In what ratio, water must be added to dilute honey costing ₹ 320 per liter so that the resulted syrup would be worth ₹ 280 per liter?

Let \(P=\left[a_{i j}\right]\) be a \(3 \times 3\) matrix and let \(Q=\left[b_{i j}\right]\) be a \(3 \times 3\) matrix,
where \(b_{i j}=2^{i+j} a_{i j}\) for \(1 \leq i, j \leq 3\).
If the determinant of \(P\) is 2 , then the determinant of the matrix \(Q\) is ___________ .

The function \(f(x)=\sin x(1+\cos x) ; x \in\left[0, \frac{3 \pi}{2}\right]\) will have:

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

\(\int \sin^3 x dx\) is equal to:

The kinetic energy of an electron in the ground state of a hydrogen atom is K . The values of its potential energy and total energy respectively are :

For an LPP: Maximize \(z=3 x+9 y; x \geq 0, y \geq 0\), the feasible region \(O A B\) is shown in the figure, then the other constraints are:

Read the passage carefully and answer the following questions
There are mainly four series of the transition metals, \(3 d\) series ( \(S c\) to Zn), \(4 d\) series (Y to Cd), \(5 d\) series ( La and Hf to Hg ) and 6d series which has \(A c\) and elements from \(R f\) to \(C n\). The two series of the inner transition metals; \(4 f\) (Ce to Lu) and \(5 f\) (Th to Lr) are known as lanthanoids and actinoids respectively. Their chemical properties are transitional between those of \(s\) and \(p\)-block elements. The transition metals (with the exception of Zn, Cd and Hg) are very hard and have low volatility. Their melting and boiling points are high. Transition metals have incomplete d subshell either in neutral atom or in their ions. With partly filled d orbitals these elements exhibit certain characteristic properties such as display of a variety of oxidation states, formation of coloured ions and entering into complex formation with a variety of ligands. The transition metals and their compounds are known for their catalytic activity.
The highest oxidation states shown by V , Fe and Ni respectively are __________ .

For an electric dipole in a non-uniform electric field with dipole moment parallel to the field, force F and torque \(T\) are:

Two parallel beams of monochromatic light pass through a long slit of width 0.2 mm one by one. The angular divergence in which most of the light is diffracted for both the beams are \( 2.25 \times 10^{-3} \text{ rad} \) and \( 4.50 \times 10^{-3} \text{ rad} \) respectively. If the wavelength of the first beam is \( \lambda_1 = 3000 \text{ Å} \), then the wavelength of the second beam is:

Which of the following bond in DNA strand is broken by the action of restriction enzyme?

The angle of minimum deviation if the refracting angle of a prism is \(2θ\), and the refractive index of the material of the prism is \(\cot θ\) is:

If \(X=11\) and \(Y=3\), then \(X \bmod Y=(X+a Y) \bmod Y\) holds: