The equation of perpendicular from the point \((1,0,1)\) to the plane \(x-y+z=4\) is

Three varieties A, B and C of rice are mixed together in the ratio 1:1:3 respectively. The price of rice A is Rs 127 per kg and that of rice B is Rs 135 per kg. If the price of the mixture is Rs. 152 per kg, then the price per kg of rice of type C is :

Two cards are drawn with replacement from a pack of 52 cards.
The probability distribution of number of aces is:

The solution of the differential equation \(\frac{d y}{d x}=\frac{3 e^{2 x}+3 e^{4 x}}{e^x+e^{-x}}\) is:

Area (in sq. units) of the region bounded by the curves \(y = - 1, y = 2 x = y^3\) and \(x = 0\) is

Match List-I with List-II

List-I (Equationof Lines)	List-II (Information)
(A) \(\vec{r}=(3 \hat{i}-2 \hat{j}+\hat{k})+\lambda(\hat{j}-2 \hat{k})\)	(I) Direction ratios are \(2,4,-1\)
(B) \(\frac{2-x}{1}=\frac{2 y+1}{4}, z=2\)	(II) Perpendicular to \(2 \hat{i}-\hat{j}+\hat{k}\)
(C) \(\frac{x}{1}=\frac{y-3}{2}=\frac{3-4 z}{2}\)	(III) Passing through the point \((3,-2,1)\)
(D) \(\vec{r}=(3 \hat{i}+2 \hat{j}+\hat{k})+\lambda(2 \hat{i}+\hat{j}-3 \hat{k})\)	(IV) Direction ratios are \(-1,2,0\)

Choose the correct answer from the options given below :

Slope of the tangent to the parabola \(y^2=x+2\) at a point in the 1st quadrant and lying on the line \(y=x\) is:

Match List I with List ||

List-I	List-II
Complex species	Isomerism
A. \([\text{Co(NH}_3\text{)}_4\text{Cl}_2\text{]}^{+}\)	I. Optical
B. \(\text{cis-[Co(en)}_2\text{Cl}_2\text{]}\)	II. Ionisation
C. \([\text{Co(NH}_3\text{)}_5\text{SO}_4\text{]Br}\)	III. Coordination
D. \([\text{Co(NH}_3\text{)}_6\text{]}[\text{Cr(CN)}_6\text{]}\)	IV. Geometrical

Choose the correct answer from the options given below:

In inorganic salt analysis, for the indicatory test of \(Cl^-\), conc. \(H_2SO_4\) is used. This is because:

The pioneer species of primary succession in water are:

The particular solution of the differential equation \(x d y=\left(2 x^2+1\right) d x, x \neq 0\), given that \(y=1\) when \(x=1\) is:

Read the passage carefully and answer
The crystal field theory (CFT) is an electrostatic model which considers the metal-ligand bond to be ionic arising purely from electrostatic interactions between the metal ion and the ligand. Ligands are treated as point charges in case of anions or dipoles in case of neutral molecules. The five d orbitals in an isolated gaseous metal atom/ion have same energy, i.e., they are degenerate. This degeneracy is maintained if a spherically symmetrical field of negative charges surrounds the metal atom/ion. However, when this negative field is due to ligands (either anions or the negative ends of dipolar molecules like NH3 and H2O) in a complex, it becomes asymmetrical and the degeneracy of the d orbitals is lifted. It results in splitting of the d orbitals. The pattern of splitting depends upon the nature of the crystal field.
Which of the following pairs exists as high-spin, tetrahedral complexes?

216 charged droplets, each of radius r and charge q, are combined to form a big drop. The ratio of potential of the big drop to the small droplet will be :