List of practice Questions

If $2(\vec a \times \vec c)+3(\vec b \times \vec c)=0$, where $\vec a=2\hat i-5\hat j+5\hat k$, $\vec b=\hat i-\hat j+3\hat k$ and $(\vec a-\vec b)\cdot\vec c=-97$, find $|\vec c \times \vec k|^2$.

\[ \left(\frac{1}{^{15}C_0}+\frac{1}{^{15}C_1}\right) \left(\frac{1}{^{15}C_1}+\frac{1}{^{15}C_2}\right) \cdots \left(\frac{1}{^{15}C_{12}}+\frac{1}{^{15}C_{13}}\right) = \frac{\alpha^{13}}{^{14}C_0\cdot {}^{14}C_1\cdot {}^{14}C_2\cdots {}^{14}C_{12}} \] If so, then find the value of \(30\alpha\).

If the product \[ \left( \frac{1}{\binom{15}{0}} + \frac{1}{\binom{15}{1}} \right) \left( \frac{1}{\binom{15}{1}} + \frac{1}{\binom{15}{2}} \right) \cdots \left( \frac{1}{\binom{15}{12}} + \frac{1}{\binom{15}{13}} \right) \] is equal to \[ \frac{\alpha^{13}}{\binom{14}{0} \binom{14}{1} \binom{14}{2} \cdots \binom{14}{12}}, \] then \( 30\alpha \) is equal to:

If \( A = \{ 1, 2, 3, 4, 5, 6 \}, B = \{ 1, 2, 3, 4, 5, 6, 7, 8, 9 \} \), then the number of strictly increasing functions from \( A \to B \) such that \( f(i) \neq i \) for \( i = 1, 2, 3, 4, 5, 6 \) is

Number of 4 letter words with or without meaning formed from the letters of the word PQRSTTUVV is:

Number of ways of distributing 16 identical oranges among 4 persons such that each one gets at least one orange is:

Let \( S \) be the number of 4-digit numbers \( abcd \), where \[ a>b>c>d \] and let \( P \) be the number of 5-digit numbers \( abcde \), where the product of digits is 20. Find \( S + P \):

If all the letters of the word 'UDAYPUR' are arranged in all possible permutations and these permutations are listed in dictionary order, then the rank of the word 'UDAYPUR' is

If all the letters of the word 'UDAYPUR' are arranged in all possible permutations and these permutations are listed in dictionary order, then the rank of the word 'UDAYPUR' is

A lift is going up to the \(10^{\text{th}}\) floor. Number of ways in which \(3\) people can exit the lift at {three different floors, if the lift will not stop at I$^{\text{st}}$, II$^{\text{nd}}$ and III$^{\text{rd}}$ floors, is: }

Grignard reagent RMgBr (P) reacts with water and forms a gas (Q). One gram of Q occupies 1.4 dm$^3$ at STP. (P) on reaction with dry ice in dry ether followed by H$_3$O$^+$ forms compound (Z). 0.1 mole of (Z) will weigh _________ g (Nearest integer).

0.25 g of an organic compound “A” containing carbon, hydrogen and oxygen was analysed using combustion method. The increase in mass of CaCl$_2$ tube and potash tube at the end of the experiment was found to be 0.15 g and 0.1837 g respectively. The percentage of oxygen in compound A is ________ % (Nearest integer).

A chromium complex with formula CrCl$_3\cdot$6H$_2$O has a spin only magnetic moment value of 3.87 BM and its solution conductivity corresponds to 1:2 electrolyte. 2.75 g of the complex solution was initially passed through a cation exchanger. The solution obtained after the process was reacted with excess of AgNO$_3$. The amount of AgCl formed in the above process is _________ g (Nearest integer).
(Given: Molar mass in g mol$^{-1}$ Cr: 52; Cl: 35.5; Ag:108; O:16; H:1)

Molar conductivity of a weak acid HQ of concentration 0.18 M was found to be \( \dfrac{1}{30} \) of the molar conductivity of another weak acid HZ with concentration 0.02 M. If \( \alpha_Q \) happened to be equal to \( \alpha_Z \), then the difference of the pK_a values of the two weak acids (pK_a(HQ) − pK_a(HZ)) is ________ (Nearest integer).
(Given: degree of dissociation \( \alpha \ll 1 \) for both weak acids, \( \lambda^\circ \): limiting molar conductivity of ions)

The half-life of $^{65}$Zn is 245 days. After $x$ days, 75% of the original activity remained. The value of $x$ in days is __________ (Nearest integer).}
(Given: $\log 3 = 0.4771$ and $\log 2 = 0.3010$)

A student has planned to prepare acetanilide from aniline using acetic anhydride. The student has started from 9.3 g of aniline. However, the student has managed to obtain 11 g of dry acetanilide. The % yield of this reaction is

The correct order of C, N, O and F in terms of second ionisation potential is

Find out the statements which are not true.
A. Resonating structures with more covalent bonds and less charge separation are more stable.
B. In electromeric effect, an unsaturated system shows +E effect with nucleophile and –E effect with electrophile.
C. Inductive effect is responsible for high melting point, boiling point and dipole moment of polar compounds.
D. The greater the number of alkyl groups attached to the doubly bonded carbon atoms, higher is the heat of hydrogenation.
E. Stability of carbanion increases with increase in s-character of the carbon carrying negative charge.

For some \(\alpha, \beta \in R\), let \(A = \begin{pmatrix} \alpha & 2 \\1 & 2 \end{pmatrix}\) and \(B = \begin{pmatrix} 1 & 1 \\ \beta & 1 \end{pmatrix}\) be such that \(A^2-4A+2I-B^2-3B+I=O\). Then \((\det(\text{adj}(A^3-B^3)))^2\) is equal to:

The unsaturated ether on acidic hydrolysis produces carbonyl compounds as shown below. Based on this, predict the solution/reagent that will help to distinguish "P" and "Q" obtained in the reaction.

Pair of species among the following having same bond order as well as paramagnetic character will be:

\(6 \int_{0}^{\pi} (\sin 3x + \sin 2x + \sin x)dx\) is equal to: