If the equation
Question: If the equation $\left(1+m^{2}\right) x^{2}+2 m c x+\left(c^{2}-a^{2}\right)=0$ has equal roots, prove that $c^{2}=a^{2}\left(1+m^{2}\right)$. Solution: Given: $\left(1+m^{2}\right) x^{2}+2 m c x+\left(c^{2}-a^{2}\right)=0$ Here, $a=\left(1+m^{2}\right), b=2 m c$ and $c=\left(c^{2}-a^{2}\right)$ It is given that the roots of the equation are equal; therefore, we have: $D=0$ $\Rightarrow\left(b^{2}-4 a c\right)=0$ $\Rightarrow(2 m c)^{2}-4 \times\left(1+m^{2}\right) \times\left(c^{2}-a^...
Read More →The diameter and height of a cylinder are measured
Question: The diameter and height of a cylinder are measured by a meter scale to be $12.6 \pm 0.1 \mathrm{~cm}$ and $34.2 \pm 0.1 \mathrm{~cm}$, respectively. What will be the value of its volume in appropriate significant figures? appropriate significant figures? proprigute(1) $4264 \pm 81 \mathrm{~cm}^{3}$(2) $4264.4 \pm 81.0 \mathrm{~cm}^{3}$(3) $4260 \pm 80 \mathrm{~cm}^{3}$(4) $4300 \pm 80 \mathrm{~cm}^{3}$Correct Option: 3 Solution: (3)...
Read More →The highest possible oxidation states of uranium and plutonium,
Question: The highest possible oxidation states of uranium and plutonium, respectively, are :6 and 76 and 47 and 64 and 6Correct Option: 1 Solution: Maximum oxidation state shown by Uranium is $+6$ and Plutonium is 7 ....
Read More →The maximum number of possible oxidation states of
Question: The maximum number of possible oxidation states of actinoides are shown by:Nobelium (No) and lawrencium (Lr)Actinium (Ac) and thorium (Th)Berkelium (Bk) and californium (Cf)Neptunium (Np) and plutonium (Pu)Correct Option: , 4 Solution:...
Read More →Contrapositive of the statement :
Question: Contrapositive of the statement : 'If a function $f$ is differentiable at $a$, then it is also continuous at $a^{\prime}$, is :(1) If a function $f$ is continuous at $a$, then it is not differentiable at $a$.(2) If a function $f$ is not continuous at $a$, then it is not differentiable at $a$.(3) If a function $f$ is not continuous at $a$, then it is differentiable at $a$(4) If a function $f$ is continuous at $a$, then it is differentiable at $a$.Correct Option: , 2 Solution: (2) Contra...
Read More →If −4 is a root of the quadratic equation
Question: If $-4$ is a root of the quadratic equation $x^{2}+2 x+4 p=0$, find the value of $k$ for which the quadratic equation $x^{2}+p x(1+3 k)+7(3+2 k)=0$ has equal roots. Solution: It is given that $-4$ is a root of the quadratic equation $x^{2}+2 x+4 p=0$. $\therefore(-4)^{2}+2 \times(-4)+4 p=0$ $\Rightarrow 16-8+4 p=0$ $\Rightarrow 4 p+8=0$ $\Rightarrow p=-2$ The equation $x^{2}+p x(1+3 k)+7(3+2 k)=0$ has equal roots. $\therefore D=0$ $\Rightarrow[p(1+3 k)]^{2}-4 \times 1 \times 7(3+2 k)=0...
Read More →Given the following two statements :
Question: Given the following two statements : $\left(S_{1}\right):(q \vee p) \rightarrow(p \leftrightarrow \sim q)$ is a tautology. $\left(S_{2}\right): \sim q \wedge(\sim p \leftrightarrow q)$ is a fallacy. Then :(1) both $\left(S_{1}\right)$ and $\left(S_{2}\right)$ are correct(2) only $\left(S_{1}\right)$ is correct(3) only $\left(S_{2}\right)$ is correct(4) both $\left(S_{1}\right)$ and $\left(S_{2}\right)$ are not correctCorrect Option: , 4 Solution: The truth table of both the statements ...
Read More →The statement that is INCORRECT about the interstitial compounds is :
Question: The statement that is INCORRECT about the interstitial compounds is :they are chemically reactive.they are very hard.they have metallic conductivity.they have high melting points.Correct Option: 1 Solution: Interstitial compounds are inert, i.e., they are chemically non-reactive....
Read More →If 3 is a root of the quadratic equation
Question: If 3 is a root of the quadratic equation $x^{2}-x+k=0$, find the value of $p$ so that the roots of the equation $x^{2}+k(2 x+k+2)+p=0$ are equal. Solution: It is given that 3 is a root of the quadratic equation $x^{2}-x+k=0$. $\therefore(3)^{2}-3+k=0$ $\Rightarrow k+6=0$ $\Rightarrow k=-6$ The roots of the equation $x^{2}+2 k x+\left(k^{2}+2 k+p\right)=0$ are equal. $\therefore D=0$ $\Rightarrow(2 k)^{2}-4 \times 1 \times\left(k^{2}+2 k+p\right)=0$ $\Rightarrow 4 k^{2}-4 k^{2}-8 k-4 p=...
Read More →The lanthanide ion that would show colour is :
Question: The lanthanide ion that would show colour is :$\mathrm{Gd}^{3+}$$\mathrm{Sm}^{3+}$$\mathrm{La}^{3+}$$\mathrm{Lu}^{3+}$Correct Option: , 2 Solution: $\mathrm{Sm}=4 f^{6} 6 s^{2}$ $\mathrm{Sm}^{3+}=4 f^{5}=$ Partially filled $f$ orbital $\therefore \mathrm{Sm}^{3+}$ will be coloured $\mathrm{Lu}^{3+}=4 f^{14}=$ colourless....
Read More →Let p, q, r be three statements such that
Question: Let $p, q, r$ be three statements such that the truth value of $(p \wedge q) \rightarrow(\sim q \vee r)$ is $\mathrm{F}$. Then the truth values of $p, q, r$ are respectively:(1) $\mathrm{T}, \mathrm{F}, \mathrm{T}$(2) T, T, T(3) $\mathrm{F}, \mathrm{T}, \mathrm{F}$(4) $\mathrm{T}, \mathrm{T}, \mathrm{F}$Correct Option: , 4 Solution: $(p \wedge q) \rightarrow(\sim q \vee r)$ $=\sim(p \wedge q) \vee(\sim q \vee r)$ $=(\sim p \vee \sim q) \vee(\sim q \vee r)$ $=(\sim p \vee \sim q \vee r)...
Read More →The sum of the total number of bonds between chromium
Question: The sum of the total number of bonds between chromium and oxygen atoms in chromate and dichromate ions is _________ . Solution: (12.00)...
Read More →The proposition
Question: The proposition $p \rightarrow \sim(p \wedge \sim q)$ is equivalent to:(1) $q$(2) $(\sim p) \vee q$(3) $(\sim p) \wedge q$(4) $(\sim p) \vee(\sim q)$Correct Option: , 2 Solution: $\therefore p \rightarrow \sim(p \wedge \sim q)$ is equivalent to $\sim p \vee q$...
Read More →If −5 is a root of the quadratic equation
Question: If $-5$ is a root of the quadratic equation $2 x^{2}+p x-15=0$ and the quadratic equation $p\left(x^{2}+x\right)+k=0$ has equal roots, find the value of $k$. Solution: It is given that $-5$ is a root of the quadratic equation $2 x^{2}+p x-15=0$. $\therefore 2(-5)^{2}+p \times(-5)-15=0$ $\Rightarrow-5 p+35=0$ $\Rightarrow p=7$ The roots of the equation $p x^{2}+p x+k=0=0$ are equal. $\therefore D=0$ $\Rightarrow p^{2}-4 p k=0$ $\Rightarrow(7)^{2}-4 \times 7 \times k=0$ $\Rightarrow 49-2...
Read More →The electronic configurations of bivalent europium and trivalent cerium are:
Question: The electronic configurations of bivalent europium and trivalent cerium are: (atomic number: $\mathrm{Xe}=54, \mathrm{Ce}=58, \mathrm{Eu}=63$ )$[\mathrm{Xe}] 4 f^{2}$ and $[\mathrm{Xe}] 4 f^{7}$$[\mathrm{Xe}] 4 f^{7}$ and $[\mathrm{Xe}] 4 f^{1}$$[\mathrm{Xe}] 4 f^{7} 6 \mathrm{~s}^{2}$ and $[\mathrm{Xe}] 4 f^{2} 6 \mathrm{~s}^{2}$$[\mathrm{Xe}] 4 f^{4}$ and $[\mathrm{Xe}] 4 f^{9}$Correct Option: , 2 Solution: $\mathrm{Eu}^{2+}:[\mathrm{Xe}] 4 f^{7} ; \mathrm{Ce}^{3+}:[\mathrm{Xe}] 4 f^...
Read More →Which of the following is a tautology?
Question: Which of the following is a tautology?(1) $(\sim \mathrm{p}) \wedge(\mathrm{p} \vee \mathrm{q}) \rightarrow \mathrm{q}$(2) $(\mathrm{q} \rightarrow \mathrm{p}) \vee \sim(\mathrm{p} \rightarrow \mathrm{q})$(3) $(\sim \mathrm{q}) \vee(\mathrm{p} \wedge \mathrm{q}) \rightarrow \mathrm{q}$(4) $(\mathrm{p} \rightarrow \mathrm{q}) \wedge(\mathrm{q} \rightarrow \mathrm{p})$Correct Option: 1 Solution: Truth table $\therefore(\mathrm{a}) \sim \mathrm{p} \wedge(\mathrm{p} \vee \mathrm{q}) \right...
Read More →The third ionization enthalpy is minimum for:
Question: The third ionization enthalpy is minimum for:CoFeNiMnCorrect Option: , 2 Solution: ${ }_{26} \mathrm{Fe}=[\mathrm{Ar}] 3 d^{6} 4 s^{2} .$ Third ionisation results into stable $d^{5}$ configuration....
Read More →Find the values of p for which the quadratic equation
Question: Find the values of $p$ for which the quadratic equation $(p+1) x^{2}-6(p+1) x+3(p+9)=0, p \neq-1$ has equal roots. Hence, find the roots of the equation. Solution: The given equation is $(p+1) x^{2}-6(p+1) x+3(p+9)=0$. This is of the form $a x^{2}+b x+c=0$, where $a=p+1, b=-6(p+1)$ and $c=3(p+9)$. $\therefore D=b^{2}-4 a c$ $=[-6(p+1)]^{2}-4 \times(p+1) \times 3(p+9)$ $=12(p+1)[3(p+1)-(p+9)]$ $=12(p+1)(2 p-6)$ The given equation will have real and equal roots ifD= 0. $\therefore 12(p+1...
Read More →In an electron microscope,
Question: In an electron microscope, the resolution that can be achieved is of the order of the wavelength of electrons used. To resolve a width of $7.5 \times 10^{-12} \mathrm{~m}$, the minimum electron energy required is close to:(1) $500 \mathrm{keV}$(2) $100 \mathrm{keV}$(3) $1 \mathrm{keV}$(4) $25 \mathrm{keV}$Correct Option: 4 Solution: (4) Using, $\lambda=\frac{h}{p}\left\{\right.$ given: $\left.\lambda=7.5 \times 10^{-12}\right\}$ $\Rightarrow \mathrm{P}=\frac{\mathrm{h}}{\lambda}$ Minim...
Read More →Consider the following reactions:
Question: Consider the following reactions: $\mathrm{NaCl}+\mathrm{K}_{2} \mathrm{Cr}_{2} \mathrm{O}_{7}+\mathrm{H}_{2} \mathrm{SO}_{4} \rightarrow(\mathrm{A})+$ Side products (Conc.) $(A)+\mathrm{NaOH} \rightarrow(\mathrm{B})+$ Side products (B) $+\mathrm{H}_{2} \mathrm{SO}_{4}+\mathrm{H}_{2} \mathrm{O}_{2} \rightarrow$ (C) $+$ Side products (dilute) The sum of the total number of atoms in one molecule each of $(A),(B)$ and $(C)$ is ______________ . Solution: (18.00)...
Read More →The contrapositive of the statement
Question: The contrapositive of the statement "If $I$ reach the station in time, then $I$ will catch the train" is :(1) If $I$ do not reach the station in time, then $I$ will catch the train.(2) If $I$ do not reach the station in time, then $I$ will not catch the train.(3) If $I$ will catch the train, then $I$ reach the station in time.(4) If $I$ will not catch the train, then $I$ do not reach the station in time.Correct Option: , 4 Solution: Contrapositive of $p \rightarrow q$ is $\sim q \right...
Read More →The pitch and the number of divisions,
Question: The pitch and the number of divisions, on the circular scale for a given screw gauge are $0.5 \mathrm{~mm}$ and 100 respectively. When the screw gauge is fully tightened without any object, the zero of its circular scale lies 3 division below the mean line. The readings of the main scale and the circular scale, for a thin sheet, are $5.5 \mathrm{~mm}$ and 48 respectively, the thickness of the sheet is:(1) $5.755 \mathrm{~mm}$(2) $5.950 \mathrm{~mm}$(3) $5.725 \mathrm{~mm}$(4) $5.740 \m...
Read More →The pitch and the number of divisions,
Question: The pitch and the number of divisions, on the circular scale for a given screw gauge are $0.5 \mathrm{~mm}$ and 100 respectively. When the screw gauge is fully tightened without any object, the zero of its circular scale lies 3 division below the mean line. The readings of the main scale and the circular scale, for a thin sheet, are $5.5 \mathrm{~mm}$ and 48 respectively, the thickness of the sheet is:(1) $5.755 \mathrm{~mm}$(2) $5.950 \mathrm{~mm}$(3) $5.725 \mathrm{~mm}$(4) $5.740 \m...
Read More →The pitch and the number of divisions,
Question: The pitch and the number of divisions, on the circular scale for a given screw gauge are $0.5 \mathrm{~mm}$ and 100 respectively. When the screw gauge is fully tightened without any object, the zero of its circular scale lies 3 division below the mean line. The readings of the main scale and the circular scale, for a thin sheet, are $5.5 \mathrm{~mm}$ and 48 respectively, the thickness of the sheet is:(1) $5.755 \mathrm{~mm}$(2) $5.950 \mathrm{~mm}$(3) $5.725 \mathrm{~mm}$(4) $5.740 \m...
Read More →Let F1 (A,B,C) =
Question: Let $F_{1}(A, B, C)=(A \wedge \sim B) \vee[\sim C \wedge(A \vee B)] \vee \sim A$ and $F_{2}(A, B)=(A \vee B) \vee(B \rightarrow \sim A)$ be two logical expressions. Then:(1) $\mathrm{F}_{1}$ is not a tautology but $\mathrm{F}_{2}$ is a tautology(2) $\mathrm{F}_{1}$ is a tautology but $\mathrm{F}_{2}$ is not a tautology(3) $\mathrm{F}_{1}$ and $\mathrm{F}_{2}$ both area tautologies(4) Both $\mathrm{F}_{1}$ and $\mathrm{F}_{2}$ are not tautologiesCorrect Option: 1 Solution: Truth table f...
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