Q1. If f(x) = x\int_{0}^{1}{f(x)}dx + \int_{0}^{2}{f(x)}dx-5 , and the area of the region bounded by y= f(x) and y=(1-x) between x=0 and x=1 is A, then 2A equals _______
Q2. If f(x) = \int x^{x+1}(logx+(logx)^2)dx and f(1) = -1, then the value of ((e^{-e}f(e)+1)e^{-1}) is _______
106yes sir, could you just show how you solved the integral? I solved it by using hit n trial .. (time was short in hand)
106Q3. MULTI ANSWER CORRECT
If a,b,c,d,c1,c2,c3....,cn are arbitrary constants, then the order of the differential equation whose solution is given by
y= (asin(x+b) + csin(x+d) + xtan^{-1}(\frac{c_{1}-c_{2}}{1+c_{1}c_{2}}) + 2xtan^{-1}(\frac{c_{2}-c_{3}}{1+c_{2}c_{3}}) + ... + (n-2)xtan^{-1}(\frac{c_{n-2}-c_{n-1}}{1+c_{n-1}c_{n-2}}) + c_{n}e^{x+c_{n-1}}
where n is a natural no. is
(a) undefined (c) 3
(b) 2 (d) 4
Q4. MULTI ANSWER
Let a1,a2,b1,b2,c1,c2 be selected from the set A= {1,2,3,.....,100}. If the roots of the equations a1x2+b1x+c1 =0 and a2x2+b2x+c2 =0 are x1, x2 and 2x1, 3x2 respectively, then the probability that (b12-4a1c1)(b22-4a2c2) < 0 is always less than
(a) 1/2 (b) 1/3 (c) 1/4 (d) 3/4
66Q1.
Let \int_0^1f(x)\ \mathrm dx =a and \int_0^2f(x)\ \mathrm dx =b
Then
f(x)=ax+b-5
Hence
a=\int_0^1f(x)\ \mathrm dx =\int_0^1(ax+b-5)\ \mathrm dx=\dfrac{a}{2}+b-5
which gives
2b - a = 10 --- (1)
Again
b=\int_0^2f(x)\ \mathrm dx =\int_0^2(ax+b-5)\ \mathrm dx=2a+2b-10
which gives
2a +b =10 ---- (2)
Solving for a and b gives a=2, b=6
Hence f(x)=2x+1
Hence A = 3/2 and so 2A = 3
341Let g(x) = x^x
Then g'(x) = x^x(1+\log x)
and \log g(x) = x \log x
The integrand is x^x(1+\log x) x \log x = g'(x) \log g(x)
Hence \int x^x(1+\log x) x \log x \ dx = \int g'(x) \log g(x) \dx = \int \log y \ dy
where y = \log g(x)
Hence the integral evaluates to g(x) (\log g(x) -1)+C
Then evaluate at x =e
106thanks anant sir, and bhatt sir
Q3,4 --> #4
Q5. Let \alpha ^{2010}+\beta ^{2010} can be expressed as a polynomial in \alpha +\beta and \alpha \beta. The sum of coefficients of the polynomial is ______
1http://targetiit.com/iit-jee-forum/posts/sum-of-coefficients-13190.html
