Given that the number 35 a 64 is divisible by 3, where a is a digit, what are the possible values of a?

The surface area of a solid metallic sphere is $2464 \mathrm{~cm}^{2}$. It is melted and recast into solid right circular cones of radius $3.5 \mathrm{~cm}$ and height $7 \mathrm{~cm}$. Calculate the number of cones recast.

A) 256
B) 128
C) 64
D) 32

The displacement of a medium in a sound wave is given by the equation $y_{1}=A \cos (a x+b t)$ where $A$, $a$, and $b$ are positive constants. The wave is reflected by an obstacle situated at $x=0$. The intensity of the reflected wave is 0.64 times that of the incident wave.

(i) What are the wavelength and frequency of the incident wave?

(ii) Write the equation for the reflected wave.

(iii) In the resultant wave formed after reflection, find the maximum and minimum values of the particle speeds in the medium.

(iv) Express the resultant wave as a superposition of a standing wave and a traveling wave. What are the positions of the antinodes of the standing wave? What is the direction of propagation of the traveling wave?

A. $y=0.8 A \cos (bt-ax)$

B. $y=0.4 A \sin(bt-ax)$

C. $y=-0.8 A \cos(bt-ax)$

D. $y=0.4 A \sin(bt+ax)$

The displacement of a medium in a sound wave is given by the equation $y_{1} = A \cos (a x+b t)$ where $A$, $a$, and $b$ are positive constants. The wave is reflected by an obstacle situated at $x=0$. The intensity of the reflected wave is 0.64 times that of the incident wave.

• What are the wavelength and frequency of the incident wave?
• Write the equation for the reflected wave.
• In the resultant wave formed after reflection, find the maximum and minimum values of the particle speeds in the medium.
• Express the resultant wave as a superposition of a standing wave and a traveling wave. What are the positions of the antinodes of the standing wave? What is the direction of propagation of the traveling wave?

Options: A: $V_{\max} = 1.8 A b$, $V_{\min} = 0 A b$ B: $V_{\max} = 0.9 Ab$, $V_{\min} = 0.1 Ab$ C: $V_{\max} = 1.1 Ab$, $V_{\min} = 0.1 Ab$ D: $V_{\max} = 1 Ab$, $V_{\min} = 0.5 Ab$

In a closed cylinder, there are $60 \mathrm{~g} \mathrm{Ne}$ and $64 \mathrm{~g} \mathrm{O}_{2}$. If the pressure of the mixture of gases in the cylinder is 30 bar, then the partial pressure of $\mathrm{O}_{2}$ in this cylinder will be (in bar).

• List - I & List - II
• (I) If $A$ and $B$ are two sets with elements 4 and 3, respectively. Then the number of onto function from $A$ to $B$ is & (P) 1
• (II) The number of points in the interval $[-\sqrt{13}, \sqrt{13}]$ at which $f(x)=\sin \left(x^{2}\right)+\cos \left(x^{2}\right)$ attains its maximum value, is & (Q) 3
• (III) In a $\triangle XYZ, \quad y^{2} \sin (2Z)+z^{2} \sin (2Y)=2yz$, where $y=15, z=8$. Then the length of inradius i & (R) 4
• (IV) $\log _{x+1}\left(4x^{3}+9x^{2}+6x+1\right)+\log _{4x+1}\left(x^{2}+2x+1\right)$ $=5$, then the number of solutions is & (S) 8
• & (T) 24
• & (U) 36

Which of the following has CORRECT pair of combination? (A) (I) $\rightarrow$ (T) (B) (II) $\rightarrow$ (S) (C) (III) $\rightarrow$ (R) (D) (IV) $\rightarrow$ (P)