Let the straight line \(x=b\) divide the area enclosed by \(y=(1-x)^2, y=0\) and \(x=0\) into two parts \(R_1(0 \leq x \leq b)\) and \(R_2(b \leq x \leq 1)\) such that \(R_1-R_2=\frac{1}{4}\). Then, \(b\) equals to

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Let \(S=S_{1} \cap S_{2} \cap S_{3}\), where

\(S_{1}=\{z \in \mathbb{C}:|z|<4\}, \quad S_{2}=\left\{z \in \mathbb{C}: \operatorname{Im}\left[\frac{z-1+\sqrt{3} i}{1-\sqrt{3} i}\right]>0\right\}\) and \(S_{3}=\{z \in \mathbb{C}: \operatorname{Re} z>0\}\).


Question:

\(\min _{z \in S}|1-3 i-\mathrm{z}|=\)

Consider a triangle \(A B C\) and let \(a, b\) and \(c\) denote the lengths of the sides opposite to vertices \(A, B\) and \(C\) respectively. Suppose \(a=6, b=10\) and the area of the triangle is \(15 \sqrt{3}\). If \(\angle A C B\) is obtuse and if \(r\) denotes the radius of the incircle of the triangle, then \(r^2\) is equal to

Let $\alpha$ and $\beta$ be real numbers such that $-\frac{\pi }{4}<\beta <0<\alpha <\frac{\pi }{4}$. If $\sin \left(\alpha +\beta \right)=\frac{1}{3}$ and $\cos \left(\alpha -\beta \right)=\frac{2}{3}$, then the greatest integer less than or equal to ${\left(\frac{\sin \alpha }{\cos \beta }+\frac{\cos \beta }{\sin \alpha }+\frac{\cos \alpha }{\sin \beta }+\frac{\sin \beta }{\cos \alpha }\right)}^2$ is

Let \(\omega=e^{i \pi / 3}\) and \(a, b, c, x, y, z\) be non-zero complex numbers such that \(a+b+c=x, a+b \omega+c \omega^2=y, a+b \omega^2+c \omega=z\).
Then, the value of \(\frac{|x|^2+|y|^2+|z|^2}{|a|^2+|b|^2+|c|^2}\) is

Let \(\omega \neq 1\) be a cube root of unity and \(S\) be the set of all non-singular matrices of the form \(\left[\begin{array}{ccc}1 & a & b \\ \omega & 1 & c \\ \omega^2 & \omega & 1\end{array}\right]\), where each of \(a, b\) and \(c\) is either \(\omega\) or \(\omega^2\). Then, the number of distinct matrices in the set \(S\) is

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Football teams \(T_{1}\) and \(T_{2}\) have to play two games against each other. It is assumed that the outcomes of the two games are independent. The probabilities of \(T_{1}\) winning, drawing and losing a game against \(T_{2}\) are \(\frac{1}{2}, \frac{1}{6}\) and \(\frac{1}{3}\), respectively. Each team gets \(3\) points for a win, \(1\) point for a draw and \(0\) point for a loss in a game. Let \(X\) and \(Y\) denote the total points scored by teams \(T_{1}\) and \(T_{2}\), respectively, after two games.


Question:

\(P(X>Y)\) is

A student is performing an experiment using a resonance column and a tuning fork of frequency \(244 s^{-1}\). He is told that the air in the tube has been replaced by another gas (assume that the column remains filled with the gas). If the minimum height at which resonance occurs is \((0.350 \pm 0.005) m\), the gas in the tube is (Useful information: \(\sqrt{167\text{ RT}}=640\text{ J}^{\frac{1}{2}}\text{ mol}^{-\frac{1}{2}} ;\) \(\sqrt{140\text{ RT}}=590\text{ J}^{\frac{1}{2}}\text{ mole}^{-\frac{1}{2}}\) . The molar masses M in grams are given in the options. Take the values of \(\sqrt{\frac{10}{\text M }}\) for each gas as given there.)

For a first order reaction, \(A \rightarrow P\), the temperature \((T)\) dependent rate constant \((k)\) was found to follow the equation, \(\log k=-(2000) / T+6.0\)
The pre-exponential factor \(A\) and the activation energy \(\left(E_a\right)\), respectively, are

A cube of unit volume contains \(35 \times 10^7\) photons of frequency \(10^{15} \mathrm{~Hz}\). If the energy of all the photons is viewed as the average energy being contained in the electromagnetic waves within the same volume, then the amplitude of the magnetic field is \(\alpha \times 10^{-9} \mathrm{~T}\). Taking permeability of free space \(\mu_0=4 \pi \times 10^{-7} \mathrm{Tm} / \mathrm{A}\), Planck's constant \(h=6 \times 10^{-34} \mathrm{Js}\) and \(\pi=\frac{22}{7}\), the value of \(\alpha\) is_______

The standard enthalpies of formation of $C{O}_2\left(g\right)$ ${\text{, H}}_2\text{O}\left(\mathit{\text{l}}\right)$ and glucose (s) at $25^{\text{o}}C$ are $-\text{400 kJ}/\text{mol}$, $-\text{300 kJ}/\text{mol}$ and $-\text{1300 kJ}/\text{mol}$, respectively. The standard enthalpy combustion per gram of glucose at $25^{\text{o}}C$ is $\left(\Delta H_f^0=-1300kJmo{l}^{-}for\text{glucose}\right)$

A particle of mass $m$ is moving in the $xy$-plane such that its velocity at a point $(x, y)$ is given as $\overrightarrow{v}=\alpha \left(y\stackrel{\wedge }{x}+2x\stackrel{\wedge }{y}\right)$ where $\alpha$ is a non-zero constant. What is the force $\overrightarrow{F}$ acting on the particle?

A line L : $y=\mathit{\text{m}}x+3$ meets $y-\mathrm{a}\mathrm{x}\mathrm{\text{is}}$ at E(0,3) and the arc of the parabola $y^2=16x\text{,}0\le y\le 6$ at the point $F\left(x_0,y_0\right)$.The tangent to the parabola at $F\left(x_0,y_0\right)$ intersects the y-axis at $G\left(0,y_1\right)$.The slope m of the line L is chosen such that the area of the triangle EFG has a local maximum.
Match List I with List II and select the correct answer using the code given below the lists :
 

	List I		List II
A.	m =	P.	$\frac{1}{2}$
B.	Maximum area of ΔEFG is	Q.	4
C.	y0​ =	R.	2
D.	y1 =	S.	1