Two parallel plate capacitors of capacitances \(2 \mu F\) and \(3 \mu F\) are joined in series and connected to a battery of \(V\) volts. The values of potential across the two capacitors V1 and V2 and energy stored in them U1 and U2 are related as :

In a 10 km race, \(A, B\) and \(C\) each run at uniform speed and get the first, second and third positions respectively.
If \(A\) beats \(B\) by 1 km and \(B\) beats \(C\) by 1 km , then by how much distance does \(A\) beat \(C\) ?

If a matrix \(A=\left[\begin{array}{cc}5 & -8 \\ -3 & 5\end{array}\right]\) then which of the following is / are TRUE?
(A) |A| = 1
(B) A is a singular matrix.
(C) \(-2 A=\left[\begin{array}{cc}10 & -16 \\ -6 & 10\end{array}\right]\)
(D) \(A I=\left[\begin{array}{ll}5 & 0 \\ 0 & 5\end{array}\right] I\) is an identity matrix of order 2.
Choose the correct answer from the options given below:

The function \(f(x)=x^x\) has a critical point at :

If set A has 5 elements and set B has 7 elements then number of one-one and onto mapping from A to B is:

Match List I with List II

LIST I	LIST II
A. \(\int_{1}^{2} \frac{\sqrt{x}}{\sqrt{3-x} + \sqrt{x}} dx\)	I. -\(\frac{1}{2}\log 3\)
B. \(\int_{1}^{2} x^2 dx\)	II. \(\frac{7}{3}\)
C. \(\int_{-4}^{-1} \frac{1}{x} dx\)	III. \(\frac{1}{2}\)
D. \(\int_{0}^{1} \frac{1}{2x-3} dx\)	IV. -\(\log 4\)

Choose the correct answer from the options given below:

A charity organization has a fund of Rs. 2, 00, 000 to provide annual grants to students. The grant amount each year is Rs. 15, 000. The fund earns an interest rate of r% per annum. If the interest earned is used entirely to provide the grants, then the annual interest rate r is:

If a young man drives his vehicle \(25 km / hr\), he has to spend Rs. 2 per km on petrol.
If he drives it at faster speed of \(40 km / hr\), the petrol cost increases to Rs. 5 per km.
He has Rs. 100 to spend on petrol and travel within one hour. The maximum distance he travels is :

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?

Arrange the following in increasing order of basic strength in aqueous solution :
(A) \(C _6 H _6 NH _2\)
(B) \(C _2 H _5 NH _2\)
(C) \(\left( C _2 H _6\right)_2 NH\)
(D) \(NH _3\)
Choose the correct answer from the options given below:

Match List - I with List - II

LIST-I (Differential Equations)	LIST-II ( Degree)
(A) \(\left(\frac{d y}{d x}\right)^3+y x=0\)	(I) 2
(B) \(e^{\frac{d y}{d x}}+y^2+y^{\prime \prime}=0\)	(II) 1
(C) \(x y y^{\prime \prime}+x\left(y^{\prime}\right)^2-y y^{\prime}=0\)	(III) Not defined
(D) \(\left(y^{\prime \prime}\right)^2+y=0\)	(IV) 3

Choose the correct answer from the options given below :

The pair(s) of co-ordination complexes/ions exhibiting the same kind of isomerism is/are :
(A) \(\left[ Cr \left( NH _3\right)_5 Cl \right] Cl _2\) and \(\left[ Cr \left( NH _3\right)_4 Cl _2\right] Cl\)
(B) \(\left[ Co \left( NH _3\right)_4 Cl _2\right]^{+}\)and \(\left[ Pt \left( NH _3\right)_2\left( H _2 O \right) Cl \right]^{+}\)
(C) \(\left[ CoBr _2 Cl _2\right]^{2-}\) and \(\left[ PtBr _2 Cl _2\right]^{2-}\)
(D) \(\left[ Pt \left( NH _3\right)_3\left( NO _3\right)\right] Cl\) and \(\left[ Pt \left( NH _3\right)_3 Cl \right] Br\)
Choose the correct answer from the options given below :

The linear magnifications of the object in the conjugate positions of a convex lens are 4 and 2, respectively. The distance between the conjugate positions is 40 cm . This convex lens of glass \((u=1.5)\) is immersed in water \((u=1.33)\). The new focal length of convex lens in water is :