If \(A=\left[\begin{array}{cc}2 & -3 \\ -4 & 7\end{array}\right]\) and \(2 A^{-1}=K I-A\), where \(K\) is a real number and \(I\) is the identity matrix of order 2 , then the value of \(K\) is :

In what ratio should a grocery shopkeeper mix two varieties of pulses worth Rs. 85 per kg and Rs. 100 per kg respectively, so as to get a mixture worth Rs. 92 per kg?

Let \(A=\left[a_{i j}\right]_{3 \times 3}\) be a matrix, defined by \(a_{i j}=\left\{\begin{array}{ll}2 i+3 j & , i < j \\ 6 & , i=j \\ 3 i-2 j & , i > j\end{array}\right.\). The number of elements in A. which are greater than 6, is

Which of the following statements are correct?
(A) \(\tan ^{-1} x+\tan ^{-1} y=\tan ^{-1}\left(\frac{x+y}{1-x y}\right), x y<1\)
(B) \(\tan ^{-1} x+\tan ^{-1} y=\pi+\tan ^{-1}\left(\frac{x+y}{1-x y}\right), x y>1\)
(C) \(\sin ^{-1} x+\cos ^{-1} x=\frac{\pi}{2}, x \in(-1,1)\)
(D) \(\tan ^{-1} x+\cot ^{-1} x=\frac{\pi}{2}, x \in R\)
(E) \(\cos ^{-1}(-x)=-\cos ^{-1} x, x \in[-1,1]\)
Choose the correct answer from the options given below:

Let \(A=\left[\begin{array}{cc}0 & 2 \alpha+1 \\ 1 & \beta\end{array}\right]\) and \(B=\left[b_{i j}\right]\) be a skew symmetric matrix of order 2 such that \(b_{12}=1\).
If \(A B=I_2\), where \(I_2\) is the identity matrix of order 2 , then :

The area of the triangle with vertices \((1,4),(2,7)\) and \((4,13)\) is:

If the matrix \(A=\left[\begin{array}{lll}0 & b & 1 \\ 2 & 0 & c \\ 3 & 4 & 0\end{array}\right]\) is a symmetric matrix, then the values of \(b\) and \(c\) are:

Evaluate \(: 3^{15} \bmod (7)\)

Which of the following statement is not correct for Butan-2-ol?

If P and Q are non-singular square matrices of the same order, then \(\left(P Q^{-1}\right)^{-1}\) equals

The values of \(b\) for which \(y=2 x^2-b x+2\) touches the \(x\)-axis are:

Read the Passage carefully and answer the the questions
The DNA fragments can be separated by a technique known as gel electrophoresis. Since DNA fragments are negatively charged molecules, they can be separated by forcing them to move towards the anode under an electric field through a medium/matrix. The separated DNA fragments can be visualised only after staining the DNA with a compound known as ethidium bromide followed by exposure to UV radiation. The repeated amplification of DNA is achieved by the use of a thermostable DNA polymerase (isolated from a bacterium, Thermus aquaticus). After completion of the biosynthetic stage, the product has to be subjected through a series of processes before it is ready for marketing as a finished product. The downstream processing and quality control testing vary from product to product. The convention for naming restriction enzymes is the first letter of the name comes from the genus and the second two letters come from the species of the prokaryotic cell from which they were isolated.
The thermostable DNA polymerase is isolated from -

The complex \(\left[\mathrm{Co}\left(\mathrm{NH}_3\right)_4\left(\mathrm{NO}_2\right)_2\right] \mathrm{Cl}\) will show.......
(A) Linkage isomerism.
(B) Ionization isomerism
(C) Coordination isomerism
(D) Geometrical isomerism
Choose the correct answer from the options given below: