Resistance are connected in a meter bridge circuit as shown in the figure. The balancing length \(l_{1}\) is \(40\,cm\). Now an unknown resistance \(x\) is connected in series with \(P\) and new balancing length is found to be \(80\,cm\) measured from the same end. Then the value of \(x\) will be \(.......\Omega\)

Rods x and y of equal dimensions but of different materials are joined as shown in figure. Temperatures of end points A and F are maintained at \( 100^{\circ}C \) and \( 40^{\circ}C \) respectively. Given the thermal conductivity of rod x is three times of that of rod y, the temperature at junction points B and E are (close to):

A block of mass \(M\) slides down on a rough inclined plane with constant velocity. The angle made by the incline plane with horizontal is \(\theta\). The magnitude of the contact force will be.

In a building there are \(15\) bulbs of \(45\; \mathrm{W}, 15\) bulbs of \(100\; \mathrm{W}, 15\) small fans of \(10 \;\mathrm{W}\) and \(2\) heaters of \(1 \;\mathrm{kW}\). The voltage of electric main is \(220\; \mathrm{V}\). The minimum fuse capacity (rated value) of the building will be: .......... \(A\)

A bullet of mass \(0.1\,kg\) moving horizontally with speed \(400\,ms ^{-1}\) hits a wooden block of mass \(3.9\,kg\) kept on a horizontal rough surface. The bullet gets embedded into the block and moves \(20\,m\) before coming to rest. The coefficient of friction between the block and the surface is \(........\) \(\left(\text { Given } g=10 \,ms ^2\right. \text { ) }\)

The least count of the main scale of a vernier callipers is \(1\, mm\). Its vernier scale is divided into \(10\) divisions and coincide with \(9\) divisions of the main scale. When jaws are touching each other, the \(7\) th division of vernier scale coincides with a division of main scale and the zero of vernier scale is lying right side of the zero of main scale. When this vernier is used to measure length of a cylinder the zero of the vernier scale between \(3.1\, cm\) and \(3.2\, cm\) and \(4^{th}\) \(VSD\) coincides with a main scale division.The length of the cylinder is \(.....cm\) (\(VSD\) is vernier scale division)

A cylindrical block of wood (density \(= 650\, kg\, m^{-3}\)), of base area \(30\,cm^2\) and height \(54\, cm\), floats in a liquid of density \(900\, kg\, m^{-3}\) . The block is depressed slightly and then released. The time period of the resulting oscillations of the block would be equal to that of a simple pendulum of length ..... \(cm\) (nearly)

The surface tension of a soap bubble is \(0.03\) N/m. The work done in increasing the diameter of bubble from \(2\) cm to \(6\) cm is \(\alpha \pi \times 10^{-4}\) J. The value of \(\alpha\) is _______. (Take \(\pi = 3.14\))

Let \(S\) be the set of all real roots of the equation, \(3^{x}\left(3^{x}-1\right)+2=\left|3^{x}-1\right|+\left|3^{x}-2\right| .\) Then \(\mathrm{S}\)

\(\lim \limits_{x \rightarrow 0} \frac{x\left(e^{\frac{\left(\sqrt{1+x^{2}+x^{4}}-1\right)}{x}}-1\right)}{\sqrt{1+x^{2}+x^{4}}-1}\) is equal to 

Given below are two statements : one is labelled as Assertion \(A\) and the other is labelled as Reason \(R\). Assertion \(A\) : When a body is projected at an angle \(45^{\circ}\), it's range is maximum. Reason \(R\) : For maximum range, the value of \(\sin 2 \theta\) should be equal to one. In the light of the above statements, choose the correct answer from the options given below :

The number of four -digit numbers strictly greater than \(4321\) that can be formed using the digits \(0, 1, 2, 3, 4, 5\) (repetition of digits is allowed) is

The term independent of \(x\) in the expansion of \(\left[\frac{x+1}{x^{2 / 3}-x^{1 / 3}+1}-\frac{x-1}{x-x^{1 / 2}}\right]^{10}, x \neq 1,\) is equal to ....... .