The vector equation of the plane which is at a distance of 6 units from the origin and has \(2,-1,2\) as the direction ratios of normal to it, is :

The area (in sq. units) of the region \(\left\{(x, y): 3 x^2 \leq y \leq|x|\right\}\) is equal to :

The equation of normal to the curve \(2 y+x^2=3\) at the point (1, 1) is:

The area bounded by \(y^2=x, y=5\) and the \(y\)-axis is:

If \(y=\frac{1}{\sqrt[3]{1-x^3}}\), then \(\frac{d y}{d x}\) is equal to :

If A = \(\left[\begin{array}{ccc}0 & 1 & -3 \\ -1 & 0 & 5 \\ 3 & -5 & 0\end{array}\right]\), then the value of \(\left|A^{2025}\right|\) is

If \(A, B\) and \(C\) are all the corner points of feasible region (bounded) of an LPP with objective function \(Z\) that needs to be maximized such that \(Z_A>Z_B\) and \(Z_C < Z_A\) (Here \(Z_A\) denotes value of \(Z\) at \(A\) ), then which of the following statements is TRUE ?

Integrating factor of the differential equation \(x \frac{d y}{d x}-y=3 x^2\) is:

Expand KVIC

The area of the region bounded by the curve \(y^2=x\) and the lines \(x=1, x=4\) is:

Read the passage carefully and answer the questions
The chemical reactions proceed at different rates which depend on various factors like concentration, temperature, etc. Rate law is an expression which relates the rate of reaction with concentration of various reacting species in the reaction. For a first order reaction, the concentration of the reactant after time \(t\) is related with its initial concentration by the relation, \(C=C_0 e^{-k t}\). The rate constants for different orders have different expressions. For a first order reaction, \(k=\frac{2.303}{t} \log \frac{a}{a-x}\), where \(a\) is the initial concentration and \(x\) is the extent of reaction.
Which plot is linear with negative slope for a first order reaction

Ethyl alcohol can be prepared from the following set of reactants and subsequent hydrolysis of adduct :

If \(y=e^{\frac{1}{2} \log \left(1+\tan ^2 x\right)}\), then \(\frac{d^2 y}{d x^2}\) is equal to :