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1. Question
English: If a + b + c = 6, ab + bc + ca = -7 and abc = -60, find a³ + b³ + c³.
Hindi: यदि a + b + c = 6, ab + bc + ca = -7 और abc = -60 है, तो a³ + b³ + c³ ज्ञात कीजिए।
Solution:
a³ + b³ + c³ - 3abc = (a + b + c)[(a + b + c)² - 3(ab + bc + ca)]
= 6 × [6² - 3 × -7]
= 6 × [36 + 21]
= 6 × 57
= 342
Now:
a³ + b³ + c³ - 3abc = 342
abc = -60
3abc = -180
a³ + b³ + c³ + 180 = 342
a³ + b³ + c³ = 162
Answer: 162
2. Question
English: If α and β are roots of x² - 9x + 14 = 0, find α³β² + α²β³.
Hindi: यदि x² - 9x + 14 = 0 के मूल α और β हैं, तो α³β² + α²β³ ज्ञात कीजिए।
Solution:
α + β = 9
αβ = 14
α³β² + α²β³ = α²β²(α + β)
= (αβ)²(α + β)
= 14² × 9
= 196 × 9
= 1764
Answer: 1764
3. Question
English: Find the least positive integer N such that N ≡ 4 mod 7, N ≡ 5 mod 11, and N ≡ 6 mod 13.
Hindi: सबसे छोटी धनात्मक संख्या N ज्ञात कीजिए जिसके लिए N ≡ 4 mod 7, N ≡ 5 mod 11, और N ≡ 6 mod 13 हो।
Solution:
Check options:
214 divided by 7 leaves remainder 4.
214 divided by 11 leaves remainder 5.
214 divided by 13 leaves remainder 6.
Answer: 214
4. Question
English: Find the equation of the circle whose diameter endpoints are (-3,8) and (7,-2).
Hindi: उस वृत्त का समीकरण ज्ञात कीजिए जिसके व्यास के अंतिम बिंदु (-3,8) और (7,-2) हैं।
Solution:
Centre is midpoint:
x-coordinate = (-3 + 7) / 2 = 2
y-coordinate = (8 - 2) / 2 = 3
Centre = (2,3)
Radius square:
= (2 + 3)² + (3 - 8)²
= 5² + (-5)²
= 25 + 25
= 50
Equation:
(x - 2)² + (y - 3)² = 50
Expand:
x² + y² - 4x - 6y - 37 = 0
Answer: x² + y² - 4x - 6y - 37 = 0
5. Question
English: Five cards are drawn from a standard deck. Find the probability that exactly 2 are kings and exactly 1 is queen.
Hindi: एक सामान्य ताश की गड्डी से 5 पत्ते निकाले जाते हैं। ठीक 2 पत्ते बादशाह और ठीक 1 पत्ता रानी होने की प्रायिकता ज्ञात कीजिए।
Solution:
Kings = 4
Queens = 4
Other cards = 44
Choose 2 kings from 4 = 6
Choose 1 queen from 4 = 4
Choose remaining 2 cards from 44 = 946
Favourable ways = 6 × 4 × 946
= 22704
Total ways to choose 5 cards from 52 = 2598960
Probability = 22704 / 2598960
= 473/54145
Answer: 473/54145
6. Question
English: If sec θ + tan θ = 5/2, find sin θ.
Hindi: यदि sec θ + tan θ = 5/2 है, तो sin θ ज्ञात कीजिए।
Solution:
sec θ + tan θ = 5/2
We know:
(sec θ + tan θ)(sec θ - tan θ) = 1
So:
sec θ - tan θ = 2/5
Now subtract:
(sec θ + tan θ) - (sec θ - tan θ)
= 5/2 - 2/5
2tan θ = 25/10 - 4/10
2tan θ = 21/10
tan θ = 21/20
This forms a right triangle:
Opposite = 21
Base = 20
Hypotenuse = 29
So:
sin θ = 21/29
Answer: 21/29
7. Question
English: A cone of radius 15 cm and height 30 cm is cut by a plane parallel to its base at 12 cm from the vertex. Find the volume of the frustum.
Hindi: 15 सेमी त्रिज्या और 30 सेमी ऊँचाई वाले शंकु को शीर्ष से 12 सेमी दूरी पर आधार के समानांतर काटा जाता है। छिन्नक का आयतन ज्ञात कीजिए।
Solution:
Original cone:
Radius = 15 cm
Height = 30 cm
Small cone height = 12 cm
By similarity:
Small cone radius = 15 × 12 / 30
= 6 cm
Volume of original cone:
= 1/3 × π × 15² × 30
= 2250π
Volume of small cone:
= 1/3 × π × 6² × 12
= 144π
Volume of frustum:
= 2250π - 144π
= 2106π cm³
Answer: 2106π cm³
8. Question
English: A boat covers 72 km downstream in 4 hours and 72 km upstream in 6 hours. Find the speed of the stream.
Hindi: एक नाव 72 किमी धारा के साथ 4 घंटे में और 72 किमी धारा के विपरीत 6 घंटे में तय करती है। धारा की गति ज्ञात कीजिए।
Solution:
Downstream speed = 72/4
= 18 km/h
Upstream speed = 72/6
= 12 km/h
Speed of stream:
= (18 - 12) / 2
= 3 km/h
Answer: 3 km/h
9. Question
English: A can complete a work in 12 days and B in 16 days. They work on alternate days starting with A. Find the total time taken.
Hindi: A किसी कार्य को 12 दिनों में और B 16 दिनों में पूरा करता है। वे A से शुरू करके बारी-बारी से काम करते हैं। कुल समय ज्ञात कीजिए।
Solution:
A’s 1-day work = 1/12
B’s 1-day work = 1/16
Work done in 2 days:
= 1/12 + 1/16
LCM = 48
= 4/48 + 3/48
= 7/48
In 12 days, 6 cycles are completed.
Work done in 12 days:
= 6 × 7/48
= 42/48
= 7/8
Remaining work:
= 1/8
Day 13 is A’s turn.
A does 1/12 work in 1 day.
Remaining work is 1/8, so time taken by A:
= 1/8 ÷ 1/12
= 12/8
= 1.5 days
Total time:
= 12 + 1.5
= 13.5 days
Answer: 13.5 days
Note: Given options do not contain exact correct answer.
10. Question
English: In how many ways can the letters of the word ENGINEER be arranged if all vowels are together?
Hindi: ENGINEER शब्द के अक्षरों को कितने तरीकों से व्यवस्थित किया जा सकता है यदि सभी स्वर साथ हों?
Solution:
Word: ENGINEER
Letters: E, N, G, I, N, E, E, R
Vowels = E, I, E, E
Consonants = N, G, N, R
Treat all vowels as 1 block.
Units are:
Vowel block, N, G, N, R
Total units = 5
N repeats 2 times.
Arrangement of units:
= 5! / 2!
= 120 / 2
= 60
Arrangement inside vowel block:
E, E, E, I
= 4! / 3!
= 4
Total arrangements:
= 60 × 4
= 240
Answer: 240
11. Question
English: Find the remainder when 8¹²³ is divided by 17.
Hindi: 8¹²³ को 17 से भाग देने पर शेषफल ज्ञात कीजिए।
Solution:
8² = 64
64 divided by 17 leaves remainder 13.
8⁴ gives remainder 16, which is same as -1.
So:
8⁸ gives remainder 1.
Now 123 divided by 8 leaves remainder 3.
So:
8¹²³ gives same remainder as 8³.
8³ = 512
512 divided by 17 leaves remainder 2.
Answer: 2
12. Question
English: Find the total number of divisors of 75600.
Hindi: 75600 के कुल भाजकों की संख्या ज्ञात कीजिए।
Solution:
75600 = 756 × 100
756 = 2² × 3³ × 7
100 = 2² × 5²
So:
75600 = 2⁴ × 3³ × 5² × 7¹
Number of divisors:
= (4 + 1)(3 + 1)(2 + 1)(1 + 1)
= 5 × 4 × 3 × 2
= 120
Answer: 120
13. Question
English: If tan θ + cot θ = 7, find tan² θ + cot² θ.
Hindi: यदि tan θ + cot θ = 7 है, तो tan² θ + cot² θ ज्ञात कीजिए।
Solution:
tan θ + cot θ = 7
Square both sides:
tan² θ + cot² θ + 2tan θ cot θ = 49
Since tan θ cot θ = 1:
tan² θ + cot² θ + 2 = 49
tan² θ + cot² θ = 47
Answer: 47
14. Question
English: Two trains of lengths 180 m and 220 m cross each other in 10 seconds in opposite directions and in 25 seconds in the same direction. Find the difference of their speeds.
