Before Seeing Solution attempt our Quiz of this Questions : Click to Attempt
1. Question
English: If x + 1/x = 5, find x³ + 1/x³.
Hindi: यदि x + 1/x = 5 है, तो x³ + 1/x³ ज्ञात कीजिए।
Solution:
x³ + 1/x³ = (x + 1/x)³ - 3(x + 1/x)
= 5³ - 3 × 5
= 125 - 15
= 110
Answer: 110
2. Question
English: If α and β are roots of x² - 10x + 8 = 0, find α²β + αβ².
Hindi: यदि x² - 10x + 8 = 0 के मूल α और β हैं, तो α²β + αβ² ज्ञात कीजिए।
Solution:
α + β = 10
αβ = 8
α²β + αβ² = αβ(α + β)
= 8 × 10
= 80
Answer: 80
3. Question
English: Find the least number which leaves remainder 3 when divided by 5, 7 and 9 but is exactly divisible by 8.
Hindi: वह सबसे छोटी संख्या ज्ञात कीजिए जो 5, 7 और 9 से भाग देने पर 3 शेष छोड़े लेकिन 8 से पूर्णतः विभाजित हो।
Solution:
LCM of 5, 7 and 9 = 315
Number = 315n + 3
It should be divisible by 8.
315n + 3 divisible by 8
315 leaves remainder 3 when divided by 8.
So:
3n + 3 divisible by 8
3(n + 1) divisible by 8
So n + 1 must be divisible by 8.
Smallest n = 7
Number = 315 × 7 + 3
= 2205 + 3
= 2208
Answer: 2208
4. Question
English: The midpoint of line joining (a,2) and (6,b) is (4,5). Find a+b.
Hindi: (a,2) और (6,b) को जोड़ने वाले रेखाखंड का मध्यबिंदु (4,5) है। a+b ज्ञात कीजिए।
Solution:
(a + 6)/2 = 4
a + 6 = 8
a = 2
(2 + b)/2 = 5
2 + b = 10
b = 8
a + b = 10
Answer: 10
5. Question
English: Three cards are drawn from a pack of 52 cards. Find probability that all are face cards.
Hindi: 52 पत्तों की गड्डी से 3 पत्ते निकाले जाते हैं। सभी के चित्र वाले पत्ते होने की प्रायिकता ज्ञात कीजिए।
Solution:
Total face cards = 12
Ways to choose 3 face cards = 12C3 = 220
Total ways = 52C3 = 22100
Probability = 220/22100
= 11/1105
Answer: 11/1105
6. Question
English: If tan θ = 3/4 and θ lies in first quadrant, find sec θ + cosec θ.
Hindi: यदि tan θ = 3/4 है तथा θ प्रथम चतुर्थांश में है, तो sec θ + cosec θ ज्ञात कीजिए।
Solution:
Opposite = 3
Adjacent = 4
Hypotenuse = 5
sec θ = 5/4
cosec θ = 5/3
sec θ + cosec θ
= 5/4 + 5/3
= 15/12 + 20/12
= 35/12
Answer: 35/12
7. Question
English: A hemisphere of radius 6 cm is melted and recast into cylinders of radius 1 cm and height 3 cm. Find number of cylinders formed.
Hindi: 6 सेमी त्रिज्या वाले अर्धगोले को पिघलाकर 1 सेमी त्रिज्या और 3 सेमी ऊँचाई वाले बेलनों में ढाला जाता है। बेलनों की संख्या ज्ञात कीजिए।
Solution:
Volume of hemisphere
= 2/3 × π × 6³
= 144π
Volume of one cylinder
= π × 1² × 3
= 3π
Number of cylinders
= 144π / 3π
= 48
Answer: 48
8. Question
English: If roots of x² - 12x + 18 = 0 are α and β, find α² + β².
Hindi: यदि x² - 12x + 18 = 0 के मूल α और β हैं, तो α² + β² ज्ञात कीजिए।
Solution:
α + β = 12
αβ = 18
α² + β²
= (α + β)² - 2αβ
= 12² - 2 × 18
= 144 - 36
= 108
Answer: 108
9. Question
English: A boat takes 3 hours downstream and 5 hours upstream for the same distance. If speed of stream is 2 km/h, find speed in still water.
