Change of Variables

Q. f(2x+3y, 2x-7y)=20x, then f(x, y) equals________

Q. If f(x) = ax2 + bx + c and f(x) – f(x – 1) = 4x + 5, then a + b equals

Q. If $f:R\to R$ be a function defined as $f\left(x\right)=e^x+\underset{0}{\overset{1}{\int }}(x+y{e}^x)f\left(y\right)dy,$ then

1. $f\left(x\right)$ is decreasing function
2. $f\left(x\right)$ is one-one function
3. $f\left(x\right)$ is onto function
4. $f\left(x\right)=0$ has only one solution

Q. A real valued function f(x) is satisfying f(x +y)= yf(x) +xf(y) +xy 2 ∀ x, y ∈ R andf(1) 3, then f(11) is equal to

Q. If x + iy = $\sqrt{\frac{a+ib}{c+id}}$, then write the value of (x² + y²)².

Q. z_1=1+i , z_2=-2+3i and z_3= ai/3 where i^2=-1 are collinear then the value of a is

Q. Let $f$ be a non-negative continuous function defined on $R$ such that $f\left(x\right)+f(x+\frac{1}{2})=6$ and $N=\underset{0}{\overset{6}{\int }}f\left(x\right)dx.$ Then

1. number of divisors of $N$ is $6.$
2. number of divisors of $N$ is $9.$
3. sum of the digits in $N$ is $9.$
4. sum of the digits in $N$ is $6.$

Q. if f(x) = x+3 / x+2 , prove that x= (2f(x) - 3) / (1-f(x))

Q. 15. If 2f(X)+f(2/x)=3x+1, then f(2) equal to

Q. †extstyle∑_{r =0}^{n-1}(C(n, r))/(C(n, r)+C(n, r-1))

Q. Find the value of∑_{r=1}^n(4r-1)5^r/(r^2+r

Q.

If for non-zero $x$, $3f\left(x\right)+4f\left(\frac{1}{x}\right)=\frac{1}{x}-10$, then ${\int }_2^{3}f\left(x\right)dx$ is equal to

1. $\frac{4}{7}ln\frac{2}{3}$
   

2. $\frac{3}{7}ln\frac{3}{2}$
   

3. $\frac{3}{7}ln\frac{2}{3}$
   

4. None of these
   

Q. The value of the integral ${\int }_{\pi }^{0}cos(\pi -x)$ will be equal to the value of which of the following integral.

1. ${\int }_0^{\pi }cos\left(x\right)dx$
2. $-{\int }_0^{\pi }cos\left(x\right)dx$
3. ${\int }_0^{\pi }sin\left(x\right)dx$
4. $-{\int }_0^{\pi }sin\left(x\right)dx$

Q. Let $f\left(x\right)=\underset{0}{\overset{x}{\int }}g\left(t\right)dt$, where $g$ is a non-zero even function. If $f(x+5)=g\left(x\right)$, then $\underset{0}{\overset{x}{\int }}f\left(t\right)dt$ equals :

1. $\underset{x+5}{\overset{5}{\int }}g\left(t\right)dt$
2. $\underset{5}{\overset{x+5}{\int }}g\left(t\right)dt$
3. $2\underset{5}{\overset{x+5}{\int }}g\left(t\right)dt$
4. $5\underset{x+5}{\overset{5}{\int }}g\left(t\right)dt$

Q. 76. If f(x)=x/(x-1) Then what is the value of expression f(a)/f(a+1)

Q.

If x+iy=(a+ib)/(a-ib), then prove x^2 + y^2 = 1. The book explanation is not clear.

How is x-iy=(a-ib)/(a+ib) found out?

Q. If $I_n=\underset{0}{\overset{\pi }{\int }}\frac{1-\cos nx}{1-\cos x}dx,$ where $n$ is whole number, then

1. $I_n,I_{n+1},I_{n+2}$ are in $G.P.$
2. $I_n,I_{n+1},I_{n+2}$ are in $A.P.$
3. $I_n=n\pi$
4. $\underset{0}{\overset{\pi /2}{\int }}\frac{{\sin }^{2}n\theta }{{\sin }^2\theta }d\theta =\frac{n\pi }{2}$

Q. f(x)=2x+|x| Then find f(2x)+f(-x) -f (x)

Q. 61. The value of x if 0 s 12x+ 3 s 3 belongs to(3) 12, 3\rbrack

Q. The value of the integral $\int \frac{dx}{3\sin x+4\cos x}$
(where $C$ is integration constant)

Q. $\int \frac{dx}{5\sin x+2\cos x+2}$ is equal to
(where $C$ is integration constant)