The projection of the vector \(\vec{a}=2 \hat{i}+3 \hat{j}+2 \hat{k}\) on the vector \(\vec{b}=\hat{i}+2 \hat{j}+\hat{k}\) is:

A parallelogram is constructed on the vectors \(\vec{a}=3 \vec{\alpha}-\vec{\beta}, \vec{b}=\vec{\alpha}+3 \vec{\beta}\). If \(|\vec{\alpha}|=|\vec{\beta}|=2\) and the angle between \(\vec{\alpha}\) and \[\vec{\beta}\] is \(\frac{\pi}{3}\), then length of the diagonal of the parallelogram is:

For what value of k, the following system have a unique solution? (where R is set of real numbers)
x + y + z = 1
2x + 3y + 4z = 3
x - y + kz = 5

In an LPP, the feasible region represented by the set off constraints \(2 x+3 y \leq 18, x+y \leq 10, x \geq 0, y \geq 0\) is

Match List I With List II

List I	List II
A. \(\frac{d^2 y}{d x^2}=\left(\frac{d y}{d x}\right)^{\frac{3}{2}}\)	I. order + degree \(=2\)
B. \(2\left(\frac{d^3 y}{d x^3}\right)^2+3\left(\frac{d^2 y}{d x^2}\right)+y\left(\frac{d y}{d x}\right)^2=e^x\)	II. order + degree \(=3\)
C. \(\frac{d y}{d x}+\frac{1}{d y / d x}=3\)	III. order + degree \(=4\)
D. \(\frac{d y}{d x}+x^2=5\)	IV. order + degree \(=5\)

In a linear programming problem, the constraints on the decision variables \(x\) and \(y\) are \(x-3 y \geq 0, y \geq 0,0 \leq x\) \(\leq 3\). The feasible region :

\(\int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \frac{d x}{1+\sqrt{\tan x}}\) is equal to:

About __________ the first cellular forms of life appeared on earth.

Answer the question on basis of passage given below :
Lyophobic sols like those of metals (Au, Ag, etc) are unstable and are easily precipitated by addition of electrolytes. However, it is observed that the addition of certain lyophilic colloids like gums, soaps, gelatine, etc. to lyophobic colloids (like a metal sol) render lyophobic colloids difficult to coagulate by the addition of electrolytes. The process is known as 'protection' and the lyophilic colloids are termed as Protective colloids. It is believed that the protective action of the lyophilic colloids is due to the covering up of the particles of the lyophobic colloid by those of the lyophilic colloid.
Evidently, smaller the gold number of a protective colloid, the greater is its protective action.
Which of the following has the highest flocculating power for a positive sol?

A conducting circular loop is placed in a uniform magnetic field of \( B = 0.5 \text{ T} \), with its plane perpendicular to the magnetic field. The radius of the loop starts shrinking at the rate of \( \frac{dr}{dt} = 4 \text{ mm/s} \). When the radius of the loop is \( r = 4 \text{ cm} \), the induced emf in the loop is

The big holes in Swiss cheese are due to:
(A) Production of large amount of \(O _2\)
(B) Production of large amount of \(CO _2\)
(C) Production of large amount of \(H _2\)
(D) Bacterium Propionibacterium sharmanii
(E) Lactobacillus casei
Choose the correct answer from the options given below:

Dimension of Permittivity is :

Match List I with List II

List I	List ll
(A) Down's syndrome	(I) 44 + XX/XY
(B) Klinefelter's syndrome	(II) 44 + XO
(C) Turner's syndrome	(III) 44 + XXY
(D) Normal human	(IV) 45 + XX/XY

Choose the correct answer from the options given below: