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Given:

x + 1/x = 1

Formula used:

(a + b)² = a² + b² + 2ab

Calculation:

x + 1/x = 1

Squaring both sides, we have –

⇒ x² + 1/x² + 2 = 1

After adding 4 on both side

⇒ x² + 1/x² + 2 + 4 = 1 + 4

⇒ x² + 1/x² + 6 = 5

Given:

The successive discount are 20%, 25% and 15%.

Formula:

Successive discount = X + Y - (XY)/100

Where,

X = First discount

Y = Second discount

Calculation:

Successive discount = X + Y - (XY)/100

⇒ 20 + 25 - 500/100

⇒ 45 - 5 = 40

Now, again

Successive discount = X + Y - (XY)/100

X = 40%, Y = 15%

⇒ 40 + 15 - (40 × 15)/100

⇒ 55 - 600/100

⇒ 55 - 6 = 49%

∴ The net discount is 49%.

 

Let, marked price = x

After first discount of 20%, price = x - x × 20/100 = 0.8x

After second discount of 25%, price = 0.8x - 0.8x × 25/100 = 0.6x

After third discount of 15%, price = 0.6x - 0.6x × 15/100 = 0.51x

∴ Net discount percentage,

\(\Rightarrow \frac{{x - 0.51x}}{x} × 100%\)

⇒ 49%

Given:

sinθ = 5/13

Concept used:

Using the concept of trigonometric ratios.

Calculation:

sinθ = 5/13

cosθ = √(1 - sin²θ)

⇒ cosθ = √(1 - 25/169)

⇒ cosθ = 12/13

tanθ = sinθ/cosθ 

⇒ tanθ = (5/13)/(12/13)

⇒ tanθ = 5/12

cotθ = 1/tanθ 

⇒ cotθ = 12/5

(cotθ - tanθ)/2cotθ

⇒ (12/5 - 5/12)/(2 × 12/5)

⇒ (119/60)/(24/5)

⇒ 119/288

∴ The value of (cotθ - tanθ)/2cotθ is 119/288.

Shortcut Trick

LCM of denominators (10, 6, 15) = 30.

Multiply the ratio by 30 → A : B : C = 3 : 5 : 2.

Total units = 3 + 5 + 2 = 10 units.

10 units = ₹25,000 ⇒ 1 unit = ₹2,500.

C's share = 2 units = 2 × ₹2,500 = ₹5,000.

∴ The correct answer is ₹5000.

Alternate Method

Given:

Total Amount = ₹25,000

Ratio of A : B : C = 1/10 : 1/6 : 1/15

Formula:

Share of a person = (Personal Ratio Part ÷ Sum of all Ratio Parts) × Total Amount

Calculations:

⇒ First, convert the fractional ratio into an integer ratio by multiplying with the LCM of 10, 6, and 15.

⇒ LCM(10, 6, 15) = 30.

⇒ A : B : C = (1/10 × 30) : (1/6 × 30) : (1/15 × 30)

⇒ A : B : C = 3 : 5 : 2

⇒ Sum of ratio parts = 3 + 5 + 2 = 10

⇒ Share of C = (2 ÷ 10) × 25000

⇒ Share of C = 1/5 × 25000

⇒ Share of C = ₹5000

∴ The correct answer is ₹5000.

Additional Information

Ratio Simplification

To convert a ratio in the form of fractions 1/a : 1/b : 1/c to integers, multiply each term by the Least Common Multiple (LCM) of a, b, and c.

Share Distribution

When a value X is divided in the ratio a:b:c, the shares are calculated as [a/(a+b+c)]×X, [b/(a+b+c)]×X, and [c/(a+b+c)]×X respectively.

Invertendo

If a : b = c : d, then b : a = d : c. This property is useful when dealing with reciprocal proportions in work or speed problems.

Given

% D = 10%

% P = 10%

Concept used

In this type of question find overall discount percent and change percentage into fraction form and with comparing and analysing solve the question

Calculation

Let marked price = MP, selling price = SP, cost price = CP, profit = P discount = D

Since 1 article is free on 4 article so discount percentage = (1/5) × 100 = 20%

Also another discount of 10% so overall discount percentage = 10 + 20 - 10 × 20/100 = 28%

If Marked Price = 100 Discount = 28 and Selling price = 72

Still earn a profit of 10% = 1/10

So if CP = 10 units, Profit = 1 unit

Selling price = 10 + 1 = 11 units

And 11 units = 72

⇒ CP (10) = 720/11

Ratio of Marked price : Cost Price = 100 : 720/11 = 55 : 36

The correct answer is "Affluent."

 Key Points

• "Wealthy" means having a lot of money, property, etc. (धनी, अमीर, संपन्‍न)
• Example: He's a very wealthy man.
• "Affluent" means having a lot of money and being wealthy. (धनवान, समृद्घ)
• Example: We live in an affluent neighbourhood.
• "Affluent" is a synonym of the word 'wealthy.'
• The correct answer is Option 1.

 Additional Information

• Option 2 - Depressed: feeling sad and unhappy. (बहुत उदास (प्रायः लंबे समय तक))
• Option 3 - Poor: having little or no money. (आरामदायक जीवन के लिए पर्याप्त धन का अभाव; ग़रीब, निर्धन)
• Option 4 - Lacking: not having enough of something. (जिसमें किसी बात की कमी हो, किसी बात में कम या न्‍यून)

The correct answer is 'was waiting'.

Key Points

• When there are two actions in progress at the same time, we use Past continuous tense to represent the action that was already in progress while the other one that happened takes Simple past tense.
• For example, 'I was doing the laundry when you rang the doorbell.'
• The subject-verb agreement indicates that the verb here will be singular.
• Correct sentence: 'He was waiting for me when I reached office.'

Hence, the correct answer is Option 3.

The correct answer is at.

Key Points​

• ​There are some fixed prepositional phrases given below:
• ​by the dint of, out of jealousy, on sale, at a discounted price, by the dozen, owing to, on a trip, for a change, etc.
• For example:
• ​There are many websites that sell designer brands at discounted prices.
• ​The phrase 'at a discounted price' means cheaper than usual.
• Thus the preposition 'at' will be the correct choice.

Correct Sentence: We are pleased to offer it to you at a discounted price.

Additional Information

• There are some verbs/nouns/adjectives which are followed by fixed preposition given below:
• ​Exonerate from, refrain from, suspicious of, interested in, inured to, accused of, a predilection for, a respite from, vexed at, etc.
• Example:
• ​After spending some time on the island, they became inured to the hardships.

The correct answer is 'Option 4' i.e. 'Resolve'.

Key Points

• The word "Hesitate" in English means to pause before saying or doing something, often because of uncertainty or insecurity (हिचकिचाना).
• Example: He hesitated for a moment before jumping into the pool.
• The word "Resolve" is an antonym of "Hesitate", meaning to come to a definite decision about; to determine (निश्चय करना).
• Example: Sarah had a firm resolve to complete her assignment on time.

Therefore, the correct answer is - 'Resolve'.

Additional Information

• 'Reform' (Option 1) means to make changes in something in order to improve it (सुधारना).
• 'Vacillate' (Option 2) similar to hesitate, it means to keep changing your opinion or thoughts, usually in a situation of uncertainty (अनिश्चित रहना).
• **'Imitate' (**Option 3) means to copy or mimic the behavior or actions of another (नकल करना).

The correct answer is 'could'.

Key Points

• In this context, "could" is the correct choice because it is used to express ability or possibility in the past, which is the purpose of this sentence.
• It indicates that the speaker had the ability to walk forty miles in a day when he was a boy.
• Let's break down the options:
• "should" - This modal verb is used to express obligation, advice, expectation, or speculation. It does not denote ability or possibility, which is required in this context.
• "can" - This modal verb is used to express present ability or possibility, not past ability or possibility, which is the focus of this sentence.
• "could" - This is the correct choice. It's often used to express an ability or possibility in the past, which fits the context of this sentence. In this case, "could" suggests that when the speaker was a boy, he had the ability to walk forty miles in a day.
• "must" - This modal verb is used to express obligation, necessity, or strong certainty. It does not express past ability or possibility, which is required in this context.