Let $f:(0,1) \rightarrow R$ be a function defined by
$f(x)=\frac{1}{1-e^{-x}}$, and $g(x)=(f(-x)-f(x))$ Consider two statements
(I) $g$ is an increasing function in $(0,1)$
(II) $g$ is one-one in $(0,1)$Then,
Let $z_1=2+3 i$ and $z_2=3+4 i$. The set $S=\left\{z \in C:\left|z-z_1\right|^2-\left|z-z_2\right|^2=\left|z_1-z_2\right|^2\right\}$ represents a
Let $y (x)=(1+x)\left(1+x^2\right)\left(1+x^4\right)\left(1+x^8\right)\left(1+x^{16}\right)$. Then $y^{\prime}-y^{\prime \prime}$ at $x=-1$ is equal to :
Consider the lines $L_1$ and $L_2$ given by
$L_1: \frac{x-1}{2}=\frac{y-3}{1}=\frac{z-2}{2} $
$ L_2: \frac{x-2}{1}=\frac{y-2}{2}=\frac{z-3}{3}$
A line $L_3$ having direction ratios $1,-1,-2$, intersects $L_1$ and $L_2$ at the points $P$ and $Q$ respectively Then the length of line segment $P Q$ is
The vector $\vec{a}=-\hat{i}+2 \hat{j}+\hat{k}$ is rotated through a right angle, passing through the y-axis in its way and the resulting vector is $\vec{b}$. Then the projection of $3 \vec{a}+\sqrt{2} \vec{b}$ on $\vec{c}=5 \hat{i}+4 \hat{j}+3 \hat{k}$ is :
A sample of gas at temperature $T$ is adiabatically expanded to double its volume The work done by the gas in the process is (given, \(\gamma=\frac{3}{2}\))
Choose the correct answer from the options given below:
But-2-yne is reacted separately with one mole of Hydrogen as shown below:
A. A is more soluble than BB. B The boiling point & melting point of $A$ are higher and lower than $B$ respectively C. A is more polar than $B$ because dipole moment of $A$ is zero D. $Br _2$ adds easily to $B$ than $A$ Identify the incorrect statements from the options given below:
The area enclosed by the closed curve $C$ given by the differential equation $\frac{d y}{d x}+\frac{x+a}{y-2}=0, y(1)=0$ is $4 \pi$.
Let $P$ and $Q$ be the points of intersection of the curve $C$ and the $y$-axis If normals at $P$ and $Q$ on the curve $C$ intersect $x$-axis at points $R$ and $S$ respectively, then the length of the line segment $R S$ is
Equivalent resistance between the adjacent corners of a regular \(n\)-sided polygon of uniform wire of resistance \(R\) would be :
The threshold frequency of a metal is \(f _0\) When the light of frequency \(2 f _0\) is incident on the metal plate, the maximum velocity of photoelectrons is \(v_1\) When the frequency of incident radiation is increased to \(5 f _0\), the maximum velocity of photoelectrons emitted is \(v_2\) The ratio of \(v_1\) to \(v_2\) is :
A light of energy 12.75 eV is incident on a hydrogen atom in its ground state. The atom absorbs the radiation and reaches to one of its excited states. The angular momentum of the atom in the excited state is \(\frac{x}{π} \times 10^{-17} eVs\). The value of x __________ is (use \(h=4.14 \times 10^{-15} eVs\), \(c=3 \times 108 ms^{-1}\)).
A certain pressure ' $P$ ' is applied to $1$ litre of water and $2$ litre of a liquid separately Water gets compressed to $0.01 \%$ whereas the liquid gets compressed to $0.03 \%$ The ratio of Bulk modulus of water to that of the liquid is $\frac{3}{x}$ The value of $x$ is ________
