If \sum_{r=0}^{5} \frac{{}^{11}C_{2r}}{2r+2} = \frac{m}{n}, \gcd(m, n) = 1, then m - n is equal to _______
Let \vec{c} be the projection vector of \vec{b} = \lambda\hat{i} + 4\hat{k}, \lambda > 0, on the vector \vec{a} = \hat{i} + 2\hat{j} + 2\hat{k}. If |\vec{a} + \vec{c}| = 7, then th
Let \text{L}_1 : \frac{x-1}{3} = \frac{y-1}{-1} = \frac{z+1}{0} and \text{L}_2 : \frac{x-2}{2} = \frac{y}{0} = \frac{z+4}{\alpha}, \alpha \in \mathbf{R}, be two lines, which inters
The product of all solutions of the equation e^{5(\log_e x)^2 + 3} = x^8, x > 0, is :
Let the triangle PQR be the image of the triangle with vertices (1, 3), (3, 1) and (2, 4) in the line x + 2y = 2. If the centroid of \Delta\text{PQR} is the point (\alpha, \beta),
Let for f(x) = 7\tan^8 x + 7\tan^6 x - 3\tan^4 x - 3\tan^2 x, \text{I}_1 = \int_0^{\pi/4} f(x)dx and \text{I}_2 = \int_0^{\pi/4} xf(x)dx. Then 7\text{I}_1 + 12\text{I}_2 is equal t
Let f(x) be a real differentiable function such that f(0) = 1 and f(x + y) = f(x)f'(y) + f'(x)f(y) for all x, y \in \mathbf{R}. Then \sum_{n=1}^{100} \log_e f(n) is equal to :
An electron is made to enter symmetrically between two parallel and equally but oppositely charged metal plates, each of 10\text{ cm} length. The electron emerges out of the electr
Given below are two statements : Statement-I : The equivalent emf of two nonideal batteries connected in parallel is smaller than either of the two emfs. Statement-II : The equival
A uniform circular disc of radius ' \text{R} ' and mass ' \text{M} ' is rotating about an axis perpendicular to its plane and passing through its centre. A small circular part of r
An amount of ice of mass 10^{-3}\text{ kg} and temperature -10^\circ\text{C} is transformed to vapour of temperature 110^\circ\text{C} by applying heat. The total amount of work re
An electron in the ground state of the hydrogen atom has the orbital radius of 5.3 \times 10^{-11}\text{ m} while that for the electron in third excited state is 8.48 \times 10^{-1
A bob of mass m is suspended at a point O by a light string of length l and left to perform vertical motion (circular) as shown in figure. Initially, by applying horizontal velocit
Given is a thin convex lens of glass (refractive index \mu) and each side having radius of curvature R. One side is polished for complete reflection. At what distance from the lens
Which of the following circuits represents a forward biased diode?
A small point of mass m is placed at a distance 2R from the centre ' O ' of a big uniform solid sphere of mass M and radius R . The gravitational force on ' m ' due to M is \text{F
A closed organ and an open organ tube are filled by two different gases having same bulk modulus but different densities \rho_1 and \rho_2, respectively. The frequency of 9^{\text{
If B is magnetic field and \mu_0 is permeability of free space, then the dimensions of (B/\mu_0) is
Given below are two statements : one is labelled as Assertion (A) and the other is labelled as Reason (R). Assertion-(A) : If Young's double slit experiment is performed in an opti
A parallel-plate capacitor of capacitance 40\mu\text{ F} is connected to a 100\text{ V} power supply. Now the intermediate space between the plates is filled with a dielectric mate
Two soap bubbles of radius 2\text{ cm} and 4\text{ cm}, respectively, are in contact with each other. The radius of curvature of the common surface, in \text{cm}, is _______.
A particle is projected at an angle of 30^\circ from horizontal at a speed of 60\text{ m/s}. The height traversed by the particle in the first second is \text{h}_0 and height trave
\text{RBr} \xrightarrow[\text{(ii) } \text{H}_2\text{O}]{\text{(i) } \text{Mg, dry ether}} \text{2-Methylbutane} The maximum number of \text{RBr} producing 2-methylbutane by above
The species which does not undergo disproportionation reaction is :
The molar solubility(s) of zirconium phosphate with molecular formula (\text{Zr}^{4+})_3(\text{PO}_4^{3-})_4 is given by relation :
The most stable carbocation from the following is :
The maximum covalency of a non-metallic group 15 element ' E ' with weakest \text{E} - \text{E} bond is :
If \lim_{x \to \infty} \left(\left(\frac{1}{e}\right) \left(\frac{1}{e} - \frac{x}{1+x}\right)\right)^x = \alpha, then the value of \frac{\log_e \alpha}{1 + \log_e \alpha} equals :
If the system of linear equations : \begin{aligned} x + y + 2z &= 6 \\ 2x + 3y + az &= a + 1 \\ -x - 3y + bz &= 2b \end{aligned} where a, b \in \mathbf{R}, has infinitely many solu
Let \alpha_\theta and \beta_\theta be the distinct roots of 2x^2 + (\cos\theta)x - 1 = 0, \theta \in (0, 2\pi). If m and M are the minimum and the maximum values of \alpha_\theta^4