Hindi: 180 मीटर और 220 मीटर लंबाई वाली दो ट्रेनें विपरीत दिशा में 10 सेकंड में और समान दिशा में 25 सेकंड में एक-दूसरे को पार करती हैं। उनकी गतियों का अंतर ज्ञात कीजिए।
Solution:
Total length = 180 + 220
= 400 m
Same direction crossing time = 25 seconds
Difference of speeds:
= 400/25
= 16 m/s
Convert into km/h:
= 16 × 18/5
= 57.6 km/h
Answer: 57.6 km/h
15. Question
English: A vessel contains pure milk. Two-seventh of the mixture is removed and replaced with water. This process is repeated 3 times. Find the remaining fraction of milk.
Hindi: एक बर्तन में शुद्ध दूध है। मिश्रण का 2/7 भाग निकालकर पानी से भर दिया जाता है। यह प्रक्रिया 3 बार दोहराई जाती है। शेष दूध का अंश ज्ञात कीजिए।
Solution:
Each time removed fraction = 2/7
Remaining fraction each time:
= 1 - 2/7
= 5/7
After 3 times:
= (5/7)³
= 125/343
Answer: 125/343
16. Question
English: Bag A contains 6 red and 4 blue balls, Bag B contains 2 red and 8 blue balls, and Bag C contains 5 red and 5 blue balls. One bag is chosen randomly and one ball drawn is red. Find probability that it came from Bag A.
Hindi: बैग A में 6 लाल और 4 नीली गेंदें हैं, बैग B में 2 लाल और 8 नीली गेंदें हैं, और बैग C में 5 लाल और 5 नीली गेंदें हैं। एक बैग यादृच्छया चुना गया और एक 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 = 6/10 = 3/5
Probability of red from B = 2/10 = 1/5
Probability of red from C = 5/10 = 1/2
Required probability:
= probability of A and red / total probability of red
= (1/3 × 3/5) / [(1/3 × 3/5) + (1/3 × 1/5) + (1/3 × 1/2)]
Cancel 1/3 from all terms:
= 3/5 / (3/5 + 1/5 + 1/2)
= 3/5 / (4/5 + 1/2)
= 3/5 / (8/10 + 5/10)
= 3/5 / 13/10
= 3/5 × 10/13
= 6/13
Answer: 6/13
17. Question
English: Find the locus of a point equidistant from (2,-5) and (8,3).
Hindi: (2,-5) और (8,3) से समान दूरी पर स्थित बिंदु का बिंदुपथ ज्ञात कीजिए।
Solution:
Let point be (x,y).
Distance from (2,-5) and (8,3) is equal.
So:
(x - 2)² + (y + 5)² = (x - 8)² + (y - 3)²
Expand:
x² - 4x + 4 + y² + 10y + 25
= x² - 16x + 64 + y² - 6y + 9
Cancel x² and y²:
-4x + 10y + 29 = -16x - 6y + 73
12x + 16y - 44 = 0
Divide by 4:
3x + 4y - 11 = 0
Answer: 3x + 4y - 11 = 0
18. Question
English: If α and β are roots of x² - 8x + 12 = 0, find the equation whose roots are α + 2/β and β + 2/α.
Hindi: यदि x² - 8x + 12 = 0 के मूल α और β हैं, तो उस समीकरण को ज्ञात कीजिए जिसके मूल α + 2/β और β + 2/α हैं।
Solution:
α + β = 8
αβ = 12
New roots are:
α + 2/β and β + 2/α
Sum of new roots:
= α + β + 2/β + 2/α
= 8 + 2(α + β)/αβ
= 8 + 2 × 8/12
= 8 + 16/12
= 8 + 4/3
= 28/3
Product of new roots:
= (α + 2/β)(β + 2/α)
= αβ + 2 + 2 + 4/αβ
= 12 + 4 + 4/12
= 16 + 1/3
= 49/3
Equation:
x² - 28/3 x + 49/3 = 0
Multiply by 3:
3x² - 28x + 49 = 0
Answer: 3x² - 28x + 49 = 0
19. Question
English: The medians of a triangle are 12 cm, 15 cm and 21 cm. Find the area of the triangle.
Hindi: एक त्रिभुज की मध्यिकाएँ 12 सेमी, 15 सेमी और 21 सेमी हैं। त्रिभुज का क्षेत्रफल ज्ञात कीजिए।
Solution:
Area of original triangle = 4/3 × area of triangle formed by medians.
Medians are 12, 15, 21.
For triangle with sides 12, 15, 21:
Semi-perimeter = (12 + 15 + 21)/2
= 48/2
= 24
Area = √[24 × 12 × 9 × 3]
= √7776
= 36√6
Original triangle area:
= 4/3 × 36√6
= 48√6 cm²
Answer: 48√6 cm²
20. Question
English: The HCF and LCM of two numbers are 18 and 540 respectively. If the sum of the numbers is 234, find their difference.
Hindi: दो संख्याओं का HCF 18 और LCM 540 है। यदि उनका योग 234 है, तो उनका अंतर ज्ञात कीजिए।
Solution:
Let numbers be 18a and 18b.
Since HCF is 18, a and b are co-prime.
LCM = 18ab
So:
18ab = 540
ab = 30
Also:
18a + 18b = 234
a + b = 13
Now two co-prime numbers with product 30 and sum 13 are 3 and 10.
Numbers are:
18 × 3 = 54
18 × 10 = 180
Difference:
= 180 - 54
= 126
Answer: 126
21. Question
English: Marked price is 75% above cost price. Successive discounts of 20% and 20% are given. Find the profit percentage.
Hindi: अंकित मूल्य क्रय मूल्य से 75% अधिक है। 20% और 20% की क्रमिक छूट दी जाती है। लाभ प्रतिशत ज्ञात कीजिए।
Solution:
Let CP = 100
Marked price = 175
After first 20% discount:
= 175 × 80/100
= 140
After second 20% discount:
= 140 × 80/100
= 112
Profit = 112 - 100
= 12
Profit percentage = 12%
Answer: 12%
22. Question
English: If sin θ + cos θ = √3/2, find sin 2θ.
Hindi: यदि sin θ + cos θ = √3/2 है, तो sin 2θ ज्ञात कीजिए।
Solution:
sin θ + cos θ = √3/2
Square both sides:
sin² θ + cos² θ + 2sin θ cos θ = 3/4
1 + sin 2θ = 3/4
sin 2θ = 3/4 - 1
= -1/4
Answer: -1/4
23. Question
English: From a point P outside a circle, tangent length is 15 cm. A secant has external segment 9 cm and internal segment x cm. Find x.
Hindi: वृत्त के बाहर स्थित बिंदु P से स्पर्शरेखा की लंबाई 15 सेमी है। एक सेकेंट का बाहरी भाग 9 सेमी और आंतरिक भाग x सेमी है। x ज्ञात कीजिए।
Solution:
Tangent square = external segment × whole secant
15² = 9 × (9 + x)
225 = 81 + 9x
9x = 144
x = 16
Answer: 16
24. Question
English: If x² - 10x + 1 = 0, find x³ + 1/x³.
Hindi: यदि x² - 10x + 1 = 0 है, तो x³ + 1/x³ ज्ञात कीजिए।
Solution:
x² - 10x + 1 = 0
Divide by x:
x - 10 + 1/x = 0
x + 1/x = 10
Now:
x³ + 1/x³ = (x + 1/x)³ - 3(x + 1/x)
= 10³ - 3 × 10
= 1000 - 30
= 970
Answer: 970
25. Question
English: How many 5-digit even numbers greater than 50000 can be formed using the digits 0 to 7 without repetition?
Hindi: 0 से 7 तक के अंकों का बिना पुनरावृत्ति उपयोग करके 50000 से बड़ी कितनी 5-अंकीय सम संख्याएँ बनाई जा सकती हैं?
Solution:
Digits available:
0, 1, 2, 3, 4, 5, 6, 7
Number must be greater than 50000.
So first digit can be:
5, 6, 7
Number must be even, so last digit can be:
0, 2, 4, 6
Case 1: First digit is 5 or 7
There are 2 choices for first digit.
Last digit can be 0, 2, 4, 6 = 4 choices.
Middle 3 places can be filled from remaining 6 digits:
6 × 5 × 4 = 120
Numbers = 2 × 4 × 120
= 960
Case 2: First digit is 6
Last digit must be even but cannot be 6.
So last digit choices:
0, 2, 4 = 3 choices.
Middle 3 places can be filled from remaining 6 digits:
6 × 5 × 4 = 120
Numbers = 1 × 3 × 120
= 360
Total numbers:
= 960 + 360
= 1320
Answer: 1320