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Distance between two points (x₁,y₁) and (x₂,y₂) is

D = √((x₂-x₁)2 + (y₂ - y₁)²)

So, given points are R(a+b, a-b), S(a-b, a+b)

D = √(((a-b)-(a+b))² + ((a+b)-(a-b))²)

D = √((-2b)² + (2b)²)

D = √(4b² + 4b²)

D = √(8b²)

D = 2√2b

Therefore, distance between them is 2√2b

Given:

Rice I costing = Rs. 60 per kg

Rice II costing = Rs. 75 per kg

Cost price of a mixture = Rs. 69 per kg

Formula used:

Quantity of cheaper/quantity of dearer = (price of dearer – mean price)/(mean price – price cheaper)

Ratio of mixture: (75 – 69)/(69 – 60)

⇒ 6/9

Given :

X has a balance of Rs. 86.54 in his bank account 

Deposited amount = Rs. 55.31 

Withdrawn amount = Rs. 84.33 

Concept used :

Final bank balance = Initial bank balance + deposited amount - withdrawn amount

Calculations :

Final bank balance = 86.54 + 55.31 - 84.33 

⇒ Rs. 57.52

∴ The balance after these transactions will be Rs. 57.52 

Correct option is A) wouldn’t

Given:

The cost price of an umbrella for Sanny = Rs. 3000

The selling price of Rohit = Rs. 4000

Formula used:

Markup price = (cost price) × (100 + % above)/100

Profit = Selling price – Cost price

Profit percentage = (Profit/cost price) × 100

Calculation:

The cost price of Rohit = the markup price of an umbrella for Sanny

⇒ The cost price of Rohit = the cost price of an umbrella for Sanny × (100 + % above)/100

⇒ The cost price of Rohit = 3000 × (100 + 20)/100

⇒ The cost price of Rohit = 30 × 120

⇒ The cost price of Rohit = Rs. 3600

Profit for Rohit = Selling price of an umbrella for Rohit – Cost price of an umbrella for Rohit

⇒ Profit for Rohit = 4000 – 3600

⇒ Profit for Rohit = 400

The profit percentage of Rohit = (400/3600) × 100

⇒ The profit percentage of Rohit = 100/9

⇒ The profit percentage of Rohit = 11.11%

∴ The profit percentage of Rohit is 11.11%. 

Given:

Amit’s Profit % = 40%

Mohan’s Loss % = 20%

C.P. for Ramesh = 2800

Formula Used:

(i) Cost price × (100 + Profit%)/100 = Selling price

(ii) Cost price × (100 - Loss%)/100 = Selling price

Calculation:

Let Amit’s C.P. be A Rs.

⇒ C.P. for Mohan = A × 140/100

⇒ S.P. for Mohan = A × (140/100) × (80/100) = C.P. of Ramesh

⇒ A × (140/100) × (80/100) = 2800

⇒ A = Rs. 2500

∴ The amount which paid by Amit for the speaker is Rs. 2500.

(D) neither (i) nor (ii)

Given:

The cost price of an umbrella for Mohan = Rs. 5000

The selling price of Rohit = Rs. 7000

Formula used:

Markup price = (cost price) × (100 + % above)/100

Profit = Selling price – Cost price

Profit percentage = (Profit/cost price) × 100

Calculation:

The cost price of Rohit = the markup price of an umbrella for Mohan

⇒ The cost price of Rohit = the cost price of an umbrella for Mohan × (100 + % above)/100

⇒ The cost price of Rohit = 5000 × (100 + 20)/100

⇒ The cost price of Rohit = 50 × 120

⇒ The cost price of Rohit = Rs. 6000

Profit for Rohit = Selling price of an umbrella for Rohit – Cost price of an umbrella for Rohit

⇒ Profit for Rohit = 7000 – 6000

⇒ Profit for Rohit = 1000

The profit percentage of Rohit = (1000/6000) × 100

⇒ The profit percentage of Rohit = 100/6

⇒ The profit percentage of Rohit = 16.66%

Given:

Cost price = Rs. 8000

Kiran earn a profit = 5%

Anupama profit = 5%

Calculation:

Selling price of an article for Kiran = 8000 + 5% of 8000

⇒ 8000 + 400

⇒ Rs. 8400

Cost price of an article for Anupama = Rs. 8400

Selling price of an article for Anupama = 8400 + 5% of 8400

⇒ 8400 + 420

⇒ Rs. 8820

Given:

The given numbers are 12, 1, 10, 1, 9, 3, 4, 9, 7, 9

We have to find the median of the given numbers.

Concept Used:

First, we have to arrange the given data as ascending order then we have to see how many numbers of data (n) is given

If n is odd then, Median = {(n/2) + 1}th term

If n is even then, Median = [(n/2)th term + {(n/2) + 1}th term]/2

Calculation:

Arranging the given data in ascending order get, 1, 1, 3, 4, 7, 9, 9, 9, 10, 12

Here the total number of data (n) = 10, which is an even number

(n/2)th term = 5th term = 7

{(n/2) + 1}th term = 6th term = 9

Required median = (7 + 9)/2 

⇒ 16/2

⇒ 8

∴ The median of the given numbers is 8.

Given:

The data S = (3, 5, 7, 8, 8, 14, 19, 93)

Formula used:

Mean = sum of number ÷ N

Median = {(N/2 + 1)^th and N/2^th}/2

Calculation

S = (3, 5, 7, 8, 8, 14, 19, 93)

Mean = 157 ÷ 8 = 19.62

Median = (4^th + 5^th)/2 = (8 + 8)/2 = 8

Hence difference = 19.62 – 8 = 11.62

Given:

The data S = {2, 3, 5, 5, 9, 15, 33, 65}

Formula used:

Mean = sum of number ÷ N

When N is even

Median = {(N/2 + 1)^th + N/2^th}/2

Calculation

S = {2, 3, 5, 5, 9, 15, 33, 65}

Mode = 5

Median = (4^th + 5^th) ÷ 2 = (5 + 9) ÷ 2 = 7

Hence difference = 7 – 5 = 2 

The given differential equation is 

(1 + xy)ydx + x(1 – xy)dy = 0

ydx + xdy + xy(ydx – xdy) = 0

d(xy) + xy(ydx – xdy) = 0

(d(xy)/(xy)) + (ydx – xdy) = 0 

d(xy)/(xy) = (xdy – ydx)

d(xy)/(xy)² = (xdy – ydx)/xy = (dy/y) – (dx/x)

Integrating, we get

– (1/xy) = log |(y/x)| + c 

Concept:

(a + b)² = a² + b² + 2ab

(a - b)2 = a2 + b2 - 2ab

Calculation:

Given:

a2 + b2 = 41 and ab = 20

(a + b)2 = 41 + (2 × 20) = 81

\((a + b) = \sqrt {81}\)

(a + b) = 9

(a - b)2 = 41 - (2 × 20) = 1

(a - b) = 1

\((a + b) ÷ (a – b) = \frac{(a+b)}{(a-b)}=\frac{9}{1}=9\)

The correct answer is 'I wish I had succeeded'.

• Option 2 gets eliminated as the word 'wish' is used in sentences that are in the past tense. 
• Option 3 gets eliminated as the word 'hope' is used to talk about future events, situations, or actions.
• Option 4 gets eliminated as the word 'desire' means 'to long or hope for'. It is also used to talk about future events. 
• We use 'wish' with the past perfect to talk about regrets from the past. 
  • ​For example, I wish I had studied harder at school.
• Therefore, from the given explanations, we can understand that option 1 is the correct answer. 

Given:

(1.001)1259 = 3.52 and (1.001)2062 = 7.85

Concept used:

X(a + b) = X(a) × X(b)

Calculation:

⇒ (1.001)3321 = (1.001)1259 + 2062 

= (1.001)1259 × (1.001)2062

= 3.52 × 7.85

= 27.64

Hence, the correct answer is "27.64".

A meeting occurs if the person arrives between 1:00 PM and 1:45 PM and the second person arrives in the next 15 minutes or if both the persons arrive between 1:45 and 2:00.

Case 1:

45/60 are favorable cases and hence the probability of first-person arriving between 1:00 and 1:45 is 3/4.

Probability of second person arriving in the next 15 min = 15/60 = 1/4

So, the probability of one person arriving between 1:00 and 1:45 and meeting the other = 3/4 × 1/4 × 2 = 3/8 (2 for choosing the first arriving friend)

Case 2: 

Both friends must arrive between 1:45 and 2:00

Probability = 1/4 × 1/4 = 1/16

So, probability of a meet = 3/8 + 1/16 = 7/16

Hence, the correct answer is 7/16.

The range of tanx is  R or (-∞ , + ∞)

Concept:

Consider a quadratic equation: ax2 + bx + c = 0.

Let, α and β are the roots.

• Sum of roots = α + β = -b/a
• Product of the roots = α × β = c/a

Calculation:

Given quadratic equation:  x2 + 6x + 4 = 0

Let  α and β are roots, then 

α + β = -6,  αβ = 4

Now,\(\frac{\alpha ^4+\beta ^4}{\alpha ^{-4}+\beta ^{-4}}\)= \(\frac{\alpha ^4+\beta ^4}{\frac {1}{\alpha ^{4}}+\frac{1}{\beta ^{4}}}\)

= \(\frac{\alpha ^4+\beta ^4}{\frac {(\alpha ^4+\beta ^4)}{\alpha^4 \beta^4 }}\)

= (αβ)⁴

= (4)⁴

= 256

Hence, option (2) is correct.

Concept:

Quadratic equation in the form of roots: x² – (sum of roots)x + product of roots = 0

In a quadratic equation surd or irrational roots occur in conjugate pairs.

For example, if one of the roots of a quadratic equation is α + √β then the other root will be its conjugate that is α − √β

Calculation:

Here, one root is 5p - √q

Since q is not a perfect square, therefore other root will be 5p +√q

Now, required quadratic equation is x2 – (sum of roots)x + product of roots = 0

⇒x² - (5p +√q + 5p - √q)x + (5p +√q)(5p -√q) = 0

⇒ x² -10px + 25p² - q = 0 

⇒ x2 -10px + 25p² - q = 0 

Hence, option (4) is correct. 

Irregular verbs are common verbs in the English language that do not follow the simple system of adding “d” or “ed” to the end of the word to form the past tense. That means the spellings can be a little tricky. Some irregular verbs follow patterns, such as drink–drank, spring–sprang, know-knew, and blow-blew but you can’t use those patterns with predictability.

One root =1+√2

Other root should be 1− √2, in this case sum of roots and product of roots will be rational leading to rational coefficients

We know that fog = f(g(x))

                                 = f(x) [ As g(x) = x ]

                                 = 3-x [ As f(x) = 3-x ]

One root of the equation x²+px+12=0 is 4

⇒16+4p+12=0

∴p=−7

as x²+px+q=0 has equal roots.

b² =4ac  ⇒p²=4q

\(\therefore q=\frac{49}{4}\)

a) It blocks sodium channels

d) All of the above

d) All of the above

c) Gingival hyperplasia

a) Diplopia, ataxia, and nausea

c) Ethosuximide

d)  Phenytoin.

b) Zonisamide 

d) All of the above

The antiseizure drug, which induces hepatic microsomal enzymes, is Phenytoin.

c)  Phenytoin.

The drug of choice for partial seizures is Carbamazepin.