Hindi: एक नाव समान दूरी तय करने में धारा के साथ 3 घंटे तथा धारा के विपरीत 5 घंटे लेती है। यदि धारा की गति 2 किमी/घंटा है, तो स्थिर जल में नाव की गति ज्ञात कीजिए।
Solution:
Let speed in still water = x
Downstream speed = x + 2
Upstream speed = x - 2
3(x + 2) = 5(x - 2)
3x + 6 = 5x - 10
16 = 2x
x = 8
Answer: 8 km/h
10. Question
English: How many arrangements of the word MATRIX are possible if vowels are never together?
Hindi: 'MATRIX' शब्द के अक्षरों को कितने प्रकार से व्यवस्थित किया जा सकता है यदि स्वर कभी साथ न हों?
Solution:
Consonants = M, T, R, X
Arrange consonants:
4! = 24
Gaps = 5
Choose 2 gaps for vowels:
5C2 = 10
Arrange vowels:
2! = 2
Total arrangements
= 24 × 10 × 2
= 480
Answer: 480
11. Question
English: Find remainder when 5¹⁰⁰ is divided by 13.
Hindi: 5¹⁰⁰ को 13 से भाग देने पर शेषफल ज्ञात कीजिए।
Solution:
5² leaves remainder 12
5⁴ leaves remainder 1
100 divisible by 4
Therefore remainder = 1
Answer: 1
12. Question
English: If x² + y² = 50 and xy = 24, find x+y.
Hindi: यदि x² + y² = 50 तथा xy = 24 है, तो x+y ज्ञात कीजिए।
Solution:
(x + y)²
= x² + y² + 2xy
= 50 + 48
= 98
x + y = √98
= 7√2
Answer: 7√2
13. Question
English: Area of triangle whose sides are 17 cm, 25 cm and 28 cm.
Hindi: 17 सेमी, 25 सेमी तथा 28 सेमी भुजाओं वाले त्रिभुज का क्षेत्रफल ज्ञात कीजिए।
Solution:
Semi-perimeter
= (17 + 25 + 28)/2
= 35
Area
= √[35 × 18 × 10 × 7]
= √44100
= 210
Answer: 210 cm²
14. Question
English: If sin θ = 4/5 and θ is acute, find tan 2θ.
Hindi: यदि sin θ = 4/5 है तथा θ न्यून कोण है, तो tan 2θ ज्ञात कीजिए।
Solution:
cos θ = 3/5
tan θ = 4/3
tan 2θ
= 2tan θ / (1 - tan² θ)
= 2 × 4/3 ÷ (1 - 16/9)
= -24/7
Answer: -24/7
15. Question
English: A can complete a work in 20 days and B in 30 days. They work together for 5 days. Remaining work will be completed by B alone in:
Hindi: A किसी कार्य को 20 दिनों में तथा B 30 दिनों में पूरा करता है। दोनों 5 दिन साथ काम करते हैं। शेष कार्य B अकेले कितने दिनों में पूरा करेगा?
Solution:
A’s 1-day work = 1/20
B’s 1-day work = 1/30
Together = 1/12
Work done in 5 days
= 5/12
Remaining work
= 7/12
Time by B
= 7/12 ÷ 1/30
= 17.5 days
Answer: 17.5 days
16. Question
English: Marked price is 60% above cost price. After successive discounts of 10% and 20%, profit percentage is:
Hindi: अंकित मूल्य क्रय मूल्य से 60% अधिक है। 10% तथा 20% की क्रमिक छूट के बाद लाभ प्रतिशत ज्ञात कीजिए।
Solution:
Let CP = 100
MP = 160
After 10% discount
= 144
After 20% discount
= 115.2
Profit = 15.2
Profit % = 15.2%
Answer: 15.2%
17. Question
English: Equation of circle with centre (3,-2) touching x-axis.
Hindi: (3,-2) केंद्र वाला तथा x-अक्ष को स्पर्श करने वाले वृत्त का समीकरण ज्ञात कीजिए।
Solution:
Radius = distance from centre to x-axis
= 2
Equation:
(x - 3)² + (y + 2)² = 4
Answer: (x - 3)² + (y + 2)² = 4
18. Question
English: Two dice are thrown. Find probability that sum is prime.
Hindi: दो पासे फेंके जाते हैं। योग के अभाज्य संख्या होने की प्रायिकता ज्ञात कीजिए।
Solution:
Prime sums:
2, 3, 5, 7, 11
Favourable outcomes = 15
Total outcomes = 36
Probability
= 15/36
= 5/12
Answer: 5/12
19. Question
English: Find total number of divisors of 864.
Hindi: 864 के कुल भाजकों की संख्या ज्ञात कीजिए।
Solution:
864 = 2⁵ × 3³
Number of divisors
= (5 + 1)(3 + 1)
= 6 × 4
= 24
Answer: 24
20. Question
English: A cone of radius 6 cm and height 8 cm is melted into spheres of radius 2 cm. Find number of spheres formed.
Hindi: 6 सेमी त्रिज्या और 8 सेमी ऊँचाई वाले शंकु को पिघलाकर 2 सेमी त्रिज्या वाले गोलों में ढाला जाता है। गोलों की संख्या ज्ञात कीजिए।
Solution:
Volume of cone
= 96π
Volume of one sphere
= 32π/3
Number of spheres
= 96π ÷ 32π/3
= 9
Answer: 9
21. Question
English: If x + y = 7 and x² + y² = 29, find x³ + y³.
Hindi: यदि x + y = 7 तथा x² + y² = 29 है, तो x³ + y³ ज्ञात कीजिए।
Solution:
Given:
x + y = 7
x² + y² = 29
First find xy:
(x + y)² = x² + y² + 2xy
7² = 29 + 2xy
49 = 29 + 2xy
2xy = 20
xy = 10
Now:
x³ + y³ = (x + y)³ - 3xy(x + y)
= 7³ - 3 × 10 × 7
= 343 - 210
= 133
Answer: 133
22. Question
English: A train crosses a platform 240 m long in 18 seconds and a pole in 8 seconds. Find speed of train.
Hindi: एक ट्रेन 240 मीटर लंबे प्लेटफॉर्म को 18 सेकंड में तथा एक खंभे को 8 सेकंड में पार करती है। ट्रेन की गति ज्ञात कीजिए।
Solution:
Let train length = L metre.
Train crosses pole in 8 seconds.
Speed = L/8
Train crosses platform in 18 seconds.
Total distance = L + 240
Speed = (L + 240)/18
So:
L/8 = (L + 240)/18
18L = 8L + 1920
10L = 1920
L = 192 metre
Speed = 192/8
= 24 m/s
Convert into km/h:
24 × 18/5 = 86.4 km/h
Answer: 86.4 km/h
23. Question
English: How many different arrangements of the letters of the word STATISTICS are possible?
Hindi: 'STATISTICS' शब्द के अक्षरों को कितने विभिन्न तरीकों से व्यवस्थित किया जा सकता है?
Solution:
Word: STATISTICS
Total letters = 10
Repeated letters:
S = 3
T = 3
I = 2
A = 1
C = 1
Total arrangements:
10! / (3! × 3! × 2!)
= 3628800 / 72
= 50400
Answer: 50400
24. Question
English: If tan θ + cot θ = 5, find tan² θ + cot² θ.
Hindi: यदि tan θ + cot θ = 5 है, तो tan² θ + cot² θ ज्ञात कीजिए।
Solution:
Given:
tan θ + cot θ = 5
Square both sides:
tan² θ + cot² θ + 2tan θ cot θ = 25
Since tan θ cot θ = 1,
tan² θ + cot² θ + 2 = 25
tan² θ + cot² θ = 23
Answer: 23
25. Question
English: A solid sphere of radius 6 cm is melted and recast into cones of radius 3 cm and height 4 cm. Find number of cones formed.
Hindi: 6 सेमी त्रिज्या वाले ठोस गोले को पिघलाकर 3 सेमी त्रिज्या तथा 4 सेमी ऊँचाई वाले शंकुओं में ढाला जाता है। बने हुए शंकुओं की संख्या ज्ञात कीजिए।
Solution:
Volume of sphere:
= 4/3 × π × 6³
= 4/3 × π × 216
= 288π
Volume of one cone:
= 1/3 × π × 3² × 4
= 1/3 × π × 9 × 4
= 12π
Number of cones:
= 288π / 12π
= 24
Answer: 24