A particle moves along the curve \(6 y=x^3+2\). The point(s) on the curve at which the \(y\)-coordinate is changing 8 times as fast as the \(x\)-coordinate are:

In a Linear Programming problem, the objective function is always :

If \(A\) and \(B\) are invertible matrices of same order, then which one of the following is NOT true?

The marginal cost (MC) and marginal revenue (MR) functions of a product are \(M C=20+\frac{x}{20}\) and \(M R=30\) respectively. If the fixed cost is 200 , then the maximum value of the profit is :

If matrix\(A_p=\left[\begin{array}{ll}p & (p+1) \\ p & (p-1)\end{array}\right], p \in \mathbb{N}\) (where \(\mathbb{N}\) is the set of natural numbers), then the value of \(\left|A_1\right|+\left|A_2\right|+\left|A_3\right|+\cdots+\left|A_{2025}\right|\) is:

The random variable X has a probability distribution \(P(X=r)=\left\{\begin{array}{ll}r k, & \text { if } r \leq 2, \\ (r-1) k, & \text { if } 2 < r \leq 4, \text { where } r \in N \cup\{0\} \text { and } k \in R . \\ 0, & \text { otherwise },\end{array}\right.\), where \(n\) is the set of natural numbers
(A) \(k=\frac{1}{9}\)
(B) \(P(2 \leq X \leq 3)=\frac{1}{2}\)
(C) \(P(X=4)=\frac{1}{3}\)
(D) \(P(X>1)=\frac{7}{8}\)
Choose the correct answer from the options given below:

A boat can row at the speed of \(16 km / hr\) in still water. If the river is flowing at \(8 km / hr\), and it takes 8 hours for a round trip, then the distance between the two places is :

A bar magnet with pole strength m and magnetic dipole moment p is cut into two identical pieces along its length.
Then for each of the two pieces

Match List-I with List-II
Let A and B be two events such that \(P(A)=0.2, P(B)=0.4, P(B \mid A)=0.5\)

List-I	List-II
(A) \(P(A \cap B)\)	(I) 0.5
(B) \(P(A \mid B)\)	(II) 0.8
(C) \(P(A \cup B)\)	(III) 0.25
(D) \(P\left(A^{\prime}\right)\)	(IV) 0.1

Choose the correct answer from the options given below:

\(\int_2^5|x-3| d x\) equals

Read the passage carefully and answer the questions:
The degeneracy of the \(d\) orbitals has been removed due to ligand electron-metal electron repulsions in the octahedral complex to yield three orbitals of lower energy, \(t_{2 g}\) set and two orbitals of higher energy, \(e_g\) set. This splitting of the degenerate levels due to the presence of ligands in a definite geometry is termed as crystal field splitting and the energy separation is denoted by \(\Delta_0\). Thus energy of the two \(e_g\) orbitals will increase by \((3 / 5) \Delta_0\) and that of three \(t_{2 g}\) will decrease by \((2 / 5) \Delta_0\). The crystal field splitting \(\Delta_{\circ}\) depends upon the field produced by the ligand and charge on the metal ion. Some ligands are able to produce strong fields, in which case the splitting will be large, whereas others produce weak fields and consequently result in small splitting of \(d\) orbitals. Relative magnitude of crystal field splitting energy \(\Delta_{\circ}\) and pairing energy, \(P\) (energy required for electron pairing in a single orbital) determine the formation of low spin ( \(\Delta_{\circ}>P\) ) or high spin ( \(\Delta_{\circ}<P\) ) complex. In tetrahedral coordination entity formation, the \(d\)-orbital splitting is inverted and is smaller as compared to the octahedral field splitting. The crystal field theory attributes the colour of the complex to \(d-d\) transition of the electron. The colour of the coordination compounds depends on the crystal field splitting. In the absence of ligand, crystal field splitting does not occur and hence the substance is colourless.
Which of the following complex is not expected to be coloured?

The number of all possible matrices of order \(3 \times 3\) with each entry 2 or 3 is:

Two identical short magnetic dipoles of magnetic moment \( 2 \text{ Am}^2 \) are placed with their axis mutually perpendicular and centers 2 m apart. The magnitude of the resultant magnetic field at the midpoint (O) between the dipoles is :