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Ultra Tough SSC CGL Tier-2 Maths Set 18 Solutions (1–10)

1. Question

English: If f(x)=x²−4x+7, find f(x+2)−f(x−2).

Hindi: यदि f(x)=x²−4x+7 है, तो f(x+2)−f(x−2) ज्ञात कीजिए।

Solution

Given:

f(x)=x²−4x+7

Now,

f(x+2)

= (x+2)²−4(x+2)+7

= x²+4x+4−4x−8+7

= x²+3

Also,

f(x−2)

= (x−2)²−4(x−2)+7

= x²−4x+4−4x+8+7

= x²−8x+19

Therefore,

f(x+2)−f(x−2)

= (x²+3)−(x²−8x+19)

= 8x−16

Answer: 8x−16

2. Question

English: The HCF and LCM of two numbers are 35 and 3150 respectively. If one number is 350, find the other number.

Hindi: दो संख्याओं का HCF 35 और LCM 3150 है। यदि एक संख्या 350 है, तो दूसरी संख्या ज्ञात कीजिए।

Solution

Product of two numbers

= HCF × LCM

= 35 × 3150

Other number

= (35 × 3150)/350

= 315

Answer: 315

3. Question

English: Find the number of perfect cube divisors of 2⁸ × 3⁷ × 5⁵.

Hindi: 2⁸ × 3⁷ × 5⁵ के पूर्ण घन भाजकों की संख्या ज्ञात कीजिए।

Solution

For a perfect cube divisor, powers must be multiples of 3.

For 2⁸:

Possible powers = 0, 3, 6

= 3 choices

For 3⁷:

Possible powers = 0, 3, 6

= 3 choices

For 5⁵:

Possible powers = 0, 3

= 2 choices

Total perfect cube divisors

= 3 × 3 × 2

= 18

Answer: 18

4. Question

English: Simplify: (√192 + √108 − √48)/√3.

Hindi: सरलीकृत कीजिए: (√192 + √108 − √48)/√3.

Solution

√192 = 8√3

√108 = 6√3

√48 = 4√3

Numerator

= 8√3 + 6√3 − 4√3

= 10√3

Now,

10√3/√3

= 10

Answer: 10

5. Question

English: Evaluate: (64^(5/6) × 81^(3/4)) / (16 × 9).

Hindi: मान ज्ञात कीजिए: (64^(5/6) × 81^(3/4)) / (16 × 9).

Solution

64^(5/6)

= (⁶√64)⁵

= 2⁵

= 32

81^(3/4)

= (⁴√81)³

= 3³

= 27

Therefore,

(32 × 27)/(16 × 9)

= 2 × 3

= 6

Answer: 6

6. Question

English: Find the least positive integer N such that N ≡ 4 (mod 7), N ≡ 5 (mod 9), N ≡ 6 (mod 11).

Hindi: सबसे छोटी धनात्मक संख्या N ज्ञात कीजिए जिसके लिए N ≡ 4 (mod 7), N ≡ 5 (mod 9), N ≡ 6 (mod 11) हो।

Solution

Let

N = 7k + 4

Using second condition:

7k + 4 ≡ 5 (mod 9)

7k ≡ 1 (mod 9)

k ≡ 4 (mod 9)

Let

k = 9m + 4

Then

N = 63m + 32

Using third condition:

63m + 32 ≡ 6 (mod 11)

8m + 10 ≡ 6 (mod 11)

8m ≡ 7 (mod 11)

Inverse of 8 modulo 11 is 7.

m ≡ 5 (mod 11)

Smallest m = 5

Therefore,

N = 63 × 5 + 32

= 347

Answer: 347

7. Question

English: If x + 1/x = 6, find x⁶ + 1/x⁶.

Hindi: यदि x + 1/x = 6 है, तो x⁶ + 1/x⁶ ज्ञात कीजिए।

Solution

Given:

x + 1/x = 6

x² + 1/x²

= 6² − 2

= 34

x³ + 1/x³

= 6³ − 3 × 6

= 216 − 18

= 198

Now,

x⁶ + 1/x⁶

= (x³ + 1/x³)² − 2

= 198² − 2

= 39204 − 2

= 39202

Answer: 39202

8. Question

English: In a GP, the first term is 2 and the fourth term is 54. Find the sum of first four terms.

Hindi: एक GP में पहला पद 2 है तथा चौथा पद 54 है। पहले चार पदों का योग ज्ञात कीजिए।

Solution

First term = 2

Fourth term

= 2r³

= 54

r³ = 27

r = 3

GP:

2, 6, 18, 54

Sum

= 2 + 6 + 18 + 54

= 80

Answer: 80

9. Question

English: How many arrangements of the word 'STATISTICS' are possible?

Hindi: 'STATISTICS' शब्द के कितने भिन्न व्यवस्थापन संभव हैं?

Solution

Total letters = 10

Repetitions:

S = 3

T = 3

I = 2

Number of arrangements

= 10!/(3! × 3! × 2!)

= 3628800/72

= 50400

Answer: 50400

10. Question

English: A box contains 5 red, 4 blue and 3 green balls. Two balls are drawn. Find the probability that both are of different colours.

