List of practice Questions

Let \(\vec{p}=2\hat{i}+\hat{j}+3\hat{k}\) and \(\vec{q}=\hat{i}-\hat{j}+\hat{k}\). If for some real numbers α, β and γ we have
\(15\hat{i}+10\hat{j}+6\hat{k}=α(2\vec{p}+\vec{q})+β(\vec{p}-2\vec{q})+γ(\vec{p}\times\vec{q})\),
then the value of γ is ________.

Let the function \(f:\R→\R\) be defined by
\(f(x)=\frac{\sin x}{e^{\pi x}}\frac{(x)^{2023}+2024x+2025}{(x^2-x+3)}+\frac{2}{e^{\pi x}}\frac{(x^{2023}+2024x+2025)}{(x^2-x+3)}\)
Then the number of solutions of f (x) = 0 in \(\R\) is __.

A bag contains N balls out of which 3 balls are white, 6 balls are green, and the remaining balls are blue. Assume that the balls are identical otherwise. Three balls are drawn randomly one after the other without replacement. For i = 1, 2, 3, let W_i, G_i and B_i denote the events that the ball drawn in the i^th draw is a white ball, green ball, and blue ball, respectively. If the probability \(P(W_1∩G_2∩B_3)=\frac{2}{5N}\) and the conditional probability \(P(B_3|W_1∩G_2)=\frac{2}{9}\), then N equals ________.

Let \(f:\R→\R\) be a function such that f(x + y) = f(x) + f(y) for all x, y ∈ \(\R\), and \(g:\R→(0,\infin)\) be a function such that g(x + y) = g(x)g(y) for all x, y ∈ \(\R\). If \(f(\frac{-3}{5})=12\) and \(g(\frac{-1}{3})=2\), then the value of \((f(\frac{1}{4})+g(-2)-8)g(0)\) is ______.

Among the following complexes, the total number of diamagnetic species is ______.
[Mn(NH₃)₆]³⁺, [MnCl₆]³⁻, [FeF₆]³⁻, [CoF₆]³⁻, [Fe(NH₃)₆]³⁺, and [Co(en)₃]³⁺
[Given, atomic number: Mn = 25, Fe = 26, Co = 27;
en = H₂NCH₂CH₂NH₂]

In the following reaction sequence, the major product P is formed.

Glycerol reacts completely with excess P in the presence of an acid catalyst to form Q. Reaction of Q with excess NaOH followed by the treatment with CaCl₂ yields Ca-soap R, quantitatively. Starting with one mole of Q, the amount of R produced in gram is ______.
[Given, atomic weight: H = 1, C = 12, N = 14, O = 16, Na = 23, Cl = 35, Ca = 40]

Among V(CO)₆, Cr(CO)₅, Cu(CO)₃, Mn(CO)₅, Fe(CO)₅, [Co(CO)₃]^3-, [Cr(CO)₄]^4-, and Ir(CO)₃, the total number of species isoelectronic with Ni(CO)₄ is _________.
[Given, atomic number: V = 23, Cr=24, Mn 25, Fe 26, Co 27, Ni 28, Cu = 29, Ir = 77]

Complete reaction of acetaldehyde with excess formaldehyde, upon heating with conc. NaOH solution, gives P and Q. Compound P does not give Tollens' test, whereas Q on acidification gives positive Tollens' test. Treatment of P with excess cyclohexanone in the presence of catalytic amount of p-toluenesulfonic acid (PTSA) gives product R.
Sum of the number of methylene groups (-CH2-) and oxygen atoms in R is _______

Consider the following volume−temperature (V−T) diagram for the expansion of 5 moles of an ideal monoatomic gas.

Considering only P-V work is involved, the total change in enthalpy (in Joule) for the transformation of state in the sequence X→Y→Z is ______.
[Use the given data: Molar heat capacity of the gas for the given temperature range, C_V,_m = 12 J K⁻¹mol⁻¹ and gas constant, R=8.3 JK⁻¹ mol⁻¹]

Two large, identical water tanks, 1 and 2, kept on the top of a building of height H, are filled with water up to height ℎ in each tank. Both the tanks contain an identical hole of small radius on their sides, close to their bottom. A pipe of the same internal radius as that of the hole is connected to tank 2, and the pipe ends at the ground level. When the water flows from the tanks 1 and 2 through the holes, the times taken to empty the tanks are t1 and t2, respectively. If \(H = ( \frac{16}{ 9} ) ℎ\), then the ratio t1/t2 is ___

A source (S) of sound has frequency\( 240 Hz\). When the observer (O) and the source move towards each other at a speed \(v\) with respect to the ground (as shown in Case 1 in the figure), the observer measures the frequency of the sound to be \(288 Hz\). However, when the observer and the source move away from each other at the same speed v with respect to the ground (as shown in Case 2 in the figure), the observer measures the frequency of sound to be \(n\) Hz. The value of \(n\) is _____.

The specific heat capacity of a substance is temperature dependent and is given by the formula C = kT, where k is a constant of suitable dimensions in SI units, and T is the absolute temperature. If the heat required to raise the temperature of 1 kg of the substance from −73 ° C to 27 ° C is nk, the value of n is ___[Given: 0 K = −273 °C.]

Let X be a random variable, and let P(X = x) denote the probability that X takes the value x. Suppose that the points (x, P(X = x)), x = 0,1,2,3,4, lie on a fixed straight line in the xy -plane, and P(X = x) = 0 for all x ∈ R - {0,1,2,3,4}. If the mean of X is \(\frac{5}{2}\) , and the variance of X is α, then the value of 24α is ______.

Let \(S=\left\{\begin{pmatrix} 0 & 1 & c \\ 1 & a & d\\ 1 & b & e \end{pmatrix}:a,b,c,d,e\in\left\{0,1\right\}\ \text{and} |A|\in \left\{-1,1\right\}\right\}\), where |A| denotes the determinant of A. Then the number of elements in S is _______.

The compound(s) having peroxide linkage is(are)