If \(A=\left[\begin{array}{ccc}2 & 3 & 1 \\ 2 & -1 & 0\end{array}\right]\) and \(B^T=\left[\begin{array}{cc}4 & 4 \\ 6 & -2 \\ 2 & 0\end{array}\right]\) then \(4 A+B\) is

Consider a random sample of 10 students having 116 cm as mean height and standard deviation as 9.798 cm .
If the suggested mean height of the students population is 110 cm , then the \(t\)-test statistic of the sample is :
Given: \(\frac{\sqrt{10}}{9.798}=0.3227\)

If \(\frac{1}{|x|-3} \leq \frac{1}{2}\), then the value of \(x\) is:

The distance of the plane \(\vec{r} \cdot(2 \hat{i}-\hat{j}+\hat{k})=3\) from the origin is :

If \(x=a \sin ^3 t\) and \(y=a \cos ^3 t\), then \(\frac{d y}{d x}\) is equal to :

If A is a skew-symmetric matrix, then which of the following statements is NOT true?
(A) A is singular if order of A is odd
(B) A is non-singular
(C) \(A^{2025}\) is a skew-symmetric matrix
(D) \(A^{2025}\) is a symmetric matrix
(E) all diagonal elements of A are zeros
Choose the correct answer from the option given below :

Two infinitely long parallel conductors carrying currents I 1 and I 2 are at distance d apart. The force \(F\) on length \(L\) of one conductor due to the other is :

The feasible region represented by the constraints of the linear programming problem :
\(x+y \leq 6, y \leq 7, x \geq 4, x, y \geq 0\) is :

If \(a, b, c\) are positive real numbers, then the least value of \((a+b+c)(a b+b c+c a)\) is :

A swimmer whose speed in swimming pool is 5 km/h, swims between two points in a river and returns back to starting point. He took 20 minutes more to cover the distance upstream than to cover downstream. If the speed of stream is 2 km/h, then the distance between two points is

Read the passage given below and answer the question.
The transition elements which give the greatest number of oxidation states occur in or near the middle of the series. Manganese, for example, exhibits all the oxidation states from +2 to +7 . The lesser number of oxidation states at extreme ends stems from either too few electrons to lose or share (Sc, Ti) on too many d-electrons (hence fewer orbitals available in which to share electrons with others) for higher valence \(( Cu , Zn )\). Thus early in the series scandium (II) is virtually unknown and titanium (IV) is more stable than Ti (III) or Ti (II). At the other end, the only oxidation state of zinc is \(+ 2\) (no d electrons are involved). The maximum oxidation states of reasonable stability correspond in value to the sum of the s - and d-electrons upto manganese ( \(Ti ^{I V} O _2, V^V O _2^{+}, Cr ^{V I} O _4^{2-}\), \(Mn ^{\text {VII }} O _4\) ) followed by a rather abrupt decrease in stability of higher oxidation states, so that the typical species to follow are \(Fe ^{I I, I I I}, Co ^{I I, I I I}, Ni ^{I I}, Cu ^{I, I I}, Zn ^{I I}\).
The variability of oxidation states, a characteristic of transition elements, arises out of incomplete filling of d-orbitals in such a way that their oxidation states differ from each other by unity, e.g. \(V ^{I I}, V^{I I I}, V^{I V}, V^V\). This is in contrast with the variability of oxidation states of non transition elements where oxidation states normally differ by a unit of two.
An interesting feature in the variability of oxidation states of the d-block elements is noticed among the groups (groups 4 through 10). Although in the p-block, the lower oxidation states are favoured by the heavier members (due to inert pair effect), the opposite is true in the groups of dblock. For example, in group 6, \(Mo ( VI )\) and \(W ( VI )\) are found to be more stable than \(Cr ( VI )\). The \(Cr ( VI )\) in the form of dichromate in acidic medium is a strong oxidising agent, whereas \(MoO _3\) and \(WO _3\) are not.
Low oxidation states are found when a complex compound has ligands capable of \(\pi\)-acceptor character in addition to the \(\sigma\)-bonding. For example, in \(Ni ( CO )_4\) and \(Fe ( CO )_5\), the oxidation state of nickel and iron is zero.
Most acidic species among the following is :

If A and B are square matrices of order n × n, then \((A-B)^2\) is equal to:
(A) \(A^2-A B-B A+B^2\)
(B) \(A^2-2 A B+B^2\)
(C) \(A^2-2 B A+B A\)
(D) \(A^2-B^2\)
(E) \(A^2+2 A B+B^2\)

Which traditional drink is made by fermenting palm sap?