Maths & Physics - Solved Questions

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Q) A particle of mass m moves on the x-axis as follows: it starts from rest at t = 0 from the point x = 0, and comes to rest at t = 1 at the...

Q)   Define a a sequence     as follows:  , if number of positive divisors of   is  odd   , if number of positive divisors of   is  even   (...

Q) Let f(x) = and f(0) = 0 . Then f(1) = ? A) = = By Applying partial fractions - = = = = BY using the statement tha...

Q) Find the number of all integer-sided isosceles obtuse-angled triangles with perimeter 2008 units . (From RMO - 2008) A) Let x be the l...

Q) Find the number of all 6-digit natural numbers such that the sum of their digits is 10 and each of the digits 0, 1, 2, 3 occurs at least ...

Q) A body weighs 6 grams when placed on one pan and 24 grams when placed on the other pan of a defective beam balance according to the readi...

Q) Find all three-term Harmonic Progressions a, b, c of strictly increasing positive integers in which a = 20, and b divides c . ( from RMO...

Q) A circle touches the x axis and also touches another fixed circle with center (0,3) and radius = 2 units. What is the locus of the cente...

Q) Find the number of triangles that can be formed with the vertexes of a 10-sided-polygon (decagon) as its vertexes ,if 1)The triangles can...

Q) How many groups can be made from 5 different green balls, 4 different blue balls and 3 different red balls, if at least 1 green ball and ...

Q) A particle starts from rest x=0 at t=0, moves on a straight-line path and again comes to rest at x=1 at t=1. If the motion is simple harm...

Q) A man is known to speak the truth 3 out of 4 times. He throws a die and reports that it is a six. Find the probability that it is actuall...

Q) Two equal resistances are connected in the gaps of a meter bridge. If the resistance in the left gap is increased by 10%, by what percent...

Q) Prove that . A) log 7 10 = 1 + log 7 (10/7) = 1 + log 7 1.428 log 11 13 = 1 + log 11 (13/11) = 1 + log 11 1.18 Let A = ...

Q) Find the coefficient of x^4 in (1 + 2x + 3x^2)^5 . A) [(1+2x)+3x^2)]^5 'r'th Term = 5Cr (1+2x)^r . (3x^2)^{5-r} To fo...

Q) Find the sum of : 1 + 1/2 + 1/3 + 1/4 .... up to infinite terms . A) 1/1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6 + 1/7 + 1/8 + ... = ( 1/1 ) + ( 1...

Q) Find the number of real pairs (x,y) satisfying and . A) y = 2xy y (1-2x) = 0..... So, either y=0 or x = 1/2 For y = 0, x = x^3 ...

Q) Evaluate the definite integral : Here, {x} is the fractional part function, defined as {x} = x - [x] ( [.] denotes greatest integer fu...

Q) Find the integral of : A) = = Now take out the 1 and you get: e^x + e^x [ f(x) + f'(x) ] ..... where f(x) =...

Q) Find the number of 5-digit numbers of the form xyzyx in which x is less than y. A) Number of ways of selecting x and y: 9C2 . 2! Out of ...

Q) If the arithmetic mean of two numbers 'a' and 'b' is 'n' times their harmonic mean, find a/b in terms of 'n...

Q) A particle is thrown upwards from the ground. It experiences constant air resistance which produces a retardation of 2 m/s^2. Find the ra...

Q) Find the integral of: dx / [(x)(1 + x^n)] A) 1 / [(x)(1 + x^n)] = (1 + x^n - x^n) / [(x)(1 + x^n)] = (1/x) - [ (x^n) / {(x)(1 + x^n)}...

Q) A car is traveling on a straight road. The maximum velocity that can be attained by the car is 24 m/s. The maximum acceleration and decce...

Q) If is a 6th root of unity, then prove that : 1) 2) A) = 1 × 1 ............. ---> (Modulus of each of the 6th roots of unity is 1) ...

Q) Find the equations of the lines parallel to x-axis between which the graph of y=     lies . A)   y=   This is of the form : | 2ab / (a^2 ...

Q) Six persons A , B , C , D , E and F  are to be seated around a circular table. Find the number of ways in which this can be done if A mus...

Q) A ball of mass 100g is projected vertically upwards from the ground with a velocity of 49 m/s. At the same time, another identical ball i...

Q) Find the remainder when 7^23 is divided by 25 . A) 7^23 = (7^22) . 7 = [ (7^2)^11] . 7 = [(50 - 1)^11] .7 Now In expansion of (50 - 1)^...

Q) Find the sum of the sequence .7 + .77 + .777 + ..... till n terms. A) .7 + .77 + .777 ... n terms = 7 (0.1 + 0.11 + 0.111....) = (7/9)*(...

Q) A fly wheel rotates about its central axis . Due to friction at the axis, it experiences angular retardation proportional to its angular ...

Q) The figure below shows the acceleration-time graph of a particle moving along a straight line. After what time does the particle acquire...

Q) How many 5- digit telephone numbers can be formed that have at least one of their digits repeated. A) (A phone number doesn't begin...

Q) There is a regular bus service between 2 towns A and B,with a bus leaving town A and B every T minutes. A cyclist moving with a speed of ...

Q) Prove that if abc=1 and a,b,c are positive real numbers . A) Rewrite the expression as (a-1 + 1/b + b - b) (b - 1 + 1/c + c - c)...

Q) Two skaters A and B of weights 40 kg wt and 60 kg wt, respectively, stand facing each other 5 m apart. They then pull a light rope stretc...

Q) If the momentum of a body increases by 0.01%, by how much will its Kinetic Energy increase? A) New momentum = m (1.0001 v) So new velocit...

Q) A body falling from a given height H hits an inclined plane in its path at a height h . As a result of this impact, the direction of the ...

Q) There are 3 groups.The 1st group contains 3 girls & and 1 boy. The 2nd group contains 2 girls and 2 boys and the 3rd group contains 1...

Q) In an increasing Geometric Progression, the sum of the first and the last term is 66, the product of the second and the last but one term...

Q) Find the number of positive integral solutions of x 4 _ y 4 = 3789108 . A) Let A = x^2 and B = y^2 A^2 - B^2 = 3789108 (A + B)(A - B...

Q) If 1 2 + 2 2 + 3 2 + ..... + 2003 2 = (2003)(4007)(334) and (1)(2003)+(2)(2002)+(3)(2001)+....(2003)(1) = (2003)(334)(x), then find...

Q) Find the sum of 1 + (1+2) + (1+2+4) +....... till n terms . A) 1 + (1+2) + (1+2+4) + ...... till n terms = 1 + [...

Q) A motorboat going downstream came across a bottle floating in the river at a point A . After 60 minutes of further motion, it turned bac...

Q) If satisfies the equation x 2 - 28x + 27 = 0, find the value of . Given that x lies between 0 and pi/2 . A) The given expressio...