List of practice Questions

Two wires of equal cross section, but one made up of steel and the other copper are joined end to end. When the combination is kept under tension, the elongations in the two wires are found to be equal. If (Y for steel = $2.0\times 10^{11}\,Nm^{-2}$ and $Y for copper = 1.1 \times 10^{11}\,Nm^{-2}$, the ratio of the lengths of the two wires is

The molecule which is linear is

If $ y = sec(tan^{-1}x) $ , then $ \frac{dy}{dx} $ at $ x = 1 $ is

For complexes $ [NiCl_4]^{2-} $ and $ Ni(CO)_4 $ which one of the following statement is true

If $ f(x) = \frac{x}{2} -1 $ then on the interval $ [0,\pi] $

Chemical formula for ?inorganic benzene? is

An ion with a charge of $ + 32 \times 10^{-19}C $ is in a region where a uniform electric field of $ 5 \times 10^4\, V/m $ is perpendicular to a uniform magnetic field of $ 0.8 \,T $ . If its acceleration is zero, then its speed must be

An ideal gas at pressure $ P $ is adiabatically compressed so that its density becomes $ n $ times the initial value. The final pressure of the gas will be $ \left(\gamma = \frac{C_{P}}{C_{V}}\right) $

A quantity of a substance in a closed system is made to undergo a reversible process from an initial volume of $ 3 \,m^3 $ and initial pressure $ 10^5\, N/m^2 $ to a final volume of $ 5\, m^3 $ . If the pressure is proportional to the square of the volume (i.e. $ p = AV^2 $ ), the work done by the substance will be

A polygon has $44$ diagonals. The number of its sides are

If origin is the centroid of a $ \Delta PQR $ with vertices $ P(2a,2,6),Q(-4,3b,- 10) $ and $ (8,14,2c) $ , then the value of $ a,b $ and $ c $ are respectively.

Given that the points $ P(3,2, - 4) $ , $ Q(5,4, - 6) $ and $ R(9,8,-10) $ are collinear, the ratio in which $ Q $ divides $ PR $ externally is

If $ A $ and $ B $ are two square matrices such that $ AB $ = $ A $ and $ BA $ = $ B $ , then

The function $ f(x) = cos^2x $ is strictly decreasing on

Consider the following propositions : p : I take medicine q : I can sleep Then, the compound statement $ \sim p \rightarrow \sim q $ means

The sum of two numbers is $10$. Their product will be maximum when they are

The maximum value of $ \frac{log x}{x} $ is

If $ \begin{bmatrix}\alpha&\beta\\ \gamma&-\alpha\end{bmatrix} $ is to be square root of the two rowed unit matrix, then $ \alpha $ , $ \beta $ and $ \gamma $ should statisfy the relation

The value of the integral $ \int_{-2}^{0}\frac{dx}{\sqrt{12-x^{2}-4x}} $ is

The integral $ \int \sqrt{16-9x^2} $ $dx$ equals

A line perpendicular to the line segment joining the points $ (1,0) $ and $ (2,3) $ divides it in the ratio $ 1 : n $ . The equation of the line is

The value of the integral $ \int_{0}^{1} $ $ \frac{e^{5log_{e}x}-e^{4log_{e}x}}{e^{log_{e}x^3}-e^{log_{e}x^2}}dx $ is

$ \int \frac{x^3dx}{1+x^4} $ equals

The position vector $\vec{r}$ of a particle of mass $m$ is given by the following equation $\vec{r}(t)=\alpha t^{3} \hat{i}+\beta t^{2} \hat{j}$ where $\alpha=\frac{10}{3} \,ms ^{-3}, \beta=5\, ms ^{-2}$ and $m =0.1 \,kg$. At $t =1\, s$, which of the following statement (s) is (are) true about the particle?

Starting from origin, a body moves along $ x $ -axis. Its velocity at any time is given by $ v = 4t^3 - 2t \,m/s $ Acceleration of the particle when it is $ 2 \,m $ away from the origin is