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In a triangle P Q R ,Sa n dT are points on Q Ra n dP R , respectively, such that Q S=3S Ra n dP T=4T Rdot Let M be the point of intersection of P Sa n dQ Tdot Determine the ratio Q M : M T using the vector method .

Let d_1a n dd_2 be the length of the perpendiculars drawn from the foci Sa n dS ' of the ellipse (x^2)/(a^2)+(y^2)/(b^2)=1 to the tangent at any point P on the ellipse. Then, S P : S^(prime)P= (a) d_1: d_2 (b) d_2: d_1 (c) d_1 ^2:d_2 ^2 (d) sqrt(d_1):sqrt(d_2)

Statement 1 : If (3, 4) is a point on a hyperbola having foci (3, 0) and (lambda,0) , the length of the transverse axis being 1 unit, then lambda can take the value 0 or 3. Statement 2 : |S^(prime)P-S P|=2a , where Sa n dS ' are the two foci, 2a is the length of the transverse axis, and P is any point on the hyperbola.

If Sa n dS ' ' are the foci of the ellipse (x^2)/(25)+(y^2)/(16)=1 , and P is any point on it, then the range of values of S PdotS ' P is (a) 9lt=f(theta)lt=16 (b) 9lt=f(theta)lt=25 (c) 16lt=f(theta)lt=25 (d) 1lt=f(theta)lt=16

Let Sa n dS ' be two foci of the ellipse (x^2)/(a^2)+(y^2)/(b^2)=1 . If a circle described on S S^(prime) as diameter intersects the ellipse at real and distinct points, then the eccentricity e of the ellipse satisfies (a) c=1/(sqrt(2)) (b) e in (1/(sqrt(2)),1) (c) e in (0,1/(sqrt(2))) (d) none of these

Let P_i and P_i ' be the feet of the perpendiculars drawn from the foci Sa n dS ' on a tangent T_i to an ellipse whose length of semi-major axis is 20. If sum_(i=0)^(10)(S P_i)(S^(prime)P_i ')=2560 , then the value of eccentricity is (a) 1/5 (b) 2/5 (c) 3/5 (d) 4/5

Co^57 decays to Fe^57 by beta^+ -emission.The resulting Fe^57 is in its excited state and comes to the ground state by emitting gamma -rays.The half-life of beta^+ -decay is 270 days and that of the gamma -emission is 10^(-8) s.A sample of Co^57 given 5.0xx10^9 gamma rays per second.How much time will elapse before the emission rate of gamma rays drops to 2.5xx10^9 per second ?