eSaral - JEE | NEET | Class 8-10 Exam Prep App

The base PQ of two equilateral triangles PQR and PQR' with side 2a lies

Question: The basePQof two equilateral trianglesPQRandPQR'with side 2alies along y-axis such that the mid-point ofPQis at the origin. Find the coordinates of the verticesRandR' of the triangles. Solution: In an equilateral triangle, the height h is given by $h=\frac{\sqrt{3} \text { (Side of the equilateral triangle) }}{2}$ Here it is given that PQ forms the base of two equilateral triangles whose side measures 2a units. The height of these two equilateral triangles has got to be $h=\frac{\sqrt{...

Five moles of an ideal gas at

Question: Five moles of an ideal gas at $293 \mathrm{~K}$ is expanded isothermally from an initial pressure of $2.1 \mathrm{MPa}$ to 1.3 MPa against at constant external 4.3 MPa. The heat transferred in this process is__________ $\mathrm{kJ}$ $\mathrm{mol}^{-1}$. (Rounded-off of the nearest integer) [Use $\mathrm{R}=8.314 \mathrm{~J} \mathrm{~mol}^{-1} \mathrm{~K}^{-1}$ ] Solution: (15) Moles $(n)=5$ $T=293 \mathrm{k}$ Process $=$ IsoT. $\rightarrow$ Irreversible $P_{\mathrm{ini}}=2.1 \mathrm{M}...

Find the point on the y-axis which is equidistant from the points A(6, 5) and B

Question: Find the point on they-axis which is equidistant from the pointsA(6, 5) andB( 4, 3). Solution: LetP(0, y) be a point on they-axis. Then as per the question, we have $A P=B P$ $\Rightarrow \sqrt{(0-6)^{2}+(y-5)^{2}}=\sqrt{(0+4)^{2}+(y-3)^{2}}$ $\Rightarrow \sqrt{(6)^{2}+(y-5)^{2}}=\sqrt{(4)^{2}+(y-3)^{2}}$ $\Rightarrow(6)^{2}+(y-5)^{2}=(4)^{2}+(y-3)^{2}$ (Squaring both sides) $\Rightarrow 36+y^{2}-10 y+25=16+y^{2}-6 y+9$ $\Rightarrow 4 y=36$ $\Rightarrow y=9$ Hence, the point on they-ax...

The reaction of cyanamide,

Question: The reaction of cyanamide, $\mathrm{NH}_{2} \mathrm{CN}_{(\mathrm{s})}$ with oxygen was run in a bomb calorimeter and $\Delta \mathrm{U}$ was found to be $-742.24 \mathrm{~kJ} \mathrm{~mol}^{-1}$. The magnitude of $\Delta \mathrm{H}_{298}$ for the reaction $\mathrm{NH}_{2} \mathrm{CN}_{(\mathrm{s})}+\frac{3}{2} \mathrm{O}_{2}(\mathrm{~g}) \rightarrow \mathrm{N}_{2(\mathrm{~g})}+\mathrm{O}_{2(\mathrm{~g})}+\mathrm{H}_{2} \mathrm{O}_{(1)}$ is__________ $\mathrm{kJ}$. (Rounded off to the ...

Let ABCD be a square of side 2a. Find the coordinates of the vertices of this square when

Question: LetABCDbe a square of side 2a. Find the coordinates of the vertices of this square when (i) A coincides with the origin andABandADare alongOXandOYrespectively.(ii) The centre of the square is at the origin and coordinate axes are parallel to the sidesABandADrespectively. Solution: The distance between any two adjacent vertices of a square will always be equal. This distance is nothing but the side of the square. Here, the side of the square ABCD is given to be 2a. (i) Since it is given...

Assuming ideal behaviour,

Question: Assuming ideal behaviour, the magnitude of $\log \mathrm{K}$ for the following reaction at $25^{\circ} \mathrm{C}$ is $\mathrm{x} \times 10^{-1}$. The value of $x$ is ________________.(Integer answer) $3 \mathrm{HC} \equiv \mathrm{CH}_{(\mathrm{g})} \rightleftharpoons \mathrm{C}_{6} \mathrm{H}_{6(\ell)}$ $\left[\right.$ Given $\left.: \Delta_{f} \mathrm{G}^{\circ}(\mathrm{HC} \equiv \mathrm{CH})=-2.04 \times 10^{5}\right] \mathrm{mol}^{-1} ; \Delta_{\mathrm{f}} \mathrm{G}^{\circ}\left(...

