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
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)
A box B1, contains 1 white ball, 3 red balls and 2 black balls. Another box B2, contains 2 white balls, 3 red balls and 4 black balls. A third box B3, contains 3 white balls, 4 red balls and 5 black balls.
Let f:R→Rbe a differentiable function with f(0)=1and satisfying the equation f(x+y)=f(x)fprime(y)+fprime(x)f(y)for all x, y∈R. Then, the value of (log)e(f(4))is _______
Consider the circle x2+y2=9 and the parabola y2=8x. They intersect at P and Q in first and 4th quadrant,respectively. Tangents to the circle at P and Q intersect the x-axis at R and tangents at the parabola at P and Q intersect the x-axis at S.
PARAGRAPH AThere are five students S1, S2, S3, S4and S5in a music class and for them there are five seats R1, R2, R3, R4and R5arranged in a row, where initially the seat Riis allotted to the student Si, i=1, 2, 3, 4, 5. But, on the examination day, the five students are randomly allotted five seats. For i=1, 2, 3, 4,let Tidenote the event that the students Siand Si+1do NOT sit adjacent to each other on the day of the examination. Then, the probability of the event T1∩T2∩T3∩T4is151(b) 101(c) 607(d) 51
A line l passing through the origin is perpendicular to the lines l1:(3+t)i^+(−1+2t)j^+(4+2t)k^,∞<t<∞,l2:(3+s)i^+(3+2s)j^+(2+s)k^,∞<t<∞ then the coordinates of the point on l2 at a distance of 17 from the point of intersection of
Let Sbe the set of all column matrices [b1b2b3]such that b1,b2,b3∈Rand the system of equations (in real variable)−x+2y+5z=b12x−4y+3z=b2x−2y+2z=b3has at least one solution. Then, which of the following system(s) (in real variables) has (have) at least one solution for each [b1b2b3]∈S?(a) x+2y+3z=b1,4y+5z=b2and x+2y+6z=b3(b) x+y+3z=b1,5x+2y+6z=b2and −2x−y−3z=b3(c) −x+2y−5z=b1,2x−4y+10z=b2and x−2y+5z=b3(d) x+2y+5z=b1,2x+3z=b2and x+4y−5z=b3