Prove the following
Question: If $P=\left[\begin{array}{ll}1 0 \\ 1 / 2 1\end{array}\right]$, then $P^{50}$ is:$\left[\begin{array}{ll}1 0 \\ 25 1\end{array}\right]$$\left[\begin{array}{ll}1 50 \\ 0 1\end{array}\right]$$\left[\begin{array}{ll}1 25 \\ 0 1\end{array}\right]$$\left[\begin{array}{ll}1 0 \\ 50 1\end{array}\right]$Correct Option: 1 Solution: $P=\left[\begin{array}{ll}1 0 \\ \frac{1}{2} 1\end{array}\right]$ $\mathrm{P}^{2}=\left[\begin{array}{ll}1 0 \\ \frac{1}{2} 1\end{array}\right]\left[\begin{array}{ll...
Read More →Solve this following
Question: Let $\overrightarrow{\mathrm{a}}=\hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}}, \overrightarrow{\mathrm{b}}$ and $\overrightarrow{\mathrm{c}}=\hat{\mathrm{j}}-\hat{\mathrm{k}}$ be three vectors such that $\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}=\overrightarrow{\mathrm{c}}$ and $\overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{b}}=1$. If the length of projection vector of the vector $\vec{b}$ on the vector $\overrightarrow{\mathrm{a}} \times \overrig...
Read More →Complete combustion of
Question: Complete combustion of $1.80 \mathrm{~g}$ of an oxygen containing compound $\left(\mathrm{C}_{\mathrm{x}} \mathrm{H}_{\mathrm{y}} \mathrm{O}_{\mathrm{z}}\right)$ gave $2.64 \mathrm{~g}$ of $\mathrm{CO}_{2}$ and $1.08 \mathrm{~g}$ of $\mathrm{H}_{2} \mathrm{O}$. The percentage of oxygen in the organic compound is:51.6363.5353.3350.33Correct Option: , 3 Solution: $\mathrm{n}_{\mathrm{c}}=\mathrm{n}_{\mathrm{co}_{2}}=\frac{2.64}{44}=0.06$ $\mathrm{n}_{\mathrm{H}}=2 \times \mathrm{n}_{\mat...
Read More →For real numbers
Question: For real numbers $\alpha$ and $\beta$, consider the following system of linear equations: $x+y-z=2, x+2 y+\alpha z=1,2 x-y+z=\beta$ If the system has infinite solutions, then $\alpha+\beta$ is equal to Solution: For infinite solutions $\Delta=\Delta_{1}=\Delta_{2}=\Delta_{3}=0$ $\Delta=\left|\begin{array}{ccc}1 1 -1 \\ 1 2 \alpha \\ 2 -1 1\end{array}\right|=0$ $\Delta=\left|\begin{array}{ccc}3 0 0 \\ 1 2 \alpha \\ 2 -1 1\end{array}\right|=0$ $\Delta=3(2+\alpha)=0$ $\Rightarrow \alpha=-...
Read More →Consider function f : A → B and
Question: Consider function $f: \mathrm{A} \rightarrow \mathrm{B}$ and $\mathrm{g}: \mathrm{B} \rightarrow \mathrm{C}(\mathrm{A}, \mathrm{B}, \mathrm{C} \subseteq \mathbf{R})$ such that (gof) $^{-1}$ exists, then:$\mathrm{f}$ and $\mathrm{g}$ both are one-one$f$ and $g$ both are onto$\mathrm{f}$ is one-one and $\mathrm{g}$ is onto$\mathrm{f}$ is onto and $\mathrm{g}$ is one-oneCorrect Option: , 3 Solution: $\therefore$ (gof) $^{-1}$ exist $\Rightarrow$ gof is bijective $\Rightarrow ' f$ ' must b...
Read More →Identify A in the given chemical reaction.
Question: Identify A in the given chemical reaction. Correct Option: , 4 Solution: $\mathrm{Mo}_{2} \mathrm{O}_{3}$ at $773 \mathrm{~K}$ temperature and $10-20$-atm pressure is aromatising agent....
