The probability that a leap year selected at random will have 53 Mondays is

If \(\int_0^1 \frac{a \sqrt{x} d x}{\left(x^{3 / 2}+1\right)}=1\), then \(a\) is equal to:

The function \(f(x)=\left\{\begin{array}{ll}\frac{x^2+2 x-3}{x-1}, & \text { if } x \neq 1 \\ 0, & \text { if } x=1\end{array}\right.\) is

The function \(f(x)=x+\cot ^{-1} x\) is increasing in the interval:

If \(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\), then \(\frac{d^2 y}{d x^2}\) is equal to

The value of \(\sin \left(2 \sin ^{-1} \frac{4}{5}\right)\) is given by :

If \(A\) be a square matrix of order 3 such that \(|A|=2\), then \(|\operatorname{adj}(2 A)|\) is equal to

Identify the correct statements.
(A) A concave mirror is preferred over a plane mirror for the purpose of shaving.
(B) The absolute refractive index of a medium can be less than unity.
(C) Snell's law gives the relative refractive index of a medium.
(D) When a convex lens is placed inside a transparent medium of refractive index greater than that of its own material, it behaves as a diverging lens.
Choose the correct answer from the options given below:

Read the given passage and answer the question given below.
The sex linked recessive disease, which shows its transmission from unaffected carrier female to some of the male progeny has been widely studied. In this disease, a single protein that is a part of the cascade of proteins involved in the clotting of blood is affected. Due to this in an affected individual a simple cut will result in non-stop bleeding. The heterozygous female (carrier) for the disease may transmit it to sons. The possibility of a female becoming a sufferer of disease is extremely rare because mother of such a female has to be at least carrier and the father should be sufferer. The family pedigree of Queen Victoria shows a number of suffering descendents as she was a carrier of the disease.
Identify the genotype of queen Victoria. ('- gene for disorder)

A proton accelerated through potential difference V has de Broglie wavelength \(\lambda\). On doubling the potential, the de Broglie wavelength of the proton :

Choose the correct match. '+' for beneficial interaction '-' for detrimental and 'O' for neutral effect.

LIst - I		LIst - II
Species A	Species B	Name of Interaction
(A) +	+	(I) Competition
(B) +	-	(II) Mutualism
(C) -	-	(III) Amensalism
(D) -	0	(IV) Predation

Choose the correct answer from the options given below:

If \(\left[\begin{array}{cc}2 & -1 \\ 1 & 0 \\ -3 & 4\end{array}\right] A=\left[\begin{array}{ccc}-1 & -8 & -10 \\ 1 & -2 & -5 \\ 9 & 22 & 15\end{array}\right]\), then matrix \(A\) is equal to:

An emf of \( 0.5 \text{ V} \) is developed in the secondary coil, when current in primary coil changes from \( 5.0 \text{ A} \) to \( 2.0 \text{ A} \) in \( 300 \text{ milliseconds} \). The mutual inductance of two coils would be :