An infinitely long straight wire carrying current I,
Question: An infinitely long straight wire carrying current I, one side opened rectangular loop and a conductor C with a sliding connector are located in the same plane, as shown in the figure. The connector has length $l$ and resistance R. It slides to the right with a velocity v. The resistance of the conductor and the self inductance of the loop are negligible. The induced current in the loop, as a function of separation $r$, between the connector and the straight wire is : $\frac{\mu_{0}}{\p...
Read More →An iron rod of volume
Question: An iron rod of volume $10^{-3} \mathrm{~m}^{3}$ and relative permeability 1000 is placed as core in a solenoid with 10 turns/cm. If a current of $0.5 \mathrm{~A}$ is passed through the solenoid, then the magnetic moment of the rod will be :$0.5 \times 10^{2} \mathrm{Am}^{2}$$50 \times 10^{2} \mathrm{Am}^{2}$$500 \times 10^{2} \mathrm{Am}^{2}$$5 \times 10^{2} \mathrm{Am}^{2}$Correct Option: , 4 Solution:...
Read More →Ten charges are placed on the circumference of a circle
Question: Ten charges are placed on the circumference of a circle of radius $\mathrm{R}$ with constant angular separation between successive charges. Alternate charges $1,3,5,7,9$ have charge $(+q)$ each, while $2,4,6,8,10$ have charge $(-q)$ each. The potential $\mathrm{V}$ and the electric field $\mathrm{E}$ at the centre of the circle are respectively: (Take $V=0$ at infinity)$\mathrm{V}=\frac{10 \mathrm{q}}{4 \pi \epsilon_{0} \mathrm{R}} ; \mathrm{E}=\frac{10 \mathrm{q}}{4 \pi \epsilon_{0} \...
Read More →A parallel plate capacitor has plate of length
Question: A parallel plate capacitor has plate of length ' $l$ ', width 'w' and separation of plates is 'd'. It is connected to a battery of emf V. A dielectric slab of the same thickness 'd' and of dielectric constant $k=4$ is being inserted between the plates of the capacitor. At what length of the slab inside plates, will be energy stored in the capacitor be two times the initial energy stored?$l / 4$$l / 2$$l / 3$$2 l / 3$Correct Option: , 3 Solution:...
Read More →In the circuit, given in the figure currents in different branches
Question: In the circuit, given in the figure currents in different branches and value of one resistor are shown. Then potential at point B with respect to the point $A$ is : $+1 \mathrm{~V}$$-1 V$$-2 \mathrm{~V}$$+2 \mathrm{~V}$Correct Option: 1 Solution:...
Read More →Prove the following
Question: If $\cos (\alpha+\beta)=\frac{3}{5}, \sin (\alpha-\beta)=\frac{5}{13}$ and $0\alpha, \beta\frac{\pi}{4}$, then $\tan (2 \alpha)$ is equal to :$\frac{21}{16}$$\frac{63}{52}$$\frac{33}{52}$$\frac{63}{16}$Correct Option: , 4 Solution:...
Read More →The greatest value of
Question: The greatest value of $\mathrm{c} \in \mathrm{R}$ for which the system of linear equations $x-c y-c z=0$ $c x-y+c z=0$ $c x+c y-z=0$ has a non-trivial solution, is :$\frac{1}{2}$$-1$02Correct Option: 1 Solution:...
Read More →Prove the following
Question: If $2 y=\left(\cot ^{-1}\left(\frac{\sqrt{3} \cos x+\sin x}{\cos x-\sqrt{3} \sin x}\right)\right)^{2}, x \in\left(0, \frac{\pi}{2}\right)$ then $\frac{d y}{d x}$ is equal to :$2 x-\frac{\pi}{3}$$\frac{\pi}{3}-x$$\frac{\pi}{6}-x$$x-\frac{\pi}{6}$Correct Option: 4, Solution:...
Read More →Prove the following
Question: $\int \frac{\sin \frac{5 x}{2}}{\sin \frac{x}{2}} d x$ is equal to : (where $\mathrm{c}$ is a constant of integration) Solution:...
Read More →All possible numbers are formed using the digits
Question: All possible numbers are formed using the digits $1,1,2,2,2,2,3,4,4$ taken all at a time. The number of such numbers in which the odd digits occupy even places is :175162160180Correct Option: , 4 Solution:...
Read More →The sum of the squares of the lengths of the chords intercepted
Question: The sum of the squares of the lengths of the chords intercepted on the circle, $x^{2}+y^{2}=16$, by the lines, $x+y=n, n \in N$, where $N$ is the set of all natural numbers, is:320160105210Correct Option: , 4 Solution:...
Read More →Prove the following
Question: Let $f:[0,2] \rightarrow \mathrm{R}$ be a twice differentiable function such that $f^{\prime \prime}(\mathrm{x})0$, for all $\mathrm{x} \in(0,2)$. If $\phi(x)=f(x)+f(2-x)$, then $\phi$ is :decreasing on $(0,2)$decreasing on $(0,1)$ and increasing on $(1,2)$increasing on $(0,2)$increasing on $(0,1)$ and decreasing on $(1,2)$Correct Option: , 2 Solution:...
Read More →The area (in sq. units) of the region
Question: The area (in sq. units) of the region $\begin{aligned} A=\{(x, y) \in R \times R \mid 0 \leq x \leq 3,0 \leq y \leq 4\\ \left.y \leq x^{2}+3 x\right\} \text { is : } \end{aligned}$$\frac{53}{6}$$\frac{59}{6}$8$\frac{26}{3}$Correct Option: , 2 Solution:...
Read More →The sum of the co-efficients of all even degree terms in
Question: The sum of the co-efficients of all even degree terms in $\mathrm{x}$ in the expansion of $\left(x+\sqrt{x^{3}-1}\right)^{6}+\left(x-\sqrt{x^{3}-1}\right)^{6},(x1)$ is equal to:32262924Correct Option: , 4 Solution:...
Read More →Let A and B be two non-null events such
Question: Let $\mathrm{A}$ and $\mathrm{B}$ be two non-null events such that $A \subset B$. Then, which of the following statements is always correct ?$\mathrm{P}(\mathrm{A} \mid \mathrm{B})=1$$\mathrm{P}(\mathrm{A} \mid \mathrm{B})=\mathrm{P}(\mathrm{B})-\mathrm{P}(\mathrm{A})$$\mathrm{P}(\mathrm{A} \mid \mathrm{B}) \leq \mathrm{P}(\mathrm{A})$$\mathrm{P}(\mathrm{A} \mid \mathrm{B}) \geq \mathrm{P}(\mathrm{A})$Correct Option: , 4 Solution:...
Read More →The sum of the solutions of the equation
Question: The sum of the solutions of the equation $|\sqrt{x}-2|+\sqrt{x}(\sqrt{x}-4)+2=0,(x0)$ is equal to :491012Correct Option: , 3 Solution:...
Read More →The sum of the series
Question: The sum of the series $2 \cdot{ }^{20} \mathrm{C}_{0}+5 \cdot{ }^{20} \mathrm{C}_{1}+8 \cdot{ }^{20} \mathrm{C}_{2}+11 .{ }^{20} \mathrm{C}_{3}+\ldots+62 . \cdot{ }^{20} \mathrm{C}_{20}$ is equal to:$2^{24}$$2^{25}$$2^{26}$$2^{23}$Correct Option: , 2 Solution:...
Read More →The sum of all natural numbers ' n ' such
Question: The sum of all natural numbers ' $n$ ' such that $100\mathrm{n}200$ and H.C.F. $(91, \mathrm{n})1$ is :3221312132033303Correct Option: , 2 Solution:...
Read More →The equation of a plane containing the line
Question: The equation of a plane containing the line of intersection of the planes $2 x-y-4=0$ and $y+2 z-4=0$ and passing through the point $(1,1,0)$ is :$x+3 y+z=4$$x-y-z=0$$x-3 y-2 z=-2$$2 \mathrm{x}-\mathrm{z}=2$Correct Option: , 2 Solution:...
Read More →Solve the following
Question: If $f(x)=\log _{e}\left(\frac{1-x}{1+x}\right),|x|1$, then $f\left(\frac{2 x}{1+x^{2}}\right)$ is equal to : $2 f(\mathrm{x})$$2 f\left(x^{2}\right)$$(f(x))^{2}$$-2 f(x)$Correct Option: 1 Solution:...
Read More →Prove the following
Question: Let $\mathrm{A}=\left(\begin{array}{cc}\cos \alpha -\sin \alpha \\ \sin \alpha \cos \alpha\end{array}\right),(\alpha \in \mathrm{R})$ such that $A^{32}=\left(\begin{array}{cc}0 -1 \\ 1 0\end{array}\right) .$ Then a value of $\alpha$ is$\frac{\pi}{16}$0$\frac{\pi}{32}$$\frac{\pi}{64}$Correct Option: , 4 Solution:...
Read More →Let O(0,0) and A(0,1) be two fixed points.
Question: Let $\mathrm{O}(0,0)$ and $\mathrm{A}(0,1)$ be two fixed points. Then the locus of a point $P$ such that the perimeter of $\triangle A O P$ is 4 , is :$8 x^{2}-9 y^{2}+9 y=18$$9 x^{2}+8 y^{2}-8 y=16$$8 x^{2}+9 y^{2}-9 y=18$$9 x^{2}-8 y^{2}+8 y=16$Correct Option: , 2 Solution:...
Read More →If S_1 and S_2 are respectively the sets of local minimum
Question: If $S_{1}$ and $S_{2}$ are respectively the sets of local minimum and local maximum points of the function, $f(\mathrm{x})=9 \mathrm{x}^{4}+12 \mathrm{x}^{3}-36 \mathrm{x}^{2}+25, \mathrm{x} \in \mathrm{R}$, then :$\mathrm{S}_{1}=\{-2,1\} ; \mathrm{S}_{2}=\{0\}$$\mathrm{S}_{1}=\{-2,0\} ; \mathrm{S}_{2}=\{1\}$$\mathrm{S}_{1}=\{-2\} ; \mathrm{S}_{2}=\{0,1\}$$S_{1}=\{-1\} ; S_{2}=\{0,2\}$Correct Option: 1 Solution:...
Read More →Prove the following
Question: If $\alpha=\cos ^{-1}\left(\frac{3}{5}\right), \beta=\tan ^{-1}\left(\frac{1}{3}\right)$, where $0\alpha, \beta\frac{\pi}{2}$, then $\alpha-\beta$ is equal to : $\sin ^{-1}\left(\frac{9}{5 \sqrt{10}}\right)$$\tan ^{-1}\left(\frac{9}{14}\right)$$\cos ^{-1}\left(\frac{9}{5 \sqrt{10}}\right)$$\tan ^{-1}\left(\frac{9}{5 \sqrt{10}}\right)$Correct Option: 1 Solution:...
Read More →If the tangents on the ellipse
Question: If the tangents on the ellipse $4 x^{2}+y^{2}=8$ at the points $(1,2)$ and $(a, b)$ are perpendicular to each other, then $\mathrm{a}^{2}$ is equal to :$\frac{64}{17}$$\frac{2}{17}$$\frac{128}{17}$$\frac{4}{17}$Correct Option: , 2 Solution:...
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