Let * be a binary operation defined on \(Q ^{+}\)(the set of positive rational numbers) by the rule \(a * b=\frac{a b}{7}\) for all \(a, b \in Q ^{+}\). Then inverse of \(2 * 7\) is:

If \(\Delta_1=\left|\begin{array}{lll}x & b & b \\ a & x & b \\ a & a & x\end{array}\right|\) and \(\Delta_2=\left|\begin{array}{ll}x & b \\ a & x\end{array}\right|\) are determinants, then :

If \(A, B\) and \(C\) are all the corner points of feasible region (bounded) of an LPP with objective function \(Z\) that needs to be maximized such that \(Z_A>Z_B\) and \(Z_C < Z_A\) (Here \(Z_A\) denotes value of \(Z\) at \(A\) ), then which of the following statements is TRUE ?

If \(X+Y=\left(\begin{array}{ll}7 & 0 \\ 2 & 5\end{array}\right)\) and \(X-Y=\left(\begin{array}{ll}3 & 0 \\ 0 & 3\end{array}\right)\), then \(X\) and \(Y\) are:

Which of the following statements are correct?
(A) If elements of a row (or column) of a determinant are multiplied with cofactors of any other row (or column), then their sum is zero.
(B) The value of determinant is equal to the sum of the product of elements of any row (or column) with their corresponding cofactors.
(C) If any two rows (or columns) of a determinant are interchanged, their sign of determinant does not change.
(D) The value of determinant remains unchanged if its rows and columns are interchanged.
(E) A square matrix A has inverse if and only if A is singular.
Choose the correct answer from the options given below:

Two ropes are tied along the two diagonals of a cubic room. Ropes are inclined to each other at :

The function, \(f(x)=x+\frac{a^2}{x}, a>0, x \neq 0\) has a local maxima at

Read the following paragraph and answer the given questions.
Cancer detection is based on biopsy and histopathological studies. Techniques like radiography (use of X-rays), CT (computed tomography) and MRI (magnetic resonance imaging) are very useful to detect cancers of the internal organs. Do you know why mother's milk is considered very essential for the new-born infant? The yellowish fluid colostrum secreted by mother during the initial days of lactation has abundant antibodies (IgA) to protect the infant. Different species of Plasmodium (P. vivax, P. malaria and P. falciparum) are responsible for different types of malaria. Of these, malignant malaria caused by Plasmodium falciparum is the most serious one and can even be fatal. The exaggerated response of the immune system to certain antigens present in the environment is called allergy. The use of drugs like anti-histamine, adrenalin and steroids quickly reduce the symptoms of allergy. Drugs like barbiturates, amphetamines, benzodiazepines, and other similar drugs, that are normally used as medicines to help patients cope with mental illnesses like depression and insomnia, are often abused. Morphine is a very effective sedative and painkiller, and is very useful in patients who have undergone surgery.
A pregnant female is advised not to discard the yellowish fluid called colostrum, secreted during the initial days of lactation, because it is rich in:

Let \(e^{\alpha y}+e^{\beta y}+\gamma x^2+\delta \log |x|+C=0\), where \(C \in R ,\) be a particular solution of the differential equation
\(x\left(e^{2 y}-1\right) d y+\left(x^2-1\right) e^y d x=0\) and passes through the point \((1,1)\).
The value of \((\alpha+\beta+\gamma+\delta-C)\) is:

Select the correct statement/s from the following:
A. Spermatogonia always undergo meiotic cell division.
B. Primary spermatocytes divide by mitotic cell division.
C. Secondary spermatocytes have 23 chromosomes and undergo second meiotic division
D. Spermatozoa are transformed into spermatids.
Choose the correct answer from the options given below:

An electric dipole of length 4 cm, when placed with its axis making an angle of \( 60^{\circ} \) with a uniform electric field, experiences a torque of \( 4\sqrt{3} \text{ N m} \). If the dipole has charges of \( \pm 8 \text{ nC} \), then the potential energy of the dipole in the electric field is:

The wavelength of light from the spectral emission line of sodium is 662 nm.
The kinetic energy at which an electron would have the same de-Broglie wavelength would be:
(Given: \( h = 6.62 \times 10^{-34} \text{ J s}, m_e = 9 \times 10^{-31} \text{ kg} \))

Which of the following is known as Hinsberg's reagent?