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    Question asked by @deepalis727

    Find the maximum value of (x - 6)^2 (11 - x)^3 for 6 < x < 11.


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    @deepalis727

    If you are comfortable with derivatives then you can check Part 1. Else, jump to Part 2 :)

    Part 1:

    f(x) = g(x)h(x)
    f'(x) = g'(x)h(x) + h'(x)g(x) where f'(x) is the derivative of f(x).
    So if f(x) = (ax + b)^m * (px + q)^n then
    f'(x) = ma(ax + b)^(m-1)(px + a)^n - np(px + q)^(n-1)(ax + b)^m
    Fox max of f(x), f'(x) = 0
    ma(ax + b)^(m-1)(px + q)^n + np(px + q)^(n-1)(ax + b)^m = 0
    ma(ax + b)^(m-1)(px + q)^n = - np(px + q)^(n-1)(ax + b)^m
    ma(ax + b)^(m-1)/(ax+b)^m = -np(px + q)^(n-1)/(px + q)^n
    ma/(ax + b) = -np/(px+q)

    Part 2:

    If f(x) = (ax + b)^m * (px + q)^n then max/min occurs at values of x for which (ax + b)/ma = - (px + q)/np
    So here, f(x) = (x - 6)^2 (11 - x)^3
    max occurs at values of x such that (x - 6)/(2 * 1) = -(11 - x)/(3 * -1)
    3x - 18 = 22 - 2x
    5x = 40
    x = 8

    max of (x-6)^2 * (11-x)^3 happens at x = 8 and max value = 2^2 * 3^3 = 108

    So, if we want to find for what of x for maximizing f(x) = (4x - 3)^5 * (18 - 2x)^4
    We need to find x such that (4x - 3)/20 = (18 - 2x)/8
    32x - 24 = 360 - 40x
    72x = 384
    x = 384/72 = 16/3
    So max for f(x) occurs at x = 16/3

    solution from @Deekonda-Saikrishna

    max occurs when there is a symmetry
    so (x-6)/2 = (11-x)/3
    so at x =8
    therefore max value =2^2 * 3^3



  • CIRCULAR MOTION

    Concept 1 : Circular Motions In Common Admission Test (CAT) Is Mostly Based On Two Or Three Objects Moving Around A Circular Track. In Such Cases The Relative Speed Becomes Effective Speed Of The Bodies. Let Us Assume The Three Objects Are Moving With A Speed Of A, B & C On A Track Of Length L. (A > B > C)

    1. If Two Bodies Are Moving In The Same Direction, Their Relative Speed Is (A-B).
    2. If Two Bodies Are Moving In The Opposite Direction, Their Relative Speed Is (A+B)
      ■Number Of Distinct Points At Which Body Meets On A Track
    3. If They Are Moving In Same Direction, They Will Meet At (A-B) Distinct Points, Where A And B Are Coprimes
    4. If They Are Moving In Opposite Directions, They Will Meet At (A+B) Distinct Points, Where A And B Are Coprimes
    5. All These Points Will Be Equidistant From Each Other.
      ●Note : If They Aren't Coprimes Then Cancel The Common Part , Make Them Coprime (A:B) And Then Use The Above Points.
      Ex 1 : If Two Bodies Are Moving In The Opposite Direction At Speed 7 And 19, Then They Will Meet At (7+19) = 26 Distinct Points On The Track.
      Ex 2 : If Two Bodies Are Moving In The Same Direction At Speed 6 And 15, (6:15 :: 2 : 5)Then They Will Meet At (5-2) = 3 Distinct Points On The Track.

    ■Concept 1 Post 2 : Number Of Distinct Points 3 Bodies Will Meet On A Circular Track .
    ●Step 1: Find Out Pairwise Distinct Meeting Points.
    ●Step 2 : The Overall Answer Will Be The Highest Common Factor Of The Pairwise Values.
    ■Example: Suppose A, B And C Are Running With Speeds Of 7, 9 And 13 On A Circular Track. A And B Are Running Clockwise, Whereas C Is Running Anti-Clockwise.
    ●A And B Will Have 9 - 7 = 2 Distinct Meeting Points.
    ●A And C Will Have 7 + 13 = 20 Distinct Meeting Points.
    ●B And C Will Have 9 + 13 = 22 Distinct Meeting Points.
    ■A, B And C Will Have HCF (2, 20, 22) = 2 Distinct Meeting Points.
    They Will Meet At 2 Distinct Points On The Track

    ■Concept Post 2 : Time When Two Objects Meets For The First Time When They Are Running Around A Circular Track In The Same Direction.
    ●Let Us Assume There Are Two Persons (A) And (B) .
    Speed Of A = 20 M/S.
    Speed Of B = 10 M/S.
    Length Of Track = 1000 M.
    ●Try To Visualise, They Will Meet The First Time When A Has Covered 1000m Distance More Than B. Hence The Relative Distance Is 1000.
    ●Now Let's Assume A And B To Be One Single Body, And Hence Everything Considered Will Be Relative,
    Relative Distance = 1000
    Relative Speed = 20-10 = 10
    ●Time Taken = Relative Distance/Relative Speed = 1000/10 = 100 S
    Hence They Will Meet For The First Time After 100 Seconds.

    ■Concept 3 : Time When Three Objects Meets For The First Time When They Are Running Around A Circular Track In The Same Direction.
    ●Now, Let Us Assume There Are 3 Persons A, B And C Running In The Same Directions On 1000 M Track .
    Speed Of A = 30 M/S
    Speed Of B = 20 M/S
    Speed Of C = 10 M/S
    ●Again The Concept Of Relativity, The Time When They Will Meet For The First Time = Time Taken By The Fastest Runner To Take One Round Over Other Runners.
    ●Time Taken By A To Take One Round Over B , 1000/(30-20) = 100 S
    ●Time Taken By A To Take One Round Over C, 1000/(30-10) = 50 S
    ●Now The Time After Which They All Three Will Meet = LCM(100,50) = 100.
    Hence All Three Persons Will Meet After 100 Seconds.

    ■Concept 4 : Time When They Will Meet For The First Time At The Starting Point.
    ●Time When They Will Meet First Time At The Starting Point = LCM(Time Taken By Each Individual To Take One Round) .
    ●Let's Assume There Are Three Persons, A, B And C With Respective Speed Of 30, 20 And 10 M/S
    ●Length Of Track = 1000 M .
    Time Taken By A = 1000/30
    Time Taken By B = 1000/20
    Time Taken By C = 1000/10
    ●LCM(100/3, 50, 100) = LCM(100,50,100)/HCF(3,1,1) = 100/1 = 100 S.
    ●Hence They All Will Meet First Time At The Starting Point After 100 S.

    ■Concept 6 : Application Of AM-GM In TSD
    ●Using AM & HM In Proportionality Relations,Quite Often In Questions We Find That The Given Speeds (Or Time Taken) Are In An Arithmetic Progression. And If Distance Covered At The Speeds Is Constant, Then Time Taken (Or Speeds) Will Be Inversely Proportional I.E. They Will Be In Harmonic Progression.
    ●For Those Who Have Forgotten, The Arithmetic Mean Of A And B Is(A + B)/2 And The Harmonic Mean Of A And B Is 2ab/(A+B)
    Though It Requires A Little Trained Eyes To Identify The Above, It Will Be Useful If You Keep A Watch For It. See The Following Data To Realise That Either Time Taken Or Speeds Are In An Arithmetic Progression.
    E.G.: 1. If I Travel At 15 Kmph, I Reach Office At 10 Am,If I Travel At 10 Kmph, I Reach Office At 10:30 Am. At What Speed Should I Travel So That I Reach Office At 10:15. Assume I Leave Home At Same Time And Take The Same Route. ( Use Am/Hm -- No Other Method)
    Solution . Leaving At Same Time And Reaching At 10 Am, 10:15 Am And 10:30 Am Suggests That The Time Travelled Are In AP. Thus, Speeds Are In HP And Required Speed Is The
    HM Of 10 & 15
    I.E.2 *10 *15/25 = 12 Kmph

    credit - iquanta



  •                             TRIANGLE                                                                   

    A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices A, B, and C is denoted.
    Area: ½ × base × height
    Perimeter: sum of side lengths of the triangle
    Number of vertices: 3
    Number of edges: 3
    Internal angle: 60° (for equilateral)
    Sum of interior angles: 180°
    Line of symmetry: 3
    0_1514558750278_07a6c396-06a3-401c-9e87-5a8d7301865c-image.png

    TYPES OF TRIANGLE
    Three types of triangles:
    Acute (all angles less than 90°),
    Right Angle (one angle is 90°),
    Obtuse (one angle is more than 90°).
    0_1514559011648_13a60a54-ce72-4987-a2a2-467e683cf57f-image.png

    IMPORTANT - Angle opposite to the larger side is always greater than angle opposite to smaller side.
    Sum of two sides is greater than third.
    So, in a ∆ABC
    AB + BC > AC,
    AB + AC > BC
    And AC + BC > AB

    ~ For acute angle triangle: - AB² + BC² > AC², AB² + AC² > AB² and BC² + AC² > AB²
    For right angle triangle: - AB² + BC² = AC², where AC is hypotenuse
    For obtuse angle triangle: - AB² + BC² < AC², where AC is the largest side

    ~ Sine Rule : if a, b,c are the sides of a triangle with opposite angles A,B,C then
    a/SinA = b/SinB = c/SinC = 2R. ( R = circum radius )

    ~Cosine Rule
    The Cosine Rule can be used in any triangle where you are trying to relate all three sides to one angle.
    Finding Sides
    If you need to find the length of a side, you need to know the other two sides and the opposite angle.
    formula:
    a^2 = b^2 + c^2 – 2bc cos(A)
    Side a is the one you are trying to find. Sides b and c are the other two sides, and angle A is the angle opposite side a .
    In case of right triangle A=90 and Cos90=0
    So a^2=b^2+c^2 which is pythagoras theorem

    ~SIMILARITY OF TRIANGLES
    AAA (angle angle angle)
    All three pairs of corresponding angles are the same.
    SSS in same proportion (side side side)
    All three pairs of corresponding sides are in the same proportion
    SAS (side angle side)
    Two pairs of sides in the same proportion and the included angle equal.

    AREA OF TRIANGLE
    Area = 1/2 * bh
    Area = 1/2* ab SinC = 1/2* ac SinB = 1/2* bc SinA
    Area = r*s ( r= inradius and s= semi perimeter ) semi perimeter = sum of sides /2
    Area = abc/4R ( R= circum radius )

    RIGHT ANGLE TRIANGLE , ISOSCELES TRIANGLE , EQUILATERAL TRIANGLE
    0_1514560493610_99b80358-12b9-4013-8e7d-270f4a67a3da-image.png

    IMP- The incenter and circumcentre lies at a point that divides height in the ratio 2 : 1. ( incenter = inradius center , circumcentre = circum radius center )

    ~ N sided POLYGON
    Sum of interior angles of n-sided polygon= (n-2) x 180°
    each angle = (n-2) x 180°/n ( n= number of sides )
    Sum of exterior angle of any polygon = 360°.
    Number of diagonals in any n sided Polygon = n(n-3)/2.
    0_1514560941480_de4e8bfc-f0ed-4e60-b378-f92c2a760e07-image.png

    CENTROID
    The centroid of a triangle is the intersection of the three medians of the triangle (each median connecting a vertex with the midpoint of the opposite side). It lies on the triangle's Euler line, which also goes through various other key points including the orthocenter and the circumcenter.
    0_1514561102866_00267ecb-64ab-4362-9695-ed4d44af44ff-image.png
    The centroid divides each median in the ratio 2:1

    INCENTRE AND CIRCUMCENTRE
    The incenter of a triangle is the center of its inscribed circle. It has several important properties and relations with other parts of the triangle, including its circumcenter, orthocenter, area, and more.

    Centre of the circle circumscribed about a triangle . Radius of which is called circumradius.
    0_1514561263383_b01ea1b7-a972-4806-ab9e-fb2981a9ce76-image.png

    ORTHOCENTER
    The orthocenter is the point where the three altitudes of a triangle intersect. A altitude is a perpendicular from a vertex to its opposite side.
    0_1514561407686_f3cc2f39-0018-4781-9af3-e68d69f22833-image.png

    IMPORTANT - PTOLEMY’S THEOREM
    AC * BD=AB * CD + BC * AD
    0_1514561505309_520e671a-96eb-4fdc-b859-f5a5d4d22358-image.png



  • How many integral triplets (x, y, z) satisfy the equation x^2 + y^2 + z^2 = 1855 ?

    solution:- Note that 1855 = 7 mod 8 while all perfect squares are 0, 1 or 4 mode8. So it is impossible for 3 squares to sum up to 7 mod8. So no solutions are there.

    Could anyone please explain how we choose number to take mod with, on both the sides.Here for example, we took mod 8..why not, mod 4? mod 4 would have given 3 a.d 3 mod 4 is possible .So , how do we arrive at a number which poses contradiction.

    How we chose 8 here?


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    @sumit-agarwal

    Legendre's three-square theorem states that a natural number (say n) can be represented as the sum of three squares of integers if and only if n is not of the form n = 4^a(8b + 7) for integers a and b.



  • Suppose real numbers x and y satisfy x^2 + 9y^2 - 4x + 6y + 4 = 0.what is the maximum value of 4x - 9y?

    Geometric approach

    We know that (x - 2)^2 + (3y + 1)^2 = 1 and we need to maximise 4x - 9y.
    Now, let's replace 3y by Y, so that know equation turns into a circle
    (x - 2)^2 + (Y + 1)^2 = 1 with center at (2, -1) and radius 1.
    And now we need to find the maximum value of 4x - 3Y which will be equal to 4x - 3Y = k (say) and will be achieved when this line is tangential to the circle.

    Doubt - what is the logic behind the point that " for the value of expression to be maximum , that line should be tangential to the circle " ..

    Could anyone help me with understanding this concept please.


  • Being MBAtious!


    @sumit-agarwal
    We are trying to maximize the distance between the line and the centre of the circle. It is also known that the circle lies on the circle so it has to be a tangent.

    Sharing a solution from @kamal_lohia sir

    x^2 + 9y^2 - 4x + 6y + 4 = 0
    i.e. {x^2 - 2(x)^(2) + 2^2} + {(3y)^2 + 2(3y)(1) + 1} = 1
    i.e. (x - 2)^2 + (3y + 1)^2 = 1

    Now we want to maximise 4x - 9y. Let 4x - 9y = k, i.e. x = (k + 9y)/4

    So eliminating x, above equation becomes: (k + 9y - 8)^2 + 16(3y + 1)^2 = 16
    i.e. 225y^2 + {18(k - 8) + 96}y + (k - 8)^2 = 0
    i.e. 225y^2 + (18k - 48)y + (k - 8)^2 = 0

    Now as y is a real number, so discriminant of the above equation should be greater than or equal to zero.
    So, (18k - 48)^2 - 4(225)(k - 8)^2 ≥ 0
    i.e. (18k - 48)^2 - (30k - 240)^2 ≥ 0
    i.e. (3k - 8)^2 - (5k - 40)^2 ≥ 0
    i.e. (-2k + 32)(8k - 48) ≥ 0
    i.e. (k - 6)(k - 16) ≤ 0
    i.e. 6 ≤ k ≤ 16.

    Thus required maximum value of 'k' is 16.



  • P and Q are two diametrically opposite points on a ciccular track. A and B start from P at speed of 20 and 40 rounds per hr respectively. C starts from Q at the same time at 10 rounds per hour. If B and C move in clockwise direction and A in anticlockwise direction ,find the time after which they meet for the third time anywhere on the track?

    Could anyone help me with the solution of this question?


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