The integral \(\int \frac{d x}{x^2\left(x^4+1\right)^{\frac{3}{4}}}\) equals __________ .

Match List-I with List-II

List-I	List-II
(A) The vectors \(\lambda \hat{i}+\hat{j}+2 \hat{k}\) and \(\hat{i}+\lambda \hat{j}+\hat{k}\) are perpendicular if \(\lambda\) is equal to	(I) 1
(B) The vectors \(3 \hat{i}+6 \hat{j}-\hat{k}\) and \(2 \hat{i}+\lambda \hat{j}-2 \hat{k}\) are collinear if \(\lambda\) is equal to	(II) -1
(C) The number of vectors of unit-length which are perpendicular to both the vectors \(\vec{a}=\hat{i}+\hat{j}+2 \hat{k}\) and \(\vec{b}=3 \hat{i}-\hat{j}+5 \hat{k}\) is	(III) \(2 / 3\)
(D) If \(|\vec{a}|=1\) and \(\vec{a}+\vec{b}=\overrightarrow{0}\), then \(|\vec{b}|\) is equal to	(IV) 2

Choose the correct answer from the options given below :

The angle between the straight lines \(\frac{x+4}{2}=\frac{y+5}{5}=\frac{z+6}{3}\) and \(\frac{x-4}{10}=\frac{y-5}{2}=\frac{z-6}{-10}\) is:

The side of an equilateral triangle increases at the rate of \(3 cm / sec\), then the rate at which the area of the triangle increases when the side is 4 cm , is :

If the objective function \(z = 4x + 3y\) has maximum value on a line joining points (3, a) and (b, 2) where \(a > 0, b > 0\) such that \(a - b = 2\) then the maximum value of z is:

If \(y=t-\frac{1}{t}\) and \(x=t+\frac{1}{t}\), then \(\frac{d y}{d x}\) is equal to

A machine has been producing steel tubes of mean inner diameter of 2 cm.
A sample of 10 tubes gives an inner diameter of 2.01 cm and a standard deviation of 0.063 cm.
Test the hypothesis that the machine is working in the proper order at 5% level of significance and answer, which of the following statements are correct?
[Given that \(t_{0.05}=2.262\)]
(A) The value of the test statistic is t = 0.476
(B) The null hypothesis is rejected at \(5 \%\) level of significance.
(C) The null hypothesis is accepted at \(5 \%\) level of significance.
(D) The degree of freedom is 10
Choose the correct answer from the options given below:

The magnitude of the magnetic field due to a short bar magnet having a magnetic moment of 0.2 \( \text{Am}^2 \), at a distance of 50 cm from the centre of the magnet on its axial line will be:

Match list-I with List -II

List-I	List-II
(A) An observed set of population selected for analysis	(I) Parameter
(B) A specific characteristic of the population	(II) Hypothesis
(C) A specific characteristic of the sample	(III) Statistic
(D) A statement made about a population parameter for testing	(IV) Sample

Choose the correct answer from the options given below:

Write the correct sequence of events performed in DNA fingerprinting technique
A. Isolation of DNA
B. Separation of DNA fragments by electrophoresis
C. Digestion of DNA using restriction endonucleases.
D. Transfer of seperated DNA fragments to synthetic membrane
Choose the correct answer from the options given below:

From the following diagram, determine the direction of propagation of an electromagnetic wave.

Which of the following statements are correct with reference to evolution ?
(A) About 500 mya invertebrates were formed.
(B) Jawless fish evolved around 350 mya.
(C) About 400 mya sea weeds existed.
(D) About 100 mya dinosaurs suddenly disappeared.
Choose the correct answer from the options given below:

A closed circuit in the form of a regular hexagon of side a carries a current I. What is the magnetic field at the centre of the hexagon?