Match List-I with List-II and select the correct option.
Step 1: Analyze the given complexes and their coordination geometry
\([NiCl_4]^{2-}\) is a tetrahedral complex, typically exhibiting \(sp^3\) hybridization.
It also has a magnetic moment of 3.87 BM.
\([Ni(CN_4)]^{2-}\) is a square planar complex, usually exhibiting \(dsp^2\) hybridization.
It shows no unpaired electrons, so the magnetic moment is 0 BM.
\([CoCl_4]^{2-}\) is an octahedral complex, usually with \(sp^3d^2\) hybridization.
Its magnetic moment is 2.82 BM.
\([Ni(H_2O)_6]^{2+}\) is a tetrahedral complex, exhibiting \(sp^3\) hybridization and a magnetic moment of 2.82 BM.
Step 2: Match the complexes with their characteristics
A.
\([NiCl_4]^{2-}\) matches with IV.
\(sp^3\), tetrahedral, 3.87 BM.
B.
\([Ni(CN_4)]^{2-}\) matches with II.
\(dsp^2\), square planar, 0 BM.
C.
\([CoCl_4]^{2-}\) matches with I.
\(sp^3d^2\), octahedral, 2.82 BM.
D.
\([Ni(H_2O)_6]^{2+}\) matches with III.
\(sp^3\), tetrahedral, 2.82 BM.
Two blocks of masses \( m \) and \( M \), \( (M > m) \), are placed on a frictionless table as shown in figure. A massless spring with spring constant \( k \) is attached with the lower block. If the system is slightly displaced and released then \( \mu = \) coefficient of friction between the two blocks.
(A) The time period of small oscillation of the two blocks is \( T = 2\pi \sqrt{\dfrac{(m + M)}{k}} \)
(B) The acceleration of the blocks is \( a = \dfrac{kx}{M + m} \)
(\( x = \) displacement of the blocks from the mean position)
(C) The magnitude of the frictional force on the upper block is \( \dfrac{m\mu |x|}{M + m} \)
(D) The maximum amplitude of the upper block, if it does not slip, is \( \dfrac{\mu (M + m) g}{k} \)
(E) Maximum frictional force can be \( \mu (M + m) g \)
Choose the correct answer from the options given below:
Let \( f : \mathbb{R} \to \mathbb{R} \) be a twice differentiable function such that \[ (\sin x \cos y)(f(2x + 2y) - f(2x - 2y)) = (\cos x \sin y)(f(2x + 2y) + f(2x - 2y)), \] for all \( x, y \in \mathbb{R}. \)
If \( f'(0) = \frac{1}{2} \), then the value of \( 24f''\left( \frac{5\pi}{3} \right) \) is: