List of practice Questions

When the temperature increases, the viscosity of

A value of \( \theta \) lying between \( 0 \) and \( \dfrac{\pi}{2} \) and satisfying \[ \begin{vmatrix} 1 + \sin^2 \theta & \cos^2 \theta & 4\sin 4\theta \\ \sin^2 \theta & 1 + \cos^2 \theta & 4\sin 4\theta \\ \sin^2 \theta & \cos^2 \theta & 1 + 4\sin 4\theta \end{vmatrix} = 0 \] is:

\[ \sum_{r=1}^{15} r^2 \left( \frac{{}^{15}C_r}{{}^{15}C_{r-1}} \right) =\ ? \]

A string of letters is to be formed by using 4 letters from all the letters of the word “MATHEMATICS”. The number of ways this can be done such that two letters are of same kind and the other two are of different kind is

If in \( \triangle ABC \), \( B = 45^\circ \), \( a = 2(\sqrt{3} + 1) \) and area of \( \triangle ABC \) is \( 6 + 2\sqrt{3} \) sq. units, then the side \( b = \ ? \)

A radar system can detect an enemy plane in one out of ten consecutive scans. The probability that it can detect an enemy plane at least twice in four consecutive scans is:

What is the pH of a 0.001 M HCl solution?

Let \( A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} \), then \( A^2 - 5A + 6I = ? \)

The equation for ²³₁₀Ne nucleus which decays by β-emission is:

The enthalpy of atomization of \({CH}_3{NH}_2(g)\) is 2313 kJ mol\(^{-1}\). If \(\Delta_{{C-H}}^{\ominus}\) and \(\Delta_{{N-H}}^{\ominus}\) are 414 and 389 kJ mol\(^{-1}\) respectively, then \(\Delta_{{C-N}}^{\ominus}\) (in kJ mol\(^{-1}\)) will be:

A body is dropped freely from a height of 8 m from the ground. If the coefficient of restitution between the body and the ground is 0.5, then the maximum height reached by the body after second impact with the ground is

If the intensities of two interfering light waves in a medium are I and 4I, then the maximum and minimum intensities are respectively:

What are X and Y respectively in the following set of reactions?
2-Bromobutane $\xrightarrow{{alc. KOH}}$ C$_4$H$_8$ (major)
C$_4$H$_8$ $\xrightarrow{{KMnO}_4/{H}^+}$ X
C$_4$H$_8$ $\xrightarrow{{Baeyer's reagent}}$ Y

Alkaline oxidative fusion of MnO$_2$ gives 'X'. The products formed when 'X' undergoes disproportionation in acid medium are
I) KMnO$_4$
II) MnO$_2$
III) O$_2$
IV) H$_2$O

The work done to take a body of mass 100 kg to the top of a building of height 20 m is (Acceleration due to gravity = 10 m s\textsuperscript{-2})

At T(K) vapour pressure of pure benzene and toluene are 500 and 200 mm Hg respectively. If they form an ideal solution, what is the mole fraction of toluene in a mixture boiling at T(K) at a total pressure of 380 mm Hg ?

At T(K), \( K_c \) value for the dissociation of \( \text{PCl}_5 \) is 0.04.
\[ \text{PCl}_5(g) \rightleftharpoons \text{PCl}_3(g) + \text{Cl}_2(g) \] The number of moles of \( \text{PCl}_5 \) to be added to a 3.0 L flask to get chlorine concentration of 0.15 mol L\(^{-1}\) is (approximately)?

For all \( n \in \mathbb{N} \), which of the following is less than or equal to \( \frac{3^n - 1}{2} \)?

The area of the region lying between the curves \( y = \sqrt{4 - x^2} \), \( y^2 = 3x \) and the Y-axis is:

A thin uniform circular disc rolls with a constant velocity without slipping on a horizontal surface. Its total kinetic energy is

Match the following:

List-I (Metal in drinking water)	List-II (Maximum prescribed concentration in ppm)
A) Mn	I) \( 0.05 \)
B) Zn	II) \( 3.0 \)
C) Cd	III) \( 0.005 \)
D) Cu	IV) \( 5.0 \)

The tangent drawn at an extremity (in the first quadrant) of latus rectum of the hyperbola $$ \frac{x^2}{4} - \frac{y^2}{5} = 1 $$ meets the x-axis and y-axis at $ A $ and $ B $ respectively. If $ O $ is the origin, find $$ (OA)^2 - (OB)^2. $$

Evaluate: $$ \int_0^1 x \sin^{-1} x \, dx $$

A wire of length 20 cm is placed horizontally on the surface of water and is gently pulled up with a force of $1.456 \times 10^{-2}\, N$ to keep the wire in equilibrium. The surface tension of water is

The maximum magnetic field produced by a current of 12 A passing through a copper wire of diameter 1.2 mm is