Questions asked by Harrison B.

• Back to user's profile
• What is #i^-1# ?
  
• Can you prove that #sqrt(3)# is irrational?
  
• Two cars collide head-on. In which of the following scenarios would you expect the occupants to suffer greater injury? If the cars stick together, or if the cars rebound?
  
• When a body falls, its momentum increases. Does this mean that the momentum is not conserved?
  
• An oxygen molecule of mass 5.32 x 10^-26 kg, moving at 6.0 x 10^4 m/s strikes a wall at 90 degrees and rebounds without loss of speed. If the duration of the impact is 10^-9 seconds, calculate..?
  
• A machine gun fires bullets at a rate of 600 per minute. If each bullet has a mass of 60g and a muzzle velocity of 400m/s, what is the force that the gunner must apply, in order to stop it from moving back?
  
• A cannon fires a cannonball 500m downrange when set at a 45 degree angle. At what velocity does the cannonball leave the cannon?
  
• Suppose #w=(z+1)/(z-1)# where #z=a+bi#, #a, b in RR#. If #|z|=1#, can you find #Re(w)#?
  
• Can you prove that #n^5 - n# is divisible by #5# for all #n in ZZ# by the principle of mathematical induction?
  
• Can you use mathematical induction to prove that #2^n > 4n# for all #n in ZZ^+ n>=5#?
  
• Can you use mathematical induction to prove that #5^n < n!# for all #n in ZZ^+, n>=12#?
  
• Can you use mathematical induction to prove that the sequence defined by #t_1=6# , #t_(n+1)= t_n/(3n# for all #n in ZZ^+# can be written as #t_n=18/(3^n(n-1)!# for all #n in ZZ^+#?
  
• Can you use mathematical induction to prove that #t_n >= t_(n-1)# for all #n in ZZ^+# for a sequence with the general term: #t_n=(3n+5)/(n+2), n in ZZ^+#?
  
• Suppose #f(x)= axln(x+b)# where #f(1)=1/2ln3# and #f'(0)=1/2ln2#. Can you find the constants #a# and #b#?
  
• A real polynomial #P(x)# is the product of real linear and real quadratic factors. Can you that if a real quadratic factor has one zero, #a+bi#, then the other zero must be the complex conjugate, #a-bi#?
  
• Can you show that #z^4 + 64# can be factorised into two real quadratic factors of the form #z^2 + az + 8# and #z^2 + bz + 8# but cannot be factorised into two real quadratic factors of the form #z^2 + bz + 16# and #z^2 + bz + 4#?
  
• If #P(x)# is divided by #(x-a)(x-b)# where #a!=b, a,b in RR#, can you prove that the remainder is: #((P(b)-P(a))/(b-a))xx(x-a)+P(a)#?
  
• Can you find the cartesian equation for the locus of points #(x, y)# if #z=x+iy# and #|z+3| + |z-3| = 8#?
  
• If an object has a displacement function #s(t)=t-ln(2t+1)# where #t# is in seconds and #t>=0#, can you find the distance travelled in the first 2 seconds?
  
• By using the binomial expansion of #(1+i)^(2n)# can you prove that: #( ""_0^(2n)) - ( ""_2^(2n)) + ( ""_4^(2n)) - ( ""_6^(2n)) + .... + (-1)^n( ""_(2n)^(2n)) = 2^ncos((npi)/2), n in ZZ^+#?
  
• By considering #1+ cistheta + cis2theta + cis3theta + .... + cisntheta# as a geometric series, can you find: #sum_(r=0)^n cosrtheta#?
  
• Can you find all quartic polynomials with real, rational coefficients having #2-isqrt(3)# and #sqrt2 +1# as two of the zeros?
  
• Can you show that #p(x)=2x^3 + 7x^2 + kx - k# is the produce of 3 linear factors, 2 of which are identical? Show that #k# can take 3 distinct values
  
• If #a# and #k# are real, for what values of #k# does #z^3 +az^2 + kz + ka = 0# have: a) one real root b) 3 real roots?
  
• The exponential function #e^x# can be defined as a power series as: #e^x=sum_(n=0)^oo x^n/(n!)=1+x+x^2/(2!)+x^3/(3!)+...# Can you use this definition to evaluate #sum_(n=0)^(oo)((0.2)^n e^-0.2)/(n!)#?
  
• In terms of the motion of the charges in a conductor, can you explain why the component of the electric field parallel to the conducting surface must be zero?
  
• Can you use #sigma^2=Sigma(x_i-mu)^2p_i# to show that #sigma^2=Sigmax_i^2p_i-mu^2#?
  
• Suppose X is a uniform discrete random variable with possible values #X=1,2,....,n.# Can you show that the variance of #X# is #(n^2-1)/12#? Hint: #1^2 + 2^2 + 3^2+ .... + n^2=(n(n+1)(2n+1))/6#
  
• A helicopter at A(6, 9, 3) moves with constant velocity. 10 minutes later, it is at B(3, 10, 2.5) Distances are in kilometres. At what angle to the horizontal does is the helicopter flying?
  
• Boat A's position is given by #x_a(t)=3-t, y_a(t)=2t-4# and boat B's position is given by #x_b(t)=4-3t, y_b(t)=3-2t#. The distance units are kilometres and the time units are hours. At what time are the boats closest to each other?
  
• If line 1 has parametric equations #x=s, y=2-s, z=-2+s, s inRR#. Line 2 has parametric eqns #x=1+3t, y=-2-2t, z=6+2t, t in RR#. Can you find shortest distance between the points? And, the coord of points where the common perpendicular meets the lines?
  
• How do I differentiate #50sum_(i=1)^x(1.03^i)#?