Choose the correct answer in each of the following questions:
Question: Choose the correct answer in each of the following questions:The sum of first 40 positive integers divisible by 6 is (a) 2460 (b) 3640 (c) 4920 (d) 4860 Solution: The positive integers divisible by 6 are 6, 12, 18, ... .This is an AP witha= 6andd= 6.Also,n= 40 (Given) Using the formula, $S_{n}=\frac{n}{2}[2 a+(n-1) d]$, we get $S_{40}=\frac{40}{2}[2 \times 6+(40-1) \times 6]$ $=20(12+234)$ $=20 \times 246$ $=4920$ Thus, the required sum is 4920.Hence, the correct answer is option C....
Read More →If A and B are two points having coordinates (−2, −2) and (2, −4) respectively,
Question: If $A$ and $B$ are two points having coordinates $(-2,-2)$ and $(2,-4)$ respectively, find the coordinates of $P$ such that $A P=\frac{3}{7} A B$. Solution: We have two points A (2,2) and B (2,4). Let P be any point which divide AB as, $\mathrm{AP}=\frac{3}{7} \mathrm{AB}$ Since, $A B=(A P+B P)$ So, $7 \mathrm{AP}=3 \mathrm{AB}$ $7 \mathrm{AP}=3(\mathrm{AP}+\mathrm{BP})$ $4 \mathrm{AP}=3 \mathrm{BP}$ $\frac{\mathrm{AP}}{\mathrm{BP}}=\frac{3}{4}$ Now according to the section formula if ...
Read More →Choose the correct answer in each of the following questions:
Question: Choose the correct answer in each of the following questions:The sum of first 20 odd natural numbers is (a) 100 (b) 210 (c) 400 (d) 420 Solution: The first 20 odd natural numbers are 1, 3, 5, ..., 39.These numbers are in AP.Here,a= 1,l= 39 andn= 20 Sum of first 20 odd natural numbers $=\frac{20}{2}(1+39) \quad\left[S_{n}=\frac{n}{2}(a+l)\right]$ $=10 \times 40$ $=400$ Hence, the correct answer is option C....
Read More →The line joining the points (2, 1) and (5, −8) is trisected at the points
Question: The line joining the points (2, 1) and (5, 8) is trisected at the pointsPandQ. If pointPlies on the line 2xy+k= 0. Find the value of k. Solution: We have two points A (2, 1) and B (5,8). There are two points P and Q which trisect the line segment joining A and B. Now according to the section formula if any point P divides a line segment joiningandin the ratio m: n internally than, $\mathrm{P}(x, y)=\left(\frac{n x_{1}+m x_{2}}{m+n}, \frac{n y_{1}+m y_{2}}{m+n}\right)$ The point P is th...
Read More →Solve the following
Question: $\mathrm{NaClO}_{3}$ is used, even in spacecrafts, to produce $\mathrm{O}_{2}$. The daily consumption of pure $\mathrm{O}_{2}$ by a person is $492 \mathrm{~L}$ at $1 \mathrm{~atm}, 300 \mathrm{~K}$. How much amount of $\mathrm{NaClO}_{3}$, in grams, is required to produce $\mathrm{O}_{2}$ for the daily consumption of a person at $1 \mathrm{~atm}, 300 \mathrm{~K}$ ? _______________ . $\mathrm{NaClO}_{3}(\mathrm{~s})+\mathrm{Fe}(\mathrm{s}) \rightarrow \mathrm{O}_{2}(\mathrm{~g})+\mathrm...
Read More →Choose the correct answer in each of the following questions:
Question: Choose the correct answer in each of the following questions:An AP 5, 12, 19, ... has 50 terms. Its last term is (a) 343 (b) 353 (c) 348 (d) 362 Solution: The given AP is 5, 12, 19, ... .Here,a= 5,d= 12 5 = 7 andn= 50Since there are 50 terms in the AP, so the last term of the AP isa50. $a_{50}=5+(50-1) \times 7 \quad\left[a_{n}=a+(n-1) d\right]$ $=5+343$ $=348$ Thus, the last term of the AP is 348.Hence, the correct answer is option C....
Read More →A graph of vapour pressure and temperature for three different liquids X, Y, and Z is shown below:
Question: A graph of vapour pressure and temperature for three different liquids $X, Y$, and $Z$ is shown below: The following inferences are made: (A) $X$ has higher intermolecular interactions compared to $\mathrm{Y}$. (B) $X$ has lower intermolecular interactions compared to $\mathrm{Y}$. (C) $Z$ has lower intermolecular interactions compared to Y. The correct inference(s) is/are:(A) and (C)(A)(B)(C)Correct Option: , 3 Solution: At a particular temperature as intermolecular force of attractio...
Read More →The line segment joining the points (3, −4) and (1, 2) is
Question: The line segment joining the points (3, 4) and (1, 2) is trisected at the points P and Q. If the coordinates of P and Q are (p, 2) and (5/3, q) respectively. Find the values ofpandq. Solution: We have two points $A(3,-4)$ and $B(1,2)$. There are two points $P(p,-2)$ and $Q\left(\frac{5}{3}, q\right)$ which trisect the line segment joining $A$ and $B$. Now according to the section formula if any point $P$ divides a line segment joining $A\left(x_{1}, y_{1}\right)$ and $B\left(x_{2}, y_{...
Read More →Choose the correct answer in each of the following questions:
Question: Choose the correct answer in each of the following questions:The 13th term of an AP is 4 times its 3rd term. If its 5th term is 16 then the sum of its first ten terms is (a) 150 (b) 175 (c) 160 (d) 135 Solution: Letabe the first term anddbe the common difference of the AP. Then, $a_{13}=4 \times a_{3}$ (Given) $\Rightarrow a+12 d=4(a+2 d) \quad\left[a_{n}=a+(n-1) d\right]$ $\Rightarrow a+12 d=4 a+8 d$ $\Rightarrow 3 a=4 d \quad \ldots(1)$ Also, $a_{5}=16$ (Given) $\Rightarrow a+4 d=16 ...
Read More →Determine the ratio in which the point (−6, a) divides
Question: Determine the ratio in which the point (6,a) divides the join ofA(3, 1) andB(8, 9). Also find the value ofa. Solution: The co-ordinates of a point which divided two points $\left(x_{1}, y_{1}\right)$ and $\left(x_{2}, y_{2}\right)$ internally in the ratio $m: n$ is given by the formula, $(x, y)=\left(\left(\frac{m x_{2}+n x_{1}}{m+n}\right),\left(\frac{m y_{2}+n y_{1}}{m+n}\right)\right)$ Here we are given that the pointP(6,a) divides the line joining the pointsA(3,1) andB(8,9) in some...
Read More →Identify the correct labels of A,B and C in the following graph from the options given below:
Question: Identify the correct labels of $\mathrm{A}, \mathrm{B}$ and $\mathrm{C}$ in the following graph from the options given below: Root mean square speed $\left(V_{r m s}\right) ;$ most probable speed $\left(\mathrm{V}_{\mathrm{mp}}\right) ;$ average speed $\left(\mathrm{V}_{\mathrm{av}}\right)$$A-V_{\mathrm{mp}} ; B-V_{\mathrm{rms}} ; C-V_{\mathrm{av}}$$A-V_{\mathrm{av}} ; B-V_{\mathrm{rms}} ; C-V_{\mathrm{mp}}$$A-V_{\mathrm{rms}} ; B-V_{\mathrm{mp}} ; C-V_{\mathrm{av}}$$\mathrm{A}-\mathrm...
Read More →Identify the correct labels of A,B and C in the following graph from the options given below:
Question: Identify the correct labels of $\mathrm{A}, \mathrm{B}$ and $\mathrm{C}$ in the following graph from the options given below: Root mean square speed $\left(V_{r m s}\right) ;$ most probable speed $\left(\mathrm{V}_{\mathrm{mp}}\right) ;$ average speed $\left(\mathrm{V}_{\mathrm{av}}\right)$$A-V_{\mathrm{mp}} ; B-V_{\mathrm{rms}} ; C-V_{\mathrm{av}}$$A-V_{\mathrm{av}} ; B-V_{\mathrm{rms}} ; C-V_{\mathrm{mp}}$$A-V_{\mathrm{rms}} ; B-V_{\mathrm{mp}} ; C-V_{\mathrm{av}}$$A-V_{\mathrm{mp}} ...
Read More →Determine the ratio in which the point P (m, 6)
Question: Determine the ratio in which the point P (m, 6) divides the join ofA(4, 3) andB(2, 8). Also, find the value ofm. Solution: The co-ordinates of a point which divided two points $\left(x_{1}, y_{1}\right)$ and $\left(x_{2}, y_{2}\right)$ internally in the ratio $m: n$ is given by the formula, $(x, y)=\left(\left(\frac{m x_{2}+n x_{1}}{m+n}\right),\left(\frac{m y_{2}+n y_{1}}{m+n}\right)\right)$ Here we are given that the pointP(m,6) divides the line joining the pointsA(4,3) andB(2,8) in ...
Read More →Choose the correct answer in each of the following questions:
Question: Choose the correct answer in each of the following questions:The 5th term of an AP is 20 and the sum of its 7th and 11th terms is 64. The common difference of the AP is (a) 4 (b) 5 (c) 3 (d) 2 Solution: Letabe the first term anddbe the common difference of the AP. Then, $a_{5}=20$ $\Rightarrow a+(5-1) d=20 \quad\left[a_{n}=a+(n-1) d\right]$ $\Rightarrow a+4 d=20 \quad \ldots .(1)$ Now, $a_{7}+a_{11}=64$ (Given) $\Rightarrow(a+6 d)+(a+10 d)=64$ $\Rightarrow 2 a+16 d=64$ $\Rightarrow a+8...
Read More →A spherical balloon of radius
Question: A spherical balloon of radius $3 \mathrm{~cm}$ containing helium gas has a pressure of $48 \times 10^{-3}$ bar. At the same temperature, the pressure, of a spherical balloon of radius $12 \mathrm{~cm}$ containing the same amount of gas will be____________ $\times 10^{-6} \mathrm{bar}$. Solution: (750) At constant temperature and number of moles $P_{1} V_{1}=P_{2} V_{2}$ $P_{1}=48 \times 10^{-3} \mathrm{bar} ; V_{1}=\frac{4}{3} \pi(3)^{3}$ $V_{2}=\frac{4}{3} \pi(12)^{3}$ $P_{2}=\frac{P_...
Read More →Show that the points A (1, 0), B (5, 3),
Question: Show that the pointsA(1, 0),B(5, 3),C(2, 7) andD(2, 4) are the vertices of a parallelogram. Solution: Let A (1, 0); B (5, 3); C (2, 7) and D (2, 4) be the vertices of a quadrilateral. We have to prove that the quadrilateral ABCD is a parallelogram. We should proceed with the fact that if the diagonals of a quadrilateral bisect each other than the quadrilateral is a parallelogram. Now to find the mid-point $\mathrm{P}(x, y)$ of two points $\mathrm{A}\left(x_{1}, y_{1}\right)$ and $\math...
Read More →Choose the correct answer in each of the following questions:
Question: Choose the correct answer in each of the following questions:The 5th term of an AP is 3 and its common difference is 4. The sum of its first 10 terms is (a) 50 (b) $-50$ (c) 30 (d) $-30$ Solution: Letabe the first term of the AP.Here,d= 4 $a_{5}=-3$ (Given) $\Rightarrow a+(5-1) \times(-4)=-3 \quad\left[a_{n}=a+(n-1) d\right]$ $\Rightarrow a-16=-3$ $\Rightarrow a=16-3=13$ Using the formula, $S_{n}=\frac{n}{2}[2 a+(n-1) d]$, we get $S_{10}=\frac{10}{2}[2 \times 13+(10-1) \times(-4)]$ $=5...
Read More →If A and B are (1, 4) and (5, 2) respectively,
Question: IfAandBare (1, 4) and (5, 2) respectively, find the coordinates ofPwhenAP/BP= 3/4. Solution: The co-ordinates of the point dividing two points $\left(x_{1}, y_{1}\right)$ and $\left(x_{2}, y_{2}\right)$ in the ratio $m: n$ is given as, $(x, y)=\left(\left(\frac{\lambda x_{2}+x_{1}}{\lambda+1}\right),\left(\frac{\lambda y_{2}+y_{1}}{\lambda+1}\right)\right)$ where, $\lambda=\frac{m}{n}$ Here the two given points areA(1,4) andB(5,2). Let pointP(x, y) divide the line joining AB in the rat...
Read More →A mixture of one mole each of
Question: A mixture of one mole each of $\mathrm{H}_{2}, \mathrm{He}$ and $\mathrm{O}_{2}$ each are enclosed in a cylinder of volume $V$ at temperature $T$. If the partial pressure of $\mathrm{H}_{2}$ is $2 \mathrm{~atm}$, the total pressure of the gases in the cylinder is :$6 \mathrm{~atm}$$38 \mathrm{~atm}$$14 \mathrm{~atm}$$22 \mathrm{~atm}$Correct Option: 1 Solution: $P_{\mathrm{gas}}=\frac{n_{\mathrm{gas}} R T}{V}$ As $n, T$ and $V$ constant so $P_{\mathrm{H}_{2}}=P_{\mathrm{O}_{2}}=P_{\mat...
Read More →Find the distance of the point (1, 2) from the mid-point
Question: Find the distance of the point (1, 2) from the mid-point of the line segment joining the points (6, 8) and (2, 4). Solution: We have to find the distance of a point A (1, 2) from the mid-point of the line segment joining P (6, 8) and Q (2, 4). In general to find the mid-point $\mathrm{P}(x, y)$ of any two points $\mathrm{A}\left(x_{1}, y_{1}\right)$ and $\mathrm{B}\left(x_{2}, y_{2}\right)$ we use section formula as, $\mathrm{P}(x, y)=\left(\frac{x_{1}+x_{2}}{2}, \frac{y_{1}+y_{2}}{2}\...
Read More →Which one of the following graphs is not correct for ideal gas?
Question: Which one of the following graphs is not correct for ideal gas? $\mathrm{d}=$ Density, $\mathrm{P}=$ Pressure, $\mathrm{T}=$ TemperatureIIIIVIIICorrect Option: , 2 Solution: For ideal gas $P V=n R T$ $P V=\frac{m}{M} R T \quad\left(\because n=\frac{m}{M}\right)$ $P M=\frac{m}{V} R T ; \quad P M=d R T ; \quad d=\left[\frac{P M}{R}\right] \frac{1}{T}$ $\Rightarrow d \propto \frac{1}{T} ; \quad d \propto P$ So, graph between $d \mathrm{Vs} T$ is not straight line....
Read More →Show that the mid-point of the line segment joining the points (5, 7) and (3, 9)
Question: Show that the mid-point of the line segment joining the points (5, 7) and (3, 9) is also the mid-point of the line segment joining the points (8, 6) and (0, 10). Solution: We have two points A (5, 7) and B (3, 9) which form a line segment and similarly C (8, 6) and D (0, 10) form another line segment. We have to prove that mid-point of AB is also the mid-point of CD. In general to find the mid-point $\mathrm{P}(x, y)$ of two points $\mathrm{A}\left(x_{1}, y_{1}\right)$ and $\mathrm{B}\...
Read More →Choose the correct answer in each of the following questions:
Question: Choose the correct answer in each of the following questions:The 7th term of an AP is 1 and its 16th term is 17. Thenth term of the AP is (a) $(3 n+8)$ (b) $(4 n-7)$ (c) $(15-2 n)$ (d) $(2 n-15)$ Solution: Letabe the first term anddbe the common difference of the AP. Then,nth term of the AP,an=a+ (n 1)dNow, $a_{7}=-1$ (Given) $\Rightarrow a+6 d=-1 \quad \ldots(1)$ Also, $a_{16}=17$ (Given) $\Rightarrow a+15 d=17 \quad \ldots(2)$ Subtracting (1) from (2), we get $(a+15 d)-(a+6 d)=17-(-1...
Read More →Find the coordinates of the points which divide
Question: Find the coordinates of the points which divide the line segment joining the points (4, 0) and (0, 6) in four equal parts. Solution: The co-ordinates of the midpoint $\left(x_{m}, y_{m}\right)$ between two points $\left(x_{1}, y_{1}\right)$ and $\left(x_{2}, y_{2}\right)$ is given by, $\left(x_{w}, y_{m}\right)=\left(\left(\frac{x_{1}+x_{2}}{2}\right),\left(\frac{y_{1}+y_{2}}{2}\right)\right)$ Here we are supposed to find the points which divide the line joiningA(4,0) andB(0,6) into 4 ...
Read More →A certain gas obeys
Question: A certain gas obeys $P\left(V_{m}-b\right)=R T$. The value of $\left(\frac{\partial Z}{\partial P}\right)_{T}$ is $\frac{x b}{R T}$. The value of $x$ is Solution: (1) $P(v-b)=R T$ $P V-P b=R T$ $\frac{P V}{R T}-\frac{P b}{R T}=1$ $Z=1+\frac{P V}{R T}$ $\frac{d z}{d p}=0+\frac{b}{R T}$ $\Rightarrow \frac{b}{R T}=\frac{x b}{R T}$ $x=1$...
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