Maths-
General
Easy
Question
be vectors of length 3,4,5 respectively.
be perpendicular to
and
is equal to :
The correct answer is: ![5 square root of 2](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAACQAAAATCAYAAAD4f6+NAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAAQ3ZOC+gAAAVlJREFUeNpjYKA+4AHi/xRgqoMIIN7PMIjAdiBOGSyOEQHi91CaIkCtOM0A4vVYxK2BeA0QfwLiX0B8AYijCTmIGgCUdkKwiB8E4khoggcBLSA+ChWjmYMUgPg1EHMQqV4eiC/R0kEVQDybRD0/aOmg80DsQYJ6S2i04XTQL2ii2wbERUjxja3gQwc6QPwciFmIdAwoWk9CEzteADLQGIhrgfgGEGsgyakA8Xyo423Q9LUDcT+RjhEE4g1A7EZqFLhBcwcM1EAdBCpnTqGpvQ/EJkSYqQR1jAq56QJbossC4n9ALA7lWwDxbSKiSwOa6LnIdQwoXdzFIfcNiJdB2ZOBuIGAWSDHryIhjTGsg6Z6Jij2gDomCIf6OUD8HWrBa7S0hg1sIUINCggF4ltA/AeI30F9Y4pHvQAQ/wXitUB8nI5VE15wGGpgwWCp2U2hDpIZTG2fEGobCACAYV3/vtY1CQAAAGJ0RVh0TWF0aE1MADxtYXRoIHhtbG5zPSJodHRwOi8vd3d3LnczLm9yZy8xOTk4L01hdGgvTWF0aE1MIj48bW4+NTwvbW4+PG1zcXJ0Pjxtbj4yPC9tbj48L21zcXJ0PjwvbWF0aD4UARYDAAAAAElFTkSuQmCC)
Related Questions to study
maths-
and
is equal to :
and
is equal to :
maths-General
maths-
If ABCDEF is a regular hexagon and if
then ![text end text lambda text is - end text](data:image/png;base64,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)
If ABCDEF is a regular hexagon and if
then ![text end text lambda text is - end text](data:image/png;base64,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)
maths-General
maths-
If
is along the angle bisector of
then -
If
is along the angle bisector of
then -
maths-General
maths-
If the solution of the differential equation ![fraction numerator d y over denominator d x end fraction equals fraction numerator 1 over denominator x c o s invisible function application y plus s i n invisible function application 2 y end fraction text end text text is end text text end text x equals c o s i n invisible function application y minus k left parenthesis 1 plus s i n invisible function application y right parenthesis comma text then end text k equals](data:image/png;base64,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)
If the solution of the differential equation ![fraction numerator d y over denominator d x end fraction equals fraction numerator 1 over denominator x c o s invisible function application y plus s i n invisible function application 2 y end fraction text end text text is end text text end text x equals c o s i n invisible function application y minus k left parenthesis 1 plus s i n invisible function application y right parenthesis comma text then end text k equals](data:image/png;base64,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)
maths-General
maths-
A curve passes through the point
Let the slope of the curve at each point (x, y) be
Then the equation of the curve is
A curve passes through the point
Let the slope of the curve at each point (x, y) be
Then the equation of the curve is
maths-General
maths-
The solution of the differential equation
![fraction numerator d y over denominator d x end fraction equals fraction numerator x plus y over denominator x end fraction](data:image/png;base64,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)
The solution of the differential equation
![fraction numerator d y over denominator d x end fraction equals fraction numerator x plus y over denominator x end fraction](data:image/png;base64,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)
maths-General
maths-
Statement-I : Integral curves denoted by the first order linear differential equation
are family of parabolas passing through the origin.
Statement-II : Every differential equation geometrically represents a family of curve having some common property.
Statement-I : Integral curves denoted by the first order linear differential equation
are family of parabolas passing through the origin.
Statement-II : Every differential equation geometrically represents a family of curve having some common property.
maths-General
maths-
A function f(x) satisfying
where x>0 , is
A function f(x) satisfying
where x>0 , is
maths-General
maths-
(where c is an arbitrary constant) is the general solution of the differential equation
then the function ![text end text phi open parentheses fraction numerator x over denominator y end fraction close parentheses text is : end text](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAE0AAAAnCAYAAABKZsWMAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAAY00gKyAAAAyRJREFUeNrtWkFkXFEUfcaIiG5qREVlk0VURc2uoiJKFxVV2UUXVTFEdFFZdFNVVVUii6ioUhFVMUJVVUSUqC6qsomqUVWluuoiZlMVWYzwe685w/f6f/67v++b3NceDvO//977/85795573zOmWFwnPvLc5xL6DRJjxI/Esud+ub8GcTw0g/URvxOrBfU/gv6PhWS0+8THBY+xhHGCQD/xJ/Gkw7M14nyK0WsZbQeIvzCeetwmrgqe/0A8EbueFgSPFeKdEIz2GUHAFReJi/h9gfhG0PYc8Yt2gw0TmznabSEacrStCNvuEk9pNhovrXqOdnPEFvFMjrZ1jKsWz4izwja8xF5giV7JMWYN46rFBnFC8PwQcR26jjXXTo7lyT5xU7PRmgIJ0I+PjRvpkjDyGhi7qdlo+8SSw3M9xJcpWm4Ns8cVJYx7GKKjbLRuvVzrv9H0jJsIXj7LxBuW3/mGJcFRq1eB0aKEAPQW3xA5vrfTt7Gj3kZFoQOuWrwiHocvmUfirM1oHKUvFzGbHybIB9ZTZ2PXZSTn2oy2jz/eK0oJYZzv7Vn3yqg0SF4+L30abZL4PqeYPtRoO9a9KvxAHOPIG7XNtI6+W0aBYdjXi2xZ11PEpwma6qpSo3Vwy7TLU17wxPoH2MfNxK7ZWK+VSI4oY1W1fL3IadOuw09Bwa8jMIyadvFvwfy5aaLRaLUEt/NXWnAUEmMPjbjgx6XoQWXiNkoJSAfIfweKENA9lrRIw8ERTaO6ggkk2L4Sdp8oOyTsXcE1S/mn4UcRojEDFaO8NLQhLOusINDY4PL3A8EqUF2EZB03LZzBC9Y9ruJ+Ne4V3JkE/agKrN0klVeeJc+te3eNbC9zzSjfWGExvCv0R5+sygqXnXoFfbA/U72Fx2hA37kiHvkW4c9ccd4EsFnMuAkH74p3pl0UHMIsk0iWVeSO6lEx7gdgGFz95YJg3Ur+szCITKViAgE7c9dDLHOYmQ3hGFzWuWcCQh+Wmssxg0nkdZLyc9UEeKiPwccNXI6PsrG2hXkwz8oxEyhYeGadiNwURlte9rPmH8YIUi9V+A3Cx7/kcOkneAAAAMd0RVh0TWF0aE1MADxtYXRoIHhtbG5zPSJodHRwOi8vd3d3LnczLm9yZy8xOTk4L01hdGgvTWF0aE1MIj48bXRleHQ+JiN4QTA7PC9tdGV4dD48bWk+JiN4M0M2OzwvbWk+PG1mZW5jZWQgc2VwYXJhdG9ycz0ifCI+PG1mcmFjPjxtaT54PC9taT48bWk+eTwvbWk+PC9tZnJhYz48L21mZW5jZWQ+PG10ZXh0PiYjeEEwO2lzJiN4QTA7OjwvbXRleHQ+PC9tYXRoPhmOcacAAAAASUVORK5CYII=)
(where c is an arbitrary constant) is the general solution of the differential equation
then the function ![text end text phi open parentheses fraction numerator x over denominator y end fraction close parentheses text is : end text](data:image/png;base64,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)
maths-General
Maths-
A function y = f (x) satisfies the condition f '(x) sin x + f (x) cos x = 1, f (x) being bounded when x
0. If ![I equals stretchy integral subscript 0 end subscript superscript there exists 2 end superscript f left parenthesis x right parenthesis d x text then end text](data:image/png;base64,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)
A function y = f (x) satisfies the condition f '(x) sin x + f (x) cos x = 1, f (x) being bounded when x
0. If ![I equals stretchy integral subscript 0 end subscript superscript there exists 2 end superscript f left parenthesis x right parenthesis d x text then end text](data:image/png;base64,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)
Maths-General
maths-
Statement-I : In throwing of two dice, let the events A, B and C be 'the first dice shows an even number', 'the second dice shows an odd numbers' and 'both the dice show an odd numbers or both the dice show an even number’ respectively. Then
and
Therefore A, B and C are mutually independent events
Statement-II : Three events A, B and C are mutually independent if and only if P(A Ç B) = P(A) . P(B), P(B Ç C) = P(B) . P(C),
= P(C) . P(A) and
= P(A) . P(B) . P(C)
Statement-I : In throwing of two dice, let the events A, B and C be 'the first dice shows an even number', 'the second dice shows an odd numbers' and 'both the dice show an odd numbers or both the dice show an even number’ respectively. Then
and
Therefore A, B and C are mutually independent events
Statement-II : Three events A, B and C are mutually independent if and only if P(A Ç B) = P(A) . P(B), P(B Ç C) = P(B) . P(C),
= P(C) . P(A) and
= P(A) . P(B) . P(C)
maths-General
maths-
The probability that 4th power of a positive integer ends in the digit 6 is:
The probability that 4th power of a positive integer ends in the digit 6 is:
maths-General
maths-
Let A, B, C be pairwise independent events with P(C)> 0 and
. Then ![P open parentheses A to the power of c end exponent intersection B to the power of c end exponent C close parentheses text is equal to: end text](data:image/png;base64,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)
Let A, B, C be pairwise independent events with P(C)> 0 and
. Then ![P open parentheses A to the power of c end exponent intersection B to the power of c end exponent C close parentheses text is equal to: end text](data:image/png;base64,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)
maths-General
maths-
Let A and B be two events such that
and
stands for complement of event A. Then events A and B are-
Let A and B be two events such that
and
stands for complement of event A. Then events A and B are-
maths-General
maths-
A random variable X has the probability distribution :
![](data:image/png;base64,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)
For the events E = {X is a prime number} and F = {X < 4}, the probability
is-
A random variable X has the probability distribution :
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAASMAAABFCAYAAAAB+oDeAAAT50lEQVR4nO2daVgUV76HX5pWUIwCBpG4IGocFRQdxQVJGMc1ihqJymgm4xL3LRPcx+SOGR0VE3WiMcYlOOpNYgYnRkdiGDQmbjFubAqCAiragMoua3dX3Q8QVxp7octO7nmfxw/gqXpPNVX/rjpV51d2sizLCAQCwTNG9aw7IBAIBCCKkUAgsBFEMRIIBDaBKEYCgcAmEMVIIBDYBKIYCQQCm0AUI4FAYBOIYiQQCGwCUYwEAoFNIIqRQCCwCUQxEggENoEoRgKBwCYQxUggENgEohgJBAKbQBQjgUBgE6gjIiIIDw+nZ8+ez6wTqamplJeX07FjR0W9hYWFxMTEEBgYqKi3tLSUkydP0r9/f0W9Op2Ow4cPM3jwYEW9siwTGRlJUFCQol6AQ4cOMWDAANRqtaLen376CS8vL5o0aaKo9/jx43Tq1AlnZ2fFnJIkER0dzZEjR6hfv77Z61Fdu3aNa9eu1WLXTEej0aDRaBT35uXlkZqaqrj33r17JCcnK+4tKysjISFBcS/A+fPnn4k3Pj4erVaruDctLY28vDzFvSkpKRQWFirq1Ol0nD9/3vKCv3fvXjk4OFh+lqxcuVJevHix4t5jx47JAQEBinsvXbokd+jQQXFvVlaW3KRJE8W9kiTJgOJeWZZlFxcXOScnR3HvsGHD5P379yvu9ff3l0+cOKGos6ioSHZycrJ4PWLMSCAQ2ASiGAkEApvApGJUcjWaTQv/xGvBY5iz/ijXK6Ak/Sj/mBPCm3/9nJgcyVr9rAY9t8/u4i/jlnKgWEHrrSiWj/0dfj0HMPkfJ7ij1CZLORxfO5GBvf3oN/VTEsoU8t4nj+glI1lyuFw5Zfk5Nk1+jeDgYEYvjOCGgruXXvMDm1esYsu/jpOuyGddwvH33yA4OLjq32jm7VFmLLfo/D9ZuWYrO7d/wKrwM+QrYn0Sk4pR/bYDmLFoPJ5p/+XY7Xq41YX6LnYUyL2ZvXAcXRsreKIlFVBWp4iEIxfJVWwnzeWb/z2N97tfsn/DCHLWj2PJIWUqYXn8UW70XsPBQxvxj1vK3/YVKeKtRCLn2w2s2XuWm0VKvUxGInPff0j3HcHIkcH8ITiA5grtXuWXtvKnyf/C7fV5TBvzEl6OCkjzf+DH3F68MW0mM2dOY1DDPKSmHtb36i7y4ZKjvDhzKuMnT8Hz8GI2XtRZ31sNKtnENxWpGvdj5fZQ6oW/xZqfUvlq7WG8F82lq9PDrSSK8/Kx6neoypWWHdrRrKGdNS2PIqnwm7CI4I7uvNBzCrOH1kNzU5li5NBlFK/7u1HXuQs9u/rSsW09RbwA0u1DbInxJcTPQTEnFTF8svUQNzQlNH95NK/18lBmTKHiHCsnbaLx/DBGedVVwgiARA8mL5/FyEH96d+3JfnlnRneW4nPW0YqOE1kdCaSTkNGsQdezz+b0RuzrI7dF7F5rppNIydwvu98RrV4bDVlkczxHUhYvLUrrB12CtYiVM40da/6mpQKuVnQgUH9GivnlwpI3v83dmjHMvG3Cj03I2Xyn22J9Jo+EGclP+tSB/zGjcbz6ieE9HqFNedKFNEWR33Mp7ea89zF9cwcE8KiiKvW/VKtQuXcGNeqP6n+6kHiPYJQpBapvZm18lUuzfwdQ6Z8SoPFGxnX9BdUjMCBtoED8LFL4sjxqzxxSe04jG1pp3ins7IPmilJWcxOzvr+hSlt7RVzSndSiE0rpejI24xecR7rn0xLZOz7lPTA6fzexeqyR2nkQ9DkBbwfcZrvFtrz8Yp/cdvql+M6rpyLRffSeBbOfZf173bm+KwZbLumt7b4IfSkH4zDPcgfJa4OQUWjtv6MGBuA7shn7Pk2gXwlh34f6YkZSJn7+eBbb7b8exa6j2YTdu7JET57tfpXe6tOyvmerdGtmT+vB+Y/b2o6Knc/Qt5ez1dbQig8cYpsa+80ukvs2rST7bNfokuX3iz89hbfLOjHYiUHsXHEZ+o0AisyyVVguEplp6J+IxccAYeOIxnSOo1LVxUcQ9Ff42BcE4b6K1OK0Kfw0ezPcF+wjcj/rsBt13RWnKpQxv0YKjtTr3PKL7Lt/QQCQ1+jXa9FbJxawcaZqzjzyFl0BVnJqQoMLMso/m7ugrPsCtfQd24wrez1ZCenKN0DHJt64NO5E1a/tFd3Yul3V7gYG0ts7I+sGdyMIe8fYXV/BceOAOluHm4B/Wlt9ZNQNe0Ce+N4NZEsCcAOlVNHfH9Tx9ri++ivHyTGbSh9FKpFaC+TkNGIZq4qHNqP570JTcnMeEbFyPimErfP7mRR0DA2F3fCq74KkHD17oRLTBh/nLCWwz/fAy07wrIRb7I10YrfKFIByUd/IOlOKicPxFrP8zDFZ1g9ciQLNv6VYN8XebG1J4O3ZCog1nNtxyQGjlvKxu2bWfu1G3MWv4yyJUFJJO58/kc6BUxg6cpVrP7KlQl/9kOJ4WTHwL+wuvMxVn96lO8//5zcUcuZ8PiYqNWQyIiM5flX+ih0iQY49mPa6yXsWbufUyf2c7AoiBlDGihlfxQxHeQXMh1EmyunJcTJiTcLZZ2Z3l/WdJB7siYpQb6cec8it3nTQbRy7rUk+UpWqdle86aD6OX86+nyXXP/wLL500GKMy/LCUka+Z7edGdtTQf59Y4w/9pQu+Dlo/Qo8rPECY/2PijwpE01qHHxbI/yn7aKRi1bKW4FqN/0N/g0fSbq+/xax5gFAsEvDFGMBAKBTaDWaDSkp6ezbdu2Z9aJM2fOUFZWpngfrly5QlZWluJejUZDXl6e4t7CwkJKS0sV98pVT/k/i32svLycXbt24eTk9PTGtcj169eJiooiOztbUW92djYHDhwgMTFRMWdZWRlarRatVkudOhbceYyIiBAD2Aoj8oyUQ+QZWR+RZyQQCH5ViGIkEAhsArNu7ZdcPsSe47fITzvFT6mlePSZTuiMQJpovmfrP7Zx0XUQw330lLV8leDuLuZXvPzTbN90HMnZHnvf8UwMaPzEuvQ5MXz54XqSemxmedBD4wIVMXwyZzVHcmXUbcYQtnIULY3tyFO9em5FrWJRWCTpDj6MeWclc/q4oUIi9+QGFiyP4JLsy4x16xjvbcrja/mc3r6J45Iz9va+jJ8YwBOpLPocYr78kPVJPdi8PAinGvtTi1705MR8yYfrk+ixeTkPPmpjljXfq78VxapFYUSmO+Az5h1WzumDm8r6XoNtimLYvekwpR4uFGhUBM6YRA8Tsu/zT29n03EJZ3t7fMdPJKCaTlfbpuQk62Z9yI/3quYc2HsxOmwVYzyNeyzdbC9YdiyZgMmrlDL/zXs7injlzcn8ef44Xkg5xA+369GkLjg625Gv7ca0eX8iKHgkrgffZfMlMx8tl3LYF7qQuJ4zmTqtLzeWTWf3rcfnl0gUFOm4eyGK+LvSI7/P2vc1V9oPYsiQIbw6vDfNjN1SY7y537D7VDsW7t7DmkHZrB27hEPFQHkC36V3Y+XXB1nvd57F732N8alDEjn7QlkY15OZU6fR98Yypu++9WSrgiJ0dy8QFX+X+70y1J9a9CIVUKS7y4WoeB581EYua7Y3l292n6Ldwt3sWTOI7LVjq/KjrO011EbHxQ8XcbjNDKaOn8xUz2gWb7xopBeknH2ELoyj58ypTOt7g2XTd/P4rmWoTf6xE2R1Hc2ESZOYNKEv9e5U4NbUuEJkideiY8lEDK9WKiI3rwJdXgbX7/w8EfYeR1Z8RPGA4XioQNW4P6u2h+KwfQ6rf0rlq7XRdFz8Z7o5ATgTGNKMg+8+tOFSMXn5xk2ylLL3sjXKg4BeTqDuSO/WMeyMuPFE911bdaZD84Y8MsOuIobNWw6RcbuCVr8fS4h/M4yd1mSMV1L5MXHJGDo386TPjLcIqqfhZrEEDr6M+uNLuDu60LVXVzp3aIPRqUNSNnu3RuER0Asn1HTs3ZqYnRFPNFO5tqJzh+Y8HONksD+16EXlSqvOHWj+iNjIZc31Sir8Ji5hTOdmePaZwVtBVflRVvcabiNLBZyOjCZT0qHJKMHD63njvEhk791KlEcAlbtWb1rH7CTihmREGx34TWHl3NEMfeUVBrYuQus7An+j5gNZ4pUsOpZMxfBEWdnAzmwHTywjS1TeLrFD9Vh5k2UJi+bLmpWhIyFLFk6hrdFrV8NPVchgZ1bnLV/GvNghS8KKrLesncEfrOs13EZGlvRU3R9U1PxwH8zZdPO8tXAsGUm1Z0ZS1ilWDfNmwOz/YdaU8Qzy6UDw9mS4F83ug43o1u1BSXbsvojNb9Xl45ETuPD7eQQ/dA5n3/q3tLr4BXuvV+bBlEXOwXdgmFEd0yYlcKWBBx51AOxxc3+OtMTL1bZ9ojiW1qfXGyH3g7nCzhqfxmiMV+XszoOMtZvkdxhM//vX4BIFyfv52z+1/GFiV+MH5bRJJFxpgEelGHs3d55LM/CsiN2jZa7m/tSi9/EwO5OWNcOrcsb94TC7/A4M7t/Y+l6DbdR4z1rFyKTZ/G7QJLY9t4SN44ydQ6ElKeEKDTw8qNy13HB/Lo3Ey1oj2uhxbuxatS/pufKfeF4I8jdyorQlXq1Fx5KpVLvHqpr2YEDXJth5jWbj3u84+XEAZz/Ygi41lkSa0PSRaw8H2r48AO/qgtbUzWnmcpmYuMoNdxy2jbRT7xjXs/JytHUc7s/UliUJnbEv42vkzZA3598P5tr8dxOCuUzylhGz8xxdlk6lzc/nrtIdUmLTKC06wtujlnPO6OCCcsq1dXB4IEbSmfrywWr6Y1WvcsuWxezkXJelTG1jr4DXcBtVo7b0efV1Avmez7/4hrg848/7y8u11HmwUiRJh1Yrm9ZGn0pkvAdDTYiBtMhrybFkIoa/Pu1UOD3XCHvA5aU+eN/LQSouoVRl/8i3vaT5mg+iOrK12qA1RxzqllNW/nPv7VGrjfvGtm/RAvfSAirz3/UU5pfg3qK5aVt3P5gry+hgLuO9EjnfbyW69TxC/R66Y6Zyxy/kbdZ/tYWQopP8aGwCmn0LWriXUlAVeK8vzKfEvYVxy9bUH2t6FVr25zC7eaF+ldEa1vYabKMn5aM5fOa+gK2R/+XvTXYzY8VJ47zY06KFO6UFRZUZXPpC8kvcadFcbVIb/bVI4tyHYnz2Wu14zTmWTKWGyiCj1VbeCZOysynr0gt1ixa4lxZS8HNnyi+y7YOLBIaOqj5oTS6kqNSd5s2qNqoii+TUXKM6pm4/lEGuKcTf0oOUQ2JqIwYO9QYKSE/O4OEst5ou3U0N5jLshYL0ZDKqxAVndxGu6cvc4FbY67NJTsl5dEWOTfHw6UwnYxPQ1O0ZOsiVlPhb6JHISUyl0cChT3gNbfBT+1Mb3sfD7GpY1hIvBekkP/ignwyzs7bXYBstlxNu0PAFF1QO7Rn/3kQ8sm4a50VN+6GDcE2Jp3LXSiS10UCGeqt5sE/X1AZAz/WDsbgNMSWStoZ13v+cn+atxNohdzUcKRK3ju1g4ycfE7aznMnvT0L1Qj/6tdFw9YbWuKC10hRSdS8x8LeVp39lR5Yx4s2txvVM3Y3QsD5cWBfOgd0biPNfw3y/uuiTw5kyfBnRxZV9LEg+yg9Jd0g9eYDY2xIWB3MZ8KJPJnzKcJZFF1N8ZjUjRy5g41+D8X3xRVp7DmZLZgP013YwaeA4lm7czua1X+M2ZzEvG302raZbaBh9Lqwj/MBuNsT5s2a+3yNeAKkgmaM/JHEn9SQHYm8DGOxPbXqrD7MzsKwlXvQkh09h+LLoGsLsrOw12MaRftNep2TPWvafOsH+g0UEzRhipBfU3UIJ63OBdeEH2L0hDv8186nctR7s04baVP4NMoiMfZ5XTIyBrH6dD2+voTYKh9xVH66mlc8u7Sa/vPrKY0FeejkrYor8xoZUowK+8vfPkceuu/TUtjXNTdPmpsvJN0wNFDMumKumuWnmebVyblqCHJd4Uy6sYcEa56Zpc+X05Bs1Lm8uNc5Ns8T7lGVrnJtmRa8s1zA3zRivoTbFmfLlhCRZU0MSmeG5aVo5Nz1ZvlGzuPo2+nz5evrdGvdJw3PTzPU+/ViycriahF6nQ6t7/IFFFe7Bq5i8JZz9KXMJbmf4a1/KOcUXVwJZ/lZHi55LULu0op3JKVeWB3OZ51Xj4uVjWSiX2oVWpostxxLvr3VZQ23qN+U3ZieRqXFp1e4p+4iBNqpGmJ+9Zq5XuZC7ai/TdJpYUpuMYKDTReIfHzpXNeblGXPorrtJgcGx2XJuZroy5u3XaCOyJAUCgRFUWyrUL/RgXGiPGhZzpGXHNjX8vwMtfdpb1jOBQPD/CrvZs2fLO3bswMPj2aQNA+Tm5iJJEs8/b+yj9bVDaWkpOTk5NG9u6iMDllFRUUFmZiaenp6KenU6HRkZGXh5eSnqlWWZ1NRU2rZtq6gXIC0tDU9PT+ztlXvZJlQG6DVs2JAGDZR908bNmzdp3Lgx9eop+PpzSSI9PZ3c3FycnU2YNfwYdpmZmfL58+dp165dLXbPNDIzM9Hr9YoXhZKSEtLT0/H29lbUW1FRQWJiIl26dFHUq9friY2NpVu3bop6Ac6ePYufn7F3vGrX27179yef0rcyly5dwsvLi/r1lXzNJ8TGxuLt7W1Z4qKJyLJMTEwMISEhFq3HTpbNnGAjEAgEtYgIVxMIBDaBKEYCgcAmEMVIIBDYBKIYCQQCm0AUI4FAYBOIYiQQCGwCUYwEAoFNIIqRQCCwCUQxEggENoEoRgKBwCYQxUggENgEohgJBAKbQBQjgUBgE4hiJBAIbAJRjAQCgU0gipFAILAJ/g/JolBC7UqsLQAAAABJRU5ErkJggg==)
For the events E = {X is a prime number} and F = {X < 4}, the probability
is-
maths-General