Questions asked by Alex Barth

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• How do you find the point of inflexion of y=xe^(-x)+3? Then the inflectional tangent?
  
• How to solve #sin(x+pi/6)# = #2sin(x-pi/6)# in radian form?
  
• Evaluate #lim _(x-> oo) (sinxcosx)/(3x) # ?
  
• If 2 curves #y=kx^n# and #y=lnx# have the same gradient exactly at x=a, find the relationship between a, k and n? Also, if these 2 curves also intersect at the point x=a, express k in terms of n?
  
• 6 equal circular discs placed so that their centres lie on the circumference of a given circle with radius (r), and each disc touches its 2 neighbours. What is the radius of a 7th disc placed in the centre which will touch each of the each existing ones?
  
• What is the positive integer values of n such that #sinx=x/n# has 69 positive solutions?
  
• How many values of x between 0.01 and 1 does the graph #sin(1/x)# cross the x-axis?
  
• For a continuous function (let's say f(x)) at a point x=c, is f(c) the limit of the function as x tends to c? Please explain.
  
• Given that a circle has an equation of x^2+y^2+2x-6y-6=0, what is the exact length of AB given that a tangent is drawn from A(6,5) and touches the circle at B?
  
• How do you find the primitive function of #f(x)=1/sqrt(3-x)# and #f(x)=(4-2x)^-2# ?
  
• What is #lim_ (x->0) (xcosx)/(sin3x)# ?
  
• What is #arcsin(sin(1/4))# ?
  
• How do you differentiate f(x)=#1/sqrt(x-4)# using first principles?
  
• What is #arctan(cospi)# ?
  
• What is #arccos(1/2)+arcsin(1/3)# in terms of pi?
  
• How to solve #(sqrtx+sqrty)/(x^(3/2)+y^(3/2))# ?
  
• What is the maximum value of #(3-cosx)/(1+cosx)# for #0< x < (2pi)#?
  
• Using mathematical induction for integers n #>=# 1, prove a + (a+d) +...+ (a+nd) = 1/2 (n+1)(2a+nd)?
  
• What is the derivative of f(x) = #sqrt(9-x^2)+4# considering the fact that the radius and tangent are perpendicular at the point of contact?
  
• For the function f(x) = #sqrtx+1/sqrtx# what are the intercepts and asymptotes ?
  
• Find #int(3x)/(x-4)^4# using substitution u=x-4. How to solve it?
  
• Find #int(5x)/(7-2x)^3# dx using substitution u=3-5x^3. How is it solved?
  
• Find #intx(3x^2-1)^5# dx using implicit substitution. How?
  
• How to prove #8cos^4x=3+4cos2x+cos4x#? Hint #cos^4x=(1/2(1+cos2x))^2)#
  
• How to prove cosec4A + cot 4A = 1/2 (cotA-tanA)?
  
• When a polynomial #P(x)# is divided by #(x^2-1)#, the remainder is #2x+3#. What is the remainder when #P(x)# is divided by #(x-1)#?
  
• Why is the derivative of lnx = 1/x? Also, why is the derivative of e^x = e^x?
  
• What is the primitive of y=e^(2lnx)?
  
• How do you differentiate #log_x3# =y ?
  
• How to find #int(10x^3-7x)/(5x^4-7x^2+8)dx# ?
  
• What is #int1/(2+sqrtx)dx# using substitution u=sqrtx ?
  
• How to verify by differentiation that #int1/(sqrt(x^2-a^2)dx# =#lnabs(x+sqrt(x^2-a^2)# +c, given that x>#absa# ?
  
• Why is it that in order to convert the change in enthalpy per mole of product into the change in enthalpy per gram of the same product, you need to divide the change in enthalpy by the molar mass? Can you provide an example, please?
  
• Why is the domain of the function #ln((2x)/(2+x))# x<-2, x>0?
  
• How to find #lim(ln(2x)/(2+x)) x-> # infinity?
  
• Prove that #limxlnx# = 0 as x approaches positive 0, given that #lim(lnx)/x# = 0 as x approaches positive infinity?
  
• What is #(lnx)^2#?
  
