Planet A has mass M and radius R.
Question: Planet $\mathrm{A}$ has mass $\mathrm{M}$ and radius $\mathrm{R}$. Planet $\mathrm{B}$ has half the mass and half the radius of Planet A. If the escape velocities from the Planets $\mathrm{A}$ and $\mathrm{B}$ are $v_{\mathrm{A}}$ and $v_{\mathrm{B}}$, respectively, then $\frac{v_{\mathrm{A}}}{v_{\mathrm{B}}}=\frac{n}{4}$. The value of $n$ is :(1) 4(2) 1(3) 2(4) 3Correct Option: 1 Solution: (1) Escape velocity of the planet $A$ is $V_{A}=\sqrt{\frac{2 G M_{A}}{R_{A}}}$ where $M_{A}$ an...
Read More →Which of the following statement(s) is (are) incorrect reason for eutrophication?
Question: Which of the following statement(s) is (are) incorrect reason for eutrophication? (A) excess usage of fertilisers (B) excess usage of detergents (C) dense plant population in water bodies (D) lack of nutrients in water bodies that prevent plant growth Choose the most appropriate answer from the options given below:(A) only(C) only(B) and (D) only(D) onlyCorrect Option: , 4 Solution: The process in which nutrient enriched water bodies support a dense plant population which kills animal ...
Read More →Reducing smog is a mixture of:
Question: Reducing smog is a mixture of:Smoke, fog and $\mathrm{O}_{3}$Smoke, fog and $\mathrm{SO}_{2}$$\mathrm{Smoke}$, fog and $\mathrm{CH}_{2}=\mathrm{CH}-\mathrm{CHO}$Smoke, fog and $\mathrm{N}_{2} \mathrm{O}_{3}$Correct Option: , 2 Solution: Reducing or classical smog is the combination of smoke, fog and $\mathrm{SO}_{2}$....
Read More →Prove the following
Question: Let $a, b, c \in \mathbf{R}$ be all non-zero and satisfy $a^{3}+b^{3}+c^{3}=2$. If the matrix $A=\left(\begin{array}{lll}a b c \\ b c a \\ c a b\end{array}\right)$ satisfies $A^{T} A=I$, then a value of $a b c$ can be $:$(1) $-\frac{1}{3}$(2) $\frac{1}{3}$(3) 3(4) $\frac{2}{3}$Correct Option: , 2 Solution: Given: $A^{T} A=I$ $\Rightarrow\left[\begin{array}{lll}a b c \\ b c a \\ c a b\end{array}\right]\left[\begin{array}{lll}a b c \\ b c a \\ c a b\end{array}\right]=\left[\begin{array}{...
Read More →An asteroid is moving directly towards the centre of the earth.
Question: An asteroid is moving directly towards the centre of the earth. When at a distance of $10 R$ ( $R$ is the radius of the earth) from the earths centre, it has a speed of $12 \mathrm{~km} / \mathrm{s}$. Neglecting the effect of earths atmosphere, what will be the speed of the asteroid when it hits the surface of the earth (escape velocity from the earth is $11.2 \mathrm{~km} / \mathrm{s}$ )? Give your answer to the nearest integer in kilometer/s Solution: $(16.00)$ Using law of conservat...
Read More →The green house gas/es is (are):
Question: The green house gas/es is (are): (A) Carbon dioxide (B) Oxygen (C) Water vapour (D) Methane Choose the most appropriate answer from the options given below:(A) and (C) only(A) only(A), (C) and (D) only(A) and (B) onlyCorrect Option: , 3 Solution: The green house gases are $\mathrm{CO}_{2}, \mathrm{H}_{2} \mathrm{O}_{(\text {vapour })}$ \ $\mathrm{CH}_{4}$...
Read More →The denominator of a fraction is 3 more than its numerator.
Question: The denominator of a fraction is 3 more than its numerator. The sum of the fraction and its reciprocal is $2 \frac{9}{10}$. Find the fraction. Solution: Let the numerator be $x$. $\therefore$ Denominator $=x+3$ $\therefore$ Original number $=\frac{x}{x+3}$ According to the question: $\frac{x}{x+3}+\frac{1}{\left(\frac{x}{x+3}\right)}=2 \frac{9}{10}$ $\Rightarrow \frac{x}{x+3}+\frac{x+3}{x}=\frac{29}{10}$ $\Rightarrow \frac{x^{2}+(x+3)^{2}}{x(x+3)}=\frac{29}{10}$ $\Rightarrow \frac{x^{2...
Read More →The type of pollution that gets increased during the day time and in the presence of
Question: The type of pollution that gets increased during the day time and in the presence of $\mathrm{O}_{3}$ is:Reducing smogOxidising smogGlobal warmingAcid rain OfficialCorrect Option: , 2 Solution: In presence of ozone $\left(\mathrm{O}_{3}\right)$, oxidising smog gets increased during the day time because automobiles and factories produce main components of the photochemcial smog (oxidising smog) results from the action of sunlight on unsaturated hydrocarbon and nitrogen oxide. Ozone is s...
Read More →Consider two solid spheres of radii
Question: Consider two solid spheres of radii $R_{1}=1 m, R_{2}=2 m$ and masses $M_{1}$ and $M_{2}$, respectively. The gravitational field due to sphere (1) and (2) are shown. The value of $\frac{m_{1}}{m_{2}}$ is: (1) $\frac{2}{3}$(2) $\frac{1}{6}$(3) $\frac{1}{2}$(4) $\frac{1}{3}$Correct Option: , 2 Solution: (2) Gravitation field at the surface $E=\frac{G m}{r^{2}}$ $\therefore E_{1}=\frac{G m_{1}}{r_{1}^{2}}$ and $E_{2}=\frac{G m_{2}}{r_{2}^{2}}$ From the diagram given in question, $\frac{E_...
Read More →Solve the following
Question: Let $\mathrm{A}=\left\{\mathrm{X}=(x, y, z)^{\mathrm{T}}: \mathrm{PX}=0\right.$ and $\left.x^{2}+y^{2}+z^{2}=1\right\}$, where $P=\left[\begin{array}{ccc}1 2 1 \\ -2 3 -4 \\ 1 9 -1\end{array}\right]$, then the set $\mathrm{A}$ :(1) is a singleton(2) is an empty set(3) contains more than two elements(4) contains exactly two elementsCorrect Option: , 4 Solution: $\because|P|=1(-3+36)-2(2+4)+1(-18-3)=0$ Given that $P X=0$ $\therefore$ System of equations $x+2 y+z=0 ; 2 x-3 y+4 z=0$ and $x...
Read More →A two-digit number is such that the product of its digits is 14.
Question: A two-digit number is such that the product of its digits is 14. If 45 is added to the number, the digits interchange their places. Find the number. Solution: Let the digits at units and tens places be $x$ and $y$, respectively. $\therefore x y=14$ $\Rightarrow y=\frac{14}{x} \quad \ldots(\mathrm{i})$ According to the question: $(10 y+x)+45=10 x+y$ $\Rightarrow 9 y-9 x=-45$ $\Rightarrow y-x=-5$ ........(ii) From (i) and (ii), we get: $\frac{14}{x}-x=-5$ $\Rightarrow \frac{14-x^{2}}{x}=...
Read More →Let S be the set of all
Question: Let $S$ be the set of all $\lambda \in \mathbf{R}$ for which the system of linear equations $2 x-y+2 z=2 \quad x-2 y+\lambda z=-4$ $x+\lambda y+z=4$ has no solution. Then the set $S$ (1) contains more than two elements.(2) is an empty set.(3) is a singleton.(4) contains exactly two elements.Correct Option: , 4 Solution: $\Delta=\left|\begin{array}{ccc}2 -1 2 \\ 1 -2 \lambda \\ 1 \lambda 1\end{array}\right|=-(\lambda-1)(2 \lambda+1)$ $\Delta_{1}=\left|\begin{array}{ccc}2 -1 2 \\ -4 -2 \...
Read More →A box weighs 196 N on a spring balance at the north pole.
Question: A box weighs $196 \mathrm{~N}$ on a spring balance at the north pole. Its weight recorded on the same balance if it is shifted to the equator is close to (Take $g=10 \mathrm{~ms}^{-2}$ at the north pole and the radius of the earth $=6400 \mathrm{~km}$ ):(1) $195.66 \mathrm{~N}$(2) $194.32 \mathrm{~N}$(3) $194.66 \mathrm{~N}$(4) $195.32 \mathrm{~N}$Correct Option: , 4 Solution: (4) Weight at pole, $w=m g=196 \mathrm{~N}$ $\Rightarrow m=19.6 \mathrm{~kg}$ Weight at equator, $w^{\prime}=m...
Read More →A two-digit number is 4 times the sum of its digits and twice the product of its digit
Question: A two-digit number is 4 times the sum of its digits and twice the product of its digit. Find the number. Solution: Let the digits at units and tens places be $x$ and $y$, respectively. $\therefore$ Original number $=10 y+x$ According to the question: $10 y+x=4(x+y)$ $\Rightarrow 10 y+x=4 x+4 y$ $\Rightarrow 3 x-6 y=0$ $\Rightarrow 3 x=6 y$ $\Rightarrow x=2 y \ldots .(\mathrm{i})$ Also, $10 y+x=2 x y$ $\Rightarrow 10 y+2 y=2.2 y . y \quad[$ From (i) $]$ $\Rightarrow 12 y=4 y^{2}$ $\Righ...
Read More →Let A be a 2x2 real matrix with entries from
Question: Let $A$ be a $2 \times 2$ real matrix with entries from $\{0,1\}$ and $|A| \neq 0$. Consider the following two statements : (P) If $A \neq I_{2}$, then $|A|=-1$ (Q) If $|A|=1$, then $\operatorname{tr}(A)=2$, where $I_{2}$ denotes $2 \times 2$ identity matrix and $\operatorname{tr}(A)$ denotes the sum of the diagonal entries of $A$. Then : (1) (P) is false and (Q) is true(2) Both (P) and (Q) are false(3) (P) is true and (Q) is false(4) Both (P) and (Q) are trueCorrect Option: , 4 Soluti...
Read More →A satellite of mass m is launched vertically upwards with an initial speed u from the surface of the earth.
Question: A satellite of mass $m$ is launched vertically upwards with an initial speed $u$ from the surface of the earth. After it reaches height $R$ ( $R=$ radius of the earth $)$, it ejects a rocket of mass $\frac{m}{10}$ so that subsequently the satellite moves in a circular orbit. The kinetic energy of the rocket is ( $G$ is the gravitational constant; $M$ is the mass of the earth):(1) $\frac{m}{20}\left(u^{2}+\frac{113}{200} \frac{G M}{R}\right)$(2) $5 m\left(u^{2}-\frac{119}{200} \frac{G M...
Read More →The standard electrode potential
Question: The standard electrode potential $\mathrm{E}^{0}$ and its temperature coefficient $\left(\frac{\mathrm{dE}^{0}}{\mathrm{dT}}\right)$ for a cell are $2 \mathrm{~V}$ and $-5 \times 10^{-4} \mathrm{VK}^{-1}$ at 300 $K$ respectively. The cell reaction is: $\mathrm{Zn}(\mathrm{s})+\mathrm{Cu}^{2+}(\mathrm{aq}) \rightarrow \mathrm{Zn}^{2+}(\mathrm{aq})+\mathrm{Cu}$ The standard reaction enthalpy $\left(\Delta_{\mathrm{r}} \mathrm{H}^{0}\right)$ at $300 \mathrm{~K}$ in $\mathrm{kJ} \mathrm{mo...
Read More →Three consecutive positive integers are such that the sum of the square of the first and the product of the other two is 46.
Question: Three consecutive positive integers are such that the sum of the square of the first and the product of the other two is 46. Find the integers Solution: Let the three consecutive positive integers bex,x+ 1 andx+ 2.According to the given condition, $x^{2}+(x+1)(x+2)=46$ $\Rightarrow x^{2}+x^{2}+3 x+2=46$ $\Rightarrow 2 x^{2}+3 x-44=0$ $\Rightarrow 2 x^{2}+11 x-8 x-44=0$ $\Rightarrow x(2 x+11)-4(2 x+11)=0$ $\Rightarrow(2 x+11)(x-4)=0$ $\Rightarrow 2 x+11=0$ or $x-4=0$ $\Rightarrow x=-\fr...
Read More →If the matrix A=
Question: If the matrix $A=\left[\begin{array}{ccc}1 0 0 \\ 0 2 0 \\ 3 0 -1\end{array}\right]$ satisfies the equation $A^{20}+\alpha A^{19}+\beta A=\left[\begin{array}{lll}1 0 0 \\ 0 4 0 \\ 0 0 1\end{array}\right]$ for some real numbers $\alpha$ and $\beta$, then $\beta-$ $\alpha$ is equal to Solution: $A^{2}=\left[\begin{array}{ccc}1 0 0 \\ 0 2 0 \\ 3 0 -1\end{array}\right]\left[\begin{array}{ccc}1 0 0 \\ 0 2 0 \\ 3 0 -1\end{array}\right]=\left[\begin{array}{ccc}1 0 0 \\ 0 4 0 \\ 0 0 1\end{arra...
Read More →Consider the following system of equations:
Question: Consider the following system of equations: $x+2 y-3 z=a$ $2 x+6 y-11 z=b$ $x-2 y+7 z=c$ where $\mathrm{a}, \mathrm{b}$ and $\mathrm{c}$ are real constants. Then the system of equations :(1) has a unique solution when $5 a=2 b+c$(2) has infinite number of solutions when $5 a=2 b+c$(3) has no solution for all $a, b$ and $c$(4) has a unique solution for all $a, b$ and $c$Correct Option: , 2 Solution: $D=\left|\begin{array}{ccc}1 2 -3 \\ 2 6 -11 \\ 1 -2 7\end{array}\right|$ $=20-2(25)-3(-...
Read More →Given the equilibrium constant :
Question: Given the equilibrium constant : $K_{C}$ of the reaction : $\mathrm{Cu}(\mathrm{s})+2 \mathrm{Ag}^{+}(\mathrm{aq}) \rightarrow \mathrm{Cu}^{2+}(\mathrm{aq})+2 \mathrm{Ag}(\mathrm{s})$ is $10 \times 10^{15}$ calculate the $\mathrm{E}_{\text {cell }}^{0}$ of this reaction at $298 \mathrm{~K}$ $\left[2.303 \frac{\mathrm{RT}}{\mathrm{F}}\right.$ at $\left.298 \mathrm{~K}=0.059 \mathrm{~V}\right]$$0.04736 \mathrm{mV}$$0.4736 \mathrm{mV}$$0.4736 \mathrm{~V}$$0.04736 \mathrm{~V}$Correct Optio...
Read More →The difference of the squares of two natural numbers is 45. The squares of the smaller number is four times the largest number.
Question: The difference of the squares of two natural numbers is 45. The squares of the smaller number is four times the largest number. Find the numbers. Solution: Let the greater number be $x$ and the smaller number be $y$. According to the question: $x^{2}-y^{2}=45 \quad \ldots($ i) $y^{2}=4 x \quad \ldots$ (ii) From (i) and (ii), we get: $x^{2}-4 x=45$ $\Rightarrow x^{2}-4 x-45=0$ $\Rightarrow x^{2}-(9-5) x-45=0$ $\Rightarrow x^{2}-9 x+5 x-45=0$ $\Rightarrow x(x-9)+5(x-9)=0$ $\Rightarrow(x-...
Read More →Two planets have masses $M$ and $16 M$ and their radii are a and 2 a,
Question: Two planets have masses $M$ and $16 M$ and their radii are $a$ and $2 a$, respectively. The separation between the centres of the planets is $10 a$. A body of mass $\mathrm{m}$ is fired from the surface of the larger planet towards the smaller planet along the line joining their centres. For the body to be able to reach the surface of smaller planet, the minimum firing speed needed is : respectively. The separation between the centres of the planets is $10 a$. A body of mass $\mathrm{m...
Read More →Let A be a symmetric matrix of order 2 with
Question: Let $\mathrm{A}$ be a symmetric matrix of order 2 with integer entries. If the sum of the diagonal elements of $\mathrm{A}^{2}$ is 1 , then the possible number of such matrices is:(1) 6(2) 1(3) 4(4) 12Correct Option: , 3 Solution: Let $A=\left[\begin{array}{ll}a b \\ b c\end{array}\right]$ $A^{2}=\left[\begin{array}{ll}a^{L} b \\ b c\end{array}\right]\left[\begin{array}{ll}a b \\ b c\end{array}\right]=\left[\begin{array}{ll}a^{2}+b^{2} a b+b c \\ a b+b c c^{2}+b^{2}\end{array}\right]$ ...
Read More →Find two natural numbers, the sum of whose squares is 25 times their sum and also equal to 50 times their difference.
Question: Find two natural numbers, the sum of whose squares is 25 times their sum and also equal to 50 times their difference. Solution: Let the two natural numbers be $x$ and $y$. According to the question: $x^{2}+y^{2}=25(x+y) \ldots$ (i) $x^{2}+y^{2}=50(x-y) \ldots$ (ii) From (i) and (ii), we get: $25(x+y)=50(x-y)$ $\Rightarrow x+y=2(x-y)$ $\Rightarrow x+y=2 x-2 y$ $\Rightarrow y+2 y=2 x-x$ $\Rightarrow 3 y=x \quad \ldots$ (iii) From (ii) and (iii), we get: $(3 y)^{2}+y^{2}=50(3 y-y)$ $\Righ...
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