If a b c are in AP then prove that a(1/b+1/c) b(1/c+1/a) c(1/a+1/b) are in AP
Question : If a b c are in AP then prove that a(1/b+1/c) b(1/c+1/a) c(1/a+1/b) are in AP Proof: Given a,b,c are in AP \[2b=a+c \] to prove that \[a(1/b+1/c),…
If a b c are in AP then prove that 1/bc 1/ca 1/ab are in AP
Q: If a b c are in AP then prove that 1/bc 1/ca 1/ab are in AP Proof: a,b,c are in AP \[2b=a+c \] Dividing this by abc both sides…
If a,b,c are in Arithmetic progression then prove that b+c, c+a, a+b are in Arithmetic progression
Question : If a,b,c are in Arithmetic progression then prove that b+c, c+a, a+b are in Arithmetic progression Solution : b+c, c+a, a+b are in Arithmetic progression proof: a,b,c are…
prove that the square of any even number is even
Question : prove that the square of any even number is even Solution : Let m be any even number. We know that any even number can be written as…
Show that the square of any odd number is odd
Question : Show that the square of any odd number is odd Solution : Let m be any odd number. We know that any odd number can be written as…
how do you find real and imaginary parts of cos(z)
Question : Find real and imaginary parts of cos(z) Solution : First we shall put \[z=x+iy\] then \[cos(z)=cos(x+iy)\]now using the formula of \[cos(A+B)=cos(A)cos(B)-sin(A)sin(B)\] we get \[cos(x+iy) = cos(x)cos(iy)-sin(x)sin(iy)\] now by…
how to find real and imaginary parts of e^(x+iy)
Question : Find real and imaginary parts of e^(x+iy) Solution : We know that Real and imaginary parts of complex Number z = a+ib are a and b respectively.So we…
What is the value of Sin(75)?
Question: What is the value of Sin(75)? Solution : We shall write \[75^0=45^0+30^0 \] And apply Sin(A+B) formula \[Sin(A+B) =Sin(A)Cos(B)+Cos(A)Sin(A)\] Therefore \[Sin(75^0) = Sin(45^0+30^0) =Sin(45^0)Cos(30^0)+Cos(45^0)Sin(30^0) \] Now we shall substitute…
how to solve ode y”=3y’-10y y(0)=1 y'(0)=-1
Question : solve ode y''=3y'-10y y(0)=1 y'(0)=-1 Solution : Given homogeneous linear differential equation \[y''=3y'-10y\] \[ y(0)=1,\ y'(0)=-1\] \[y''-3y'+10y=0 \] (1) So it’s auxiliary equation is \[\lambda^2-3\lambda+10=0\] \[\lambda^2-5\lambda+2\lambda+10=0\] \[\lambda\left(\lambda-5\right)+2\left(\lambda-5\right)=0\] \[\left(\lambda+2\right)\left(\lambda-5\right)=0\]…
how to solve differential equation y”+2y’-8y=0
Question: Solve differential equation y''+2y'-8y=0 , y(0)=1, y'(1)=1 Given homogeneous linear differential equation \[y''=-2y'+8y\] \[\ y(0)=1,\ y'(0)=1 \] \[y''+2y'-8y=0 \] (1) So it’s auxiliary equation is \[ \lambda^2+2\lambda-8=0\] \[\lambda^2+4\lambda-2\lambda-8=0\] \[\lambda\left(\lambda+4\right)-2\left(\lambda+4\right)=0\]…