The derivative of \((\log x)^x\) with respect to log \(x\) is:

If A and B are two matrices of order \(4 \times m\) and \(4 \times n\), respectively and \(m=n\),
then the order of matrix \((5 A-2 B)\) is:

The degree of the differential equation \(\left(1+\frac{d y}{d x}\right)^3=\left(\frac{d^2 y}{d x^2}\right)^2\) is :

Match List-I with List-II

List-I (Example)	List-II (Time-series component)
(A) Labour strike	(I) Secular-trend
(B) Continuous decline in death rate	(II) Seasonal
(C) Rise in prices before Diwali	(III) Cyclical
(D) Rise and fall of the share-market	(IV) Irregular

Choose the correct answer from the options given below :

Let \(y(x)=a(x+1) \log (x+1)+b x+5\) be the solution of the differential equation \(e^{\frac{d y}{d x}}=x+1, y(0)=5\), then the value of \((a+b)\) is :

The area of the region enclosed by \(y=x+2\) and \(y=x^2+x-2\) is:

For \(x \in\left(0, \frac{\pi}{2}\right), \int \frac{\sin x+\cos x}{\sqrt{\sin 2 x}} d x\) is equal to

For \(x>e\),\(\int \frac{d x}{x-\sqrt{x}}\) is equal to

Match List-I with List-II

List-I  ( Sources)	List-II  (Vitamin name)
(A) Fish liver oil	(I) Vitamin C
(B) Citrus fruits	(II) Vitamin A
(C) Exposure to sunlight	(III) Vitamin \(B _{12}\)
(D) Curd	(IV) Vitamin D

Choose the correct answer from the options given below:

In aldehydes, the carbonyl group is bonded to a carbon and hydrogen, while in ketones it is bonded to two carbon atoms. The carbonyl compounds in which carbonyl group is bonded to oxygen are known as carboxylic acid and their derivatives.
The hybridisation of the \(C\) in the carbonyl group is:

Read the following paragraph and answer the given questions.
Bio-fertilisers are organisms that enrich the nutrient quality of the soil like cyanobacteria, while there are organisms that help to control plant diseases and pests. The very familiar beetle, the ladybird and dragonflies are useful to get rid of aphids and mosquitoes, respectively. An example of microbial biocontrol agents that can be introduced to control butterfly caterpillars is certain bacteria, which are available in sachets in the form of dry spores. These spores are mixed with water and sprayed onto vulnerable plants. Trichoderma species that are very common in the root ecosystem, are very effective biocontrol agents of several plant pathogens.Baculoviruses attack insects and other arthropods. Usage of these viruses is desirable when an ecological sensitive area is being treated.
Which one of the following is not an advantage of Nucleopolyhedrovirus as biocontrol agents?

If \(y=a+b(x-2022)\) is a straight line trend using the least square method for the following data

Year(x)	2020	2021	2022	2023	2024
Profit(Rs. '000)(y)	2	3	4	5	2

Then the value of \(\frac{a}{b}\) is:

Formic acid does not respond to