• How do you find the primitive of #5^-x# given that you have to use the identity a=#e^(lna)# to express it as a power of e?
  
• How do you find #int(e^-x)/(1+e^-x)dx#?
  
• How do you find the domain and range of y = inverse cos(2x-1)?
  
• How do you find the domain and range of inverse cos(e^x)?
  
• How do you find the domain and range of ln (inverse sinx)?
  
• How do you show that inverse tan (1/x) = #pi/2-theta#, given that x is a positive number and #theta# = inverse tanx?
  
• How do you find the domain of inverse sin #sqrt(x/(1-x)# ?
  
• How do you find the derivative of #cos^-1(-1/x)# ?
  
• How do you evaluate by completing the square #int_-1^0dx/sqrt(1-2x-x^2)#?
  
• Why is the odd monic polynomial of least degree with a triple root of x=-2 and a single root of x=1 : P(x) =# x(x-1)(x+1)(x-2)^3(x+2)^3# ?
  
• If #alpha#, #beta# and #gamma# are the roots of the equation #x^3+6x+1=0#, how do you find the polynomial whose roots are #alpha beta# , #beta gamma# and #alpha gamma#?
  
• The roots of the polynomial equation #2x^3-8x^2+3x+5=0# are #alpha#, #beta# and #gamma#. What is the polynomial equation with roots #alpha^2#, #beta^2# and #gamma^2#?
  
• Let f(x) = #x^3+5x^2+17x-10#. The equation f(x) = 0 has only one real root. So how do you sow the root lies between 0 and 2?
  
• How do you show that the equation x^3-12x+10=0 has 3 real roots and determine 2 consecutive integers between which each of the roots lie?
  
• If small non-polar molecules were mixed with water, would it dissolve or not?
  
• From a pack of 9 cards numbered 1,2,3.......9 , three cards are drawn at random and laid on a table from left to right. What is the probability that the digits are drawn in descending order (by just using permutation)?
  
• Four identical apples and four identical pears are arranged in a line. How many ways can this be done?
  
• Given there are 6 boys and 2 girls arranged in a row, what is the probability there are at least 3 boys separating the 2 girls?
  
• How do you find the principle argument of #4(-cos(pi/3)+isin(pi/3))#?
  
• If z = 2(#cospi/4+isinpi/4#), how do you write (-z) in mod-arg form?
  
• From complex numbers, how is the value of #i^3#= -i?
  
• Given there is f(x) = #(3x)/(x^2+1)#=c. How many roots does f(x) has for a ) c>#3/2# b) c=#3/2# c) 0<c<#3/2#?
  
• How will you show?
  
• Pressure (p) and volume (v) given in eqn #pv^1.4=k# where k is constant. At certain instant pressure is #(25gm)/(cm^3# and volume is #32cm^3#. If volume is increasing at rate of #(5cm^3)/s#, how do you find the rate at which pressure is changing?
  
• There's a school of 1000 fish, the number P infected with a disease at time t years is given by P = #1000/(1+ce^-(1000t)# where c is a constant. How do you show that eventually all the fish will become infested?
  
• How do you simplify #(x^5+y^5)/(x^3+y^3)#?
  
• The height h metres of a ball above the ground at time t secs is h = 5t(4-t) How do you find the average velocities of the ball during the first and third seconds and the average speed? Can you also explain the difference btwn velocity/speed/acceleration?
  
• A hall has 'n' doors. Suppose that 'n' people each choose any door at random to enter the hall. a) In how many ways can this be done? b) What is the probability that at least 1 door will not be chosen by any of the people?
  
• 6 keys are to be placed on a keyring. What is the number of ways of arranging the keys if there are 3 identical locker keys and 2 identical car keys?
  
• The acceleration of particle P is moving in a straight line eqn a = 3x(x-4) where x(m) is the displacement from origin O and t is time in secs. Initially, the particle is at O and its velocity is #4sqrt(2)# m/s. How to prove that #v^2=2(x^3-6x^2+16)#?
  
• A certain particle moves along the x-axis shown in the eqn #t=2x^2-5x+3# where x is measured in cm and t in seconds. Initially, the particle is 1.5cm to the right of the origin. How do you prove the velocity v cm/s is given by #v=1/(4x-5)#?
  