Read More →'A' and 'B' in the following reactions are :
Question: 'A' and 'B' in the following reactions are : Correct Option: , 3 Solution:...
Read More →The number of real solutions of the equation,
Question: The number of real solutions of the equation, $x^{2}-|x|-12=0$ is:2314Correct Option: 1 Solution: $|x|^{2}-|x|-12=0$ $(|x|+3)(|x|-4)=0$ $|x|=4 \Rightarrow x=\pm 2$...
Read More →Which of the following equation depicts
Question: Which of the following equation depicts the oxidizing nature of $\mathrm{H}_{2} \mathrm{O}_{2}$ ?$\mathrm{KIO}_{4}+\mathrm{H}_{2} \mathrm{O}_{2} \rightarrow \mathrm{KIO}_{3}+\mathrm{H}_{2} \mathrm{O}+\mathrm{O}_{2}$$2 \mathrm{I}^{-}+\mathrm{H}_{2} \mathrm{O}_{2}+2 \mathrm{H}^{+} \rightarrow \mathrm{I}_{2}+2 \mathrm{H}_{2} \mathrm{O}$$\mathrm{I}_{2}+\mathrm{H}_{2} \mathrm{O}_{2}+2 \mathrm{OH}^{-} \rightarrow 2 \mathrm{I}^{-}+2 \mathrm{H}_{2} \mathrm{O}+\mathrm{O}_{2}$$\mathrm{Cl}_{2}+\m...
Read More →Solve this following
Question: Let $A=\left\{(x, y) \in \mathbf{R} \times \mathbf{R} \mid 2 x^{2}+2 y^{2}-2 x-2 y=1\right\}$ $B=\left\{(x, y) \in \mathbf{R} \times \mathbf{R} \mid 4 x^{2}+4 y^{2}-16 y+7=0\right\}$ and $C=\left\{(x, y) \in \mathbf{R} \times \mathbf{R} \mid x^{2}+y^{2}-4 x-2 y+5 \leq r^{2}\right\}$ Then the minimum value of $|\mathrm{r}|$ such that $\mathrm{A} \cup \mathrm{B} \subseteq \mathrm{C}$ is equal to $\frac{3+\sqrt{10}}{2}$$\frac{2+\sqrt{10}}{2}$$\frac{3+2 \sqrt{5}}{2}$$1+\sqrt{5}$Correct Opt...
Read More →Prove the following
Question: If $|\overrightarrow{\mathrm{a}}|=2,|\overrightarrow{\mathrm{b}}|=5$ and $|\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}|=8$, then $|\overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{b}}|$ is equa to :6435Correct Option: 1 Solution: $|\overrightarrow{\mathrm{a}}|=2,|\overrightarrow{\mathrm{b}}|=5$ $|\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}|=|\overrightarrow{\mathrm{a}}||\overrightarrow{\mathrm{b}}| \sin \theta=\pm 8$ $\sin \theta=\pm \frac...
Read More →The number of distinct real roots of
Question: The number of distinct real roots of $\left|\begin{array}{lll}\sin x \cos x \cos x \\ \cos x \sin x \cos x \\ \cos x \cos x \sin x\end{array}\right|=0 \quad$ in the interval $-\frac{\pi}{4} \leq x \leq \frac{\pi}{4}$ is: 4123Correct Option: , 2 Solution: $\left|\begin{array}{lll}\sin x \cos x \cos x \\ \cos x \sin x \cos x \\ \cos x \cos x \sin x\end{array}\right|=0, \frac{-\pi}{4} \leq x \leq \frac{\pi}{4}$ Apply : $\mathrm{R}_{1} \rightarrow \mathrm{R}_{1}-\mathrm{R}_{2} \ \mathrm{R}...
