A test particle is moving in circular orbit in the gravitational field
Question: A test particle is moving in circular orbit in the gravitational field produced by a mass density $r(r)=\frac{\mathrm{K}}{r^{2}}$. Identify the correct relation between the radius $R$ of the particle's orbit and its period $\mathrm{T}$ :(1) $\mathrm{T} / \mathrm{R}$ is a constant(2) $\mathrm{T}^{2} / \mathrm{R}^{3}$ is a constant(3) $\mathrm{T} / \mathrm{R}^{2}$ is a constant(4) TR is a constantCorrect Option: 1 Solution: (1) $F=\frac{G M m}{r}=\int a \frac{\rho(d V) m}{r^{2}}$ $=m G \...
Read More →Thermal power plants can lead to :
Question: Thermal power plants can lead to :Acid rainBlue baby syndromeOzone layer depletionEutrophicationCorrect Option: Solution: Burning of fossil fuels (which contain sulphur and nitrogenous matter) such as coal and oil in power stations and furnaces produce sulphur dioxide and nitrogen oxides which causes acid rain....
Read More →If the system of equations
Question: If the system of equations $x+y+z=2 \quad 2 x+4 y-z=63 x+2 y+\lambda z=\mu$ has infinitely many solutions, then :(1) $\lambda+2 \mu=14$(2) $2 \lambda-\mu=5$(3) $\lambda-2 \mu=-5$(4) $2 \lambda+\mu=14$Correct Option: , 4 Solution: $\left|\begin{array}{ccc}1 1 1 \\ 2 4 -1 \\ 3 2 \lambda\end{array}\right|=0 \quad[\because$ Equation has many solutions $]$ $\Rightarrow-15+6+2 \lambda=0 \Rightarrow \lambda=\frac{9}{2}$ $\therefore D_{Z}=\left|\begin{array}{llc}1 1 2 \\ 2 4 6 \\ 3 2 2 \mu\end...
Read More →The statement that is not true about ozone is :
Question: The statement that is not true about ozone is :in the stratosphere, CFCs release chlorine free radicals (Cl) which reacts with $\mathrm{O}_{3}$ to give chlorine dioxide radicals.in the atmosphere, it is depleted by CFCs.in the stratosphere, it forms a protective shield against UV radiation.it is a toxic gas and its reaction with $\mathrm{NO}$ gives $\mathrm{NO}_{2}$.Correct Option: , 2 Solution: In presence of sunlight CFC's molecule divides and release chlorine free radical, which rea...
Read More →Some students planned a picnic. The total budget for food was ₹2000.
Question: Some students planned a picnic. The total budget for food was ₹2000. But, 5 students failed to attend the picnic and thus the cost for food for each member increased by ₹20. How many students attended the picnic and how much did each student pay for the food? Solution: Letxbe the number of students who planned a picnic. $\therefore$ Original cost of food for each member $=₹ \frac{2000}{x}$ Five students failed to attend the picnic. So, (x 5) students attended the picnic. $\therefore$ N...
Read More →If the system of equations
Question: If the system of equations $x-2 y+3 z=9,2 x+y+z=b$ $x-7 y+a z=24$, has infinitely many solutions, then $a-b$ is equal to__________. Solution: For infinitely many solutions, $\Delta=\Delta_{1}=\Delta_{2}=\Delta_{3}=0$ $\Delta=\left|\begin{array}{ccc}1 -2 3 \\ 2 1 1 \\ 1 -7 a\end{array}\right|=0$ $\Rightarrow(a+7)-2(1-2 a)+3(-15)=0$ $\Rightarrow a=8$ $\Delta_{3}=\left|\begin{array}{ccc}1 -2 9 \\ 2 1 b \\ 1 -7 24\end{array}\right|=0$ $\Rightarrow(24+7 b)-2(b-48)+9(-15)=0$ $\Rightarrow b=3...
Read More →A solid sphere of mass ' M ' and radius 'a'
Question: A solid sphere of mass ' $M$ ' and radius 'a' is surrounded by a uniform concentric spherical shell of thickness $2 \mathrm{a}$ and mass $2 \mathrm{M}$. The gravitational field at distance '3a' from the centre will be:(1) $\frac{2 \mathrm{GM}}{9 a^{2}}$(2) $\frac{\mathrm{GM}}{9 a^{2}}$(3) $\frac{\mathrm{GM}}{3 a^{2}}$(4) $\frac{2 \mathrm{GM}}{3 a^{2}}$Correct Option: , 3 Solution: (3) $E_{P}=\frac{G M}{(3 a)^{2}}+\frac{G(2 M)}{(3 a)^{2}}=\frac{G M}{3 a^{2}}$ For a part on the surface o...
Read More →The presence of ozone in troposphere :
Question: The presence of ozone in troposphere :generates photochemical smogProtects us from the UV radiationProtects us from the X-ray radiationProtects us from greenhouse effectCorrect Option: , 2 Solution: The presence of ozone in troposphere protect earth from ultra violet rays...
Read More →In a class test, the sum of Kamal's marks in Mathematics and English is 40.
Question: In a class test, the sum of Kamal's marks in Mathematics and English is 40. Had he got 3 marks more in Mathematics and 4 marks less in English, the product of the marks would have been 360. Find his marks in two subjects separately. Solution: Let the marks of Kamal in mathematics and english be $x$ and $y$, respectively. According to the question: $x+y=40 \quad \ldots$ (i) Also, $(x+3)(y-4)=360$ $\Rightarrow(x+3)(40-x-4)=360 \quad[$ From (i) $]$ $\Rightarrow(x+3)(36-x)=360$ $\Rightarro...
Read More →Prove the following
Question: If $A=\left[\begin{array}{cc}\cos \theta i \sin \theta \\ i \sin \theta \cos \theta\end{array}\right],\left(\theta=\frac{\pi}{24}\right)$ and $A^{5}=\left[\begin{array}{ll}a b \\ c d\end{array}\right]$, where $i=\sqrt{-1}$, then which one of the following is not true? (1) $0 \leq a^{2}+b^{2} \leq 1$(2) $a^{2}-d^{2}=0$(3) $a^{2}-c^{2}=1$(4) $a^{2}-b^{2}=\frac{1}{2}$Correct Option: , 4 Solution: $\because A=\left[\begin{array}{cc}\cos \theta i \sin \theta \\ i \sin \theta \cos \theta\end...
Read More →Given below are two statements :
Question: Given below are two statements : Statement $\mathrm{I}$ : The $\mathrm{pH}$ of rain water is normally $\sim 5.6$. Statement II : If the $\mathrm{pH}$ of rain water drops below $5.6$, it is called acid rain. In the light of the above statements, choose the correct answer from the option given below.Statement I is false but Statement II is trueBoth statement I and statement II are trueBoth statement I and statement II are falseStatement I is true but statement II is falseCorrect Option: ...
Read More →Given below are two statements:
Question: Given below are two statements: Statement-I : An allotrope of oxygen is an important intermediate in the formation of reducing smog. Statement-II : Gases such as oxides of nitrogen and sulphur present in troposphere contribute to the formation of photochemical smog. In the light of the above statements, choose the correct answer from the options given below:Statement I and Statement II are trueStatement I is true about Statement II is falseBoth Statement I and Statement II are falseSta...
Read More →A rocket has to be launched from earth in such a way that it never returns.
Question: A rocket has to be launched from earth in such a way that it never returns. If $\mathrm{E}$ is the minimum energy delivered by the rocket launcher, what should be the minimum energy that the launcher should have if the same rocket is to be launched from the surface of the moon? Assume that the density of the earth and the moon are equal and that the earth's volume is 64 times the volume of the moon.(1) $\frac{\mathrm{E}}{64}$(2) $\frac{E}{32}$(3) $\frac{\mathrm{E}}{4}$(4) $\frac{\mathr...
Read More →300 apples are distributed equally among a certain number of students
Question: 300 apples are distributed equally among a certain number of students. Had there been 10 more students, each would have received one apple less. Find the number of student. Solution: Let the total number of students be $x$. According to the question: $\frac{300}{x}-\frac{300}{x+10}=1$ $\Rightarrow \frac{300(x+10)-300 x}{x(x+10)}=1$ $\Rightarrow \frac{300 x+3000-300 x}{x^{2}+10 x}=1$ $\Rightarrow 3000=x^{2}+10 x$ $\Rightarrow x^{2}+10 x-3000=0$ $\Rightarrow x^{2}+(60-50) x-3000=0$ $\Rig...
Read More →Let S be the set of all integer solutions,
Question: Let $S$ be the set of all integer solutions, $(x, y, z)$, of the system of equations $x-2 y+5 z=0$ $-2 x+4 y+z=0$ $-7 x+14 y+9 z=0$ such that $15 \leq x^{2}+y^{2}+z^{2} \leq 150$. Then, the number of elements in the set $S$ is equal to__________. Solution: The given system of equations $x-2 y+5 z=0$.....(1) $-2 x+4 y+z=0$......(2) $-7 x+14 y+9 z=0$.........(3) From equation, $2 \times$ (1) $+$ (2) $\Rightarrow z=0$ Put $z=0$ in equation (1), we get $x=2 y$ $\because 15 \leq x^{2}+y^{2}...
Read More →Given below are two statements :
Question: Given below are two statements : Statement I : The value of the parameter "Biochemical Oxygen Demand (BOD)" is important for survival of aquatic life. Statement II : The optimum value of BOD is $6.5 \mathrm{ppm}$. In the light of the above statements, choose the most appropriate answer from the options given below.Both Statement I and Statement II are falseStatement $\mathrm{I}$ is false but Statement II is trueStatement I is true but Statement II is falseBoth Statement I and Statement...
Read More →A teacher on attempting to arrange the students for mass drill in the form of a solid square found that 24 students were left.
Question: A teacher on attempting to arrange the students for mass drill in the form of a solid square found that 24 students were left. When he increased the size of the square by one student, he found that he was short of 25 students. Find the number of students. Solution: Let there be $x$ rows. Then, the number of students in each row will also be $x$. $\therefore$ Total number of students $=\left(x^{2}+24\right)$ According to the question: $(x+1)^{2}-25=x^{2}+24$ $\Rightarrow x^{2}+2 x+1-25-...
Read More →Four identical particles of mass M
Question: Four identical particles of mass $\mathrm{M}$ are located at the corners of a square of side ' $a$ '. What should be their speed if each of them revolves under the influence of others' gravitational field in a circular orbit circumscribing the square ? (1) $1.35 \sqrt{\frac{\mathrm{GM}}{a}}$(2) $1.16 \sqrt{\frac{\mathrm{GM}}{a}}$(3) $1.21 \sqrt{\frac{\mathrm{GM}}{a}}$(4) $1.41 \sqrt{\frac{\mathrm{GM}}{a}}$Correct Option: , 2 Solution: (2) $A C=a \sqrt{2} \quad \because r=\frac{A C}{2}=...
Read More →The gas released during anaerobic degradation of vegetation may lead to:
Question: The gas released during anaerobic degradation of vegetation may lead to:Global warming and cancerAcid rainCorrosion of metalsOzone holeCorrect Option: 1 Solution: Biogas is the mixtrue of gases produced by the breakdown of organic matter in the absence of oxygen (anaerobically), primary consisting of methane and carbondioxide. Biogas can be produced from raw material such as agricultural waste, manure, municiple waste, plant material, sewage, green waste or good waste. Due to release o...
Read More →Let A be a 3x3 matrix such that adj A=
Question: Let $A$ be a $3 \times 3$ matrix such that $\operatorname{adj} A=\left[\begin{array}{ccc}2 -1 1 \\ -1 0 2 \\ 1 -2 -1\end{array}\right]$ and $B=\operatorname{adj}(\operatorname{adj} A)$. If $|A|=\lambda$ and $\left|\left(B^{-1}\right)^{T}\right|=\mu$, then the ordered pair, $(|\lambda|, \mu)$ is equal to :(1) $\left(3, \frac{1}{81}\right)$(2) $\left(9, \frac{1}{9}\right)$(3) $(3,81)$(4) $\left(9, \frac{1}{81}\right)$Correct Option: 1 Solution: $|\operatorname{adj} A|=|A|^{2}=9 \quad\lef...
Read More →The sum of a number and its reciprocal is
Question: The sum of a number and its reciprocal is $2 \frac{1}{30}$. Find the number. Solution: Let the required number bex.According to the given condition, $x+\frac{1}{x}=2 \frac{1}{30}$ $\Rightarrow \frac{x^{2}+1}{x}=\frac{61}{30}$ $\Rightarrow 30 x^{2}+30=61 x$ $\Rightarrow 30 x^{2}-61 x+30=0$ $\Rightarrow 30 x^{2}-36 x-25 x+30=0$ $\Rightarrow 6 x(5 x-6)-5(5 x-6)=0$ $\Rightarrow(5 x-6)(6 x-5)=0$ $\Rightarrow 5 x-6=0$ or $6 x-5=0$ $\Rightarrow x=\frac{6}{5}$ or $x=\frac{5}{6}$ Hence, the req...
Read More →Given below are two statements :
Question: Given below are two statements : Statement I : Non-biodegradable wastes are generated by the thermal power plants. Statement II : Bio-degradable detergents leads to eutrophication. In the light of the above statements, choose the most appropriate answer from the option given belowBoth statement I and statement II are falseStatement I is true but statement II is falseStatement I is false but statement II is trueBoth statement I and statement II are true.Correct Option: , 4 Solution: Non...
Read More →The satements that are TRUE :
Question: The satements that are TRUE : (A) Methane leads to both global warming and photochemical smog (B) Methane is generated from paddy fields (C) Methane is a stronger global warming gas than $\mathrm{CO}_{2}$ (D) Methane is a part of reducing smog Choose the most appropriate answer from the options given below :$(\mathrm{A}),(\mathrm{B}),(\mathrm{C})$ only(A) and (B) only$(\mathrm{B}),(\mathrm{C}),(\mathrm{D})$ only(A), (B), (D) onlyCorrect Option: 1 Solution: Methane leads to both global ...
Read More →Prove the following
Question: If $\Delta=\left|\begin{array}{ccc}x-2 2 x-3 3 x-4 \\ 2 x-3 3 x-4 4 x-5 \\ 3 x-5 5 x-8 10 x-17\end{array}\right|=A x^{3}+B x^{2}+C x+D$ then $B+C$ is equal to:(1) $-1$(2) 1(3) $-3$(4) 9Correct Option: , 3 Solution: $\Delta=\left|\begin{array}{ccc}x-2 2 x-3 3 x-4 \\ 2 x-3 3 x-4 4 x-5 \\ 3 x-5 5 x-8 10 x-17\end{array}\right|$ $\Rightarrow \Delta=\left|\begin{array}{ccc}x-2 x-1 x-1 \\ 2 x-3 x-1 x-1 \\ 3 x-5 2 x-3 5 x-9\end{array}\right| \quad\left[\begin{array}{l}C_{3} \rightarrow C_{3}-C...
Read More →The numerator of a fraction is 3 less than its denominator.
Question: The numerator of a fraction is 3 less than its denominator. If 1 is added to the denominator, the fraction is decreased by $\frac{1}{15}$. Find the fraction. Solution: Let the denominator of the required fraction bex.Numerator of the required fraction =x 3 $\therefore$ Original fraction $=\frac{x-3}{x}$ If 1 is added to the denominator, then the new fraction obtained is $\frac{x-3}{x+1}$. According to the given condition, $\frac{x-3}{x+1}=\frac{x-3}{x}-\frac{1}{15}$ $\Rightarrow \frac{...
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