Find points on the x-axis, each of which is at a distance of 10 units from the point A

Question: Find points on thex-axis, each of which is at a distance of 10 units from the pointA(11, 8). Solution: LetP(x, 0) be the point on thex-axis. Then as per the question, we have $A P=10$ $\Rightarrow \sqrt{(x-11)^{2}+(0+8)^{2}}=10$ $\Rightarrow(x-11)^{2}+8^{2}=100$ (Squaring both sides) $\Rightarrow(x-11)^{2}=100-64=36$ $\Rightarrow x-11=\pm 6$ $\Rightarrow x=11 \pm 6$ $\Rightarrow x=11-6,11+6$ $\Rightarrow x=5,17$ Hence, the points on thex-axis are (5, 0) and (17, 0)....

On which axis do the following points lie?

Question: On which axis do the following points lie? (a) $\mathrm{P}(5,0)$ (b) $Q(0-2)$ (c) $R(-4,0)$ (d) $S(0,5)$ Solution: According to the Rectangular Cartesian Co-ordinate system of representing a point(x, y), If $x0, y0$ then the point lies in the $1^{\text {st }}$ quadrant If $x0, y0$ then the point lies in the $2^{\text {nd }}$ quadrant If $x0, y0$ then the point lies in the $3^{\text {rd }}$ quadrant If $x0, y0$ then the point lies in the $4^{\text {th }}$ quadrant But in case If $x=0, y...

During which of the following processes,

Question: During which of the following processes, does entropy decrease? (A) Freezing of water to ice at $0^{\circ} \mathrm{C}$ (B) Freezing of water to ice at $-10^{\circ} \mathrm{C}$ (C) $\mathrm{N}_{2}(\mathrm{~g})+3 \mathrm{H}_{2}(\mathrm{~g}) \rightarrow 2 \mathrm{NH}_{3}(\mathrm{~g})$ (D) Adsorption of $\mathrm{CO}(\mathrm{g})$ and lead surface (E) Dissolution of $\mathrm{NaCl}$ in water(A), (B), (C) and (D) only(B) and (C) only(A) and (E) only(A), (C) and (E) onlyCorrect Option: 1 Soluti...

Find the coordinates of the point on x-axis which is equidistant from the points

Question: Find the coordinates of the point onx-axis which is equidistant from the points (2, 5) and (2, 3). Solution: Let the point on thex- axis be (x,0). We have $\mathrm{A}(-2,5)$ and $\mathrm{B}(2,-3)$ $\mathrm{AX}=\mathrm{BX}$ $\mathrm{AX}^{2}=\mathrm{BX}^{2} \quad \ldots \ldots(1)$ Using distance formula : $d=\sqrt{\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1}\right)^{2}}$ For AX $\mathrm{AX}=\sqrt{(x-(-2))^{2}+(0-5)^{2}}$ $\mathrm{AX}^{2}=(x+2)^{2}+(-5)^{2}$ $\mathrm{AX}^{2}=x^{2}+4 x+2...

A (3, 2) and B (−2, 1) are two vertices of a triangle ABC whose

Question: $A(3,2)$ and $B(-2,1)$ are two vertices of a triangle $A B C$ whose centroid $G$ has the coordinates $\left(\frac{5}{3},-\frac{1}{3}\right)$. Find the coordinates of the third vertex $C$ of the triangle. Solution: We have to find the co-ordinates of the third vertex of the given triangle. Let the co-ordinates of the third vertex be $(x, y)$. The co-ordinates of other two vertices are $A(3,2)$ and $C(-2,1)$ The co-ordinate of the centroid is $\left(\frac{5}{3},-\frac{1}{3}\right)$ We kn...

If the point A(0, 2) is equidistant from the points B(3, p) and C(p, 5), find the value of p.

Question: If the pointA(0, 2) is equidistant from the pointsB(3,p) andC(p, 5), find the value ofp. Also, find the length ofAB. Solution: As per the question $A B=A C$ $\Rightarrow \sqrt{(0-3)^{2}+(2-p)^{2}}=\sqrt{(0-p)^{2}+(2-5)^{2}}$ $\Rightarrow \sqrt{(-3)^{2}+(2-p)^{2}}=\sqrt{(-p)^{2}+(-3)^{2}}$ Squaring both sides, we get $(-3)^{2}+(2-p)^{2}=(-p)^{2}+(-3)^{2}$ $\Rightarrow 9+4+p^{2}-4 p=p^{2}+9$ $\Rightarrow 4 p=4$ $\Rightarrow p=1$ Now, $A B=\sqrt{(0-3)^{2}+(2-p)^{2}}$ $=\sqrt{(-3)^{2}+(2-1...

