Let s, t, rbe non-zero complex numbers and Lbe the set of solutions z=x+iy (x, y∈R, i=−1)of the equation sz+tz+r=0, where z=x−iy. Then, which of the following statement(s) is (are) TRUE?If Lhas exactly one element, then ∣s∣=∣t∣(b) If ∣s∣=∣t∣, then Lhas infinitely many elements(c) The number of elements in
Let f:R→Rand g:R→Rbe two non-constant differentiable functions. If fprime(x)=(e(f(x)−g(x)))gprime(x)for all x∈R, and f(1)=g(2)=1, then which of the following statement(s) is (are) TRUE?f(2)<1−(log)e2(b) f(2)>1−(log)e2(c) g(1)>1−(log)e2(d) g(1)<1−(log)e2
Let F(x)=∫xx2+6π[2cos2t.dt] for all x∈R and f:[0,21]→[0,∞) be a continuous function.For a∈[0,21], if F'(a)+2 is the area of the region bounded by x=0,y=0,y=f(x) and x=a, then f(0) 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)
The sides of a right angled triangle are in arithmetic progression. If the triangle has area 24, then what is the length of its smallest side?
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
Consider two straight lines, each of which is tangent to both the circle x2+y2=21and the parabola y2=4x. Let these lines intersect at the point Q. Consider the ellipse whose center is at the origin O(0, 0)and whose semi-major axis is OQ. If the length of the minor axis of this ellipse is 2, then which of the following statement(s) is (are) TRUE?For the ellipse, the eccentricity is 21and the length of the latus rectum is 1(b) For the ellipse, the eccentricity is 21and the length of the latus rectum is 21(c) The area of the region bounded by the ellipse between the lines x=21and x=1is 421(π−2)(d) The area of the region bounded by the ellipse between the lines x=21and x=1is 161(π−2)