List of practice Questions

Consider two GPs $2,2^2, 2^3, \ldots$ and $4,4^2$, $4^3, \ldots$ of 60 and $n$ terms respectively. If the geometric mean of all the $60+ n$ terms is $(2)^{\frac{225}{8}}$, then $\displaystyle\sum_{ k =1}^{ n } k ( n - k )$ is equal to :

Given below are two statements :
Statement I: Davisson-Germer experiment establishes the wave nature of electrons.
Statement II: If electrons have wave nature, they can interfere and show diffraction.
In the light of the above statements choose the correct answer from the option given below :

The half life of a radioactive substance is 5 years. After x years, a given sample of the radioactive substance gets reduced to 6.25% of its initial value. The value of x is ______.

If the sum of the first ten terms of the series
$\frac{1}{5} + \frac{2}{65} + \frac{3}{325} + \frac{4}{1025} + \frac{5}{2501}$+… is $\frac{m}{n}$,
where m and n are co-prime numbers, then m + n is equal to __________.

In the figure, a very large plane sheet of positive charge is shown. $P_1$ and $P_2$ are two points at distance $l $ and $2l$ from the charge distribution. If $σ $ is the surface charge density, then the magnitude of electric fields $E_1$ and $E_2$ and $P_1$ and $P_2$ respectively are

The value of $ tan^{-1}(\frac{cos\frac{15}{4}-1}{sin(\frac{π}{4})} )$ is equal to:

The value of the integral
$\frac{48}{\pi^4} \int_{0}^{\pi}(\frac{3\pi x^2}{2} - x^3) \frac{ \sin(x)}{1 + \cos^2x} \, dx$
is equal to ________

Let the lines$ \begin{array}{l} \frac{x-1}{\lambda}=\frac{y-2}{1}=\frac{z-3}{2}\ \text{and}\ \frac{x+26}{-2}=\frac{y+18}{3}=\frac{z+28}{\lambda}\end{array}$be coplanar and P be the plane containing these two lines. Then which of the following points does NOT lie on P?

If the length of the perpendicular drawn from the point P(a, 4, 2), a> 0 on the line
$\frac{x+1}{2} = \frac{y-3}{3} = \frac{z-1}{1}$ is $2\sqrt6$ units and $Q(α1, α2, α3)$
is the image of the point P in this line, then
$\alpha + \sum_{i=1}^{3} \alpha_i$
is equal to :

The difference in the reaction of phenol with bromine in chloroform and bromine in water medium is due to :

Let $f : R \rightarrow R$ be a continuous function such that $f (3x)- f(x)= x$. If $f(8)=7$, then $f (14)$ is equal to :

If $f(x)=\begin{cases} x+a, & x \leq 0 \\ |x-4|, & x\gt0\end{cases}$ and
$g(x)= \begin{cases}x+1 & , x\lt0 \\ (x-4)^2+b, & x \geq 0\end{cases}$
are continuous on $R$, then $(gof) (2)+(fog)(-2)$ is equal to:

If the minimum value of $f(x)=\frac{5 x^2}{2}+\frac{\alpha}{x^5}, x>0$, is 14 , then the value of $\alpha$ is equal to:

If force $\vec F=3\hat i+4\hat j−2\hat k$ acts on a particle having position vector $2\hat i+\hat j+2\hat k$ then, the torque about the origin will be

The pair, in which ions are isoelectronic with $Al^{3+}$ is:

Phenol on reaction with dilute nitric acid, gives two products. Which method will be most efficient for large scale separation?

Among the following species
N₂,N₂⁺,N₂⁻,N₂²⁻,O₂,O₂⁺,O₂⁻,O₂²⁻
the number of species showing diamagnetism is ________.

A tower PQ stands on a horizontal ground with base Q on the ground. The point R divides the tower in two parts such that QR = 15 m. If from a point A on the ground the angle of elevation of R is 60° and the part PR of the tower subtends an angle of 15° at A, then the height of the tower is :

The number of solutions of the equation sin x = cos² x in the interval (0, 10) is _____.

The value of
$\lim_{{n \to \infty}} 6\tan\left\{\sum_{{r=1}}^{n} \tan^{-1}\left(\frac{1}{{r^2+3r+3}}\right)\right\}$
is equal to :

The major product in the reaction is:

The major product in the reaction with product2-Methyl prop-1-ene

If α, β, γ, δ are the roots of the equation x⁴ + x³ + x² + x + 1 = 0, then α²⁰²¹ + β²⁰²¹ + γ²⁰²¹ + δ²⁰²¹ is equal to

In the following reaction:

The compound A and B respectively are:

If K₁ and K₂ are the thermal conductivities, L₁ and L₂ are the lengths and A₁ and A₂ are the cross sectional areas of steel and copper rods respectively such that $\frac{K_2}{K_1}=9,\frac{A_1}{A_2}=2,\frac{L_1}{L_2}=2.$

Then, for the arrangement as shown in the figure, the value of temperature T of the steel-copper junction in the steady state will be

lf the work function of a metal is 6.63 × 10^–19 J, the maximum wavelength of the photon required to remove a photoelectron from the metal is ______ nm. (Nearest integer)
[Given : h = 6.63 × 10^–34 J s, and c = 3 × 10⁸ m s^–1]