The general solution of the differential equation \(x d y+e^{-y} d x=x e^{x-y} d x\) is (given C is constant of integration)

The solution of the differential equation \(\frac{d y}{d x}-\frac{y}{x}=2 \log _e x\)

Let \(f(x)=\log _e(\sin x), x \in(0, \pi)\), then which of the following statements is/are TRUE?
(A) \(f(x)\) is increasing on \((0, \pi / 2)\)
(B) \(f ( x )\) is decreasing on \(( \pi / 2 , \pi)\)
(C) \(f(x)\) is increasing on \((0, \pi)\)
(D) \(f(x)\) is decreasing on \((0, \pi)\)
Choose the correct answer from the options given below :

Let \(A=\left[a_{i j}\right]_{2 \times 4}\) and \(B=\left[b_{i j}\right]_{4 \times 2}\), then \(|3 A B|\) is equal to

If \(A=\left[\begin{array}{lll}1 & 1 & 1 \\ 1 & 0 & 2 \\ x & 1 & 1\end{array}\right]\) and \(A^{-1}=\frac{1}{4}\left[\begin{array}{ccc}-2 & 0 & y \\ 5 & -2 & -1 \\ 1 & 2 & -1\end{array}\right]\), then values of \(x\) and \(y\) are:

Which of the following functions f(x) are differentiable at \(x = 0\)?
(A) \(|x|\)
(B) \(| x - 1|\)
(C) [x],where [t] denotes the greatest integer ≤ t
(D) \(| x + 1|\)
(E) \(x^2\)
Choose the correct answert from the option given below :

A die is thrown three times. If the first throw is a five, the probability of getting 14 as the sum is:

Evaluate the integral \(\int\left(x^4+x^2+1\right) d\left(x^2\right)\), where C is the constant of integration

Positive carbylamine test is shown by :

Which of the following is NOT correct in the case of the output obtained from a full wave rectifier?

For a heavily doped n-type semiconductor, Fermi level lies:

The process of extracting the separated bond from the gel matrix is called

Read the passage carefully. Attempt the questions.
An overwhelming majority (99 per cent) of animals and nearly all plants cannot maintain a constant internal environment. Their body temperature changes with the ambient temperature. In aquatic animals, the osmotic concentration of the body fluids change with that of the ambient air and water osmotic concentration. These animals and plants are simply conformers. Considering the benefits of a constant internal environment to the organism, we must ask why these conformers had not evolved to become regulators. Recall the human analogy we used above; much as they like, how many people can really afford an air conditioner? Many simply 'sweat it out' and resign themselves to suboptimal performance in hot summer months. Very small animals are rarely found in polar regions. During the course of evolution, the costs and benefits of maintainin a constant internal environment are taken into consideration. Some species have evolved the ability to regulate, but only over a limited range of environmental conditions, beyond which they simply conform.
If the stressful external conditions are localised or remain only for a short duration, the organism has five other alternatives for survival.
Identify the statements which support the reason why small animals are rarely found in polar regions.
A. Small animals have larger surface area relative to their volume
B. Small animals have larger volume relative to their surface area
C. Shrews and humming birds are not found in polar region
D. Small animals have smaller surface area relative to their volume