The value of \(\int_1^2 \frac{\sqrt{x}}{\sqrt{3-x}+\sqrt{x}} d x\) is:

The area (in sq. units) of the region bounded by \(y=2 x+3\), the \(x\)-axis, and the ordinates \(x=-2, x=2\) is equal to

The points on the curve \(\frac{x^2}{16}+\frac{y^2}{25}=1\) at which tangents are parallel to \(x\)-axis are:

\(\int_0^{\frac{\pi}{2}} \frac{\cos x}{(1+\sin x)(2+\sin x)} d x=\)

If \(A\) is any square matrix of order 3 and \(B=\left[\begin{array}{ccc}\sin \theta & \cos \theta & 0 \\ -\cos \theta & \sin \theta & 0 \\ 0 & 0 & a\end{array}\right]\), \(a\) is any constant, then \(|A B|\) is equal to:

A machine costing Rs. \(2,00,000\) has a useful life of 5 years. The estimated scrap value is Rs. 20,000 . By using straight line method, the annual depreciation is

Area lying in the first quadrant and bounded by the circle \(x^2+y^2=4\) and the lines x = 0 and x = 2 is:

Electromagnetic wave travels with speed of \(2.0 \times 10^8 m / s\) in any medium of relative permeability ( \(\mu_r\) ) 1.5. Find the relative permittivity of the medium ( \(\epsilon_r\) )

Let \(A=\{1,2,3\}\), then the possible equivalence relations on \(A\) are :
(A) \(\{(1,1),(2,2),(3,3)\}\)
(B) \(\{(1,1),(2,2),(3,3),(1,2),(2,1)\}\)
(C) \(\{(1,1),(2,2),(3,3),(1,2),(2,1),(1,3)\}\)
(D) \(\{(1,1),(2,2),(3,3),(1,3),(3,1)\}\)
Choose the correct answer from the options given below :

Read the following passage carefully and answer the given questions.
Theory of Evolution
The evolution of life is a gradual process through which simple organisms are transformed into complex life forms over billions of years. It involves genetic changes, adaptation to the environment, and the emergence of diversity in species. Fossil records provide evidence of extinct life forms and transitional stages. Evolution is influenced by factors such as mutation, genetic recombination, and environmental pressure, shaping the biodiversity we see today. Charles Darwin and Alfred Russel Wallace were pioneering naturalists who developed the theory of evolution by natural
selection. The voyage of the H.M.S. Beagle was significant in the development of Darwin's theory of evolution as-

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]

For a photoelectric effect:
(A) the photoelectric current is directly proportional to the intensity of the incident radiation.
(B) the maximum kinetic energy of the emitted photoelectrons increases linearly with the frequency of the incident radiation.
(C) for a given frequency of the incident radiation, the stopping potential increases linearly with the intensity of the incident radiation.
(D) the photoelectric emission is an instantaneous process without any apparent time lag.
Choose the correct answer from the options given below:

A battery of 200 V charges a capacitor to \( 10 \times 10^{-6} \, C \). Now the battery is disconnected, and the plate separation is doubled. The potential of the capacitor will be