Read More →Which of the following reaction/s
Question: Which of the following reaction/s will not give p-aminoazobenzene? A onlyB onlyC only$\mathrm{A}$ and $\mathrm{B}$Correct Option: , 2 Solution: In basic or neutral medium $\mathrm{N}-\mathrm{N}$ coupling favourable while in slightly acidic medium $\mathrm{C}-$ $\mathrm{N}$ coupling favourable....
Read More →Consider the elements Mg. Al, S, P and Si, the correct increasing order of their first ionization enthalpy is.
Question: Consider the elements $\mathrm{Mg} . \mathrm{Al}, \mathrm{S}, \mathrm{P}$ and $\mathrm{Si}$, the correct increasing order of their first ionization enthalpy is. $\mathrm{Al}\mathrm{Mg}\mathrm{Si}\mathrm{S}\mathrm{P}$(1) $\mathrm{Mg}\mathrm{Al}\mathrm{Si}\mathrm{S}\mathrm{P}$(2) $\mathrm{Al}\mathrm{Mg}\mathrm{Si}\mathrm{S}\mathrm{P}$(3) $\mathrm{Mg}\mathrm{Al}\mathrm{Si}\mathrm{P}\mathrm{S}$(4) $\mathrm{Al}\mathrm{Mg}\mathrm{S}\mathrm{Si}\mathrm{P}$Correct Option: , 2 Solution: In gener...
Read More →The probability that a randomly selected 2-digit number belongs
Question: The probability that a randomly selected 2-digit number belongs to the $\operatorname{set}\left\{\mathrm{n} \in \mathrm{N}:\left(2^{\mathrm{n}}-2\right)\right.$ is a multiple of 3$\}$ is equal to$\frac{1}{6}$$\frac{2}{3}$$\frac{1}{2}$$\frac{1}{3}$Correct Option: , 3 Solution: Total number of cases $={ }^{90} \mathrm{C}_{1}=90$ Now, $2^{\mathrm{n}}-2=(3-1)^{\mathrm{n}}-2$ ${ }^{n} C_{0} 3^{n}-{ }^{n} C_{1} \cdot 3^{n-1}+\ldots+(-1)^{n-1} \cdot{ }^{n} C_{n-1} 3+(-1)^{n} \cdot{ }^{n} C_{n...
Read More →If [x] be the greatest integer less than or equal to x,
Question: If $[x]$ be the greatest integer less than or equal to $x$, then $\sum_{n=8}^{100}\left[\frac{(-1)^{n} n}{2}\right]$ is equal to:04-22Correct Option: , 2 Solution: $\sum_{n=8}^{100}\left[\frac{(-1)^{n} \cdot n}{2}\right]$ $=4-5+5-6+6+\ldots-50+50=4$...
Read More →According to molecular theory,
Question: According to molecular theory, the species among the following that does not exist is:$\mathrm{He}_{2}^{+}$$\mathrm{He}_{2}^{-}$$\mathrm{Be}_{2}$$\mathrm{O}_{2}^{2-}$Correct Option: , 3 Solution: According to M.O.T. If bond order of chemical species is zero then that chemical species does not exist....
Read More →Let a, b and c be distinct positive numbers.
Question: Let $a, b$ and $c$ be distinct positive numbers. If the vectors $a \hat{i}+a \hat{j}+c \hat{k}, \hat{i}+\hat{k}$ and $c \hat{i}+c \hat{j}+b \hat{k}$ are co-planar, then $\mathrm{c}$ is equal to:$\frac{2}{\frac{1}{a}+\frac{1}{b}}$$\frac{\mathrm{a}+\mathrm{b}}{2}$$\frac{1}{a}+\frac{1}{b}$$\sqrt{\mathrm{ab}}$Correct Option: , 4 Solution: Because vectors are coplanar Hence $\left|\begin{array}{lll}\mathrm{a} \mathrm{a} \mathrm{c} \\ 1 0 1 \\ \mathrm{c} \mathrm{c} \mathrm{b}\end{array}\righ...
Read More →Solve this following
Question: Let $\alpha, \beta$ be two roots of the equation $x^{2}+(20)^{1 / 4} x+(5)^{1 / 2}=0 .$ Then $\alpha^{8}+\beta^{8}$ is equal to 1010050160Correct Option: , 3 Solution: $\left(x^{2}+\sqrt{5}\right)^{2}=\sqrt{20} x^{2}$ $x^{4}=-5 \Rightarrow x^{8}=25$ $\alpha^{8}+\beta^{8}=50$...
Read More →The product formed in the first step of the reaction of
Question: The product formed in the first step of the reaction of $\mathrm{Mg} / \mathrm{Et}_{2} \mathrm{O}\left(\mathrm{Et}=\mathrm{C}_{2} \mathrm{H}_{5}\right)$ is :Correct Option: , 4 Solution:...
Read More →The value of the integral
Question: The value of the integral $\int_{-1}^{1} \log \left(x+\sqrt{x^{2}+1}\right) d x$ is:20-11Correct Option: , 2 Solution: Let $I=\int_{-1}^{1} \log \left(x+\sqrt{x^{2}+1}\right) d x$ $\because \log \left(x+\sqrt{x^{2}+1}\right)$ is an odd function $\therefore I=0$...
Read More →Given below are two statements:
Question: Given below are two statements: Statement $\mathrm{I}: \mathrm{CeO}_{2}$ can be used for oxidation of aldehydes and ketones. Statement II : Aqueous solution of $\mathrm{EuSO}_{4}$ is a strong reducing agent. In the light of the above statements, choose the correct answer from the options given below:Statement I is false but statement II is trueStatment I is true but statement IIis falseBoth statement I and statement II are trueBoth statement I and statement II are falseCorrect Option: ...
Read More →The lowest integer which is greater than
Question: The lowest integer which is greater than $\left(1+\frac{1}{10^{100}}\right)^{10^{100}}$ is___________.3421Correct Option: 1 Solution: Let $P=\left(1+\frac{1}{10^{100}}\right)^{10^{100}}$ Let $x=10^{100}$ $\Rightarrow P=\left(1+\frac{1}{x}\right)^{x}$ $\Rightarrow P=1+(x)\left(\frac{1}{x}\right)+\frac{(x)(x-1)}{\lfloor 2} \cdot \frac{1}{x^{2}}$ $+\frac{(x)(x-1)(x-2)}{\lfloor 3} \cdot \frac{1}{x^{3}}+\ldots$ (upto $10^{100}+1$ terms) $\Rightarrow P=1+1+\left(\frac{1}{12}-\frac{1}{12 x^{2...
Read More →Solve this following
Question: Let $P$ and $Q$ be two distinct points on a circle which has center at $\mathrm{C}(2,3)$ and which passes through origin $\mathrm{O}$. If $\mathrm{OC}$ is perpendicular to both the line segments $\mathrm{CP}$ and $\mathrm{CQ}$, then the set $\{\mathrm{P}, \mathrm{Q}\}$ is equal to$\{(4,0),(0,6)\}$$\{(2+2 \sqrt{2}, 3-\sqrt{5}),(2-2 \sqrt{2}, 3+\sqrt{5})\}$$\{(2+2 \sqrt{2}, 3+\sqrt{5}),(2-2 \sqrt{2}, 3-\sqrt{5})\}$$\{(-1,5),(5,1)\}$Correct Option: , 4 Solution: $\tan \theta=-\frac{2}{3}$...
Read More →Solve this following
Question: Let $f: \mathbf{R} \rightarrow \mathbf{R}$ be a function such that $f(2)=4$ and $f^{\prime}(2)=1 .$ Then, the value of $\lim _{x \rightarrow 2} \frac{x^{2} f(2)-4 f(x)}{x-2}$ is equal to : 481612Correct Option: , 4 Solution: Apply L'Hopital Rule $\lim _{x \rightarrow 2}\left(\frac{2 x f(2)-4 f^{\prime}(x)}{1}\right)$ $=\frac{4(4)-4}{1}=12$...
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