\(\int e^x \sec x(1+\tan x) d x\) equals:

The domain of the function \(\sin ^{-1}(4 x-1)\) is:

The general solution of the differential equation \(\frac{d y}{d x}+4 x y^2=0\) is :

A machine costing ₹ \(36000\) has an effective life of 5 years with scrap value of ₹ \( 5000\) following a linear method of depreciation. Which are correct?
(A) The value of the machine after 1 year is ₹ \(31000\)
(B) The value of the machine after 2 years is ₹ \(23600\)
(C) The value of the machine after 3 years is ₹ \(18400\)
(D) The value of the machine after 4 years is ₹ \(11200\)
Choose the correct answer from the options given below :

Match List I with List II

List - I	List - II
(A) \(|\vec{a} \times \hat{i}|^2+|\vec{a} \times \hat{j}|^2+|\vec{a} \times \hat{k}|^2\)=	(I) \(|\vec{a}|^2\)
(B) \(\frac{(\vec{a} \cdot \vec{b})^2+|\vec{a} \times \vec{b}|^2}{|\vec{b}|^2}=\)	(II) \(\sqrt{3}|\vec{a}|\)
(C) \(\vec{a}, \vec{b}\) and \(\vec{c}\) are three mutually perpendicular vectors of equal magnitude then \(|\vec{a}+\vec{b}+\vec{c}|=\)	(III) \(\frac{1}{2}|\vec{a} \times \vec{b}|\)
(D) Area of parallelogram with diagonals \(\vec{a}\) and \(\vec{b}\)	(IV) \(2|\vec{a}|^2\)

Choose the correct answer from the options given below:

If the sum of the lengths of the hypotenuse and one of the other two sides of a right angled triangle is fixed, then the area of the triangle is maximum, when the angle between them is :

The angle between the line \(\vec{r}=(\hat{i}+2 \hat{j}-\hat{k})+\lambda(\hat{i}-\hat{j}+\hat{k})\) and the plane \(\vec{r} \cdot(2 \hat{i}-\hat{j}+\hat{k})=4\) is :

Which of the following is the correct order of second ionization enthalpy?

A TV tower has a height of 160 m. How much population is covered by the television broadcast, if the average population density around the tower is 1000 per km^2? Given radius of the earth is 6400 km.

In a binomial distribution, the probability of getting a success is \(\frac{1}{3}\) and the standard deviation is 2 . Then its mean is :

Read the passage carefully and answer the given questions.
Recombinant DNA Technology
Asexual reproduction preserves the genetic information, while sexual reproduction permits variation. Traditional hybridization procedures used in plants and animals breeding very often lead to inclusion and multiplication of undesirable genes along with the desired genes. The techniques of genetic engineering, which include creation of recombinant DNA, use of gene cloning and gene transfer, overcome this limitation and allow us to isolate and introduce only one or a set of desirable genes without introducing genes into the target organism.
Which one of the following options is correct for working of restriction enzymes?

Match List - I with List - II

LIST-I (Differential Equations)	LIST-II ( Degree)
(A) \(\left(\frac{d y}{d x}\right)^3+y x=0\)	(I) 2
(B) \(e^{\frac{d y}{d x}}+y^2+y^{\prime \prime}=0\)	(II) 1
(C) \(x y y^{\prime \prime}+x\left(y^{\prime}\right)^2-y y^{\prime}=0\)	(III) Not defined
(D) \(\left(y^{\prime \prime}\right)^2+y=0\)	(IV) 3

Choose the correct answer from the options given below :

A coil of 500 turns is threaded by a magnetic flux of \( 7 \times 10^{-5} \text{ Wb} \). If the flux changes to \( 3 \times 10^{-5} \text{ Wb} \) in 0.01 s, what average induced emf is developed in the coil? If resistance offered by the coil is \( 10 \, \Omega \), then the flow of average induced current will be :