List of practice Questions

A Carnot engine has efficiency of $50 \%$ .If the temperature of sink is reduced by $40^{\circ} C$, its efficiency increases by $30 \%$ .The temperature of the source will be :

A uniform metal chain of mass m and length ‘$L$’ passes over a massless and frictionless pully. It is released from rest with a part of its length ‘$l$’ is hanging on one side and rest of its length '$L – l$' is hanging on the other side of the pully. At a certain point of time, when $l = L/x$, the acceleration of the chain is $g/2$. The value of $x$ is ______.

If the system of linear equations
2x + 3y – z = –2
x + y + z = 4
x – y + |λ|z = 4λ – 4
where λ ∈ R, has no solution, then

If the initial velocity in horizontal direction of a projectile is unit vector $\hat i$ and the equation of trajectory is $y = 5x(1 – x)$. The y component vector of the initial velocity is ______ $\hat j$.
(Take $g=10\ m/s^2$)

Let f(x)=$\left\{\begin{matrix} |4x^2-8x+5|, & if\,8x^2-6x+1\geq0 \\ |4x^2-8x+5|, & if\,8x^2-6x+1<0 \end{matrix}\right.$ where [α] denotes the greatest integer less than or equal to α.Then the number of points in R where f is not differentiable is

What is the molar mass of solute when 2.3 gram non-volatile solute dissolved in 46 gram benzene at 30°C? (Relative lowering of vapour pressure is 0.06 and molar mass of benzene is 78 gram mol^–1)

If $ \alpha, \beta, \gamma \in [0, \pi] $ and if $ \alpha, \beta, \gamma $ are in AP, then \[ \frac{\sin \alpha - \sin \gamma}{\cos \gamma - \cos \alpha} \] {is equal to:}

Let a function f: ℝ → ℝ be defined as :
$\begin{array}{l}f(x) = \left\{\begin{matrix}\int_{0}^{x}(5-|t-3|)dt, & x>4 \\x^2 + bx, & x \le4 \\\end{matrix}\right.\end{array}$
where b ∈ ℝ. If f is continuous at x = 4 then which of the following statements is NOT true?

Let $\alpha, \beta$ be the roots of the equation $x^2 - 4\lambda x + 5 = 0$ and α, γ be the roots of the equation $x^2 - (3\sqrt{2} + 2\sqrt{3})x + 7 + 3\lambda\sqrt{3} = 0$. If $\beta + \gamma = 3\sqrt{2}$ then $(\alpha + 2\beta + \gamma)^2$ is equal to _______.

If α, β are the roots of the equation
$x^2-(5+3^{\sqrt{log_35}}-5^{\sqrt{log_53}})+3(3^{(log_35)^{\frac{1}{3}}}-5^{(log_53)^{\frac{2}{3}}}-1) = 0$
then the equation, whose roots are α + 1/β and β + 1/α , is

When a certain amount of solid A is dissolved in 100 g of water at 25°C to make a dilute solution, the vapour pressure of the solution is reduced to one-half of that of pure water. The vapour pressure of pure water is 23.76 mmHg. The number of moles of solute A added is _________. (Nearest Integer).

A √(34) m long ladder weighing 10 kg leans on a frictionless wall. Its feet rest on the floor 3 m away from the wall as shown in the figure. If F_f and F_w are the reaction forces of the floor and the wall, then ratio of F_w/F_f will be:
(Use g = 10 m/s²)