Hindi: एक डिब्बे में 5 लाल, 4 नीली और 3 हरी गेंदें हैं। दो गेंदें निकाली जाती हैं। दोनों के भिन्न रंग की होने की प्रायिकता ज्ञात कीजिए।

Solution

Total balls

= 5 + 4 + 3

= 12

Total ways

= 12C2

= 66

Same colour ways:

Red-red

= 5C2

= 10

Blue-blue

= 4C2

= 6

Green-green

= 3C2

= 3

Total same colour ways

= 19

Different colour ways

= 66 − 19

= 47

Probability

= 47/66

Answer: 47/66

11. Question

English: A can complete a work in 24 days, B in 36 days and C in 72 days. Working together, in how many days will they complete the work?

Hindi: A कार्य को 24 दिनों में, B 36 दिनों में और C 72 दिनों में पूरा करता है। तीनों मिलकर कार्य कितने दिनों में पूरा करेंगे?

Solution

A’s 1-day work:

= 1/24

B’s 1-day work:

= 1/36

C’s 1-day work:

= 1/72

Together 1-day work:

= 1/24 + 1/36 + 1/72

LCM = 72

= 3/72 + 2/72 + 1/72

= 6/72

= 1/12

So they complete the work in:

= 12 days

Answer: 12 days

12. Question

English: Pipe A fills a tank in 16 hours and Pipe B fills it in 24 hours. Pipe C can empty it in 48 hours. All are opened together. Find the time taken to fill the tank.

Hindi: पाइप A टंकी को 16 घंटे में तथा पाइप B 24 घंटे में भरता है। पाइप C उसे 48 घंटे में खाली करता है। तीनों को साथ खोलने पर टंकी कितने समय में भरेगी?

Solution

A’s 1-hour work:

= 1/16

B’s 1-hour work:

= 1/24

C’s 1-hour work:

= -1/48

Together 1-hour work:

= 1/16 + 1/24 - 1/48

LCM = 48

= 3/48 + 2/48 - 1/48

= 4/48

= 1/12

So tank will be filled in:

= 12 hours

Answer: 12 hours

13. Question

English: Find the smaller angle between the hour and minute hands at 8:20.

Hindi: 8:20 बजे घंटे और मिनट की सुई के बीच छोटा कोण ज्ञात कीजिए।

Solution

Minute hand angle:

= 20 × 6

= 120°

Hour hand angle:

= 8 × 30 + 20 × 0.5

= 240 + 10

= 250°

Difference:

= 250° - 120°

= 130°

Answer: 130°

14. Question

English: In a 600 m race, A beats B by 60 m. In a 540 m race, B beats C by 90 m. By how many metres will A beat C in a 1200 m race?

Hindi: 600 मीटर की दौड़ में A, B को 60 मीटर से हराता है। 540 मीटर की दौड़ में B, C को 90 मीटर से हराता है। 1200 मीटर की दौड़ में A, C को कितने मीटर से हराएगा?

Solution

When A runs 600 m, B runs 540 m.

So:

A : B = 600 : 540

= 10 : 9

When B runs 540 m, C runs:

= 540 - 90

= 450 m

So:

B : C = 540 : 450

= 6 : 5

Now combine:

A : B = 10 : 9

B : C = 6 : 5

Make B equal.

LCM of 9 and 6 = 18

A : B = 20 : 18

B : C = 18 : 15

So:

A : C = 20 : 15

= 4 : 3

In 1200 m race, when A runs 1200 m, C runs:

= 1200 × 3/4

= 900 m

A beats C by:

= 1200 - 900

= 300 m

Answer: 300 m

15. Question

English: A mixture contains milk and water in ratio 5:3. If 16 litres water is added, ratio becomes 5:5. Find initial quantity of mixture.

Hindi: एक मिश्रण में दूध और पानी का अनुपात 5:3 है। यदि 16 लीटर पानी मिलाया जाए तो अनुपात 5:5 हो जाता है। मिश्रण की प्रारंभिक मात्रा ज्ञात कीजिए।

Solution

Let initial milk and water be:

Milk = 5x

Water = 3x

After adding 16 litres water:

Water = 3x + 16

New ratio:

5x : (3x + 16) = 5 : 5

So milk = water.

5x = 3x + 16

2x = 16

x = 8

Initial quantity:

= 5x + 3x

= 8x

= 8 × 8

= 64 litres

Answer: 64 litres

16. Question

English: The sides of a triangle are 17 cm, 25 cm and 28 cm. Find its area.

Hindi: एक त्रिभुज की भुजाएँ 17 सेमी, 25 सेमी और 28 सेमी हैं। उसका क्षेत्रफल ज्ञात कीजिए।

Solution

Semi-perimeter:

= (17 + 25 + 28)/2

= 70/2

= 35

Area:

= √[35 × (35 - 17) × (35 - 25) × (35 - 28)]

= √[35 × 18 × 10 × 7]

= √44100

= 210 cm²

Answer: 210 cm²

17. Question

English: Find the equation of the line passing through (2,3) and parallel to 3x−4y+5=0.

Hindi: (2,3) से गुजरने वाली तथा 3x−4y+5=0 के समानांतर रेखा का समीकरण ज्ञात कीजिए।

Solution

Parallel line will have same x and y coefficients.

So required line:

3x - 4y + c = 0

It passes through (2,3).

Put x = 2 and y = 3:

3 × 2 - 4 × 3 + c = 0

6 - 12 + c = 0

c = 6

Therefore:

3x - 4y + 6 = 0

Answer: 3x−4y+6=0

18. Question

English: If tanθ = 8/15, find secθ + cosθ.

Hindi: यदि tanθ = 8/15 है, तो secθ + cosθ ज्ञात कीजिए।

Solution

tanθ = 8/15

So take:

Perpendicular = 8

Base = 15

Hypotenuse = 17

cosθ = 15/17

secθ = 17/15

Now:

secθ + cosθ

= 17/15 + 15/17

= (289 + 225)/255

= 514/255

Answer: 514/255

19. Question

English: A sum amounts to ₹12100 in 2 years and ₹13310 in 3 years at compound interest. Find the principal.

Hindi: एक राशि चक्रवृद्धि ब्याज पर 2 वर्षों में ₹12100 तथा 3 वर्षों में ₹13310 हो जाती है। मूलधन ज्ञात कीजिए।

Solution

Amount after 2 years = ₹12100

Amount after 3 years = ₹13310

One-year growth factor:

= 13310/12100

= 1.10

So rate = 10%

Now amount after 2 years:

P × 1.10² = 12100

P × 1.21 = 12100

P = 12100/1.21

= 10000

Answer: ₹10000

20. Question

English: If f(n)=2f(n−1)+1 and f(1)=2, find f(6).

Hindi: यदि f(n)=2f(n−1)+1 तथा f(1)=2 है, तो f(6) ज्ञात कीजिए।

Solution

f(1) = 2

f(2) = 2 × 2 + 1

= 5

f(3) = 2 × 5 + 1

= 11

f(4) = 2 × 11 + 1

= 23

f(5) = 2 × 23 + 1

= 47

f(6) = 2 × 47 + 1

= 95

Answer: 95

21. Question

English: If roots of x²−px+81=0 are positive and differ by 18, find p.

Hindi: यदि x²−px+81=0 के मूल धनात्मक हैं और उनका अंतर 18 है, तो p ज्ञात कीजिए।

Solution

Let roots be a and b.

Given:

ab = 81

a - b = 18

Now:

(a + b)² = (a - b)² + 4ab

= 18² + 4 × 81

= 324 + 324

= 648

So:

a + b = √648

= 18√2

In equation:

x² - px + 81 = 0

Sum of roots = p

So:

p = 18√2

Answer: 18√2

22. Question

English: Average of 30 observations is 42. If one observation 54 was wrongly written as 45, find correct average.

Hindi: 30 प्रेक्षणों का औसत 42 है। यदि एक प्रेक्षण 54 को गलत रूप से 45 लिखा गया हो, तो सही औसत ज्ञात कीजिए।

Solution

Wrong total:

= 30 × 42

= 1260

Correction needed:

= 54 - 45

= 9

Correct total:

= 1260 + 9

= 1269

Correct average:

= 1269/30

= 42.3

Answer: 42.3

23. Question

English: A cube is painted on all faces and cut into 512 equal cubes. How many cubes have exactly two faces painted?

Hindi: एक घन को सभी सतहों पर रंगकर 512 बराबर छोटे घनों में काटा गया। ठीक दो सतहों पर रंग वाले छोटे घनों की संख्या ज्ञात कीजिए।

Solution

512 = 8³

So cube is cut into 8 parts along each edge.

Small cubes with exactly two faces painted are on edges but not on corners.

Formula:

= 12 × (n - 2)

Here n = 8

= 12 × (8 - 2)

= 12 × 6

= 72

Answer: 72

24. Question

English: If x+y=12 and xy=32, find x⁵+y⁵.

Hindi: यदि x+y=12 तथा xy=32 है, तो x⁵+y⁵ ज्ञात कीजिए।

Solution

Given:

x + y = 12

xy = 32

First:

x² + y² = (x + y)² - 2xy

= 12² - 2 × 32

= 144 - 64

= 80

Now:

x³ + y³ = (x + y)³ - 3xy(x + y)

= 12³ - 3 × 32 × 12

= 1728 - 1152

= 576

Now:

x⁵ + y⁵ = (x² + y²)(x³ + y³) - xy(x + y)

= 80 × 576 - 32 × 12

= 46080 - 384

= 45696

Answer: 45696

25.English: If α and β are the roots of x² − 12x + 20 = 0, find the value of α³ + β³.

Hindi: यदि α और β, x² − 12x + 20 = 0 के मूल हैं, तो α³ + β³ का मान ज्ञात कीजिए।

Solution

Given equation:

x² − 12x + 20 = 0

Using the relationship between roots and coefficients:

α + β = 12

αβ = 20

We know:

α³ + β³ = (α + β)³ − 3αβ(α + β)

Substituting the values:

α³ + β³

= 12³ − 3 × 20 × 12

= 1728 − 720

= 1008

Answer

1008