$A$ six-faced unbiased die is thrown twice and the sum of the numbers appearing on the upper face is observed to be $7$. The probability that the number $3$ has appeared at least once is:

Which one of the following plots is correct for a first order reaction?

$A$ plane $\pi$ makes intercepts $3$ and $4$ respectively on $Z$-axis and $X$-axis. If $\pi$ is parallel to $Y$-axis,then its equation is:

$A$ telescope has an objective lens of focal length $150 \ cm$ and an eyepiece of focal length $5 \ cm$. If a $50 \ m$ tall tower at a distance of $1 \ km$ is observed through this telescope in normal setting,the angle formed by the image of the tower is $\theta$,then $\theta$ is close to..... (in $^{\circ}$)

$A$ transverse sinusoidal wave moves along a string in the positive $x-$ direction at a speed of $10\, cm/s$. The wavelength of the wave is $0.5\, m$ and its amplitude is $10\, cm$. At a particular time $t$,the snapshot of the wave is shown in the figure. The velocity of point $P$ when its displacement is $5\, cm$ is:

Each pair forms an ideal solution except: