If the line x cos α + y sin α = 2√3 is tangent to the ellipse \(\frac{x^2}{16} + \frac{y^2}{8} = 1\) and α is an acute angle then α =
A straight line parallel to the line y = √3 x passes through Q(2,3) and cuts the line 2x + 4y - 27 = 0 at P. Then the length of the line segment PQ is
If the radical center of the given three circles x2 + y2 = 1, x2 + y2 -2x - 3 =0 and x2 + y2 -2y - 3 = 0 is C(α,β) and r is the sum of the radii of the given circles, then the circle with C(α,β) as center and r as radius is
A random variable X has the following probability distribution
For the events E = {x/x is a prime number} and F = {x/x <4} then P(E ∪ F)
If the angle between the pair of tangents drawn to the circle $ x^2 + y^2 - 2x + 4y + 3 = 0 $ from the point $(6, -5)$ is \(\theta\) than \(\cot \theta\) =
The radius of a circle touching all the four circles (x ± λ)2 + (y ± λ)2 = λ2 is
If (h,k) is the image of the point (3,4) with respect to the line 2x - 3y -5 = 0 and (l,m) is the foot of the perpendicular from (h,k) on the line 3x + 2y + 12 = 0, then lh + mk + 1 = 2x - 3y - 5 = 0.
The angle between the circles \(x^2+y^2−4x−6y−3=0\), \(x^2+y^2+8x−4y+11=0\) is \(\frac{\pi}{2}\), then the value of K is?
If the angle between the asymptotes of a hyperbola is 30° then its eccentricity is
A tangent PT is drawn to the circle x2 + y2 = 4 at the point P(√3, 1). If a straight line L which is perpendicular to PT is a tangent to the circle (x- 3)2 + y2 = 1, then a possible equation of L is
If a point P moves so that the distance from (0,2) to P is \(\frac{1}{√2 }\) times the distance of P from (-1,0), then the locus of the point P is
Let d be the distance between the parallel lines 3x - 2y + 5 = 0 and 3x - 2y + 5 + 2√13 = 0. Let L1 = 3x - 2y + k1 = 0 (k1 > 0) and L2 = 3x - 2y + k2 = 0 (k2 > 0) be two lines that are at the distance of \(\frac{4d}{√13}\) and \(\frac{3d}{√13}\) from the line 3x - 2y + 5y = 0. Then the combined equation of the lines L1 = 0 and L2 = 0 is:
5 persons entered a lift cabin in the cellar of a 7-floor building apart from cellar. If each of the independently and with equal probability can leave the cabin at any floor out of the 7 floors beginning with the first, then the probability of all the 5 persons leaving the cabin at different floors is
If sin y = sin 3t and x = sin t, then \(\frac{dy}{dx}\) =
Which of the following does not form a buffer solution?
The ratio of lone pair of electrons to bond pair of electrons in ozone molecule is:
Ziegler-Natta Catalyst is used in the manufacture of
The aromatic compound/species with maximum number of x - electrons is
XeF4 + O2F2 → X + O2 X + H2O → Y +2HF The shape of the molecules of X and Y respectively are:
The following molecule with the structure acts as
The number of electrons with (n+1) values equal to 3,4 and 5 in an element with atomic number (z) 24 are respectively (n = principal quantum number and l = azimuthal quantum number)
Which one of the following has the same number of atoms as are in 6g of H2O
A current of 15.0 amperes is passed through a solution of CrCl2, for 45 minutes. The volume of Cl2 , (in I) obtained at the anode at 1 atm and 273 K is around (IF=96500 Cmol-1, At. wt. of Cl=35.5, R=0.082 L-atmK-1 mol-1)
Consider the following statements about the oxides of halogens A. At room temperature, OF2; is thermally stable B. Order of stability of oxides of halogens is I > Br > Cl C. I2O5 is used in the estimation of CO D. ClO2; is used as a bleaching agent The correct statements are