Let A be a non-singular square matrix of order n, then
Match List - I with List - II

List - I	List - II
(A) A (adj A)	(I) \(\frac{1}{|A|}\)
(B) |adj A|	(II) \(|A|^n\)
(C) \(\left|A^{-1}\right|\)	(III) \(|A| I\)
(D) \(|A(\operatorname{adj} A)|\)	(IV) \(|A|^{n-1}\)

Choose the correct answer from the options given below:

In which of the following interval the function \(f(x)=x^x, x>0\) is strictly increasing?

The integral \(\int e^x\left(\frac{x-1}{2 x^2}\right)\)dx is equal to

For an objective function \(z=a x+b y\) where \(a, b>0\).
The corner points of feasible region determined by a set of constraints are \((0,20),(10,10),(30,30),(0,40)\).
The condition on \(a\) and \(b\) such that max \(z\) occurs at both the points \((30,30)\) and \((0,40)\) is:

For the differential equation \(x \frac{d y}{d x}+3 y=x^2 \log _e x\) which of the following statements are TRUE?
(A) Product of order and degree is 1
(B) Integrating factor is \(x^3\)
(C) Integrating factor is \(3 x\)
(D) General solution is \(y=\frac{x^3}{36}\left(6 \log _e|x|-1\right)+C x^{-3}\), C is an arbitrary constant
Choose the correct answer from the options given below:

The vector equation of a line passing through the point with position vector \(2 \hat{i}-\hat{j}+\hat{k}\) and parallel to the line joining the points with position vectors \(-\hat{i}+4 \hat{j}+\hat{k}\) and \(\hat{i}+2 \hat{j}+2 \hat{k}\) is:

The function \(f ( x )=\log (\sin x )\) is ___________

Read the passage carefully and answer
Carbohydrates are essential for both plants and animals. Some common examples of carbohydrates are glucose, fructose, sucrose, glycogen, cellulose and starch. They form a major portion of our food. Honey has been used for a long time as an instant source of energy in Ayurvedic medicine. Carbohydrates are used as storage molecules in both plants and animals. The cell wall of bacteria and plants is made of cellulose. We use cellulose in the form of wood for making furniture and as cotton fiber for clothing. They also provide raw material for textiles, paper, lacquers, and breweries. Carbohydrates are classified as Monosaccharides, Oligosaccharides and Polysaccharides on the basis of their behavior towards hydrolysis.
Monosaccharides cannot be hydrolyzed further to simpler units whereas oligosaccharides can yield 2-10 units of monosaccharides on hydrolysis. Polysaccharides, which are non-sugars yield a large number of monosaccharide units on hydrolysis.
Answer the following questions about carbohydrates:
Identify a component of starch from among the following.

Match List-I with List-II

List-I (Types of inheritance)	List-II (Examples)
(A) Pleiotropy	(I) ABO blood grouping in human
(B) Polygenic inheritance	(II) Snapdragon plant
(C) Incomplete dominance	(III) Phenylketonuria
(D) Co-dominance	(IV) Human skin colour

Choose the correct answer from the options given below:

If it is 7:00 pm currently in the clock, what will the clock show (in am or pm) after 674 hours?

The slope of the normal to the curve \(y=2 x^2-4\) at \(P(1,-2)\) is:

Read the following passage carefully and answer the given questions.
Theory of Evolution
The evolution of life is a gradual process through which simple organisms are transformed into complex life forms over billions of years. It involves genetic changes, adaptation to the environment, and the emergence of diversity in species. Fossil records provide evidence of extinct life forms and transitional stages. Evolution is influenced by factors such as mutation, genetic recombination, and environmental pressure, shaping the biodiversity we see today. Charles Darwin and Alfred Russel Wallace were pioneering naturalists who developed the theory of evolution by natural selection.
The study of fossils in different sedimentary layers indicates ______.

For \(y \neq 0\), the particular solution of the differential equation \(2 y e^{x / y} d x+\left(y-2 x e^{x / y}\right) d y=0\) at the point \((1,1)\) is