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if u talking bout the values of sin 0, 15, or any 'x' degrees, the easiest way would be 'use a scientific calculator'  ....
jokes apart... sin 0 = 0
for calculating sin 15 ,
we knw value sin 30 = 1/2
=> sin 2(15) = 0.5
=> 2 (sin 15 ) (sqrt (1-sin215 ) ) = 0.5 Taking sin 15 = t
=> 2t sqrt (1-t2 ) = 0.5
solve it for 't' and u'll get the value
for sin 18, use sin 90 = 1
=>sin 5(18)= 1
use the formula for sin 5x , in terms of sin x, and find the value
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[y]=3[x]/2 for [y] is an integer=> [x] is even. For 0<=x<1 [x]=0 => [y]=0, 0<=y<1 For 2<=x<3 [x]=2
=> [y]=3, 3<=y<4 For 4<=x<5 [x]=4
=> [y]=6, 6<=y<7 For 6<=x<7 => [x]=6
=> [y] = 9 => 9<=y<10
Similarly for negative values . Thus the function is valid for all x such that 2n<=x<2n+1 , whr n is any integer
When the graph is plotted with these values, there will be infinite number of bounded regions. Hence the area bounded = infinity
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sumiran has solved the question perfectly :)
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are you sure, the question is correct ??
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preparation for any exam requires, clear concepts of the subject's topic + practicing the question.
for aieee, u shud be well versed, the CBSE's syllabus for 11n 12th, study the chapters thouroughly from any standard book, and then try to solve as much questions, as possible...check for your accuracy as well as your speed!!
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Elecric Field inside a conductor is always zero
charge always preside on the surface of the body...so when the electric field is calculated, according to guassian law , since there is no charge inside the conductore, no electric field is present...
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h = ut + 1/2 gt2
here u = 0 (freely dropped)
if h is the total height then,
h = 1/2 gt2 ....................(A)
and,
16h/25= 1/2 g(t-1)2 ....................(B)
Solving (A) and (B) simultaneously, we can find the value of both 'h' and 't'
t = 5 sec
h = 122.5 m
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equation of ellipse : x2 + 4y2= 4
=> dy/dx = -x/4y
assume, that the normal is at point (h,k)
slope of normal at (h,k) = -4k/h = -8/3
=> k/h = 2/3
=> k = 2h/3
h2 + 4k2= 4 => h2 + 4(2h/3)2= 4
=> 9h2 + 16h2= 36
=> 25h2= 36
=> h = 6/5 = 1.2
k= 2h/3 = 0.8
now since u know the point (1.2, 0.8) and the slope = -8/3
=> (y - 0.8)/(x-1.2) = -8/3
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Let after time 't' hrs, the two were at the shortest distance.
Then, distance travelled by car A = 30 t
distance left, for reaching the junction = 15-30t
similarly,
distance travelled by car B = 10 t
distance left, for reaching the junction = 25-10t
therefore the shortest distance , s2= (15-30t)2+ (25-10t)2 differentiating 's' wrt 't' and putting ds/dt = 0
2(15-30t) (-30) + 2 (25-10t) (-10) = 0
=> 45 (2t-1) - 5 (5-2t) = 0
=> t = 0.7 hrs
thus, s2= (15-30X0.7)2+ (25-10X0.7)2
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v2= u2+2as
=> u2/4 = u2+ 2a X 2.5 (velocity in cm/sec )
=> 5a = -3u2/4 => a = -3u2/20 cm/sec2
Now,
v2= u2+2as
=> 0 = u2+ 2a X s (s = total distance penetrated )
=> u2 -3u2/10 X s = 0
=> s = 10/3 = 3.33 cm
distance further penetrated = 3.33-2.5 = 0. 9 (approx)
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whats the question dear :?
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a1+a5+a10+a15+a20+a24=225 => a + (a+4d) + (a+9d) + (a+10d) + (a+19d) + (a+23d) = 225
=> 6a + 69 d = 225=> 2a + 23 d = 75 sum of 24 terms = 2a+ 23d = 75
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tan(pie/4+1/2cos inverse a/b)+tan(pie/4-1/2cosinverse a/b) Let 1/2cos inverse a/b = t tan(pie/4+1/2cos inverse a/b)+tan(pie/4-1/2cosinverse a/b) = tan (pie/4 + t) + tan (pie/4-t) = tan pie/4 (1+ tan^2 t)/ (1- tan^2 t) = (1+ tan^2 t)/ (1- tan^2 t) = sec 2t = sec 2 (1/2cos inverse a/b ) = 2b/a
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practice from any authentic book, with less no. of errors. U can have either of arihant, TMH or any gud ones available in the market!
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beautifully solved pritam!
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a1/3 + b1/3 - 1 = 49 + 20 (6)1/3
=> a1/3 + b1/3 = 50 + 20 (6)1/3 comparing, a = 503
b = 6*203
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f(x)=ax2+bxt+c
f(5)+3f(2)=0
=>[25a+ 5bt +c] + 3[4a+2bt+c] = 0
=>37a + 11bt + 4c= 0 ....................(a)
f(3)=0
=>9a+3bt +c=0 Multilpying both sides by '4'
=>36a + 12bt + 4c = 0 .....................(b)
(b)-(a)
a - bt =0
=>bt =a
=>c=-12a
thrfore, f(x)= ax2+ax-12a thus f(x) = 0
=> a (x2+x-12) = 0
=> x = 3, -4
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the sum of all the pages, including the page torn S = n(n+1)/2
thus, n (n+1)/2 >= 15000
the minimum integral value of 'n', for which the above equation is true is n =173, for which S= 15051
since, the pages torn, wil have consecutive numbering,
p+(p+1)= 15051-15000
=> 2p+1= 51 => p = 25
pages torn = 25,26
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2sinx + 2sin3x=siny............r u sure , its 2 sin x and not 3 sinx???
if its 2 sin x, the question is getting un-necessarily lengthy, n then also, the answer is not guranteed.
for 3 sin x, the solution given by jagdish, is perfect :)
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a rectangular hyperbola is given by xy=c2
differentiating the equation, we have the differential equation: y + y'x =0
degree=1, order =1
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