After completion of the reactions (I and II) , the organic compound(s) in the reaction mixtures is(are)
In R', consider the planes P1,y=0 and P2:x+z=1. Let P3, be a plane, different from P1, and P2, which passes through the intersection of P1, and P2. If the distance of the point (0,1,0) from P3, is 1 and the distance of a point (α,β,γ) from P3 is 2, then which of the following relation is (are) true ?
Let f:[21,1]→R (the set of all real numbers) be a positive, non-constant, and differentiable function such that fprime(x)<2f(x)andf(21)=1 . Then the value of ∫211f(x)dx lies in the interval (a)(2e−1,2e) (b) (3−1,2e−1)(c)(2e−1,e−1) (d) (0,2e−1)
Column 1,2 and 3 contains conics, equations of tangents to the conics and points of contact, respectively.Column I, Column 2, Column 3I, x2+y2=a, (i), my=m2x+a, (P), (m2a,m2a)II, x2+a2y2=a, (ii), y=mx+am2+1, (Q), (m2+1−ma,m2+1a)III, y2=4ax, (iii), y=mx+a2m2−1, (R), (a2m2+1−a2m,a2m2+11)IV, x2−a2y2=a2, (iv), y=mx+a2m2+1, (S), (a2m2+1−a2m,a2m2+1−1)If a tangent to a suitable conic (Column 1) is found to be y=x+8and its point of contact is (8,16), then which of the followingoptions is the only CORRECT combination?(III) (ii) (Q) (b) (II) (iv) (R)(I) (ii) (Q) (d) (III) (i) (P)
Let a and b be two unit vectors such that a.b=0 For some x,y∈R, let c=xa+yb+(a×b If (∣c∣=2 and the vector c is inclined at same angle α to both a and b then the value of 8cos2α is
Let the curve C be the mirror image of the parabola y2=4x with respect to the line x+y+4=0. If A and B are the points of intersection of C with the line y=−5, then the distance between A and B is
For a non-zero complex number z, let arg(z)denote theprincipal argument with π<arg(z)≤πThen, whichof the following statement(s) is (are) FALSE?arg(−1,−i)=4π,where i=−1(b) The function f:R→(−π,π],defined by f(t)=arg(−1+it)for all t∈R, iscontinuous at all points of R, where i=−1(c) For any two non-zero complex numbers z1and z2, arg(z2z1)−arg(z1)+arg(z2)is an integer multiple of 2π(d) For any three given distinct complex numbers z1, z2and z3, the locus of the point zsatisfying the condition arg((z−z3)(z2−z1)(z−z1)(z2−z3))=π, lies on a straight line
Consider the cube in the first octant with sides OP,OQ and OR of length 1, along the x-axis, y-axis and z-axis, respectively, where O(0,0,0) is the origin. Let S(21,21,21) be the centre of the cube and T be the vertex of the cube opposite to the origin O such that S lies on the diagonal OT. If p=SP,q=SQ,r=SR and t=ST then the value of ∣(p×q)×(r×(t)∣is