Correct Answer: $\frac{25}{7}$

Solution : We know if $\operatorname{cosec}x + \cot x = a$, then $\operatorname{cosec} x - \cot x = \frac{1}{a}$
Given, $\operatorname{cosec} A+\cot A=7$ ........(1)
⇒ $\operatorname{cosec} A-\cot A=\frac17$ .........(2)
Solving these equations to get the value of $\operatorname{cosec} A$
(1) + (2), we get,
⇒ $\operatorname{cosec} A+\cot A+\operatorname{cosec} A-\cot A=7+\frac{1}{7}$
⇒ $2\operatorname{cosec} A = \frac{49+1}{7}$
⇒ $2\operatorname{cosec} A = \frac{50}{7}$
$\therefore \operatorname{cosec} A = \frac{25}{7}$
Hence, the correct answer is $\frac{25}{7}$.

Correct Answer: Sculptor-Chisel

Solution : Let's check each option –
First option: Scalpel - Surgeon; The scalpel is an essential tool used by surgeons.
Second option: Axe - Woodcutter; An axe is a tool used by woodcutter.
Third option: Sword - Warrior; A sword is used by a warrior.
Fourth option: Sculptor - Chisel; a sculptor is a person who carves figures or forms of clay, stone, metal, wood, etc. 

So, the fourth option does not follow the pattern as followed by the other given pairs.
Hence, the fourth option is correct.

Correct Answer: 117

Solution : Given:
A stands for ÷, B stands for ×, C stands for +, D stands for – 
13 B 145 A 29 C 69 D 17 = ?

Changing the sign as per the instruction and solving the equation, we will get –
= 13 × 145 ÷ 29 + 69 – 17
= 13 × 5 + 69 – 17
= 65 + 69 – 17
= 117

So, 117 is the required answer. Hence, the fourth option is correct.

Correct Answer: $8 \frac{4}{7}$ hours

Solution : Given: Aditya can complete a grill painting job in 20 hours.
Vinay can complete the same work in 15 hours.
Let the total work be LCM of (20, 15) = 60 units
⇒ Efficiency of Aditya = $\frac{60}{20} = 3$ units/hour
⇒ Efficiency of Vinay = $\frac{60}{15} = 4$ units/hour
⇒ Efficiency of Aditya + Vinay = 3 + 4 = 7 units/ hour
If they work together, time taken = $\frac{60}{7} = 8\frac{4}{7}$
Hence, the correct answer is $8 \frac{4}{7}$ hours.

Correct Answer: Pink head duck

Solution : The correct option is Pink head duck.

The extinct species among the options you provided is the Pink-headed duck. The Pink-headed duck is a species of waterfowl that is believed to be extinct, with the last confirmed sightings dating back to the 1940s. The other species, Hornbill, Andaman wild pig and Andaman teal, are not extinct.

Correct Answer: 4 hours

Solution : P can fill a tank in 5 hours.
Q can fill the same tank in 10 hours.
R can empty the same tank in 20 hours.
P's per hour efficiency = $\frac{1}{5}$
Q's per hour efficiency = $\frac{1}{10}$
R's per hour efficiency = $- \frac{1}{20}$ (negative because it is emptying the tank)
Total efficiency of P, Q, and R working together = $\frac{1}{5} + \frac{1}{10} − \frac{1}{20}$ = $\frac{5}{20}$ = $\frac{1}{4}$
$\therefore$ Together they will take 4 hours to fill the tank.
Hence, the correct answer is 4 hours.

Correct Answer: Nil

Solution : Let the amount deposited in the bank and post office be Rs. $x$ and Rs. $(300000-x)$ respectively.
We know, interest = $\frac{PRT}{100}$, where $P$ is principal, and $R$ is the rate of interest per annum for $T$ years.
So, Interest gained from a 10% interest rate from Rs. $x$ in a year = Rs. $0.1x$
And Interest gained from a 6% interest rate from Rs. 3 Lacs in a year = Rs. $0.06(300000-x)$
Given, 1 month of interest income = 2000
So, 1 year interest income = 2000(12)= Rs. 24,000
Now, Total interest in a year = Interest from the bank + Interest from the post office
⇒ $24000 = 0.06(300000-x) + 0.1x$
⇒ $24000-18000 = 0.04x$
⇒ $x = 1,50,000$
⇒ $(300000 - x) = 1,50,000$
So, the Amount deposited = Rs. $1,50,000$ each
and difference = 0
Hence, the correct answer is 'Nil'.

Correct Answer: 361 – 9786

Solution : Let's check each option –
First option: 225 : 5733; 225 = 15² and 5733 on reversing it gives 3375 = 15³
Second option: 169 : 7912; 169 = 13² and 7912 on reversing it gives 2197 = 13³
Third option: 361 : 9786; 361 = 19² and 9785 on reversing it gives 5879 ≠ 6859 = 19³
Fourth option: 289 : 3194; 289 = 17² and 3194 on reversing it gives 4913 = 17³

So, only the third option is different from the other three options as 9785 on reversing, it gives 9586 ≠ 19³. Hence, the third option is correct.​​​​​​

Correct Answer: 45$^\circ$

Solution : We have to find: the measure of the central angle of the arc whose length is 11 cm and the radius of the circle is 14 cm.
We know that,
Length of arc $= \frac{\theta}{360^\circ}\times 2\pi r$
$⇒11= \frac{\theta}{360^\circ}\times 2\times \frac {22}{7}\times 14$
$\therefore\theta =45^\circ$
Hence, the correct answer is 45$^\circ$.

Correct Answer: 1032 cm²

Solution : Length = 18 cm
Width = 12 cm
Height = 10 cm
The formula for the total surface area of a cuboidal box is:
Total surface area = 2 × (Length × Width + Length × Height + Width × Height)
⇒ Total surface area = 2 × (18 × 12 + 18 × 10 + 12 × 10)
⇒ Total surface area = 2 × (216 + 180 + 120)
⇒ Total surface area = 2 × 516
⇒ Total surface area = 1032 cm²
Hence, the correct answer is 1032 cm².