Find the third vertex of a triangle, if two of its vertices are

Question: Find the third vertex of a triangle, if two of its vertices are at (3, 1) and (0, 2) and the centroid is at the origin. Solution: We have to find the co-ordinates of the third vertex of the given triangle. Let the co-ordinates of the third vertex be. The co-ordinates of other two vertices are (3, 1) and (0, 2) The co-ordinate of the centroid is (0, 0) We know that the co-ordinates of the centroid of a triangle whose vertices are $\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right),\lef...

In Fig. 14.36, a right triangle BOA is given C is the mid-point of the hypotenuse AB.

Question: In Fig. 14.36, a right triangle BOA is given C is the mid-point of the hypotenuse AB. Show that it is equidistant from the vertices O, A and B. Solution: We have a right angled triangle $\Delta \mathrm{BOA}$, right angled at $\mathrm{O}$. Co-ordinates are $\mathrm{B}(0,2 b) ; \mathrm{A}(2 a, 0)$ and $\mathrm{C}(0,0)$. We have to prove that mid-point C of hypotenuse AB is equidistant from the vertices. In general to find the mid-point $\mathrm{P}(x, y)$ of two points $\mathrm{A}\left(x_...

If the point A(x, 2) is equidistant from the points B(8, − 2) and C(2, − 2),

Question: If the pointA(x, 2) is equidistant from the pointsB(8, 2) andC(2, 2), find the value ofx. Also, find the length ofAB. Solution: As per the question $A B=A C$ $\Rightarrow \sqrt{(x-8)^{2}+(2+2)^{2}}=\sqrt{(x-2)^{2}+(2+2)^{2}}$ Squaring both sides, we get $(x-8)^{2}+4^{2}=(x-2)^{2}+4^{2}$ $\Rightarrow x^{2}-16 x+64+16=x^{2}+4-4 x+16$ $\Rightarrow 16 x-4 x=64-4$ $\Rightarrow x=\frac{60}{12}=5$ Now, $A B=\sqrt{(x-8)^{2}+(2+2)^{2}}$ $=\sqrt{(5-8)^{2}+(2+2)^{2}} \quad(\because x=2)$ $=\sqrt{...

If (−2, 3), (4, −3) and (4, 5) are the mid-points of the sides of a triangle,

Question: If (2, 3), (4, 3) and (4, 5) are the mid-points of the sides of a triangle, find the coordinates of its centroid. Solution: Let $\triangle \mathrm{ABC}$ be ant triangle such that $\mathrm{P}(-2,3) ; \mathrm{Q}(4,-3)$ and $\mathrm{R}(4,5)$ are the mid-points of the sides $\mathrm{AB}, \mathrm{BC}, \mathrm{CA}$ respectively. We have to find the co-ordinates of the centroid of the triangle. Let the vertices of the triangle be $\mathrm{A}\left(x_{1}, y_{1}\right) ; \mathrm{B}\left(x_{2}, y...

For the following Assertion and Reason, the correct option is:

Question: For the following Assertion and Reason, the correct option is: Assertion: For hydrogenation reactions, the catalytic activity increases from Group 5 to Group 11 metals with maximum activity shown by Group 7-9 elements. Reason: The reactants are most strongly adsorbed on group 7-9 elements.The assertion is true, but the reason is false.Both assertion and reason are false.Both assertion and reason are true and the reason is the correct explanation for the assertion.Both assertion and rea...

As per Hardy-Schulze formulation,

Question: As per Hardy-Schulze formulation, the flocculation values of the following for ferric hydroxide sol are in the order:$\mathrm{K}_{3}\left[\mathrm{Fe}(\mathrm{CN})_{6}\right]\mathrm{K}_{2} \mathrm{CrO}_{4}\mathrm{KBr}=\mathrm{KNO}_{3}=\mathrm{AlCl}_{3}$$\mathrm{~K}_{3}\left[\mathrm{Fe}(\mathrm{CN})_{6}\right]\mathrm{K}_{2} \mathrm{CrO}_{4}\mathrm{AlCl}_{3}\mathrm{KBr}\mathrm{KNO}_{3}$$\mathrm{AlCl}_{3}\mathrm{K}_{3}\left[\mathrm{Fe}(\mathrm{CN})_{6}\right]\mathrm{K}_{2} \mathrm{CrO}_{4}...

For Freundlich adsorption isotherm,

Question: For Freundlich adsorption isotherm, a plot of $\log (x / \mathrm{m})$ ( $y$-axis) and $\log \mathrm{p}(x$-axis) gives a straight line. The intercept and slope for the line is $0.4771$ and 2, respectively. The mass of gas, adsorbed per gram of adsorbent if the initial pressure is $0.04 \mathrm{~atm}$, is________$\times 10^{-4} \mathrm{~g}$. $(\log 3=0.4771)$ Solution: (48) Freundlich adsorption isotherm : $\frac{x}{m}=k_{p}^{1 / n}$ $\Rightarrow \log \frac{x}{m}=\log k+\frac{1}{n} \log ...

Kraft temperature is the temperature:

Question: Kraft temperature is the temperature:below which the aqueous solution of detergents starts freezing.below which the formation of micelles takes place.above which the aqueous solution of detergents starts boiling.above which the formation of micelles takes place.Correct Option: 1 Solution: Above Kraft temperature the formation of micelles takes place and the conc. above which micelle formation become appreciable is called critical micelles conc....

Find the values of x for which the distance between the points P(x, 4) and Q(9, 10) is 10 units.

Question: Find the values ofxfor which the distance between the pointsP(x, 4) andQ(9, 10) is 10 units. Solution: The given points areP(x, 4) andQ(9, 10). $\therefore P Q=\sqrt{(x-9)^{2}+(4-10)^{2}}$ $=\sqrt{(x-9)^{2}+(-6)^{2}}$ $=\sqrt{x^{2}-18 x+81+36}$ $=\sqrt{x^{2}-18 x+117}$ $\because P Q=10$ $\therefore \sqrt{x^{2}-18 x+117}=10$ $\Rightarrow x^{2}-18 x+117=100 \quad$ (Squaring both sides) $\Rightarrow x^{2}-18 x+17=$ $\Rightarrow x^{2}-17 x-x+17=0$ $\Rightarrow x(x-17)-1(x-17)=0$ $\Rightarr...

Adsorption of a gas follows Freundlich adsorption isotherm.

Question: Adsorption of a gas follows Freundlich adsorption isotherm. If $x$ is the mass of the gas adsorbed on mass $m$ of the adsorbent, the correct plot of $\frac{x}{m}$ versus $p$ is :Correct Option: 1 Solution: As adsorption is exothermic process, hence $\frac{x}{m}$ decreases with increase of temperature....

Find all possible values of y for which the distance between the points

Question: Find all possible values of $y$ for which the distance between the points $A(2,-3)$ and $B(10, y)$ is 10 units. Solution: The given points are $A(2,-3)$ and $B(10, y)$. $\therefore A B=\sqrt{(2-10)^{2}+(-3-y)^{2}}$ $=\sqrt{(-8)^{2}+(-3-y)^{2}}$ $=\sqrt{64+9+y^{2}+6 y}$ $\because A B=10$ $\therefore \sqrt{64+9+y^{2}+6 y}=10$ $\Rightarrow 73+y^{2}+6 y=100 \quad$ (Squaring both sides) $\Rightarrow y^{2}+6 y-27=0$ $\Rightarrow y^{2}+9 y-3 y-27=0$ $\Rightarrow y(y+9)-3(y+9)=0$ $\Rightarrow(...

Find all possible values of x for which the distance between the points

Question: Find all possible values ofxfor which the distance between the pointsA(x, 1) andB(5, 3) is 5 units. Solution: Given AB = 5 unitsTherefore, (AB)2= 25 units $\Rightarrow(5-x)^{2}+\{3-(-1)\}^{2}=25$ $\Rightarrow(5-x)^{2}+(3+1)^{2}=25$ $\Rightarrow(5-x)^{2}+(4)^{2}=25$ $\Rightarrow(5-x)^{2}+16=25$ $\Rightarrow(5-x)^{2}=25-16$ $\Rightarrow(5-x)^{2}=9$ $\Rightarrow(5-x)=\pm \sqrt{9}$ $\Rightarrow 5-x=\pm 3$ $\Rightarrow 5-x=3$ or $5-x=-3$ $\Rightarrow x=2$ or 8 Therefore,x = 2 or 8....

If G be the centroid of a triangle ABC and P be any other point in the plane,

Question: If $\mathrm{G}$ be the centroid of a triangle $\mathrm{ABC}$ and $\mathrm{P}$ be any other point in the plane, prove that $\mathrm{PA}^{2}+\mathrm{PB}^{2}+\mathrm{PC}^{2}=\mathrm{GA}^{2}+\mathrm{GB}^{2}+\mathrm{GC}^{2}+3 \mathrm{GP}^{2}$. Solution: Let $\triangle \mathrm{ABC}$ be any triangle whose coordinates are $\mathrm{A}\left(x_{1}, y_{1}\right) ; \mathrm{B}\left(x_{2}, y_{2}\right) ; \mathrm{C}\left(x_{3}, y_{3}\right)$. Let $\mathrm{P}$ be the origin and $\mathrm{G}$ be the cent...

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