List of practice Questions

A monochromatic light of wavelength 6000 \AA \ coming from a star is detected in a 100 inch telescope. The limit of resolution of the telescope is approximately

25 capacitors each of capacitance 1 $\mu$F are connected in series to a battery of 100 V. The total charge stored on capacitors is

One end of the steel rod is clamped to the roof and the other end is attached to a mass of 1000 kg as shown in the figure. The length of the rod is 50 cm and its cross-sectional area is 1000 mm$^2$. The change in the length of the rod due to the weight of the mass is \underline{\hspace{2cm}} .

A circular film of a liquid has an area of 10 cm$^2$. If the work done in making its radius two times the initial radius is $8 \times 10^{-3$ J, the surface tension of the liquid is $\left(1+\frac{1}{a}\right)$ Nm$^{-1}$. The value of $a$ is}

A cylindrical tank having large diameter is filled with water to a height H. A hole of cross-sectional area 5 cm$^2$ in the tank allows water to drain out. If the water drains out at the rate of $2 \times 10^{-3$ m$^3$s$^{-1}$, then the value of H is} (acceleration due to gravity = $10$ ms$^{-2}$)

A gas at 37 $^\circ$C is compressed adiabatically to half of its volume, then the final temperature of the gas is (Ratio of specific heat capacities of the gas is 1.5)

An object of mass 20 kg. is displayed by \(x = 5t^2\) m (here t is time) by the application of a force. Then the ratio of the work done in times 3 s and 5 s is

A car is moving along a straight line is brought to a stop within a distance of 200 m and in time 10 s. The initial speed of the car is

A ball is thrown upward from the top of a building at an angle of 30\(^\circ\) to the horizontal and with an initial speed of 20 m s\(^{-1}\). If the ball strikes the ground after 3 s, then the height of the building is (acceleration due to gravity = 10 ms\(^{-2}\))

A particle is moving on a circular path with a constant speed v. It's change of velocity as it moves from A to B in the figure is

A box of mass 2 kg is placed on a inclined plane that makes 30\(^\circ\) with the horizontal. The coefficient of friction between the box and inclined plane is 0.2. A force F is applied on the box perpendicular to the incline to prevent the box from sliding down. The minimum value of F is (acceleration due to gravity = 10 m s\(^{-2}\))

\( \int_{-\pi/4}^{\pi/4} \cos^8 x \ dx = \)

If \( f(x) = \sin(\tan^{-1} x) \), then \( \int_0^1 xf''(x)dx = \)

If an open cylinder of given surface area has maximum volume then its radius is

If \(x+y=k, x>0, y>0\) then \(x^2+y^2\) is minimum if

If \( \int \frac{x^8+4}{x^4-2x^2+2} dx = Ax^5+Bx^3+Cx+K \), then \(5A+3B+C = \)

If \( f(x) = \begin{cases 4 & -\infty<x<-\sqrt{5}
x^2-1 & -\sqrt{5} \le x \le \sqrt{5}
4 & \sqrt{5}<x<\infty \end{cases} \). If k is the number of points where f(x) is not differentiable then k -- 2 =}

Assertion (A) : \( f(x)=|x| \) is differentiable at \( x=a \neq 0 \) and continuous but not differentiable at \( x=0 \) Reason (R) : If a function is differentiable at a point then it is continuous at that point. But converse is not true. .

The minimum distance of a point on the curve \( y = x^2 - 4 \) from the origin is

If the centroid of a triangle with vertices \((4, p, -3), (-1, -1, 2)\) and \((3, 5, -8)\) is given by mid point of \((1, 4, -2)\) and \((q, 2, -4)\), then \(p^2+q^2 = \)

A ray makes angles \( \frac{\pi}{3} \) and \( \frac{\pi}{4} \) with Y and Z-axes respectively. Then the value of the sine of the angle made by the ray with X-axis is

A plane meets the X, Y, Z -- axes in A, B, C respectively. If the centroid of the triangle ABC is \((2, -3, 5)\) then the perpendicular distance from origin to the given plane is

\( \lim_{x \to a} \frac{x^3+a^3}{x+a} = 7 \implies a = \)

\( \lim_{x \to \frac{\pi}{2}} \frac{1-\tan\frac{x}{2}}{1+\tan\frac{x}{2}} \cdot \frac{1-\sin x}{(\pi-2x)^3 = \)}

The length of the latus rectum of the parabola \( y^2-4x-4y+8=0 \) is