if

Question: If $\int \frac{d \theta}{\cos ^{2} \theta(\tan 2 \theta+\sec 2 \theta)}=\lambda \tan \theta+2 \log _{\mathrm{e}}|f(\theta)|+\mathrm{C}$ where $\mathrm{C}$ is a constant of integration, then the ordered pair $(\lambda, f(\theta))$ is equal to:(1) $(1,1-\tan \theta)$(2) $(-1,1-\tan \theta)$(3) $(-1,1+\tan \theta)$(4) $(1,1+\tan \theta)$Correct Option: , 3 Solution: $I=\int \frac{d \theta}{\cos ^{2} \theta(\tan 2 \theta+\sec 2 \theta)}$ $=\int \frac{\sec ^{2} \theta}{\frac{1+\tan ^{2} \th...

The process that is NOT endothermic in nature is :

Question: The process that is NOT endothermic in nature is :$\operatorname{Ar}(g)+e^{-} \rightarrow \operatorname{Ar}^{-}(g)$$\mathrm{H}(\mathrm{g})+\mathrm{e}^{-} \rightarrow \mathrm{H}^{-}(\mathrm{g})$$\mathrm{O}^{-}(\mathrm{g})+\mathrm{e}^{-} \rightarrow \mathrm{O}^{2-}(\mathrm{g})$$\mathrm{Na}(\mathrm{g}) \rightarrow \mathrm{Na}^{+}(\mathrm{g})+\mathrm{e}^{-}$Correct Option: , 2 Solution: - Electron gaining enthalpy (EGE) of $\mathrm{H}(\mathrm{g})$ is negative while that of $\operatorname{A...

Two small spheres each of mass 10 mg

Question: Two small spheres each of mass $10 \mathrm{mg}$ are suspended from a point by threads $0.5 \mathrm{~m}$ long. They are equally charged and repel each other to a distance of $0.20 \mathrm{~m}$. Then charge on each of the sphere is $\frac{a}{21} \times 10^{-8} \mathrm{C}$. The value of ' $\mathrm{a}$ ' will be Solution: $T \sin \theta=\frac{k q^{2}}{r^{2}}$ $T \cos \theta=m g$ $\tan \theta=\frac{\mathrm{kq}^{2}}{\mathrm{mgr}^{2}}$ $q^{2}=\frac{\tan \theta m g r^{2}}{k}$ $\because \tan \t...

The integral

Question: The integral $\int \frac{d x}{(x+4)^{8 / 7}(x-3)^{6 / 7}}$ is equal to: (where $\mathrm{C}$ is a constant of integration)(1) $\left(\frac{x-3}{x+4}\right)^{1 / 7}+\mathrm{C}$(2) $-\left(\frac{x-3}{x+4}\right)^{-1 / 7}+\mathrm{C}$(3) $\frac{1}{2}\left(\frac{x-3}{x+4}\right)^{3 / 7}+\mathrm{C}$(4) $-\frac{1}{13}\left(\frac{x-3}{x+4}\right)^{-13 / 7}+\mathrm{C}$Correct Option: 1 Solution: $I=\int \frac{d x}{(x+4)^{8 / 7}(x-3)^{6 / 7}}$ $=\int\left(\frac{x-3}{x+4}\right)^{\frac{-6}{7}} \fr...

The ionic radii of

Question: The ionic radii of $\mathrm{O}^{2-}, \mathrm{F}^{-}, \mathrm{Na}^{+}$and $\mathrm{Mg}^{2+}$ are in the order:$\mathrm{F}^{-}\mathrm{O}^{2-}\mathrm{Na}^{+}\mathrm{Mg}^{2+}$$\mathrm{O}^{2-}\mathrm{F}^{-}\mathrm{Na}^{+}\mathrm{Mg}^{2+}$$\mathrm{Mg}^{2+}\mathrm{Na}^{+}\mathrm{F}^{-}\mathrm{O}^{2-}$$\mathrm{O}^{2-}\mathrm{F}^{-}\mathrm{Mg}^{2+}\mathrm{Na}^{+}$Correct Option: , 2 Solution:...

Solve each of the following quadratic equations:

Question: Solve each of the following quadratic equations: $\frac{3 x-4}{7}+\frac{7}{3 x-4}=\frac{5}{2}, \quad x \neq \frac{4}{3}$ Solution: $\frac{3 x-4}{7}+\frac{7}{3 x-4}=\frac{5}{2}, \quad x \neq \frac{4}{3}$ $\Rightarrow \frac{(3 x-4)^{2}+49}{7(3 x-4)}=\frac{5}{2}$ $\Rightarrow \frac{9 x^{2}-24 x+16+49}{21 x-28}=\frac{5}{2}$ $\Rightarrow \frac{9 x^{2}-24 x+65}{21 x-28}=\frac{5}{2}$ $\Rightarrow 18 x^{2}-48 x+130=105 x-140$ $\Rightarrow 18 x^{2}-153 x+270=0$ $\Rightarrow 2 x^{2}-17 x+30=0$ $...

if

Question: If $f^{\prime}(x)=\tan ^{-1}(\sec x+\tan x),-\frac{\pi}{2}x\frac{\pi}{2}$, and $f(0)=0$, then $f(1)$ is equal to:(1) $\frac{\pi+1}{4}$(2) $\frac{1}{4}$(3) $\frac{\pi-1}{4}$(4) $\frac{\pi+2}{4}$Correct Option: 1 Solution: $f^{\prime}(x)=\tan ^{-1}(\sec x+\tan x)$ $=\tan ^{-1}\left(\frac{1+\sin x}{\cos x}\right)=\tan ^{-1}\left(\frac{1-\cos \left(\frac{\pi}{2}+x\right)}{\sin \left(\frac{\pi}{2}+x\right)}\right)$ $=\tan ^{-1}\left(\frac{2 \sin ^{2}\left(\frac{\pi}{4}+\frac{x}{2}\right)}{2...

The elements with atomic numbers 101 and 104 belong to,

Question: The elements with atomic numbers 101 and 104 belong to, respectively:Group 11 and Group 4Actinoids and Group 6Actinoids and Group 4Group 6 and ActinoidsCorrect Option: , 3 Solution: ${ }_{89} \mathrm{AC} \longrightarrow{ }_{103} \mathrm{Lr}$ Belongs to actinoids series and they all belongs to $3^{\mathrm{rd}}$ group. So atomic no. 101 element is actinoids and atomic number 104 element belongs to $4^{\text {th }}$ group....

Question: The elements with atomic numbers 101 and 104 belong to, respectively:Group 11 and Group 4Actinoids and Group 6Actinoids and Group 4Group 6 and ActinoidsCorrect Option: , 3 Solution: ${ }_{89} \mathrm{AC} \longrightarrow{ }_{103} \mathrm{Lr}$ Belongs to actinoids series and they all belongs to $3^{\mathrm{rd}}$ group. So atomic no. 101 element is actinoids and atomic number 104 element belongs to $4^{\text {th }}$ group....

A charge ' q ' is placed at one corner of a cube as shown in figure.

Question: A charge ' $q$ ' is placed at one corner of a cube as shown in figure. The flux of electrostatic field $E$ though the shaded area is: (1) $\frac{q}{48 \varepsilon_{0}}$(2) $\frac{q}{8 \varepsilon_{0}}$(3) $\frac{q}{24 \varepsilon_{0}}$(4) $\frac{q}{4 \varepsilon_{0}}$Correct Option: , 3 Solution: (3) $\phi=\frac{q}{24 \varepsilon_{0}}$ $\phi_{T}=\left(\frac{q}{24 \varepsilon_{0}}+\frac{q}{24 \varepsilon_{0}}\right) \times \frac{1}{2}$ $\phi_{T}=\frac{q}{24 \varepsilon_{0}}$...

Solve each of the following quadratic equations:

Question: Solve each of the following quadratic equations: $\frac{x+3}{x-2}-\frac{1-x}{x}=4 \frac{1}{4}, x \neq 2,0$ Solution: Given : $\frac{x+3}{x-2}-\frac{1-x}{x}=4 \frac{1}{4}$ $\Rightarrow \frac{(x+3)}{(x-2)}-\frac{(1-x)}{x}=\frac{17}{4}$ $\Rightarrow \frac{x(x+3)-(1-x)(x-2)}{(x-2) x}=\frac{17}{4}$ $\Rightarrow \frac{x^{2}+3 x-\left(x-2-x^{2}+2 x\right)}{x^{2}-2 x}=\frac{17}{4}$ $\Rightarrow \frac{x^{2}+3 x+x^{2}-3 x+2}{x^{2}-2 x}=\frac{17}{4}$ $\Rightarrow \frac{2 x^{2}+2}{x^{2}-2 x}=\frac...

if

Question: If $\int \frac{\cos x d x}{\sin ^{3} x\left(1+\sin ^{6} x\right)^{2 / 3}}=f(x)\left(1+\sin ^{6} x\right)^{1 / \lambda}+c$ where $c$ is(1) $-\frac{9}{8}$(2) 2(3) $\frac{9}{8}$(4) $-2$Correct Option: , 4 Solution: Let $I=\int \frac{\cos x d x}{\sin ^{3} x\left(1+\sin ^{6} x\right)^{2 / 3}}$ $=f(x)\left(1+\sin ^{6} x\right)^{1 / \lambda}+c$ ...(i) If $\sin x=t$ then, $\cos x d x=d t$ $I=\int \frac{d t}{t^{3}\left(1+t^{6}\right)^{\frac{2}{3}}}=\int \frac{d t}{t^{7}\left(1+\frac{1}{t^{6}}\r...

The five successive ionization enthalpies of an element are

Question: The five successive ionization enthalpies of an element are $800,2427,3658,25024$ and $32824 \mathrm{~kJ} \mathrm{~mol}^{-1}$. The number of valence electrons in the element is :5432Correct Option: , 3 Solution: As difference in $3^{\text {rd }}$ and $4^{\text {th }}$ ionisation energies is high so atom contains 3 valence electrons....

Solve each of the following quadratic equations:

Question: Solve each of the following quadratic equations: $\frac{1}{2 a+b+2 x}=\frac{1}{2 a}+\frac{1}{b}+\frac{1}{2 x}$ Solution: $\frac{1}{2 a+b+2 x}=\frac{1}{2 a}+\frac{1}{b}+\frac{1}{2 x}$ $\Rightarrow \frac{1}{2 a+b+2 x}-\frac{1}{2 x}=\frac{1}{2 a}+\frac{1}{b}$ $\Rightarrow \frac{2 x-2 a-b-2 x}{2 x(2 a+b+2 x)}=\frac{2 a+b}{2 a b}$ $\Rightarrow \frac{-(2 a+b)}{4 x^{2}+4 a x+2 b x}=\frac{2 a+b}{2 a b}$ $\Rightarrow 4 x^{2}+4 a x+2 b x=-2 a b$ $\Rightarrow 4 x^{2}+4 a x+2 b x+2 a b=0$ $\Righta...

Among the statements (I - IV), the correct ones are :

Question: Among the statements (I - IV), the correct ones are : (I) Be has smaller atomic radius compared to $\mathrm{Mg}$. (II) Be has higher ionization enthalpy than $\mathrm{Al}$. (III) Charge/radius ratio of Be is greater than that of Al: (IV) Both Be and Al form mainly covalent compounds.(I), (II) and (IV)(I), (III) and (IV)(II), (III) and (IV)(I), (II) and (III)Correct Option: 1 Solution: Charge/ radius ratio of Be and Al is same because of diagonal relationship. Remaining statements are c...

The integral

Question: The integral $\int_{1}^{2} e^{x} \cdot x^{x}\left(2+\log _{e} x\right) d x$ equals:(1) $e(4 e+1)$(2) $4 e^{2}-1$(3) $e(4 e-1)$(4) $e(2 e-1)$Correct Option: 1 Solution: $I=\int_{1}^{2} e^{x} x^{x}\left(2+\log _{e} x\right) d x$ $I=\int_{1}^{2} e^{x} x^{x}\left[1+\left(1+\log _{e} x\right)\right] d x$ $=\int_{1}^{2} e^{x}\left[x^{x}+x^{x}\left(1+\log _{e} x\right)\right] d x$ $\because \int e^{x}\left(f(x)+f^{\prime}(x)\right) d x=e^{x} f(x)+c$ $\therefore I=\left[e^{x} x^{x}\right]_{1}^...

512 identical drops of mercury are charged to a potential

Question: 512 identical drops of mercury are charged to a potential of $2 \mathrm{~V}$ each. The drops are joined to form a single drop. The potential of this drop is___ V. Solution: (128) Let charge on each drop $=\mathrm{q}$ radius $=r$ $v=\frac{k q}{r}$ $2=\frac{k q}{r}$ radius of bigger $\frac{4}{3} \pi R^{3}=512 \times \frac{4}{3} \pi r^{3}$ $R=8 r$ $v=\frac{k(512) q}{R}=\frac{512}{8} \frac{k q}{r}=\frac{512}{8} \times 2$ $=128 \mathrm{~V}$...

The atomic number of the element unnilennium is :

Question: The atomic number of the element unnilennium is :109102108119Correct Option: 1 Solution: un $=1$ nil $=0$ enn $=9$ So, atomic number $=109$...

if

Question: If $\mathrm{I}_{1}=\int_{0}^{1}\left(1-x^{50}\right)^{100} d x$ and $\mathrm{I}_{2}=\int_{0}^{1}\left(1-x^{50}\right)^{101} d x$ such that $\mathrm{I}_{2}=\alpha \mathrm{I}_{1}$ then $\alpha$ equals to :(1) $\frac{5049}{5050}$(2) $\frac{5050}{5049}$(3) $\frac{5050}{5051}$(4) $\frac{5051}{5050}$Correct Option: , 3 Solution: $I_{2}=\int_{0}^{1}\left(1-x^{50}\right)^{101} d x=\int_{0}^{1}\left(1-x^{50}\right)\left(1-x^{50}\right)^{100} d x$ $I_{2}=\int_{0}^{1}\left(1-x^{50}\right)^{100} d...

Solve each of the following quadratic equations:

Question: Solve each of the following quadratic equations: (i) $\frac{1}{x-1}-\frac{1}{x+5}=\frac{6}{7}, \quad x \neq 1,-5$ (ii) $\frac{1}{2 x-3}+\frac{1}{x-5}=1 \frac{1}{9}, \quad x \neq \frac{3}{2}, 5$ Solution: (i) $\frac{1}{x-1}-\frac{1}{x+5}=\frac{6}{7}, \quad x \neq 1,-5$ $\Rightarrow \frac{x+5-x+1}{(x-1)(x+5)}=\frac{6}{7}$ $\Rightarrow \frac{6}{x^{2}+4 x-5}=\frac{6}{7}$ $\Rightarrow x^{2}+4 x-5=7$ $\Rightarrow x^{2}+4 x-12=0$ $\Rightarrow x^{2}+6 x-2 x-12=0$ $\Rightarrow x(x+6)-2(x+6)=0$ ...

Three elements X, Y and Z are in the 3 period of the periodic table.

Question: Three elements $X, Y$ and $Z$ are in the $3^{\text {rd }}$ period of the periodic table. The oxides of X, Y and Z, respectively, are basic, amphoteric and acidic. The correct order of the atomic numbers of $X, Y$ and $Z$ is :$\mathrm{Z}\mathrm{Y}\mathrm{X}$$\mathrm{X}\mathrm{Y}\mathrm{Z}$$XZY$$YXZ$Correct Option: , 2 Solution: On moving left to right in a period, the acidic character of oxides increases. $3^{\text {rd }}$ period element oxides. Acidic character $\propto$ Atomic No. So,...

if

Question: If $\int \frac{\cos \theta}{5+7 \sin \theta-2 \cos ^{2} \theta} d \theta=A \log _{e}|B(\theta)|+C$, where $C$ is a constant of integration, then $\frac{B(\theta)}{A}$ can be :(1) $\frac{2 \sin \theta+1}{\sin \theta+3}$(2) $\frac{2 \sin \theta+1}{5(\sin \theta+3)}$(3) $\frac{5(\sin \theta+3)}{2 \sin \theta+1}$(4) $\frac{5(2 \sin \theta+1)}{\sin \theta+3}$Correct Option: , 4 Solution: Let $\sin \theta=t \Rightarrow \cos \theta d \theta=d t$ $\int \frac{\cos \theta}{5+7 \sin \theta-2 \cos...

The electric field in a region is given by

Question: The electric field in a region is given by $\vec{E}=\left(\frac{3}{5} E_{0} \hat{i}+\frac{4}{5} E_{0} \hat{j}\right)^{\frac{N}{c}}$. The ratio of flux of reported field through the rectangular surface of area $0.2 \mathrm{~m}^{2}$ (parallel to $\mathrm{y}-\mathrm{z}$ plane ) to that of the surface of area $0.3 \mathrm{~m}^{2}$ (parallel to $\mathrm{x}-\mathrm{z}$ plane ) is a : $\mathrm{b}$, where $\mathrm{a}=$ (round off to nearest integer) [Here $\hat{\mathrm{i}}, \hat{\mathrm{j}}$ a...

if

Question: If $\int\left(e^{2 x}+2 e^{x}-e^{-x}-1\right) e^{\left(e^{x}+e^{-x}\right)} d x=g(x) e^{\left(e^{x}+e^{-x}\right)}+c$, where $c$ is a constant of integeration, then $g(0)$ is equal to:(1) $e$(2) $e^{2}$(3) 1(4) 2Correct Option: , 4 Solution: $\int\left(e^{2 x}+2 e^{x}-e^{-x}-1\right) \cdot e^{\left(e^{x}+e^{-x}\right)} d x$ $I=\int\left(e^{2 x}+e^{x}-1\right) \cdot e^{\left(e^{x}+e^{-x}\right)} d x+\int\left(e^{x}-e^{-x}\right) e^{\left(e^{x}+e^{-x}\right)} d x$ $=\int e^{x}\left(e^{x}...

Solve each of the following quadratic equations:

Question: Solve each of the following quadratic equations: $\frac{3}{x+1}-\frac{1}{2}=\frac{2}{3 x-1}, \quad x \neq-1, \frac{1}{3}$ Solution: $\frac{3}{x+1}-\frac{1}{2}=\frac{2}{3 x-1}, \quad x \neq-1, \frac{1}{3}$ $\Rightarrow \frac{3}{x+1}-\frac{2}{3 x-1}=\frac{1}{2}$ $\Rightarrow \frac{9 x-3-2 x-2}{(x+1)(3 x-1)}=\frac{1}{2}$ $\Rightarrow \frac{7 x-5}{3 x^{2}+2 x-1}=\frac{1}{2}$ $\Rightarrow 3 x^{2}+2 x-1=14 x-10 \quad$ (Cross multiplication) $\Rightarrow 3 x^{2}-12 x+9=0$ $\Rightarrow x^{2}-4...