The sum of all values of \theta \in [0, 2\pi] satisfying 2\sin^2\theta = \cos 2\theta and 2\cos^2\theta = 3\sin\theta is
Let \text{E} : \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1, a > b and \text{H} : \frac{x^2}{A^2} - \frac{y^2}{B^2} = 1. Let the distance between the foci of \text{E} and the foci of \tex
If \int e^x \left( \frac{x \sin^{-1} x}{\sqrt{1 - x^2}} + \frac{\sin^{-1} x}{(1 - x^2)^{3/2}} + \frac{x}{1 - x^2} \right) dx = g(x) + C, where C is the constant of integration, the
Let f(x) = \int_0^{x^2} \frac{t^2 - 8t + 15}{e^t} dt, x \in \mathbf{R}. Then the numbers of local maximum and local minimum points of f, respectively, are :
Let P(4, 4\sqrt{3}) be a point on the parabola y^2 = 4ax and PQ be a focal chord of the parabola. If M and N are the foot of perpendiculars drawn from P and Q respectively on the d
Given are statements for certain thermodynamic variables, (A) Internal energy, volume (V) and mass (M) are extensive variables. (B) Pressure (P), temperature (T) and density ( \rho
A rectangular metallic loop is moving out of a uniform magnetic field region to a field free region with a constant speed. When the loop is partially inside the magnate field, the
Given below are two statements. One is labelled as Assertion (A) and the other is labelled as Reason (R). Assertion (A) : A simple pendulum is taken to a planet of mass and radius,
A series LCR circuit is connected to an alternating source of emf E. The current amplitude at resonant frequency is I_0. If the value of resistance R becomes twice of its initial v
A tube of length L is shown in the figure. The radius of cross section at the point (1) is 2\text{ cm} and at the point (2) is 1\text{ cm}, respectively. If the velocity of water e
A force \vec{F} = 2\hat{i} + b\hat{j} + \hat{k} is applied on a particle and it undergoes a displacement \hat{i} - 2\hat{j} - \hat{k}. What will be the value of b, if work done on
A proton is moving undeflected in a region of crossed electric and magnetic fields at a constant speed of 2 \times 10^5\text{ ms}^{-1}. When the electric field is switched off, the
A parallel plate capacitor of area A = 16\text{ cm}^2 and separation between the plates 10\text{ cm}, is charged by a DC current. Consider a hypothetical plane surface of area A_0
What amount of bromine will be required to convert 2\text{ g} of phenol into 2,4,6-tribromophenol? (Given molar mass in \text{g}\text{mol}^{-1} of \mathrm{C}, \mathrm{H}, \mathrm{O
The value of (\sin 70^\circ)(\cot 10^\circ \cot 70^\circ - 1) is
Marks obtains by all the students of class 12 are presented in a freqency distribution with classes of equal width. Let the median of this grouped data be 14 with median class inte
If the function f(x) = \begin{cases} \frac{2}{x}\{\sin(k_1+1)x + \sin(k_2-1)x\}, & x 0 \end{cases} is continuous at x = 0, then k_1^2+k_2^2 is equal to
Consider a circular disc of radius 20\text{ cm} with centre located at the origin. A circular hole of radius 5\text{ cm} is cut from this disc in such a way that the edge of the ho
The motion of an airplane is represented by velocity-time graph as shown below. The distance covered by airplane in the first 30.5\text{ second} is ______ \text{km}.
Two particles are located at equal distance from origin. The position vectors of those are represented by \vec{\mathrm{A}} = 2\hat{\mathrm{i}} + 3n\hat{\mathrm{j}} + 2\hat{\mathrm{
Given below are the atomic numbers of some group 14 elements. The atomic number of the element with lowest melting point is :
Identify \mathrm{A}, \mathrm{B} and \mathrm{C} in the given below reaction sequence \mathrm{A} \xrightarrow{\mathrm{HNO}_3} \mathrm{Pb}\left(\mathrm{NO}_3\right)_2 \xrightarrow{\ma
Given below are two statements : Statement (I) : The boiling points of alcohols and phenols increase with increase in the number of C-atoms. Statement (II) : The boiling points of
A massless spring gets elongated by amount x_1 under a tension of 5\text{ N}. Its elongation is x_2 under the tension of 7\text{ N}. For the elongation of \left(5 x_1-2 x_2\right),
The equation of a transverse wave travelling along a string is y(x, t)=4.0 \sin \left[20 \times 10^{-3} x+600 t\right] \mathrm{mm}, where x is in \mathrm{mm} and t is in second. Th
Let f be a differentiable function such that 2(x + 2)^2 f(x) - 3(x + 2)^2 = 10 \int_0^x (t + 2) f(t) dt, x \ge 0. Then f(2) is equal to ______.
Which of the following figure represents the relation between Celsius and Fahrenheit temperatures ?
The increase in pressure required to decrease the volume of a water sample by 0.2\% is \text{P} \times 10^5\text{Nm}^{-2}. Bulk modulus of water is 2.15 \times 10^9\text{Nm}^{-2}.
The compounds that produce \text{CO}_2 with aqueous \text{NaHCO}_3 solution are: A. Benzoic acid B. Phenol C. 2,4,6-trinitrophenol (Picric acid) D. Cyclohexanecarboxylic acid E. 4-
A bar magnet has total length 2l = 20\text{ units} and the field point P is at a distance \text{d} = 10\text{ units} from the centre of the magnet. If the relative uncertainty of l