If \(\theta\) is the angle between the vectors \(2\hat{i} - 2\hat{j} + 4\hat{k}\) and \(3\hat{i} + \hat{j} + 2\hat{k}\), then value of \(\sin \theta\) is:

If A and B are two events such that \(P(A) = 0.2, P(B) = 0.4\) and \(P(A \cup B) = 0.5\) then value of \(P(A/B)\) is:

The slope of the normal to the curve \(x=a \cos ^3 \theta, y=a \sin ^3 \theta\) at \(\theta=\frac{\pi}{4}\) is :

The value of \(\tan ^2\left(\sec ^{-1} 2\right)\) is:

The interval in which the function \(f(x)=10-6 x-2 x^2\) is decreasing is :

The probability that it will rain on any particular day is \(50 \%\). The probability that it rains only on the first 4 days of the week is :

Which of the following statements is true for \(f(x)=(x-1)^3(x-2)^2\)?
(A) f(x) decreases on \(\left(-\infty, \frac{8}{5}\right)\)
(B) f(x) increases on \(\left[\frac{8}{5}, 2\right]\)
(C) f(x) has point of inflection at x = 1
(D) f(x) decreases on \((2, \infty)\)
Choose the correct answer from the options given below:

Pick the correct structure for carbolic acid among the following compounds:

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, \(n s, n p\) or \(n s, n p, n d\) 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.
Select the incorrect statement about the Crystal field splitting in octahedral complexes.

The de Broglie wavelength associated with a proton moving with a speed of 1 m/s will be :
[Plank's Constant \( = 6.63 \times 10^{-34} \text{ J s}, M_p = 1.6 \times 10^{-27} \text{ kg} \)]

Match List - I with List - II.

List - I	List - II
(A) Pollination between two flowers on different plants	(I) Syngamy
(B) Fusion of male gamete with egg cell	(II) Cleistogamous
(C) Pollination between two flowers of the same plant	(III) Xenogamy
(D) Closed flowers	(IV) Geitonogamy

Choose the correct answer from the options given below:

Match List-I with List-II

List-I (Hormones)	List-II (Role/functions)
(A) LH	(I) Causes stronger uterine contractions
(B) FSH	(II) Essential for maintenance of the endometrium during menstrual cycle
(C) Oxytocin	(III) Acts on Leydig cells and stimulates synthesis and secretion of androgens
(D) Progester one	(IV) Acts on Sertoli cells and stimulates secretion of some factors which help in the process of spermiogenesis

Choose the correct answer from the options given below:

The formation of pomato is an example of: