Solve this following
Question: The population $\mathrm{P}=\mathrm{P}(\mathrm{t})$ at time ' $\mathrm{t}$ ' of a certain species follows the differential equation $\frac{\mathrm{dP}}{\mathrm{dt}}=0.5 \mathrm{P}-450 .$ If $\mathrm{P}(0)=850$, then the time at which population becomes zero is : $\log _{\mathrm{e}} 18$$\log _{\mathrm{e}} 9$$\frac{1}{2} \log _{c} 18$$2 \log _{\mathrm{e}} 18$Correct Option: , 4 Solution: $\frac{\mathrm{dP}}{\mathrm{dt}}=0.5 \mathrm{P}-450$ $\Rightarrow \int_{0}^{t} \frac{d p}{P-900}=\int_...
Read More →Let α be the angle between the lines whose direction cosines
Question: Let $\alpha$ be the angle between the lines whose direction cosines satisfy the equations $l+\mathrm{m}-\mathrm{n}=0$ and $l^{2}+\mathrm{m}^{2}-\mathrm{n}^{2}=0$. Then the value of $\sin ^{4} \alpha+\cos ^{4} \alpha$ is :$\frac{3}{4}$$\frac{3}{8}$$\frac{5}{8}$$\frac{1}{2}$Correct Option: , 3 Solution: $\mathrm{n}=\ell+\mathrm{m}$ Now, $\ell^{2}+\mathrm{m}^{2}=\mathrm{n}^{2}=(\ell+\mathrm{m})^{2}$ $\Rightarrow 2 \ell \mathrm{m}=0$ If $\ell=0 \Rightarrow \mathrm{m}=\mathrm{n}=\pm \frac{1...
Read More →The value of
Question: The value of $\lim _{h \rightarrow 0} 2\left\{\frac{\sqrt{3} \sin \left(\frac{\pi}{6}+h\right)-\cos \left(\frac{\pi}{6}+h\right)}{\sqrt{3} h(\sqrt{3} \cosh -\sinh )}\right\}$ is$\frac{4}{3}$$\frac{2}{\sqrt{3}}$$\frac{3}{4}$$\frac{2}{3}$Correct Option: 1 Solution: $L=\lim _{h \rightarrow 0} 2\left(\frac{\sqrt{3}\left(\frac{1}{2} \cosh +\frac{\sqrt{3}}{2} \sinh \right)-\left(\frac{\sqrt{3}}{2} \cosh -\frac{\sinh }{2}\right)}{(\sqrt{3} h)(\sqrt{3})}\right)$ $\mathrm{L}=\lim _{\mathrm{h} \...
Read More →The equation of the plane passing through the
Question: The equation of the plane passing through the point $(1,2,-3)$ and perpendicular to the planes $3 x+y-2 z=5$ and $2 x-5 y-z=7$, is $3 x-10 y-2 z+11=0$$6 x-5 y-2 z-2=0$$11 x+y+17 z+38=0$$6 x-5 y+2 z+10=0$Correct Option: , 3 Solution: Normal vector : So drs of normal to the required plane is $11,1,17$ plane passes through $(1,2,-3)$ So eq $^{\mathrm{n}}$ of plane : $11(x-1)+1(y-2)+17(z+3)=0$ $\Rightarrow \quad 11 x+y+17 z+38=0$...
Read More →The sum of the infinite series
Question: The sum of the infinite series $1+\frac{2}{3}+\frac{7}{3^{2}}+\frac{12}{3^{3}}+\frac{17}{3^{4}}+\frac{22}{3^{5}}+\ldots$ is equal to$\frac{13}{4}$$\frac{9}{4}$$\frac{15}{4}$$\frac{11}{4}$Correct Option: 1 Solution: $\mathrm{S}=1+\frac{2}{3}+\frac{7}{3^{2}}+\frac{12}{3^{3}}+\frac{17}{3^{4}}+\ldots$ $\frac{S}{3}=\frac{1}{3}+\frac{2}{3^{2}}+\frac{7}{3^{3}}+\frac{12}{3^{4}}+\ldots$ $\frac{2 S}{3}=1+\frac{1}{3}+\frac{5}{3^{2}}+\frac{5}{3^{3}}+\frac{5}{3^{4}}+\ldots+$ up to infinite terms $\...
Read More →The equation of the line through the point (0,1,2)
Question: The equation of the line through the point $(0,1,2)$ and perpendicular to the line $\frac{x-1}{2}=\frac{y+1}{3}=\frac{z-1}{-2}$ is :$\frac{x}{3}=\frac{y-1}{4}=\frac{z-2}{3}$$\frac{x}{3}=\frac{y-1}{-4}=\frac{z-2}{3}$$\frac{x}{3}=\frac{y-1}{4}=\frac{z-2}{-3}$$\frac{x}{-3}=\frac{y-1}{4}=\frac{z-2}{3}$Correct Option: , 4 Solution: $\frac{x-1}{2}=\frac{y+1}{3}=\frac{z-1}{-2}=r$ $\Rightarrow \mathrm{P}(\mathrm{x}, \mathrm{y}, \mathrm{z})=(2 \mathrm{r}+1,3 \mathrm{r}-1,-2 \mathrm{r}+1)$ Since...
Read More →A man is walking on a straight line. The arithmetic mean of the reciprocals of the intercepts of this line on the coordinate axes
Question: A man is walking on a straight line. The arithmetic mean of the reciprocals of the intercepts of this line on the coordinate axes is $\frac{1}{4} .$ Three stones A, B and C are placed at the points $(1,1),(2,2)$ and $(4,4)$ respectively. Then which of these stones is / are on the path of the man? A onlyConlyAll the threeB onlyCorrect Option: , 4 Solution: Let the line be $y=m x+c$ $\mathrm{x}$-intercept : $-\frac{\mathrm{c}}{\mathrm{m}}$ y-intercept : c A.M of reciprocals of the interc...
Read More →In the circle given below,
Question: In the circle given below, let $\mathrm{OA}=1$ unit, $\mathrm{OB}=13$ unit and $\mathrm{PQ} \perp \mathrm{OB}$. Then, the area of the triangle PQB (in square units) is $24 \sqrt{2}$$24 \sqrt{3}$$26 \sqrt{3}$$26 \sqrt{2}$Correct Option: , 2 Solution: $\mathrm{PA}=\mathrm{AQ}=\lambda$ $\mathrm{OA} \cdot \mathrm{AB}$ $=\mathrm{AP} \cdot \mathrm{AQ}$.3 $\Rightarrow 1.12=\lambda . \lambda$ $\Rightarrow \lambda=2 \sqrt{3}$ Area $\triangle \mathrm{PQB}=\frac{1}{2} \times 2 \lambda \times \mat...
Read More →Let f, g: N → N such that f(n+1)=f(n)+f(1)
Question: Let $f, g: N \rightarrow N$ such that $f(n+1)=f(n)+f(1)$ $\forall \mathrm{n} \in \mathrm{N}$ and $g$ be any arbitrary function. Which of the following statements is NOT true?If fog is one-one, then $g$ is one-oneIf $\mathrm{f}$ is onto, then $\mathrm{f}(\mathrm{n})=\mathrm{n} \forall \mathrm{n} \in \mathrm{N}$$\mathrm{f}$ is one-oneIf $g$ is onto, then fog is one-oneCorrect Option: , 4 Solution: $f(n+1)-f(n)=f(1)$ $\Rightarrow \mathrm{f}(\mathrm{n})=\mathrm{nf}(1)$ $\Rightarrow \mathrm...
Read More →The value of
Question: The value of $\sum_{n=1}^{100} \int_{n-1}^{n} e^{x-[x]} d x$, where $[x]$ is the greatest integer $\leq \mathrm{x}$, is$100(\mathrm{e}-1)$$100(1-\mathrm{e})$$100 \mathrm{e}$$100(1+e)$Correct Option: 1 Solution: $\sum_{n=1}^{100} \int_{n-1}^{n} e^{\{x\}} d x$, period of $\{x\}=1$ $\sum_{n=1}^{100} \int_{0}^{1} e^{\{x\}} d x=\sum_{n=1}^{100} \int_{0}^{1} e^{x} d x$ $\sum_{n=1}^{100}(e-1)=100(e-1)$...
Read More →The statement among the following that is a tautology is :
Question: The statement among the following that is a tautology is : $A \vee(A \wedge B)$$A \wedge(A \vee B)$$\mathrm{B} \rightarrow[\mathrm{A} \wedge(\mathrm{A} \rightarrow \mathrm{B})]$$[\mathrm{A} \wedge(\mathrm{A} \rightarrow \mathrm{B})] \rightarrow \mathrm{B}$Correct Option: , 4 Solution: $(\mathrm{A} \wedge(\mathrm{A} \rightarrow \mathrm{B})) \rightarrow \mathrm{B}$ $=(A \wedge(\sim A \vee B)) \rightarrow B$ $=((A \wedge \sim A) \vee(A \wedge B)) \rightarrow B$ $=(A \wedge B) \rightarrow ...
Read More →Prove the following
Question: If $0\theta, \phi\frac{\pi}{2}, x=\sum_{n=0}^{\infty} \cos ^{2 n} \theta, y=\sum_{n=0}^{\infty} \sin ^{2 n} \phi$ and $z=\sum_{n=0}^{\infty} \cos ^{2 n} \theta \cdot \sin ^{2 n} \phi$ then :$x y-z=(x+y) z$$x y+y z+z x=z$$x y z=4$$x y+z=(x+y) z$Correct Option: , 4 Solution: $x=\frac{1}{1-\cos ^{2} \theta} \Rightarrow \sin ^{2} \theta=\frac{1}{x}$ Also, $\cos ^{2} \theta=\frac{1}{y} \ 1-\sin ^{2} \theta \cos ^{2} \theta=\frac{1}{z}$ So, $1-\frac{1}{x} \times \frac{1}{y}=\frac{1}{z} \Righ...
Read More →In a increasing geometric series,
Question: In a increasing geometric series, the sum of the second and the sixth term is $\frac{25}{2}$ and the product of the third and fifth term is 25 . Then, the sum of $4^{\text {th }}, 6^{\text {th }}$ and $8^{\text {th }}$ terms is equal to30263532Correct Option: , 3 Solution: $\mathrm{a}, \mathrm{ar}, \mathrm{ar}^{2}, \ldots$ $\mathrm{T}_{2}+\mathrm{T}_{6}=\frac{25}{2} \Rightarrow \operatorname{ar}\left(1+\mathrm{r}^{4}\right)=\frac{25}{2}$ $a^{2} r^{2}\left(1+r^{4}\right)^{2}=\frac{625}{...
Read More →When a missile is fired from a ship, the
Question: When a missile is fired from a ship, the probability that it is intercepted is $\frac{1}{3}$ and the probability that the missile hits the target, given that it is not intercepted, is $\frac{3}{4}$. If three missiles are fired independently from the ship, then the probability that all three hit the target, is :$\frac{1}{27}$$\frac{3}{4}$$\frac{1}{8}$$\frac{3}{8}$Correct Option: , 3 Solution: Required probability $=\left(\frac{2}{3} \times \frac{3}{4}\right)^{3}=\frac{1}{8}$...
Read More →Solve the Following Questions
Question: Let $A$ 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 is41612Correct Option: 1 Solution: $A=\left(\begin{array}{ll}a b \\ b c\end{array}\right)$ $\mathrm{a}, \mathrm{b}, \mathrm{c} \in \mathrm{I}$ $A^{2}=\left(\begin{array}{ll}a b \\ b c\end{array}\right)\left(\begin{array}{ll}a b \\ b c\end{array}\right)=\left(\begin{array}{cc}a^{2}+b^{2} b(a+c) \\ b(a+c) b^{2}+c^{2}\end{a...
Read More →A fair coin is tossed
Question: A fair coin is tossed a fixed number of times. If the probability of getting 7 heads is equal to probability of getting 9 heads, then the probability of getting 2 heads is$\frac{15}{2^{13}}$$\frac{15}{2^{12}}$$\frac{15}{2^{8}}$$\frac{15}{2^{14}}$Correct Option: 1 Solution: Let the coin be tossed n-times $\mathrm{P}(\mathrm{H})=\mathrm{P}(\mathrm{T})=\frac{1}{2}$ $P(7$ heads $)={ }^{n} C_{7}\left(\frac{1}{2}\right)^{n-7}\left(\frac{1}{2}\right)^{7}=\frac{{ }^{n} C_{7}}{2^{n}}$ $\mathrm{...
Read More →Solve the Following Question
Question: If $\overrightarrow{\mathrm{a}}$ and $\overrightarrow{\mathrm{b}}$ are perpendicular, then $\vec{a} \times(\vec{a} \times(\vec{a} \times(\vec{a} \times \vec{b})))$ is equal to$\overrightarrow{0}$$\frac{1}{2}|\vec{a}|^{4} \vec{b}$$\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}$$|\vec{a}|^{4} \vec{b}$Correct Option: , 4 Solution: $\overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{b}}=0$ $\vec{a} \times(\vec{a} \times \vec{b})=(\vec{a} \cdot \vec{b}) \vec{a}-(\vec{...
Read More →A prism of angle A=1° has a refractive
Question: A prism of angle $\mathrm{A}=1^{\circ}$ has a refractive index $\mu=1.5$. A good estimate for the minimum angle of deviation (in degrees) is close to N/ 10. Value of $\mathrm{N}$ is___________. Solution: $\delta_{\min }=(\mu-1) \mathrm{A}$ $=(1.5-1) 1$ $=0.5$ $\delta_{\min }=\frac{5}{10}$ $\mathrm{N}=5$...
Read More →A body of mass 2kg is driven by an engine
Question: A body of mass $2 \mathrm{~kg}$ is driven by an engine delivering a constant power $1 \mathrm{~J} / \mathrm{s}$. The body starts from rest and moves in a straight line. After 9 seconds, the body has moved a distance (in $\mathrm{m}$ )_______. Solution: $P=\operatorname{mav}$ $\mathrm{m} \frac{\mathrm{dv}}{\mathrm{dt}} \mathrm{v}=\mathrm{P}$ $\int_{0}^{v} v d v=\frac{P}{m} \int_{0}^{t} d t$ $\frac{\mathrm{v}^{2}}{2}=\frac{\mathrm{Pt}}{\mathrm{m}} \Rightarrow \mathrm{v}=\left(\frac{2 \ma...
Read More →A thin rod of mass 0.9kg and length
Question: A thin rod of mass $0.9 \mathrm{~kg}$ and length $1 \mathrm{~m}$ is suspended, at rest, from one end so that it can freely oscillate in the vertical plane. A particle of move $0.1 \mathrm{~kg}$ moving in a straight line with velocity $80 \mathrm{~m} / \mathrm{s}$ hits the rod at its bottom most point and sticks to it (see figure). The angular speed (in $\mathrm{rad} / \mathrm{s}$ ) of the rod immediately after the collision will be Solution: '$\overrightarrow{\mathrm{L}}_{\mathrm{i}}=\...
Read More →Nitrogen gas is at 300° C temperature.
Question: Nitrogen gas is at $300^{\circ} \mathrm{C}$ temperature. The temperature (in $\mathrm{K}$ ) at which the rms speed of a $\mathrm{H}_{2}$ molecule would be equal to the rms speed of a nitrogen molecule, is__________. (Molar mass of $\mathrm{N}_{2}$ gas $28 \mathrm{~g}$ ) Solution: $\mathrm{V}_{\mathrm{rms}}=\sqrt{\frac{3 \mathrm{RT}}{\mathrm{M}}}$ $\mathrm{V}_{\mathrm{N}_{2}}=\mathrm{V}_{\mathrm{H}_{2}}$' $\sqrt{\frac{3 R T_{N_{2}}}{M_{N_{2}}}}=\sqrt{\frac{3 R T_{H_{2}}}{M_{H_{2}}}}$' $...
Read More →The surface of a metal is illuminated alternately
Question: The surface of a metal is illuminated alternately with photons of energies $\mathrm{E}_{1}=4 \mathrm{eV}$ and $\mathrm{E}_{2}=2.5 \mathrm{eV}$ respectively. The ratio of maximum speeds of the photoelectrons emitted in the two cases is 2 . The work function of the metal in $(\mathrm{eV})$ is_________. Solution: $\mathrm{E}_{1}=\phi+\mathrm{K}_{1} \ldots(1)$ $\mathrm{E}_{2}=\phi+\mathrm{K}_{2}$.......(2) $E_{1}-E_{2}=K_{1}-K_{2}$ Now $\frac{\mathrm{V}_{1}}{\mathrm{~V}_{2}}=2$ $\frac{\mat...
Read More →The correct match between the entries in column I and column II are:
Question: The correct match between the entries in column I and column II are: (a)-(ii), (b)-(i), (c)-(iv), (d)-(iii)(a)-(i), (b)-(iii), (c)-(iv), (d)-(ii)(a)-(iii), (b)-(ii), (c)-(i), (d)-(iv)(a)-(iv), (b)-(ii), (c)-(i), (d)-(iii)Correct Option: , 4 Solution: Energes of given Radiation can have The following relation $\mathrm{E}_{\gamma \text {-Rays }}\mathrm{E}_{\mathrm{X}-\mathrm{Rays}}\mathrm{E}_{\text {microwave }}\mathrm{E}_{\mathrm{AM} \text { Radiowaves }}$' $\therefore \lambda_{\gamma \...
Read More →Two Zener diodes (A and B) having breakdown voltages
Question: Two Zener diodes (A and B) having breakdown voltages of $6 \mathrm{~V}$ and $4 \mathrm{~V}$ respectively, are connected as shown in athe circuit below. The output voltage $\mathrm{V}_{0}$ variation with input voltage linearly increasing with time, is given by: $\left(\mathrm{V}_{\text {input }}=0 \mathrm{~V}\right.$ at $\left.\mathrm{t}=0\right)$ (figures are qualitative) Correct Option: , 4 Solution: Till input voltage Reaches $4 \mathrm{~V}$ No zener is in Breakdown Region So $\mathr...
Read More →A spaceship in space sweeps stationary interplanetary dust.
Question: A spaceship in space sweeps stationary interplanetary dust. As a result, its mass increases at a rate $\frac{\mathrm{dM}(\mathrm{t})}{\mathrm{dt}}=\mathrm{bv}^{2}(\mathrm{t})$, where $\mathrm{v}(\mathrm{t})$ is its instantaneous velocity. The instantaneous acceleration of the satellite is:$-\frac{2 b v^{3}}{M(t)}$$-\frac{\mathrm{bv}^{3}}{2 \mathrm{M}(\mathrm{t})}$$-b v^{3}(t)$$-\frac{\mathrm{bv}^{3}}{\mathrm{M}(\mathrm{t})}$Correct Option: , 4 Solution: $\frac{\mathrm{dm}(\mathrm{t})}{...
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