• A toy rocket is driven vertically upwards from the origin at rest, and it rises with an acceleration of #a=(9-5t)g ms^-2# for the first 2 secs and thereafter an acceleration of #a=-g ms^-2#. How to find the max speed and max height reached?
  
• Why is it that you divide the angle value in the trig function by #2pi# to get the period?
  
• How do you find the period of #2sin(2x) +1#?
  
• How do you find the centre of motion of a particle moving in simple harmonic motion of period 8hrs and amplitude 6m. When t=3hrs and x=4m?
  
• Show by maths induction that for integers greater than 5, #4^n>n^4#?
  
• Why does neutralisation of any strong acid in an aqueous solution by a strong base always result in a heat of reaction approx. -57kJ/mol?
  
• How to prove #ab+cd<=sqrt((a^2+c^2)(b^2+d^2))# given a, b, c, d, are any four positive numbers?
  
• How to prove #(a+5b)(a+2b)>=9b(a+b)# given a, b, c, d, are any 4 positive numbers?
  
• Say it's given logx. Is this log base e of x or log base 10 of or both somehow?
  
• From differentiating #xe^x#, how do you deduce #int_0^2xe^x# from that?
  
• How do you find y given #dy/dx=2^(-x)# and #y = 1/(2log2# when x=1?
  
• How to find the primitive of #sqrt(x)e^(xsqrt(x))#?
  
• How do you find the primitive of #e^(2logx)#?
  
• How do you evaluate the sum from n=3 to n=10 for #(n+2^n)#?
  
• From the word 'propriety', how do you find all the possible words / different arrangements of all the letters? (answer is 90720)
  
• A poker hand consisting of 5 cards, is dealt from pack of 52 cards. Find the probability of obtaining: a) 1 pair b) 2 pairs c) 3 of a kind d) 4 of a kind e) a flush f) a royal flush?
  
• The acceleration of a particle moving along the x-axis is given by a = #2x^3-10x#. a) if u = 3 show that the particle oscillates within the interval #-1<=x<=1# b) Why is it not simple harmonic motion c) if u=6 describe the motion?
  
• If a monic cubic polynomial is divided by #x^2-9# leaving a remainder of x+8 and when divided by x leaves a remainder of -4, how do you find this polynomial?
  
• For x^3-8x+1=0, it has a solution which is approx -3. How do you find a better approximation to 2 decimal places using the half-interval method?
  
• How do you show that the equation x-inverse tan(3x)=0 has a solution lying between x=1 and x=2? (Not sure if you need to use half-interval or newton's method??)
  
• Consider the circle #x^2+y^2-2x-14y+25=0#. For what values of m is the line y=mx tangent to this circle?
  
• How to find the domain and range of #"arccos"(e^x)#?
  
• How to evaluate #int_0^2(e^(2x)/(e^x-1))#? Is there a way of not directly doing substitution?
  
• The polynomial #8x^3-36x^2+22x+21=0# has roots which form an arithmetic progression. How do you find the roots?
  
• How to evaluate #int_1^2(e^(2x)/(e^x-1))# ? Can you solve it without substitution and somehow use chain rule?
  
• If a particle moves with acceleration = #10x-2x^3# and v= 0 at x=1, is the motion simple harmonic? And how do you describe the motion?
  
• How to evaluate lim (as x approaches 0) #(5xcos2x)/(sinx)#?
  
• How do you find the equation of the chord of contact to the parabola #x^2=8y# from the point (3,-2)?
  
• How do you solve #abs(2x-1)<3x+3#?
  
• If A(-3,1), B(8,-2) and C(4,16) are vertices of a triangle and AM is a median of this triangle, how do you show that the coordinates of M is (6,7)?
  
• A pack of 36 cards includes 20 numbered cards from 6 to 10 inclusive, 4 aces and 12 picture cards. If a hand of 5 cards is selected at random, how do you find the probability of receiving at least 2 aces?
  
• How do you evaluate #int(1-x)/(1-sqrtx)dx#?
  
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