Phrases Previous Year Questions (PYQs) – Others

Find the smallest number which when divided by 9, 10, 15 and 20 leaves remainders 4, 5, 10 and 15 respectively.

✅ Solution:

Let the number be $x$.

• $x\equiv 4\mathrm{mod}9\Rightarrow x-4$ divisible by 9
• $x\equiv 5\mathrm{mod}10\Rightarrow x-5$ divisible by 10
• $x\equiv 10\mathrm{mod}15\Rightarrow x-10$ divisible by 15
• $x\equiv 15\mathrm{mod}20\Rightarrow x-15$ divisible by 20

So, $x+5$ is divisible by LCM of 9, 10, 15, 20

LCM = $2^2\cdot 3^2\cdot 5=180$

$x+5=180\times 2=360\Rightarrow x=355$

Final Answer: $\boxed{{\displaystyle 355}}$

Vector Field:

$$\overrightarrow{A}=(2x+1)\hat{i}+(x^2-6y)\hat{j}+(x{y}^2+3z)\hat{k}$$

Divergence:

$$\mathrm{\nabla }\cdot \overrightarrow{A}=2-6+3=-1\ne 0$$

Not solenoidal ❌

Curl:

$$\mathrm{\nabla }\times \overrightarrow{A}=(2xy)\hat{i}-(y^2)\hat{j}+(2x)\hat{k}\ne \overrightarrow{0}$$

Not conservative ❌

Final Answer:

$\overrightarrow{A}$ is neither conservative nor solenoidal.

Vector Sink Field Analysis

Given vector field:

$$\overrightarrow{A}=(2x+1)\hat{i}+(x^2-6y)\hat{j}+(x{y}^2+3z)\hat{k}$$

Divergence:

$$\mathrm{\nabla }\cdot \overrightarrow{A}=2-6+3=-1$$

✅ Conclusion:

The divergence is negative at every point, so $\overrightarrow{A}$ is a sink field.