The differentiation of \(\frac{x^3}{1-x^3}\) w.r.t. \(x^3\) is:

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:

Match List-l with List-ll

List-l (Statement)	List-ll (Its value)
(A) Degree of the differential equation \(\frac{d^2 y}{d x^2}=e^{d y / d x}\) is	(I) 2
(B) Order of the differential equation \(\left(\frac{d y}{d x}\right)^2+\frac{d^3 y}{d x^3}=0\) is	(II) not defined
(C) Degree of the differential equation \(\frac{d^2 y}{d x^2}+\left(\frac{d y}{d x}\right)^2-5 x^2=0\)	(III) 3
(D) If \(p\) is the order and \(q\) is the degree of the differential equation \(\frac{d y}{d x}+3 y=e^x\), then \(p+q\) is	(IV) 1

Choose the correct answer from the options given below :

Match List-I with List-II

List-I	List-II
(A) The maximum value of \(f(x)=\sin (3 x)+6\)	(I) 2
(B) The maximum value of \(f(x)=-|x+2|+4\)	(II) 5
(C) The maximum value of \(f(x)=(3 x+1)^2+5\)	(III) 7
(D) The maximum value of \(f(x)=2 \cos x+4\)	(IV) 4

Choose the correct answer from the options given below:

The maximum value of the objective function \(Z=2 x+y\) of an LPP, subject to the constraints \(x \leq 6, y \leq 2, x-y \leq 0\), \(x \geq 0, y \geq 0\), is:

\(\int \frac{(x-3) e^x}{(x-1)^3} d x\) is equal to

The corner points of the feasible region of LPP \(x+y \leq 200, y \geq 4 x, x \geq 20, x \geq 0, y \geq 0\) are :
(A) \((20,180)\)
(B) \((40,180)\)
(C) \((20,80)\)
(D) \((40,80)\)
(E) \((40,160)\)

Read the given paragraph and answer the question:
Humans have 23 pairs of chromosomes. 22 pairs are exactly same in both males and females; these are the autosomes. A pair of X-chromosome are present in the female, whereas the presence of an X and Y chromosome are determinant of the male characteristic.
A number of disorders in human beings have been found to be associated with the inheritance of changed or altered genes or chromosomes. Broadly, genetic disorder may be grouped into two categories-Mendelian disorders and Chromosomal disorders.
A haemophilic man marries a normal woman. What is the possibility that their child have haemophilia?

If \(\vec{a}=\hat{i}+\hat{j}+\hat{k}, \vec{b}=\hat{j}-\hat{k}\) and \(\vec{c}=\hat{i}+\hat{j}+\hat{k}\) such that \(\vec{r} \times \vec{b}=\vec{c} \times \vec{b}\) and \(\vec{r} \cdot \vec{a}=0\), then \(\vec{r}\) is :

The probability distribution of a random variable \(x\) is, \(P(x)=\frac{k}{2^x}, x=0,1,2,3\). Then
Match List-I with List-II

List-I	List-II
(A) k	(I) \(\frac{2}{15}\)
(B) \(P ( x =1)\)	(II) \(\frac{1}{5}\)
(C) \(P (1<x<3)\)	(III) \(\frac{8}{15}\)
(D) \(P(x \geq 2)\)	(IV) \(\frac{4}{15}\)

Choose the correct answer from the options given below :

Read the following passage carefully and answer the given questions.
During anaerobic sludge digestion, bacteria produce a mixture of gases such as methane, hydrogen sulphide and carbon dioxide. These gases form biogas and can be used as source of energy as it is inflammable. The bioactive molecule, cyclosporin A, that is used as an immunosuppressive agent in organ-transplant patients, is produced by the fungus Trichoderma polysporum. You must have noticed that bottled fruit juices bought from the market are clearer as compared to those made at home. This is because the bottled juices are clarified by the use of pectinases and proteases. Statins produced by the yeast Monascus purpureus have been commercialised as blood-cholesterol lowering agents. It acts by competitively inhibiting the enzyme responsible for synthesis of cholesterol. The BOD test measures the rate of uptake of oxygen by micro-organisms in a sample of water and thus, indirectly, BOD is a measure of the organic matter present in the water.
BOD is a measure of-

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.
According to Darwin, the term 'Fitness' in the context of natural selection refers to:

A pipe can fill a tank in 2 hours. Because of a leak in the tank, the pipe takes 2 hours 40 minutes to fill the tank. The time the leakage takes to empty the full tank is :