A list of species having the formula XZ4 is given below.
Defining shape on the basis of the location of X and Z atoms, the total number of species having a square planar shape is
Let f:RRbe a continuous odd function, which vanishes exactly at one point and f(1)=21˙Suppose that F(x)=∫−1xf(t)dtforallx∈[−1,2]andG(x)=∫−1xt∣f(f(t))∣dtforallx∈[−1,2]I˙G(x)f(lim)x1(F(x))=141,Then the value of f(21)is
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)The tangent to a suitable conic (Column 1) at (3,21)is found to be 3x+2y=4,then which of the following options is the only CORRECT combination?(IV) (iii) (S) (b) (II) (iii) (R)(II) (iv) (R) (d) (IV) (iv) (S)
Word of length 10 are formed using the letters A,B,C,D,E,F,G,H,I,J. Let xbe the number of such words where no letter is repeated; and let ybe the number of such words where exactly one letter is repeated twice and no other letter is repeated. The, 9xy=
The number of real solutions of the equation sin−1(i=1∑∞xi+1−xi=1∑∞(2x)i)=2π−cos−1(i=1∑∞(−2x)i−i=1∑∞(−x)i)lying in the interval (−21,21)is ____. (Here, the inverse trigonometric function =sin−1xand cos−1xassume values in [2π,2π]and [0, π], respectively.)
PARAGRAPH XLet Sbe the circle in the xy-plane defined by the equation x2+y2=4.(For Ques. No 15 and 16)Let E1E2and F1F2be the chords of Spassing through the point P0(1, 1)and parallel to the x-axis and the y-axis, respectively. Let G1G2be the chord of Spassing through P0and having slope −1. Let the tangents to Sat E1and E2meet at E3, the tangents to Sat F1and F2meet at F3, and the tangents to Sat G1and G2meet at G3. Then, the points E3, F3and G3lie on the curvex+y=4(b) (x−4)2+(y−4)2=16(c) (x−4)(y−4)=4(d) xy=4
The function y=f(x) is the solution of the differential equation dxdy+x2−1xy=1−x2x4+2x in (−1,1) satisfying f(0)=0. Then ∫2323f(x)dx is
For how many values, of p, the circle x2+y2+2x+4y−p=0and the coordinate axes have exactly three common points?