The probability of a shooter hitting the target in one shot is \(\frac{1}{4}\)
The minimum number of shots needed so that the probability of hitting the target at least once is greater than \(\frac{7}{16}\) is:

If the matrix \(\left[\begin{array}{ccc}0 & 7 & -12 \\ -7 & 0 & -5 \\ 2 a & 5 & 3 b\end{array}\right]\) is skew-symmetric, then the value of \((4 a+3 b)\) is :

Let \(A=\left[\begin{array}{ccc}1 & \sin \theta & 1 \\ -\sin \theta & 1 & \sin \theta \\ -1 & -\sin \theta & 1\end{array}\right]\), where \(0 \leq \theta \leq 2 \pi\), then which of the following are true?
(A) \(|A|=2+2 \sin ^2 \theta\)
(B) \(|A|=2+\sin ^2 \theta\)
(C) minimum value of \(|A|\) is 1
(D) maximum value of \(|A|\) is 4.
Choose the correct answer from the options given below:

If \(A=\left[\begin{array}{ll}1 & 0 \\ 0 & 0\end{array}\right], B=\left[\begin{array}{ll}0 & 0 \\ 3 & 0\end{array}\right]\) then

If \(M=\left[\begin{array}{c}2 \\ -1 \\ 3\end{array}\right]\) and \(N=\left[\begin{array}{lll}7 & 1 & -4\end{array}\right]\), then \((M N)^T\) will be equal to:

The principal value of \(\cot ^{-1}\left(-\frac{1}{\sqrt{3}}\right)\) is

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

The structure in the ovary from which corpus luteum formed is known as:

The function \(f(x)=x^3\), is increasing for:

Arrange the following steps of replication of retrovirus in correct sequence
A. Incorporation of viral DNA into host genome.
B. Formation of viral DNA by reverse transcription.
C. Infection of normal cell by virus and introduction of viral RNA into host
D. Production of new viral RNA by infected cells.
Choose the correct answer from the options given below:

Answer the question on the basis of passage given below :
Aldehydes and ketones represent two essential classes of organic compounds
which are extensively used in the synthesis of a wide variety of other organic molecules.
Due to the inherent polarity of the central carbonyl group, these substances readily undergo nucleophilic addition reactions.
Broadly speaking, these addition reactions can be categorized into two types:
one in which simple addition of the nucleophilic reagent occurs across the \(> C = O\) bond, and the second type where the initial addition is subsequently followed by the elimination of a water molecule.
A variety of ammonia derivatives, including hydroxylamine, hydrazine, phenylhydrazine,
2, 4-dinitroplhenylhdrasine, and semicarbaside, all belong to this second category of reactions.
These specific derivatives are fundamentally used for the identification and definitive characterization of aldehydes and ketones. Significantly, both types of nucleophilic addition reactions are strongly influenced by both steric and electronic factors.
As a general rule, aliphatic aldehydes exhibit higher reactivity compared to aromatic aldehydes.
The aliphatic aldehydes demonstrate the ability to reduce both Fehling's solution and Tollen's reagent, whereas aromatic aldehydes are limited to reducing only Tollen's reagent.
In a similar manner to aldehydes, aromatic ketones are observed to be less reactive than aliphatic ketones towards all nucleophilic addition reactions.
In a separate context, phenols and carborylic acids both exhibit acidic properties in nature.
Both classes of compounds readily dissolve in NaOH solution and cause blue litmus to turn red.
However, carboxylic acids stand out as much stronger acids than phenols and thus can decompose sodium bicarbonate \(\left( NaHCO _3\right)\) with the distinct evolution of \(CO _2\) gas, a reaction which phenols do not undergo.
It is important to note that both electron- donating and electron- withdrawing substituents significantly influence the acid strength of both aliphatic as well as aromatic carborylic acids.
Furthermore, unlike all other common aliphatic acids, formic acid uniquely possesses reducing properties and simultaneously does not display the typical reactions characteristic of the alkyl group.
Which of the following compound most readily undergoes nucleophilic addition reaction?

Which of the following is not based on total internal reflection?

Which of the following is/are NOT the step(s) used in separation and isolation of DNA fragments by gel electrophoresis technique:
(A) Staining with ethedium bromide
(B) Exposure to UV Radiation
(C) Cutting DNA fragments into small pieces by using restriction endonuclease
(D) Denaturation of DNA fragments
Choose the correct answer from the options given below :