If \(x, y\) and \(z\) are non-zero distinct numbers, then \(\left|\begin{array}{ccc}x+y & y+z & z+x \\ z & x & y \\ 1 & 1 & 1\end{array}\right|\) is equal to

The values of ' \(a\) ' and ' \(b\) ' such that the function defined by is a continuous function :
\(f(x)=\left\{\begin{array}{ll}3 & x \leq 5 \\ a x+b & 5 < x \leq 15 \\ 18 & x > 15\end{array}\right.\)

The corner points of a bounded feasible region are \(( 0 , 5 ),( 6 , 1 ),( 1 7 , 2 )\) and \(( 4 , 2 9 )\).
If the maximum value of the objective function \(z=p x+q y\), where \(p>0\) and \(q>0\), occurs at two points \((17,2)\) and \((4,29)\), then the relation between \(p\) and \(q\) is :

The value of the definite integral \(I=\int_{-1}^1 \frac{1}{1+\sqrt{e^x}} d x\) is :

If a random variable \(X\) has the following probability distribution :

X	0	1	2	3
P(X)	K	\(\frac{K}{2}\)	\(\frac{K}{4}\)	\(\frac{K}{8}\)

then,
Match List-I with List-II

List-I	List-II
(A) The value of \(K\) is	(I) \(2 / 15\)
(B) \(P(0<X<2)\) is	(II) \(1 / 15\)
(C) \(P(1<X<3)\) is	(III) \(8 / 15\)
(D) \(P(X>2)\) is	(IV) \(4 / 15\)

Choose the correct answer from the options given below :

The number of arbitrary constants in general solution of a differential equation of third order is :

The linear programming problem:
Maximize \(Z = x + y\),
subject to the constraints \(x-y \leq-1,-x+y \leq 0, x, y \geq 0\) has

Carbonyl carbon in acetone is \(\mathrm{sp}^2\) hybridized. The C-O-C bond angle and the geometry of acetone is.......

Read the passage carefully and answer the Questions
Carbonyl compounds such as aldehydes, ketones, and carboxylic acids play an important role in the biochemical process of plants and animal kingdon. Many aldehydes and ketones find applications in dye, perfumery and other industries.
The carbonyl carbon atom is \(s p^2\)-hybridized forming three sigma and one pi-bond with oxygen.
C-O bonds are polar bond due to high electronegativity of oxygen than carbon.
Hence the carbonyl carbon behaves as an electrophilic center and oxygen as a nucleophilic centre.
Primary and secondary alcohols yield aldehydes and ketones respectively upon oxidation with variety of oxidixing agents.
The boiling points of aldehydes and ketones differ from hydrocarbons and ethers.
Aldehydes and ketones undergo nucleophilic addition reaction due to presence of electrophilic carbonyl carbon.
Nucleophile and a proton add across the carbon oxygen double bond.
Formalin which is used to preserve biological specimen is an aqueous solution of

Cu exist as Cu\(^+\) and Cu\(^{2+}\) ion. Which of the following statements is correct?

A coil having area A is placed in a magnetic field. The field is perpendicular to the plane of the coil. When the magnetic field changes from \( B_0 \) to \( 4B_0 \) in a time interval t, the magnitude of emf induced in the coil will be :

The corner points of the feasible region determined by the system of linear inequalities are \((0, 0), (0, 4), (4, 0), (2, 4)\) and \((0, 5)\). If the maximum value of \(Z = ax + by\) where \(a, b > 0\) occurs at both \((2, 4)\) and \((4, 0)\), then

Ionization potential of a hydrogen atom is 13.6 V. Hydrogen atom in the ground state is excited by monochromatic radiation of photon energy 12.75 eV. The number of spectral lines emitted by hydrogen atom will be