For a point P in the plane, let $d_1\left(P\right)$ and $d_2\left(P\right)$ be the distances of the point P from the lines $x-y=0$ and $x+y=0$ respectively. The area of the region R consisting of all points P lying in the first quadrant of the plane and satisfying $2 \le d_1\left(P\right)+d_2\left(P\right) \le 4,$ is ________

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Read the following passage and answer the questions.
For every function \(f(x)\) which is twice differentiable, these will be good approximation of \(\int_a^b f(x) d x \cong\left(\frac{b-a}{2}\right)\{f(a)+f(b)\}\). Now, if we take \(c=\frac{a+b}{2}\), then using above again, we get \(\int_a^b f(x) d x=\int_a^c f(x) d x+\int_c^b f(x) d x \cong \frac{b-a}{4}\{f(a)+f(b)+2 f(c)\}\) and so on.
We get approximation for value of \(\int_a^b f(x) d x\).Question:
If \(f^{\prime \prime}(x) < 0, \forall x \in(a, b), c(c, f(c))\) is point of maxima where \(c \in(a, b)\), then \(f^{\prime}(c)\) is

An ellipse intersects the hyperbola \(2 x^2-2 y^2=1\) orthogonally. The eccentricity of the ellipse is reciprocal to that of the hyperbola. If the axes of the ellipse are along the coordinate axes, then

Let \((x, y)\) be such that \(\sin ^{-1}(a x)+\cos ^{-1}(y)+\cos ^{-1}(b x y)=\frac{\pi}{2}\). Match the statements in Column I with the values in Column II.

Let $\text{P}=\left[\begin{array}{ccc}3& -1& -2\\ 2& 0& \text{α}\\ 3& -5& 0\end{array}\right]$ , where $\text{α}\in \text{R},$ Suppose $\text{Q}=\left[{\text{q}}_{\text{i}\text{j}}\right]$ is a matrix such that$PQ = kI$, where $\text{k}\in \text{R},\text{k}\ne 0$ and$I$ is the identity matrix of order 3. If ${\text{q}}_{23}=-\frac{\text{k}}{8}$ and $\text{det}\left(\text{Q}\right)=\frac{{\text{k}}^2}{2}$ then 

Let $f:[1.\infty )\to \mathrm{ℝ}$ be a differentiable function such that $f\left(1\right)=\frac{1}{3}$ and $3{\int }_1^{x}f\left(t\right)dt=xf\left(x\right)-\frac{x^3}{3},x\in [1,\infty )$. Let $e$ denote the base of the natural logarithm. Then the value of $f\left(e\right)$ is

Let $\overline{z}$ denote the complex conjugate of a complex number $z$ and let $i=\sqrt{-1}$. In the set of complex numbers, the number of distinct roots of the equation $\overline{z}-z^2=i\left(\overline{z}+z^2\right)$ is _______.

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p-amino-N, N-dimethylaniline is added to a strongly acidic solution of \(X\). The resulting solution is treated with a few drops of aqueous solution of \(Y\) to yield blue colouration due to the formation of methylene blue. Treatment of the aqueous solution of \(Y\) with the reagent potassium hexacyanoferrate(II) leads to the formation of an intense blue precipitate. The precipitate dissolves an excess addition of the reagent. Similarly, treatment of the solution of \(Y\) with the solution of potassium hexacyanoferrate(III) leads to a brown colouration due to the formation of \(Z\).
Question:
The compound \(Y\), is

Let \(L_1\) be the line of intersection of the planes given by the equation- \(2 x+3 y+z=4 \text { and } x+2 y+z=5 .\)
Let \(L_2\) be the line passing through the point \(P(2,-1,3)\) and parallel to \(L_1\). Let \(M\) denote the plane given by the equation- \(2 x+y-2 z=6\)
Suppose that the line \(L_2\) meets the plane \(M\) at the point \(Q\). Let \(R\) be the foot of the perpendicular drawn from \(P\) to the plane \(M\).
Then which of the following statements is (are) TRUE?

The number of water molecule(s) directly bonded to the metal centre in \(\mathrm{CuSO}_4 \cdot 5 \mathrm{H}_2 \mathrm{O}\) is

A uniform electric field, $\overrightarrow{E}=-400\sqrt{3}\hat{\mathrm{j}}\mathrm{N}{\mathrm{C}}^{-1}$ is applied in a region. A charged particle of mass $m$ carrying positive charge $q$ is projected in this region with an initial speed of $2\sqrt{10}\times {10}^6\mathrm{m}{\mathrm{s}}^{-1}$. This particle is aimed to hit a target $T$, which is$5\mathrm{m}$ away from its entry point into the field as shown schematically in the figure. Take $\frac{q}{m}={10}^{10}\mathrm{C}{\mathrm{kg}}^{-1}$. Then

\(\mathrm{MgSO}_4\) on reaction with \(\mathrm{NH}_4 \mathrm{OH}\) and \(\mathrm{Na}_2 \mathrm{HPO}_4\) forms a white crystalline precipitate. What is its formula?

A person in a lift is holding a water jar, which has a small hole at the lower end of its side. When the lift is at rest, the water jet coming out of the hole hits the floor of the lift at a distance d of 1.2 m from the person. In the following, state of the lift's motion is given in List I and the distance where the water jet hits the floor of the lift is given in List II. Match the statements from List I with those in List II and select the correct answer using the code given below the lists.
 

	List I		List II
A.	Lift is accelerating vertically up.	$P.$	$\mathrm{d}=1.2\mathrm{ }\mathrm{m}$
B.	Lift is accelerating vertically down with an acceleration less than the gravitational acceleration.	$Q.$	$\mathrm{d}>1.2\mathrm{ }\mathrm{m}$
C.	Lift is moving vertically up with constant speed.	$R.$	$\mathrm{d}<1.2\mathrm{ }\mathrm{m}$
D.	Lift is falling freely.	$S.$	No water leaks out of the jar