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Two plane harmonic sound waves are expressed by the equations.
\(\begin{aligned}
& y_1(x, t)=A \cos (0.5 \pi x-100 \pi t) \\
& y_2(x, t)=A \cos (0.46 \pi x-92 \pi t)
\end{aligned}\)
(All parameters are in MKS)
Question :
How many times does an observer hear maximum intensity in one second?

An optical bench has $1.5\mathrm{ }\mathrm{m}$ long scale having four equal divisions in each $\mathrm{c}\mathrm{m}.$ While measuring the focal length of a convex lens, the lens is kept at $75\mathrm{ }\mathrm{c}\mathrm{m}$ mark of the scale and the object pin is kept at $45\mathrm{ }\mathrm{c}\mathrm{m}$ mark. The image of the object pin on the other side of the lens overlaps with image pin that is kept at $135\mathrm{ }\mathrm{c}\mathrm{m}$ mark. In this experiment, the percentage error in the measurement of the focal length of the lens is_____

A diatomic ideal gas is compressed adiabatically to \(\frac{1}{32}\) of its initial volume. If the initial temperature of the gas is \(T_i\) \(T_i\) (in kelvin) and the final temperature is \(a T_i\), the value of \(a\) is

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When liquid medicine of density \(\rho\) is to be put in the eye, it is done with the help of a dropper. As the bulb on the top of the dropper is pressed, a drop forms at the opening of the dropper. We wish to estimate the size of the drop.
We first assume that the drop formed at the opening is spherical because that requires a minimum increase in its surface energy. To determine the size, we calculate the net vertical force due to the surface tension \(T\) when the radius of the drop is \(R\). When this force becomes smaller than the weight of the drop, the drop gets detached from the dropper.
Question :
If \(r=5 \times 10^{-4} \mathrm{~m}, \rho=10^3 \mathrm{~kg} \mathrm{~m}^{-3}\), \(g=10 \mathrm{~ms}^{-2}, T=0.11 \mathrm{Nm}^{-1}\), the radius of the drop when it detaches from the dropper is approximately

A uniform rod of length \(L\) and mass \(M\) is pivoted at the centre. Its two ends are attached to two springs of equal spring constants \(k\). The springs are fixed to rigid supports as shown in the figure, and rod is free to oscillate in the horizontal plane. The rod is gently pushed through a small angle \(\theta\) in one direction and released. The frequency of oscillation is

A point source S is placed at the bottom of a transparent block of height 10 mm and refractive index 2.72. It is immersed in a lower refractive index liquid as shown in the figure. It is found that the light emerging from the block to the liquid forms a circular bright spot of diameter 11.54 mm on the top of the block. The refractive index of the liquid is

Parallel rays of light of intensity $\mathrm{I}=912\mathrm{ }\mathrm{W}{\mathrm{m}}^{-2}$ are incident on a spherical black body kept in surroundings of temperature 300 K. Take Stefan-Boltzmann constant $\mathrm{\sigma }=5.7\times {10}^{-8}\mathrm{ }\mathrm{W}{\mathrm{m}}^{-2}\mathrm{ }{\mathrm{K}}^{-4}$ and assume that the energy exchange with the surroundings is only through radiation. The final steady state temperature of the black body is close to

Match the integrals in Column I with the values in Column II.

In a $∆XYZ$ let $x,y,z$,  be the lengths of sides opposite to the angles, $X, Y, Z$, respectively and $2s=x+y+z.$ If $\frac{s-x}{4}=\frac{s-y}{3}=\frac{s-z}{2}$ and area of incircle of the triangle XYZ is $\frac{8\pi }{3},$ then

Let \(m\) be the smallest positive integer such that the coefficient of \(x^2\) in the expansion of \((1+x)^2 +(1+x)^3+\ldots+(1+x)^{49}\) \(+~(1+m x)^{50}\) is \((3 n+1){ }^{51} C_3\) for some positive integer n . Then the value of \(n\) is

Let \(\mathbb{R}\) denote the set of all real numbers. Define the function \(f: \mathbb{R \rightarrow \mathbb { R }}\) by
\(f(\mathrm{x})=\left\{\begin{array}{cc}2-2 x^2-x^2 \sin \frac{1}{x} & \text { if } x \neq 0 \\ 2 & \text { if } x=0\end{array}\right.\)
Then which one of the following statements is TRUE ?

In a triangle the sum of two sides is x and the product of the same two sides is y. If ${\mathrm{x}}^2-{\mathrm{c}}^2=\mathrm{y},$ (where c is the third side of the triangle) then the ratio of the inradius to the circumradius of the triangle is

A small block of mass of \(0.1 \mathrm{~kg}\) lies on a fixed inclined plane PQ which makes an angle \(\theta\) with the horizontal. A horizontal force of \(1 \mathrm{~N}\) acts on the block through its centre of mass as shown in the figure.
The block remains stationary if (take \(g=10 \mathrm{~m} / \mathrm{s}^{2}\) )