" 30."(m)/(n)x^(2)+(n)/(m)=1-2x

Solve by factorization: (m)/(n)x^(2)+(n)/(m)=1-2x

The area of the parallelogram formed by the lines y=m x ,y=x m+1,y=n x ,a n dy=n x+1 equals. (|m+n|)/((m-n)^2) (b) 2/(|m+n|) 1/((|m+n|)) (d) 1/((|m-n|))

The area of the parallelogram formed by the lines y=m x ,y=x m+1,y=n x ,a n dy=n x+1 equals. (a) (|m+n|)/((m-n)^2) (b) 2/(|m+n|) 1/((|m+n|)) (d) 1/((|m-n|))

If 2 cos alpha = x + (1)/(x) and 2 cos beta = y + (1)/(y) , show that (x^(m))/(y^(n))- (y^(n))/(x^(m))= 2i sin (m alpha - n beta)

lim_(x rarr0)((2^(m)+x)^((1)/(m))-(2^(n)+x)^((1)/(n)))/(x) is equal to (1)/(m2^(m))-(1)/(n2^(n)) (b) (1)/(m2^(m))+(1)/(n2^(n))(1)/(m2^(-m))-(1)/(n2^(-n))( d) (1)/(m2^(-m))+(1)/(n2^(-n))

("lim")_(xto0)((2^m+x)^(1/m)-(2^n+x)^(1/n))/x is equal t o (a) 2 (1/(m2^m)-1/(n2^n))' (b) (1/(m2^m)+1/(n2^n)) (c) 1/(m2^(-m))-1/(n2^(-n)) (d) 1/(m2^(-m))+1/(n2^(-n))

("lim")_(xvec 0)((2^m+x)^(1/m)-(2^n+x)^(1/n))/xi se q u a lto 1/(m2^m)-1/(n2^n) (b) 1/(m2^m)+1/(n2^n) 1/(m2^(-m))-1/(n2^(-n)) (d) 1/(m2^(-m))+1/(n2^(-n))

The mth term of an arithmetic progression is x and nth term is y.Then the sum of the first (m+n) terms is: a.(m+n)/(2)[x-y+(x+y)/(m+n)] b.(1)/(2)[(x+y)/(m+n)+(x-y)/(m-n)]c(1)/(2)[(x+y)/(m+n)-(x-y)/(m-n)]d(m+n)/(2)[x+y+(x-y)/(m-n)]

Let I_(m","n)= int sin^(n)x cos^(m)x dx . Then , we can relate I_(n ","m) with each of the following : (i) I_(n-2","m) " " (ii) I_(n+2","m) (iii) I_(n","m-2) " " (iv) I_(n","m+2) (v) I_(n-2","m+2)" " I_(n+2","m-2) Suppose we want to establish a relation between I_(n","m) and I_(n","m-2) , then we get P(x)=sin^(n+1)x cos^(m-1)x ...(i) In I_(n","m) and I_(n","m-2) the exponent of cos x in m and m-2 respectively, the minimum of the two is m - 2, adding 1 to the minimum we get m-2+1=m-1 . Now, choose the exponent of sin x for m - 1 of cos x in P(x). Similarly, choose the exponent of sin x for P(x)=(nH)sin^(n)x cos^(m)x-(m-1)sin^(n+2) x cos^(m-2)x . Now, differentiating both the sides of Eq. (i), we get =(n+1)sin^(n)x cos^(m)x-(m-1)sin^(n)x(1-cos^(2)x)cos^(m-2)x =(n+1)sin^(n)x cos^(m)x-(m-1)sin^(n)x cos^(m-2)x+(m-1)sin^(n)x cos^(n)x =(n+m)sin^(n)x cos^(m)x-(m-1)sin^(n)x cos^(m-2)x Now, integrating both the sides, we get sin^(n+1)x cos^(m-1)x=(n+m)I_(n","m)-(m-1)I_(n","m-2) Similarly, we can establish the other relations. The relation between I_(4","2) and I_(2","2) is