If \(|\vec{a}|=3\) and \(|\vec{b}|=4\), then a value of \(\lambda\) for which \(\vec{a}+\lambda \vec{b}\) and \(\vec{a}-\lambda \vec{b}\) are perpendicular is :

The shortest distance between the lines \(\vec{r}=(\hat{i}+2 \hat{j}+3 \hat{k})+\lambda(2 \hat{i}+3 \hat{j}+4 \hat{k})\) and \(\vec{r}=(2 \hat{i}+4 \hat{j}+5 \hat{k})+\mu(3 \hat{i}+4 \hat{j}+5 \hat{k})\) is equal to :

The system of equations:
\(x+y+z=5\)
\(x+2 y+3 z=9\)
\(x+3 y+\lambda z=\mu\) has a unique solution, if

If \(A=\left[a_{i j}\right]_{3 \times 2}\), where \(a_{i j}=i+j\), then

(A) A is a square matrix
(B) \(a_{21}+a_{32}=8\)
(C) Number of elements in A is 6
(D) Transpose of \(\mathrm{A}=\left[\begin{array}{ll}2 & 3 \\ 3 & 4 \\ 4 & 5\end{array}\right]\)
Choose the correct answer from the options given below:

If \(x\) is in the 1 st Quadrant, then the value of \(\tan ^{-1}\left(\frac{\sqrt{1+\cos x}+\sqrt{1-\cos x}}{\sqrt{1+\cos x}-\sqrt{1-\cos x}}\right)\) is :

The shortest distance between the lines \(\vec{r}=(\hat{i}+2 \hat{j}+3 \hat{k})+\lambda(2 \hat{i}+3 \hat{j}+4 \hat{k})\) and \(\vec{r}=(2 \hat{i}+4 \hat{j}+5 \hat{k})+\mu(4 \hat{i}+6 \hat{j}+8 \hat{k})\) is equal to

The curve \(y = f ( x )\) is normal probability curve, then which of the following statements are correct?
(A) mean, median and mode of the distribution coincide.
(B) the area bounded by the curve \(y = f ( x )\) and \(x\)-axis is one unit.
(C) The curve is symmetrical about the line \(x =\mu\), where \(\mu\) is the mean.
(D) y -axis is an asymptote to the curve.
Choose the correct answer from the options given below :

How many haploid cells will be formed after first meiotic division of one primary spermatocyte ?

Read the following paragraph and answer the given questions.
Repeated amplification is achieved by the use of a thermostable DNA polymerase (isolated from a bacterium, Thermus aquaticus), which remain active during the high temperature induced denaturation of double stranded DNA. The separated DNA fragments can be visualised only after staining the DNA with a compound known as ethidium bromide followed by exposure to UV radiation. The separated bands of DNA are cut out from the agarose gel and extracted from the gel piece. The convention for naming these enzymes is the first letter of the name comes from the genus and the second two letters come from the species of the prokaryotic cell from which they were isolated, e.g., EcoRI comes from Escherichia coli RY 13. In EcoRI, the letter 'R' is derived from the name of strain. Roman numbers following the names indicate the order in which the enzymes were isolated from that strain of bacteria. Stanley Cohen and Herbert Boyer accomplished this in 1972 by isolating the antibiotic resistance gene by cutting out a piece of DNA from a plasmid which was responsible for conferring antibiotic resistance. Since the DNA is enclosed within the membranes, we have to break the cell open to release DNA, RNA, protein, polysaccharides etc. The RNA can be removed by treatment with ribonuclease whereas proteins can be removed by treatment with protease. Other molecules can be removed by appropriate treatments and purified DNA ultimately precipitates out after the addition of chilled ethanol.

In agarose gel electrophoresis, the separated DNA fragments can be visualized with the help of compound known as:

Hind II is a restriction endonuclease which always cuts DNA molecule at a particular point by recognizing:

The rate of change of the area of an equilateral triangle with respect to its side when its side \(=2\) is :

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 \(Z = 4x + 3y\).
Compare the quantity in Column A and Column B
Column A : Column B :
Maximum of Z 325

The shortest wavelength in the Balmer series of a hydrogen atom is (Given: \(R=1.1 \times 10^7 m^{-1}\) )