If \(\vec{p}\) and \(\vec{q}\) are two unit vectors such that \(|\vec{p}+\vec{q}|=\sqrt{2}\), then which of the following are correct?
(A) \(|\vec{p}|=|\vec{q}|=1\)
(B) \(\vec{p}\) and \(\vec{q}\) are orthogonal vectors
(C) \(\vec{p}\) and \(\vec{q}\) are collinear vectors
(D) \((4 \vec{p}-\vec{q}) \cdot(2 \vec{p}+\vec{q})=7\)
Choose the correct answer from the options given below :

Let \(A\) be a square matrix of order 2 such that \(\left[\begin{array}{ll}2 & 1 \\ 3 & 2\end{array}\right] A\left[\begin{array}{cc}-3 & 2 \\ 5 & -3\end{array}\right]=\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]\) Then \(A\) is :

The largest interval, in which the function \(f(x)=x^3+2 x^2-1\) is increasing, is:

The values of \(\lambda\) for which the system of equation \(x+2 y+z=14,-x+y+z=10, x+\lambda y+z=2\) has unique solution is :

If \(f(x)=\left\{\begin{array}{ll}\frac{\tan \left(\frac{\pi}{4}-x\right)}{\cot 2 x}, & x \neq \frac{\pi}{4} \\ K, & x=\frac{\pi}{4}\end{array}\right.\) is continuous at \(x=\frac{\pi}{4}\), then the value of \(K\) is :

The values of ' \(a\) ' and ' \(b\) ' if the equation of straight line trend by least square method is given by \(y=a+b x\) such that \(\sum x=0, \sum y=84, \sum x y=108, \sum x^2=70\) for 6 observation as are :

The general solution of the differential equation \(x\left(\frac{d y}{d x}\right)=y+x \tan \left(\frac{y}{x}\right)\) is

The process of evolution of different species in a given geographical area starting from a point and dispersing to other areas is called as:

If three points \(A \left(a_1, b_1\right), B \left(a_2, b_2\right)\) and \(C \left(a_3, b_3\right)\) are collinear and D is the determinant of their coordinates with a column of 1s, then:

Read the passage carefully and answer the Questions:
According to Valence Bond theory, the metal atom or ion under the influence of ligands can use its (n - 1)d, ns, np or ns, np, nd orbitals for hybridisation to yield a set of equivalent orbitals of definite geometry such as octahedral, tetrahedral, square planar and so on. These hybridised orbitals are allowed to overlap with ligand orbitals that can donate electron pairs for bonding. While the VB theory, to a larger extent, explains the formation, structures and magnetic behaviour of coordination compounds, it suffers from the many shortcomings. The Crystal field theory (CFT) is an electrostatic model which considers the metal-ligand bond to be ionic arising purely from electrostatic interactions between the metal ion and the ligand. The five d orbitals in an isolated gaseous metal atom/ion have same energy, i.e., they are degenerate. This degeneracy is maintained if a spherically symmetrical field of negative charges surrounds the metal atom/ion. However, when this negative field is due to ligands in a complex, it becomes asymmetrical and the degeneracy of the d orbitals is lifted. It results in splitting of the d orbitals. The pattern of splitting depends upon the nature of the crystal field.
Which of the following represents the correct pair?

A network of four \( 9 \mu F \) capacitors is connected to a 500 V supply, as shown. In the context of this, identify the correct statements.

(A) \( C_4 \) is connected in parallel to the combination of \( C_1, C_2 \) and \( C_3 \).
(B) The equivalent capacitance of the combination \( C_1, C_2, C_3 \) and \( C_4 \) is \( 12 \mu F \).
(C) The charge on \( C_4 \) is \( 4.5 \times 10^{-3} \text{ C} \).
(D) The charge stored on \( C_1 \) and \( C_4 \) is the same.
Choose the correct answer from the options given below:

Industrial melanism in peppered moths is an example of:

The correct sequence of attachment of sugar, base and phosphoric acid in a nucleic acid is
(A) Formation of phosphodiester linkage
(B) Attachment of base to 1' position of sugar
(C) Selection of appropriate base and sugar
(D) Attachment of phosphoric acid
Choose the correct answer from the options given below: