Consider the function
Question: Consider the function $f(x)=\frac{P(x)}{\sin (x-2)}, \quad x \neq 2$ $=7 \quad, \quad x=2$ Where $\mathrm{P}(\mathrm{x})$ is a polynomial such that $\mathrm{P}^{\prime \prime}(\mathrm{x})$ is always a constant and $\mathrm{P}(3)=9$. If $\mathrm{f}(\mathrm{x})$ is continuous at $x=2$, then $P(5)$ is equal to________. Solution: $f(x)= \begin{cases}\frac{P(x)}{\sin (x-2)}, x \neq 2 \\ 7, x=2\end{cases}$ $P^{\prime \prime}(x)=$ const. $\Rightarrow P(x)$ is a 2 degree polynomial $f(x)$ is c...
Read More →Solve this following
Question: Let $S=\{1,2,3,4,5,6,7\} .$ Then the number of possible functions $\mathrm{f}: \mathrm{S} \rightarrow \mathrm{S}$ such that $\mathrm{f}(\mathrm{m} \cdot \mathrm{n})=\mathrm{f}(\mathrm{m}) \cdot \mathrm{f}(\mathrm{n})$ for every $m, n \in S$ and $m \cdot n \in S$ is equal to Solution: $\mathrm{F}(\mathrm{mn})=\mathrm{f}(\mathrm{m}) \cdot \mathrm{f}(\mathrm{n})$ Put $\mathrm{m}=1 \mathbf{f}(\mathbf{n})=\mathbf{f}(\mathbf{1}) \cdot \mathrm{f}(\mathrm{n}) \Rightarrow \mathrm{f}(1)=1$ Put $...
Read More →In the following reaction the reason why meta-nitro product also formed is :
Question: In the following reaction the reason why meta-nitro product also formed is : low temperature$-\mathrm{NH}_{2}$ group is highly meta-directiveFormation of anilinium ion$-\mathrm{NO}_{2}$ substitution always takes place at meta-positionCorrect Option: , 4 Solution: Aniline on protonation gives anilinium ion which is meta directing. So considerable amount of meta product is formed....
Read More →Let a plane P pass through the point
Question: Let a plane P pass through the point $(3,7,-7)$ and contain the line, $\frac{x-2}{-3}=\frac{y-3}{2}=\frac{z+2}{1}$. If distance of the plane $P$ from the origin is $d$, then $d^{2}$ is equal to Solution: $\overrightarrow{\mathrm{BA}}=(\hat{\mathrm{i}}+4 \hat{\mathrm{j}}-5 \hat{\mathrm{k}})$ $\overrightarrow{\mathrm{BA}}=(\hat{\mathrm{i}}+4 \hat{\mathrm{j}}-5 \hat{\mathrm{k}})$ $\overrightarrow{\mathrm{BA}} \times \vec{\ell}=\overrightarrow{\mathrm{n}}=\left|\begin{array}{ccc}\hat{\math...
Read More →Let n ∈ N and [x] denote the greatest integer
Question: Let $n \in \mathbf{N}$ and $[x]$ denote the greatest integer less than or equal to $x$. If the sum of $(n+1)$ terms ${ }^{\mathrm{n}} \mathrm{C}_{0}, 3 \cdot{ }^{\mathrm{n}} \mathrm{C}_{1}, 5 \cdot{ }^{\mathrm{n}} \mathrm{C}_{2}, 7 \cdot{ }^{\mathrm{n}} \mathrm{C}_{3}, \ldots \ldots$ is equal to $2^{100} \cdot 101$ then $2\left[\frac{\mathrm{n}-1}{2}\right]$ is equal to_________. Solution: $1 .{ }^{\mathrm{n}} \mathrm{C}_{0}+3 .{ }^{\mathrm{n}} \mathrm{C}_{1}+5 .{ }^{\mathrm{n}} \mathr...
Read More →Compound(s) which will liberate
Question: Compound(s) which will liberate carbon dioxide with sodium bicarbonate solution is/are: B onlyC only$\mathrm{B}$ and $\mathrm{C}$ onlyA and B onlyCorrect Option: , 3 Solution: equilibrium favours forward direction and $\mathrm{CO}_{2} \uparrow$ is librated. Equilibrium favours forward direction and $\mathrm{CO}_{2} \uparrow$ is librated. Equilibrium favours back word direction and $\mathrm{CO}_{2} \uparrow$ is not librated....
Read More →Which of the glycosidic linkage
Question: Which of the glycosidic linkage between galactose and glucose is present in lactose?C-1 of galactose and C-4 of glucoseC-1 of glucose and C-6 of galactoseC-1 of glucose and C-4 of galactoseC-1 of galactose and C-6 of glucoseCorrect Option: 1 Solution: In lactose linkage is formed between $C_{1}$ of galactose and $\mathrm{C}_{4}$ of gluocse....
Read More →In which of the following pairs,
Question: In which of the following pairs, the outer most electronic configuration will be the same?$\mathrm{Cr}^{+}$and $\mathrm{Mn}^{2+}$$\mathrm{Ni}^{2+}$ and $\mathrm{Cu}^{+}$$\mathrm{Fe}^{2+}$ and $\mathrm{Co}^{+}$$\mathrm{V}^{2+}$ and $\mathrm{Cr}^{+}$Correct Option: 1, Solution: Option $-1 \quad \mathrm{Mn}^{+2}[\mathrm{Ar}] 3 \mathrm{~d}^{5}, \mathrm{Cr}^{+}[\mathrm{Ar}] 3 \mathrm{~d}^{5}$ Option $-2 \quad \mathrm{Ni}^{+2}[\mathrm{Ar}] 3 \mathrm{~d}^{8}, \mathrm{Cu}^{+}[\mathrm{Ar}] 3 \m...
Read More →What is the final product (major) 'A' in the given reaction ?
Question: What is the final product (major) 'A' in the given reaction ? Correct Option: 1 Solution:...
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:Both statement I and Statement II are falseStatement I is true but Statement II is falseBoth Statement I and Statement II are true...
Read More →The plots of radial distribution
Question: The plots of radial distribution functions for various orbitals of hydrogen atom against ' $r$ ' are given below: The correct plot for $3 \mathrm{~s}$ orbital is:(B)(A)(D)(C)Correct Option: , 3 Solution: Number of radial nodes $=\mathrm{n}-\ell-1$ $=3-0-1=2$ Therefor corresponding graph is (D) Hence answer is (3)...
Read More →Which of the following are isostructural pairs ?
Question: Which of the following are isostructural pairs ? A. $\mathrm{SO}_{4}^{2-}$ and $\mathrm{CrO}_{4}^{2-}$ B. $\mathrm{SiCl}_{4}$ and $\mathrm{TiCl}_{4}$ C. $\mathrm{NH}_{3}$ and $\mathrm{NO}_{3}^{-}$ D. $\mathrm{BCl}_{3}$ and $\mathrm{BrCl}_{3}$ $\mathrm{BCl}_{3}$ and $\mathrm{BrCl}_{3}$C and D onlyA and B onlyA and $C$ onlyB and C onlyCorrect Option: , 2 Solution: Isostructural means same structure...
Read More →Let y=y(x) be the solution of the differential equation
Question: Let $y=y(x)$ be the solution of the differential equation $x d y=\left(y+x^{3} \cos x\right) d x$ with $y(\pi)=0$, then $y\left(\frac{\pi}{2}\right)$ is equal to:$\frac{\pi^{2}}{4}+\frac{\pi}{2}$$\frac{\pi^{2}}{2}+\frac{\pi}{4}$$\frac{\pi^{2}}{2}-\frac{\pi}{4}$$\frac{\pi^{2}}{4}-\frac{\pi}{2}$Correct Option: 1 Solution: $x d y=\left(y+x^{3} \cos x\right) d x$ $x d y=y d x+x^{3} \cos x d x$ $\frac{x d y-y d x}{x^{2}}=\frac{x^{3} \cos x d x}{x^{2}}$ $\frac{d}{d x}\left(\frac{y}{x}\right)...
Read More →Solve this following
Question: Let $\mathrm{F}:[3,5] \rightarrow \mathbf{R}$ be a twice differentiable function on $(3,5)$ such that $F(x)=e^{-x} \int_{3}^{x}\left(3 t^{2}+2 t+4 F^{\prime}(t)\right) d t$ If $\mathrm{F}^{\prime}(4)=\frac{\alpha \mathrm{e}^{\beta}-224}{\left(\mathrm{e}^{\beta}-4\right)^{2}}$, then $\alpha+\beta$ is equal to Solution: $\mathrm{F}(3)=0$ $e^{x} F(x)=\int_{3}^{x}\left(3 t^{2}+2 t+4 F^{\prime}(t)\right) d t$ $e^{x} F(x)+e^{x} F^{\prime}(x)=3 x^{2}+2 x+4 F^{\prime}(x)$ $\left(e^{x}-4\right)...
Read More →Prove the following
Question: If ${ }^{n} P_{r}={ }^{n} P_{r+1}$ and ${ }^{n} C_{r}={ }^{n} C_{r-1}$, then the value of $r$ is equal to:1423Correct Option: , 3 Solution: ${ }^{\mathrm{n}} \mathrm{P}_{\mathrm{r}}={ }^{\mathrm{n}} \mathrm{P}_{\mathrm{r}+1} \Rightarrow \frac{\mathrm{n} !}{(\mathrm{n}-\mathrm{r}) !}=\frac{\mathrm{n} !}{(\mathrm{n}-\mathrm{r}-1) !}$ $\Rightarrow(\mathrm{n}-\mathrm{r})=1$ ..........(1) ${ }^{n} \mathrm{C}_{r}={ }^{n} \mathrm{C}_{r-1}$ $\Rightarrow \frac{n !}{r !(n-r) !}=\frac{n !}{(r-1) ...
Read More →Let the equation of the pair of lines,
Question: Let the equation of the pair of lines, $\mathrm{y}=\mathrm{px}$ and $y=q x$, can be written as $(y-p x)(y-q x)=0$. Then the equation of the pair of the angle bisectors of the lines $x^{2}-4 x y-5 y^{2}=0$ is:$x^{2}-3 x y+y^{2}=0$$x^{2}+4 x y-y^{2}=0$$x^{2}+3 x y-y^{2}=0$$x^{2}-3 x y-y^{2}=0$Correct Option: , 3 Solution: $\frac{x^{2}-y^{2}}{1-(-5)}=\frac{x y}{-2}$ $\frac{x^{2}-y^{2}}{6}=\frac{x y}{-2}$ $\Rightarrow x^{2}-y^{2}=-3 x y$ $\Rightarrow x^{2}+3 x y-y^{2}=0$...
Read More →Solve this following
Question: Let $f(x)=\left|\begin{array}{ccc}\sin ^{2} x -2+\cos ^{2} x \cos 2 x \\ 2+\sin ^{2} x \cos ^{2} x \cos 2 x \\ \sin ^{2} x \cos ^{2} x 1+\cos 2 x\end{array}\right|, x \in[0, \pi]$ Then the maximum value of $f(x)$ is equal to Solution: $\left|\begin{array}{ccc}-2 -2 0 \\ 2 0 -1 \\ \sin ^{2} x \cos ^{2} x 1+\cos 2 x\end{array}\right|\left(\begin{array}{l}R_{1} \rightarrow R_{1}-R_{2} \\ \ R_{2} \rightarrow R_{2}-R_{3}\end{array}\right)$ $-2\left(\cos ^{2} x\right)+2\left(2+2 \cos 2 x+\si...
Read More →Al2O3 was leached with alkali to get X. The solution of X on passing of gas Y, forms Z. X, Y and Z respectively are :
Question: $\mathrm{Al}_{2} \mathrm{O}_{3}$ was leached with alkali to get $\mathrm{X}$. The solution of $\mathrm{X}$ on passing of gas $\mathrm{Y}$, forms $\mathrm{Z}$. X, $Y$ and $Z$ respectively are :$\mathrm{X}=\mathrm{Na}\left[\mathrm{Al}(\mathrm{OH})_{4}\right], \mathrm{Y}=\mathrm{SO}_{2}, \mathrm{Z}=\mathrm{Al}_{2} \mathrm{O}_{3}$$\mathrm{X}=\mathrm{Na}\left[\mathrm{Al}(\mathrm{OH})_{4}\right], \mathrm{Y}=\mathrm{CO}_{2}, \mathrm{Z}=\mathrm{Al}_{2} \mathrm{O}_{3} \cdot \mathrm{XH}_{2} \mat...
Read More →The correct statement about
Question: The correct statement about $\mathrm{B}_{2} \mathrm{H}_{6}$ is:Terminal B-H bonds have less p-character when compared to bridging bonds.The two B-H-B bonds are not of same lengthAll B-H-B angles are of $120^{\circ}$Its fragment, $\mathrm{BH}_{3}$, behaves as a Lewis baseCorrect Option: 1 Solution: - $\quad \theta_{2}\theta_{1}, \therefore \mathrm{B}-\mathrm{H}$ (terminal) having less $\mathrm{p}-$ character as compare to bridge bond. - Both B-H-B bridge bond having same bond length. - ...
Read More →If a tangent to the ellipse
Question: If a tangent to the ellipse $x^{2}+4 y^{2}=4$ meets the tangents at the extremities of its major axis at B and $\mathrm{C}$, then the circle with $\mathrm{BC}$ as diameter passes through the point :$(\sqrt{3}, 0)$$(\sqrt{2}, 0)$$(1,1)$$(-1,1)$Correct Option: 1 Solution: $\frac{x^{2}}{4}+\frac{y^{2}}{1}=1$ Equation of tangent is $(\cos \theta) x+2 \sin \theta y=2$ $\mathrm{B}\left(-2, \frac{1+\cos \theta}{\sin \theta}\right), \quad \mathrm{C}\left(2, \frac{1-\cos \theta}{\sin \theta}\ri...
Read More →Let the domain of the function
Question: Let the domain of the function $f(x)=\log _{4}\left(\log _{5}\left(\log _{3}\left(18 x-x^{2}-77\right)\right)\right)$ be $(a, b)$ Then the value of the integral $\int_{a}^{b} \frac{\sin ^{3} x}{\left(\sin ^{3} x+\sin ^{3}(a+b-x)\right)} d x$ is equal to Solution: For domain $\log _{5}\left(\log _{3}\left(18 x-x^{2}-77\right)\right)0$ $\log _{3}\left(18 x-x^{2}-77\right)1$ $18 x-x^{2}-773$ $x^{2}-18 x+800$ $x \in(8,10)$ $\Rightarrow \mathrm{a}=8$ and $\mathrm{b}=10$ $I=\int_{a}^{b} \fra...
Read More →Solve this following
Question: If $\log _{3} 2, \log _{3}\left(2^{x}-5\right), \log _{3}\left(2^{x}-\frac{7}{2}\right)$ are in an arithmetic progression, then the value of $x$ is equal to Solution: $2 \log _{3}\left(2^{x}-5\right)=\log _{3} 2+\log _{3}\left(2^{x}-\frac{7}{2}\right)$ Let $2^{x}=t$ $\log _{3}(t-5)^{2}=\log _{3} 2\left(t-\frac{7}{2}\right)$ $(\mathrm{t}-5)^{2}=2 \mathrm{t}-7$ $\mathrm{t}^{2}-12 \mathrm{t}+32=0$ $(\mathrm{t}-4)(\mathrm{t}-8)=0$ $\Rightarrow 2^{x}=4$ or $2^{x}=8$ $X=2$ (Rejected) Or $x=3...
Read More →Which of the following ore is concentrated using group 1 cyanide salt ?
Question: Which of the following ore is concentrated using group 1 cyanide salt ?SphaleriteCalamineSideriteMalachiteCorrect Option: 1 Solution: Sphalerite ore : $\mathrm{ZnS}$ Calamine ore : $\mathrm{ZnCO}_{3}$ Siderite ore : $\mathrm{FeCO}_{3}$ Malachite ore : $\mathrm{Cu}(\mathrm{OH})_{2} . \mathrm{CuCO}_{3}$ It is possible to separate two sulphide ores by adjusting proportion of oil to water or by using 'depressants'. In case of an ore containing ZnS and $\mathrm{PbS}$, the depressant used is...
Read More →Let x be a random variable such that the probability
Question: Let $x$ be a random variable such that the probability function of a distribution is given by $\mathrm{P}(\mathrm{X}=0)=\frac{1}{2}, \mathrm{P}(\mathrm{X}=\mathrm{j})=\frac{1}{3^{\mathrm{j}}}(\mathrm{j}=1,2,3, \ldots, \infty)$ Then the mean of the distribution and $\mathrm{P}(\mathrm{X}$ is positive and even) respectively are:$\frac{3}{8}$ and $\frac{1}{8}$$\frac{3}{4}$ and $\frac{1}{8}$$\frac{3}{4}$ and $\frac{1}{9}$$\frac{3}{4}$ and $\frac{1}{16}$Correct Option: , 2 Solution: mean $=...
Read More →Which one of the following reactions
Question: Which one of the following reactions will not form acetaldehyde?Correct Option: , 4 Solution:...
Read More →