List of practice Questions

A building of 25 meters casts a shadow 15 meters long on the ground. What will be the length of the shadow cast by a building 40 meters tall?

A flat belt pulley has a circumference of 40 cms and turns at 60 rpm. It rotates another pulley of 1.2 meters circumference through a belt having 7.2 meters in length. If the driven pulley has revolved 600 times, what is the time the driver pulley rotated?

There is a pole in a lake, one half of the pole is buried in soil, another one third is submerged in water, and 2 meters are out of the water. What is the total length of the pole?

Local time of a place is related to the value of.

The front and rear wheels of a tractor have different diameters in the proportion of 1:2. What will be the speed of the front wheel, if the rear wheel moves at 40 km/hr on the road?

A number of children are standing evenly spaced around a circle. The third child is directly opposite to the 28th child. How many children are there altogether?

Which of these ions is expected to be coloured in aqueous solution?

Which of the following ligands is tetradentate?

What is the EAN of \( [\text{Al}(\text{C}_4\text{O}_4)_3]^{3-} \)?

Which of the following is paramagnetic ?

The solution of the differential equation \[ (1 + y^2) \, \frac{dy}{dx} = e^{-(x - y)} \] is

If the gradient of the tangent at any point \( (x, y) \) of a curve passing through the point \( (1, \frac{\pi}{4}) \) is \[ \left| \frac{dy}{dx} \right| = \frac{1}{x} \cdot \left| \log \left( \frac{y}{x} \right) \right| \] then the equation of the curve is

In the expansion of \( a + bx \), the coefficient of \( x^r \) is

If \( n = 1999 \), then \( \sum_{i=1}^{1999} \log x_i \) is equal to

\( P \) is a fixed point \( (a, a, a) \) on a line through the origin equally inclined to the axes, then any plane through \( P \) perpendicular to \( OP \), makes intercepts on the axes, the sum of whose reciprocals is equal to

For which of the following values of \( m \), the area of the region bounded by the curve \( y = x - x^2 \) and the line \( y = mx \) equals 5?

If \( R \to R \) be such that \( f(1) = 3 \) and \( f'(1) = 6 \), then \( f(x) \) is equal to

If \( f(x) = \left\{ \begin{array}{ll} 1 + \left| \sin x \right|, & \text{for } -\pi \leq x<0
e^{x/2}, & \text{for } 0 \leq x<\pi
\end{array} \right. \) then the value of \( a \) and \( b \), if \( f \) is continuous at \( x = 0 \), are respectively

The domain of the function \[ f(x) = \frac{1}{\log(1 - x)} + \sqrt{x + 2} \] is

\( \int (x + 1)(x - x^2) e^x \, dx \) is equal to

If \( f(x) = x - \lfloor x \rfloor \), for every real number \( x \), where \( \lfloor x \rfloor \) is the integral part of \( x \), then \[ \int f(x) \, dx \] is equal to

The value of the integral \[ \int_1^\infty \frac{x+1}{|x-1|} \left( \frac{x-1}{x+1} \right)^{1/2} \, dx \] is

If a tangent having slope \( \frac{-4}{3} \) to the ellipse \[ \frac{x^2}{18} + \frac{y^2}{32} = 1 \] intersects the major and minor axes in points A and B respectively, then the area of \( \triangle OAB \) is equal to

The locus of mid points of tangents intercepted between the axes of ellipse \[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \] is

If \( P \) is a double ordinate of hyperbola \[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \] such that \( OPQ \) is an equilateral triangle, \( O \) being the centre of the hyperbola, then the eccentricity \( e \) of the hyperbola satisfies