Let the equations of lines be \(L_1: \vec{r}_1=\vec{a}_1+\lambda \vec{b}_1\) and \(L_2: \vec{r}_2=\vec{a}_2+\lambda \vec{b}_2\)
Then the shortest distance between \(L_1\) and \(L_2\) is

\(\int_{\pi / 6}^{\pi / 3} \frac{1}{1+\sqrt{\cot x}} d x\) is :

The solution of \(y^{\prime}-y^{\prime \prime}=2 x\) is:
A. \(y=x^2+2 x+2\)
B. \(y=x^2+2 x+1\)
C. \(y=x+2\)
D. \(y=x^2-2 x+1\)
Choose the correct answer from the options given below :

An LPP defined by the objective function Z = 2x + 2y has to be maximized.
The corner points are (0,0), (5, 0), (8, 2), (4, 6) and (0,8).
The maximum value occurs at :

Match List-I with List-II

List-I	List-II
(A) The equation of the line passing through the points \((-1,0,2)\) and \((3,4,6)\) is	(I) \(\vec{r}=(-\hat{i}+2 \hat{k})+\lambda(4 \hat{i}+4 \hat{j}+4 \hat{k})\)
(B) The vector equation of the line \(\frac{x-1}{2}=\frac{y+1}{3}=\frac{4-z}{5}\) is	(II) \(\vec{r}=(\hat{i}+3 \hat{j}+2 \hat{k})+\lambda(2 \hat{i}+\hat{j}+3 \hat{k})\)
(C) The equation of the line passing through \((1,3,2)\) and parallel to \(\frac{x+1}{2}=\frac{y-1}{1}=\frac{z+1}{3}\) is	(III) \(\vec{r}=(-\hat{i}+2 \hat{j}+3 \hat{k})+\lambda(2 \hat{i}+\hat{j}+4 \hat{k})\)
(D) The equation of the line along \(2 \hat{i}+\hat{j}+4 \hat{k}\) and passing through \((-1,2,3)\) is	(IV) \(\vec{r}=(\hat{i}-\hat{j}+4 \hat{k})+\lambda(2 \hat{i}+3 \hat{j}-5 \hat{k})\)

where \(\lambda\) is an arbitrary constant.
Choose the correct answer from the options given below:

1It is given that 3% of items manufactured by an industry are defective. The probability that a packet of 250 items contains one defective item is: [Given: \(e^{-7.5}\) 0.000553]

The area of the region bounded by the curve \(y^2=x\) and the lines \(x=1, x=4\) is:

Which of the following is not correct about the Central Limit Theorem?

Arrange the following geological periods in latest to oldest duration of existence.
(A) Triassic
(B) Carboniferous
(C) Cretaceous
(D) Permian
Choose the correct answer from the options given below:

Name the reagent required to convert cyclohexene into hexane-1, 6-dioic acid.

Answer the question on the basis of passage given below :
There are some deposits of nitrates and phosphates in the earth’s crust. Nitrates are more soluble in water. Nitrates are difficult to reduce under laboratory conditions, but microbes do it easily. Ammonia forms a large number of complexes with transition metals. Hybridisation easily explains the ease of sigma donation capability of NH₃ and PH₃. Phosphine is an inflammable gas, prepared from white phosphorus.
Which of the following is the correct statement?
A. Between NH₃ and PH₃, NH₃ is a better electron donor because the lone pair of electrons occupy spherical 's' orbital and is less directional.
B. Between NH₃ and PH₃, NH₃ is a better electron donor because the lone pair of electrons occupies sp³ orbital and is more directional.
C. Between NH₃ & PH₃, PH₃ is a better electron donor because the lone pair of electrons occupies sp² orbital and is more directional.
D. Between NH₃ & PH₃, PH₃ is a better electron donor because the lone pair of electrons occupies 's' orbital and is less directional.
Choose the most appropriate answer from the options given below:

If \(A=\left[\begin{array}{ccc}a & 4 & -5 \\ d & b & -6 \\ 5 & e & c\end{array}\right]\) is a skew symmetric matrix, then value of \(a+b+c+d+e\) is equal to

A car starts from a point at time \(t = 0\) seconds and stops at Q.
The distance \(x\) in meters covered by it in \(t\) seconds is given by
\(x = t^2 \left( 2 - \frac{t}{3} \right)\).
The time taken by it to reach Q is: