A function \(f: R \rightarrow R\) defined by \(f(x)=\frac{x}{x^2+1}\) is (where \(R\) is the set of real numbers) :

The angle between the lines
\(l_1: \frac{x+1}{1}=\frac{2-y}{2}=\frac{z-1}{1}\) and \(l_2: \frac{x-1}{4}=\frac{2 y-4}{6}=\frac{z-1}{2}\), is:

The following system of equations:
x + y - z = 7
\(4 x+\lambda y-\lambda z=3\)
3x + 2y - 4x = 5
does not possess a solution if the value of \(\lambda\) is:

\(\int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} \frac{d x}{1+\cos 2 x}\) is equal to

Determinant is unaltered if :

The total cost \(C(x)\) in Rupees associated with the production of \(x\) units of an item is given by
\(C(x)=0.007 x^3+26 x^2+15 x+400\) 
The marginal cost when 10 items are produced is:

One root of the equation \(\left|\begin{array}{ccc}3 x-8 & 3 & 3 \\ 3 & 3 x-8 & 3 \\ 3 & 3 & 3 x-8\end{array}\right|=0\) is:

Manganese ions on reaction with peroxydisulphate ions form :

The current through a \(4 / 3 \Omega\) external resistance connected to a parallel combination of two cells with emfs of 2 V and 1 V , internal resistances \(1 \Omega\) and \(2 \Omega\) respectively, is :

During lockdown, Amit was suffering from heart problem. After proper check-up it was deducted that he has \(45\%\) chance of having heart attack, but yoga and meditation assumed to reduce the chance of heart attack by \(25\%\) and \(20\%\) respectively. He has option to choose any of the two options with equal probabilities.
If Amit goes through one of the two options selected at random and suffers a heart attack, then the probability that he had selected yoga is:

A bar magnet is cut into two equal halves along a plane parallel to the magnetic axis. The one parameter which remains unchanged 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.
How can we express the (acidity constant) \(K_{ a }\) of RCOOH ?

For the objective function \(Z = 3x + 5y\) subject to constraints \(x + 3y ≥ 3, x + y ≥ 2, x ≥ 0, y ≥ 0\):