Match List - I with List - II. Match the integrating factors :

List - I (Differential Equation)	List - II (Integrating factor)
(A) \(\frac{d y}{d x}+3 y=e^{-2 x}\)	(I) \(\frac{1}{x}\)
(B) \(x \frac{d y}{d x}+y=3 x^2\)	(II) \(e^{-x}\)
(C) \(x \frac{d y}{d x}-y=3 x^2\)	(III) \(x\)
(D) \(\frac{d y}{d x}-y=x\)	(IV) \(e^{3 x}\)

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If \(5 x+y \leq 100, x+y \leq 60, x \geq 0, y \geq 0\). Then one of the corner points of the feasible region is:

\(\int \frac{(x-1) e^x}{x^2} d x\), \(x > 0\) equals (where C is an arbitrary constant)

Match List-I with List-II

List-I (Differential Equations)	List-II (Order and degree)
(A) \(\frac{d y}{d x}+e^y=0\)	(I) order 2, degree not defined
(B) \(\frac{d^2 y}{d x^2}=\left[1+\left(\frac{d y}{d x}\right)^2\right]^{3 / 2]}\)	(II) order 2, degree 1
(C) \(\left(\frac{d^2 y}{d x^2}\right)^2+e^{\left(\frac{\text { 临 })}{}\right.}=0\)	(III) order 1, degree 1
(D) \(\frac{d^2 y}{d x^2}+x \frac{d y}{d x}-2 y=\log x ; x>0\)	(IV) order 2, degree 2

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If \(f:[3, \infty) \rightarrow A\) defined by \(f(x)=x^2\) is an onto function, then \(A\) is:

If A and B are symmetric matrices of the same order, then which one of the following is true?

If A is any \(m \times n\) matrix and B is a matrix such that AB and BA are both defined, then B is a matrix of order :

If the function
\(f(x)=\left\{\begin{array}{ll}a x+2, & x \leq 1 \\x^2+3 x+b, & x>1\end{array}\right.\)
is differentiable at \(x=1\), then the value of \((2 a+b)\) is

Read the passage carefully and answer the Questions
Carboxylic acids are the earliest organic compounds isolated from nature. These are also known as organic acids but are weaker acids as compared to mineral acids. However, they are more acidic than alcohols and phenols because of greater resonance stabilization of the parent acid and its conjugate base. The acidic strength is determined by the \(K _a\), which is called the acidity constant. The greater the value of \(K _a\), the greater the tendency of the acid to ionize and hence the stronger the acid. The acidic strength of the saturated aliphatic carboxylic acid depends upon the inductive effect and its position with respect to - COOH groups. Electron-donating substituents tend to decrease whereas electron-withdrawing substituents tend to increase the acidic strength. The acidic strength of the aromatic carboxylic acid, on the other hand depends upon both the inductive effect and the resonance effect of the substituents.
\(p K_a\) of chloroacetic acid is lower than acetic acid because of

Identify the correct statements among the following:
(A) The gravitational force is much stronger than the Coulomb force.
(B) The nuclear force between two nucleons falls rapidly to zero as their distance is more than a few femtometers.
(C) The nuclear force depends on the electric charge.
(D) The nuclear force is much stronger than the coulomb force.
Choose the correct answer from the options given below:

Eli Lilly, in 1983, developed human insulin using rDT by combining chain A and chain B with:

The process of copying genetic information from one strand of DNA into RNA is termed as

The largest open interval in which the function \(f(x)=4 x^3-5 x^2-8 x+12\) increases, is: