The solution of the differential equation \(x \frac{d y}{d x}+2 y=x^2\) is: ( \(C\) is constant of integration)

\(\int(x-1) e^{-x} d x=\)

The values of \(\lambda\) for which the following system of equations has unique solution:
\(\begin{array}{l}\lambda x+3 y-z=1 \\x+2 y+z=2 \\-\lambda x+y+2 z=-1\end{array}\)

The corner points of the feasible region determined by the system of linear constraints are \((0,0),(0,40),(20,40),(60,20),(60,0)\).
The objective function is \(z=4 x+3 y\).
Compare the quantity in Column - A and Column - B.

Column - A	Column - B
Maximum value of z	350

The interval in which where the function \(f(x)=x^3-3 x^2+4 x+1, x \in R\) is increasing, is:

The function \(f(x)=\frac{x-2}{x+1}, x \neq-1\) is increasing when (Where \(R\) is a set of real numbers:

Match List-I with List-II

List-I	List-II
(A) Degree of this differential equation \(\frac{d^4 y}{d x^4}+2 \log _e\left(\frac{d^3 y}{d x^3}\right)=0\)	(I) 1
(B) Order of this differential equation \(e^{\left(\frac{dy}{dx}\right)^3}+3 y\left(\frac{d^2 y}{d x^2}\right)^3=0\)	(II) 4
(C) Degree of \(\frac{d^4 y}{d x^4}+\left(\frac{d y}{d x}\right)^2=0\)	(III) not defined
(D) Order of the differential equation \(2 \frac{d^4 y}{d x^4}+\left(\frac{d^2 y}{d x^2}\right)^5=0\)	(IV) 2

Choose the Correct answer from the options given below :

Question based on the following passage: P-Block elements are placed in group 13 to 18 in the periodic table. Group 15 is nitrogen family.
Nitrogen can react with hydrogen, oxygen, halogens etc. It shows multiple oxidation states in its various oxides i.e. \(N_2O, N_2O_3, N_2O_5\) etc. Important allotropic forms of phosphorus are white, red and black phosporus. Various compounds of phosphorus are phosphine, \(PCl_3, PCl_5\) and oxoacids of phosphorus. Oxidation states \(+1\) to \(+5\) are seen in various oxoacids of phosphorus. Group 17 elements are collectively known as halogens. Bond dissociation enthalpy of \(F_2\) is smaller than \(Cl_2\).
The gases present in group 18 are chemically unreactive, they form only few compounds. Neil Bartlett (in 1962) observed the reaction of a noble gas. Since then, many noble gas compounds have been synthesised.
Explosive nature of fluorine (\(F_2\)) is due to:

The function \(f(x)=\left\{\begin{array}{ll}\frac{x^2+2 x-3}{x-1}, & \text { if } x \neq 1 \\ 0, & \text { if } x=1\end{array}\right.\) is

Which of the following compounds can be used as antifreeze in automobile radiators?

If \(\left[\begin{array}{cc}x-y & t \\ 2 x-y & w\end{array}\right]=\left[\begin{array}{cc}-1 & 2 \\ 0 & 1\end{array}\right]\), then \(2 x+y+3 t+w\) is equal to

Answer the question on the basis of passage given below :
For the transition elements
Sc \(\quad\) Ti \(\quad\) V \(\quad\) Cr \(\quad\) Mn\(\quad\) Fe \(\quad\) Co \(\quad\) Ni \(\quad\) Cu \(\quad\) Zn
Atomic numbers :
21 \(\quad\) 22 \(\quad\) 23 \(\quad\) 24 \(\quad\) 25 \(\quad\) 26 \(\quad\) 27 \(\quad\) 28 \(\quad\) 29 \(\quad\) 30 \(\quad\)
Which transition metal ion \(\text{M}^{2+}/\text{M}\) has standard electrode potential \(\text{E}^{\circ} = +0.34 \text{ V}\) ?

At what temperature a solution containing 5.60 g of glucose per 1000 g of water would boil at 1.013 bar?
(\(K_b\) for water is 0.52 K kg \(\text{mol}^{-1}\)).