Four distinct points (2 \mathrm{k}, 3 \mathrm{k}),(1,0),(0,1) and (0,0) lie on a circle for k equal to :
If the shortest distance of the parabola y^2=4 x from the centre of the circle x^2+y^2-4 x-16 y+64=0 is d, then d^2 is equal to :
The length of the chord of the ellipse \frac{x^2}{25}+\frac{y^2}{16}=1, whose mid point is \left(1, \frac{2}{5}\right), is equal to:
If a=\lim _{x \rightarrow 0} \frac{\sqrt{1+\sqrt{1+x^4}}-\sqrt{2}}{x^4} and b=\lim _{x \rightarrow 0} \frac{\sin ^2 x}{\sqrt{2}-\sqrt{1+\cos x}}, then the value of a b^3 is :
Let a_1, a_2, \ldots, a_{10} be 10 observations such that \sum_{k=1}^{10} a_k=50 and \sum_{\forall k<j} a_k \cdot a_j=1100. Then the standard deviation of a_1, a_2, \ldots, a_{10}
Let S=\{1,2,3, \ldots, 10\}. Suppose M is the set of all the subsets of S, then the relation R=\{(A, B): A \cap B \neq \phi ; A, B \in M\} is :
Consider the matrix f(x)=\begin{bmatrix}\cos x & -\sin x & 0 \\ \sin x & \cos x & 0 \\ 0 & 0 & 1\end{bmatrix}. Given below are two statements : Statement I:f(-x) is the inverse of
The function \mathrm{f}: \mathbb{N}-\{1\} \rightarrow \mathbb{N}; defined by \mathrm{f}(\mathrm{n})= the highest prime factor of n, is :
Consider the function \mathrm{f}(\mathrm{x})= \begin{cases}\frac{\mathrm{a}\left(7 \mathrm{x}-12-\mathrm{x}^2\right)}{\mathrm{b}\left|\mathrm{x}^2-7 \mathrm{x}+12\right|} &, \mathr
If \int_0^1 \frac{1}{\sqrt{3+x}+\sqrt{1+x}} \mathrm{d} x=a+b \sqrt{2}+c \sqrt{3}, where a, b, c are rational numbers, then 2 a+3 b-4 c is equal to :
Let x=x(t) and y=y(t) be solutions of the differential equations \frac{\mathrm{d} x}{\mathrm{~d} t}+\mathrm{a} x=0 and \frac{\mathrm{d} y}{\mathrm{~d} t}+\mathrm{b} y=0 respectivel
If \vec{a}=\hat{i}+2 \hat{j}+\hat{k}, \vec{b}=3(\hat{i}-\hat{j}+\hat{k}) and \vec{c} be the vector such that \vec{a} \times \vec{c}=\vec{b} and \vec{a} \cdot \vec{c}=3, then \vec{a
The distance, of the point (7,-2,11) from the line \frac{x-6}{1}=\frac{y-4}{0}=\frac{z-8}{3} along the line \frac{x-5}{2}=\frac{y-1}{-3}=\frac{z-5}{6}, is :
Let the set of all a \in \mathbb{R} such that the equation \cos 2 x+a \sin x=2 a-7 has a solution be [p, q] and r=\tan 9^{\circ}-\tan 27^{\circ}-\frac{1}{\cot 63^{\circ}}+\tan 81^{
Let f(x)=x^3+x^2 f^{\prime}(1)+x f^{\prime \prime}(2)+f^{\prime \prime \prime}(3), x \in R. Then f^{\prime}(10) is equal to
Let the area of the region \left\{(x, y): x-2 y+4 \geq 0, x+2 y^2 \geq 0, x+4 y^2 \leq 8, y \geq 0\right\} be \frac{m}{n}, where m and n are coprime numbers. Then \mathrm{m}+\mathr
A fair die is tossed repeatedly until a six is obtained. Let X denote the number of tosses required and let \mathrm{a}=\mathrm{P}(\mathrm{X}=3), \mathrm{b}=\mathrm{P}(\mathrm{X} \g
Given below are two statements: Statement (I) : Planck's constant and angular momentum have the same dimensions. Statement (II) : Linear momentum and moment of force have the same
An electric charge 10^{-6}\text{ }\mu\text{C} is placed at origin (0, 0)\text{ m} of \mathrm{X}-\mathrm{Y} co-ordinate system. Two points P and Q are situated at (\sqrt{3}, \sqrt{3
A wire of resistance R and length L is cut into 5 equal parts. If these parts are joined parallelly, then resultant resistance will be :
A proton moving with a constant velocity passes through a region of space without any change in its velocity. If \vec{E} and \vec{B} represent the electric and magnetic fields resp
A rectangular loop of length 2.5\text{ m} and width 2\text{ m} is placed at 60^{\circ} to a magnetic field of 4\text{ T}. The loop is removed from the field in 10\text{ sec}. The a
A plane electromagnetic wave propagating in x-direction is described by E_y = (200\text{ V m}^{-1}) \sin[1.5 \times 10^7 t - 0.05x]; The intensity of the wave is : (\text{Use } \ep
If the refractive index of the material of a prism is \cot(\frac{A}{2}), where A is the angle of prism then the angle of minimum deviation will be
A convex lens of focal length 40\text{ cm} forms an image of an extended source of light on a photoelectric cell. A current I is produced. The lens is replaced by another convex le
Which of the following circuits is reverse - biased ?
Position of an ant ( \mathrm{S} in metres) moving in \mathrm{Y}-\mathrm{Z} plane is given by \mathrm{S}=2 t^2 \hat{\mathrm{j}}+5 \hat{\mathrm{k}} (where t is in second). The magnit
Four particles, each of mass 1\text{ kg} are placed at four corners of a square of side 2\text{ m}. The moment of inertia of the system about an axis perpendicular to its plane and
If average depth of an ocean is 4000\text{ m} and the bulk modulus of water is 2 \times 10^9\text{ N m}^{-2}, then fractional compression \frac{\Delta V}{V} of water at the bottom
A particle executes simple harmonic motion with an amplitude of 4\text{ cm}. At the mean position, velocity of the particle is 10\text{ cm s}^{-1}. The distance of the particle fro