है यदि सभी स्वर साथ हों?
Solution:
Word: BALLOON
Vowels = A, O, O
Consonants = B, L, L, N
Treat vowels as 1 block.
Units are:
Vowel block, B, L, L, N
Total units = 5
L repeats 2 times.
Arrangements of units:
= 5! / 2!
= 120 / 2
= 60
Arrangements inside vowel block A, O, O:
= 3! / 2!
= 3
Total arrangements:
= 60 × 3
= 180
Answer: 180
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11. Question
English: Find the remainder when 6¹²³ is divided by 13.
Hindi: 6¹²³ को 13 से भाग देने पर शेषफल ज्ञात कीजिए।
Solution:
Check powers:
6² = 36
36 divided by 13 leaves remainder 10.
6³ = 10 × 6 = 60
60 divided by 13 leaves remainder 8.
6⁴ = 8 × 6 = 48
48 divided by 13 leaves remainder 9.
6⁶ gives remainder 12, which is same as -1.
So 6¹² gives remainder 1.
Now:
123 divided by 12 leaves remainder 3.
So remainder of 6¹²³ is same as 6³.
6³ = 216
216 divided by 13 leaves remainder 8.
Answer: 8
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12. Question
English: Find the total number of divisors of 10800.
Hindi: 10800 के कुल भाजकों की संख्या ज्ञात कीजिए।
Solution:
10800 = 108 × 100
108 = 2² × 3³
100 = 2² × 5²
So:
10800 = 2⁴ × 3³ × 5²
Number of divisors:
= (4 + 1)(3 + 1)(2 + 1)
= 5 × 4 × 3
= 60
Answer: 60
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13. Question
English: The HCF and LCM of two numbers are 15 and 1260 respectively. If their sum is 435, find their difference.
Hindi: दो संख्याओं का HCF 15 और LCM 1260 है। यदि उनका योग 435 है, तो उनका अंतर ज्ञात कीजिए।
Solution:
Let numbers be 15a and 15b.
Since HCF is 15, a and b are co-prime.
LCM = 15ab
So:
15ab = 1260
ab = 84
Also:
15a + 15b = 435
a + b = 29
Two co-prime numbers with product 84 and sum 29 are 4 and 21.
Numbers are:
15 × 4 = 60
15 × 21 = 315
Difference:
= 315 - 60
= 255
Answer: 255
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14. Question
English: Bag A has 7 red and 3 blue balls, Bag B has 4 red and 6 blue balls, and Bag C has 1 red and 9 blue balls. One bag is chosen randomly and one ball drawn is red. Find probability it came from Bag A.
Hindi: बैग A में 7 लाल और 3 नीली गेंदें हैं, बैग B में 4 लाल और 6 नीली गेंदें हैं, और बैग C में 1 लाल और 9 नीली गेंदें हैं। एक बैग यादृच्छया चुना गया और लाल गेंद निकली। उसके बैग A से आने की प्रायिकता ज्ञात कीजिए।
Solution:
Probability of choosing each bag = 1/3
Probability of red from A = 7/10
Probability of red from B = 4/10
Probability of red from C = 1/10
Required probability:
= (1/3 × 7/10) / [(1/3 × 7/10) + (1/3 × 4/10) + (1/3 × 1/10)]
Cancel 1/3:
= 7/10 / (7/10 + 4/10 + 1/10)
= 7/10 / 12/10
= 7/12
Answer: 7/12
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15. Question
English: Find the locus of a point equidistant from (-4,3) and (6,-1).
Hindi: (-4,3) और (6,-1) से समान दूरी पर स्थित बिंदु का बिंदुपथ ज्ञात कीजिए।
Solution:
Let the point be (x,y).
Distance from (-4,3) and (6,-1) is equal.
So:
(x + 4)² + (y - 3)² = (x - 6)² + (y + 1)²
Expand:
x² + 8x + 16 + y² - 6y + 9
= x² - 12x + 36 + y² + 2y + 1
Cancel x² and y²:
8x - 6y + 25 = -12x + 2y + 37
20x - 8y - 12 = 0
Divide by 4:
5x - 2y - 3 = 0
Answer: 5x - 2y - 3 = 0
16. Question
English: If α and β are roots of x² - 6x + 8 = 0, find the equation whose roots are α + 3/β and β + 3/α.
Hindi: यदि x² - 6x + 8 = 0 के मूल α और β हैं, तो उस समीकरण को ज्ञात कीजिए जिसके मूल α + 3/β और β + 3/α हैं।
Solution:
α + β = 6
αβ = 8
New roots are:
α + 3/β and β + 3/α
Sum of new roots:
= α + β + 3/β + 3/α
= 6 + 3(α + β)/αβ
= 6 + 3 × 6/8
= 6 + 18/8
= 6 + 9/4
= 33/4
Product of new roots:
= (α + 3/β)(β + 3/α)
= αβ + 3 + 3 + 9/αβ
= 8 + 6 + 9/8
= 14 + 9/8
= 121/8
Equation:
x² - 33/4x + 121/8 = 0
Multiply by 8:
8x² - 66x + 121 = 0
Answer: 8x² - 66x + 121 = 0
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17. Question
English: The medians of a triangle are 15 cm, 18 cm and 21 cm. Find the area of the triangle.
Hindi: एक त्रिभुज की मध्यिकाएँ 15 सेमी, 18 सेमी और 21 सेमी हैं। त्रिभुज का क्षेत्रफल ज्ञात कीजिए।
Solution:
Area of original triangle = 4/3 × area of triangle formed by medians.
Triangle formed by medians has sides 15, 18 and 21.
Semi-perimeter:
= (15 + 18 + 21)/2
= 54/2
= 27
Area of median triangle:
= √[27 × (27 - 15) × (27 - 18) × (27 - 21)]
= √[27 × 12 × 9 × 6]
= √17496
= 54√6
Area of original triangle:
= 4/3 × 54√6
= 72√6
Answer: 72√6 cm²
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18. Question
English: Marked price is 80% above cost price. Successive discounts of 25% and 10% are given. Find the profit percentage.
Hindi: अंकित मूल्य क्रय मूल्य से 80% अधिक है। 25% और 10% की क्रमिक छूट दी जाती है। लाभ प्रतिशत ज्ञात कीजिए।
Solution:
Let CP = 100
Marked price = 180
After 25% discount:
= 180 × 75/100
= 135
After 10% discount:
= 135 × 90/100
= 121.5
Selling price = 121.5
Profit = 121.5 - 100
= 21.5
Profit percentage = 21.5%
Answer: 21.5%
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19. Question
English: A cube is painted on all faces and cut into 125 equal small cubes. How many small cubes have exactly two faces painted?
Hindi: एक घन की सभी सतहों को रंगा गया और उसे 125 बराबर छोटे घनों में काटा गया। कितने छोटे घनों की ठीक दो सतहें रंगी होंगी?
Solution:
125 = 5³
So cube is cut into 5 parts along each edge.
Small cubes with exactly 2 faces painted lie on edges but not at corners.
Formula:
12 × (n - 2)
Here n = 5
= 12 × (5 - 2)
= 12 × 3
= 36
Answer: 36
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20. Question
English: From a point P outside a circle, tangent length is 18 cm. A secant has external segment 12 cm and internal segment x cm. Find x.
Hindi: वृत्त के बाहर स्थित बिंदु P से स्पर्शरेखा की लंबाई 18 सेमी है। एक सेकेंट का बाहरी भाग 12 सेमी और आंतरिक भाग x सेमी है। x ज्ञात कीजिए।
Solution:
Tangent square = external segment × whole secant
18² = 12(12 + x)
324 = 144 + 12x
12x = 180
x = 15
Answer: 15
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21. Question
English: If x² - 14x + 1 = 0, find x² + 1/x².
Hindi: यदि x² - 14x + 1 = 0 है, तो x² + 1/x² ज्ञात कीजिए।
Solution:
x² - 14x + 1 = 0
Divide by x:
x - 14 + 1/x = 0
x + 1/x = 14
Now:
x² + 1/x² = (x + 1/x)² - 2
= 14² - 2
= 196 - 2
= 194
Answer: 194
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22. Question
English: Two dice are thrown. Find the probability that their product is divisible by 4.
Hindi: दो पासे फेंके जाते हैं। उनके गुणनफल के 4 से विभाज्य होने की प्रायिकता ज्ञात कीजिए।
Solution:
Total outcomes = 36
Product must be divisible by 4.
Count favourable outcomes:
If one die is 4, product is always divisible by 4.
Pairs with 4:
(4,1), (4,2), (4,3), (4,4), (4,5), (4,6) = 6
(1,4), (2,4), (3,4), (5,4), (6,4) = 5 more
Total = 11
If both numbers are even but neither is 4:
Possible even numbers excluding 4 are 2 and 6.
Pairs:
(2,2), (2,6), (6,2), (6,6) = 4
Total favourable outcomes:
= 11 + 4
= 15
Probability:
= 15/36
= 5/12
Answer: 5/12
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23. Question
English: If (x + y)/(x - y) = 5/2, find (x² + y²)/xy.
Hindi: यदि (x + y)/(x - y) = 5/2 है, तो (x² + y²)/xy ज्ञात कीजिए।
Solution:
(x + y)/(x - y) = 5/2
2(x + y) = 5(x - y)
2x + 2y = 5x - 5y
7y = 3x
x/y = 7/3
Let x = 7 and y = 3
Now:
(x² + y²)/xy
= (7² + 3²) / (7 × 3)
= (49 + 9) / 21
= 58/21
Answer: 58/21
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24. Question
English: Alloy A contains 60% copper and Alloy B contains 35% copper. In what ratio should they be mixed to get an alloy containing 50% copper?
Hindi: मिश्रधातु A में 60% तांबा और मिश्रधातु B में 35% तांबा है। 50% तांबा वाली मिश्रधातु प्राप्त करने के लिए उन्हें किस अनुपात में मिलाया जाए?
Solution:
Using allegation:
A = 60%
B = 35%
Mean = 50%
Ratio A : B
= (50 - 35) : (60 - 50)
= 15 : 10
= 3 : 2
Answer: 3 : 2
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25. Question
English: In an AP, the 17th term is 42 and the 42nd term is 17. Find the sum of the first 59 terms.
Hindi: एक AP में 17वाँ पद 42 और 42वाँ पद 17 है। पहले 59 पदों का योग ज्ञात कीजिए।
Solution:
Let first term = a and common difference = d.
17th term:
a + 16d = 42
42nd term:
a + 41d = 17
Subtract:
25d = -25
d = -1
Now:
a + 16(-1) = 42
a - 16 = 42
a = 58
Sum of first 59 terms:
S = 59/2 × [2a + 58d]
= 59/2 × [2 × 58 + 58 × -1]
= 59/2 × [116 - 58]
= 59/2 × 58
= 59 × 29
= 1711
Answer: 1711