If `(a_1)/(a_2) = (b_1)/(b_2) ne (c_1)/(c_2)`, then the equations `a_1x + b_1y + c_1 =0 and a_2x + b_2y + c_2=0` are _________. (consistent/inconsiste

Correct Answer - A We have `(1)/(2)|{:(x_1,,y_1,,1),(x_2,,y_2,,1),(x_3,,y_3,,1):}|=|{:(a_1,,b_1,,1),(a_2,,b_2,,1),(a_3,,b_3,,1):}|or (1)/(2)|{:(x_1,,y_1,,1),(x_2,,y_2,,1),(x_3,,y_3,,1):}|=(1)/(2)|{:(a_1,,b_1,,1),(a_2,,b_2,,1),(a_3,,b_3,,1):}|` Hence, the area of triangle with vertices `(x_1,y_1),(x_2,y_2),(x_3,x_3)` is the same as the area of triangle with vertices `(a_1,b_1),(a_2,b_2),(a_3,b_3)`. Hence, the two triangles...

Correct Answer - D Let (h,k) be the point on the locus. Then by the given conditions, `(h-a_1)^1+(k-b_1)^2=(h-a_2)^2+(k-b_2)^2` or `2h(a_1-a_2)+2k(b_1-b_2)+a_2^2+b_2^2-a_1^2=0` `h(a_1-a_2 )+k(b_1-b_2)+(1)/(2)(a_2^2+b_2^2-a_1^2-b_1^2)=0` .....(1) Also, since (h,k) lies on the given locus,...

`(1+x+x^(2)+"….."+x^(p))^(n)=a_(0) + a_(1)x+"…."+a_(np)x^(np)` Differentiating both side w.r.t. , we get `n(1+x+x^(2)+"……"+x^(p))^(n-1)(1+2x+"….."+px^(p-1))` `= a_(1)+2a_(2)x+"….."+npa_(np)x^(np-1)` Now put `x = 1` `:. a_(1) + 2a_(2) + "......." np a_(np) = n(p+1)^(n-1)(1+2+"...."+p)` `= (n(p+1)^(n).p)/(2)`

`.^(m)C_(4)-.^(m+n)C_(1).^(m)C_(3)+.^(m+n)C_(2).^(m)C_(2)-.^(m+n)C_(3).^(m)C_(1)+.^(m+n)C_(4)` `= .^(m+n)C_(0).^(m)C_(4)-.^(m+n)C_(1).^(m)C_(3)+.^(m+n)C_(2).^(m)C_(2)-.^(m+n)C_(3).^(m)C_(1)+.^(m+n)C_(4).^(m)C_(0)` = Coefficient of `x^(4)` in `(1+x)^(m+n)(1-x)^(m)` = Coefficient of `x^(4)` in `(1-x)^(m)(1+x)^(n)` = Coefficient of `x^(4)` in `[1-.^(m)C_(1)x^(2)+.^(m)C_(2)x^(4)-"....."][1+.^(n)C_(1)x+.^(n)C_(2)x^(2)+"....."+.^(n)C_(n)x^(n)]` `= .^(n)C_(4)-.^(m)C_(1) xx .^(n)C_(2)+.^(m)C_(2)`

Correct Answer - 31 `(1+x-2x^(2))^(6) = 1+a_(1)x+a_(2)x^(2)+"…."+a_(12)x^(12)` Putting `x = 1` and `x = - 1` and adding the results, we get `64 = 2(1+a_(2)+a_(4)+"….")` `:. a_(2)+a_(4)+a_(6)+"…."a_(12) = 31`

Correct Answer - B Put `x = i` `(1+i)^(5) = (a_(0) - a_(2) + a_(4)) + i(a_(1)-a_(3)+a_(5))` `rArr |1+i|^(5) = |(a_(0) - a_(2) + a_(4)) + i(a_(1) - a_(3) + a_(5))|`...

Correct Answer - B `(f(x))/(1-x) = b_(0) + b_(1)x+b_(2)x^(2)+"…."+b_(n)x^(n)+"….."` `rArr a_(0) + a_(1)x +a_(2)x^(2) + "….." + a_(n)x^(n) + "…."` `= (1-x)(b_(0) + b_(1)x+b_(2)x^(2) + "….." + b_(n)x^(n)+ "…..")` Comparing the...

Correct Answer - C Let the probability that a man aged x dies in a year p. Thus the probability that a man aged x does not die in a year...

Correct Answer - A `(a)` First arrange `12` persons `A_(4),A_(5),….A_(15)` in `"^(15)P_(12)` ways There remains `3` places. Keep `A_(1)` in the first place and arrange `A_(2)`, `A_(3)` in the remaining two...

Correct Answer - A `(a)` Putting `x=1` and `-1` and adding `a_(0)+a_(2)+…+a_(50)=(3^(25)+1)/(2)` `=((1+2)^(25)+1)/(2)` `=("^(25)C_(0)+^(25)C_(1)*2+^(25)C_(2)*2^(2)+^(25)C_(25)*2^(25)+1)/(2)` `=(2[1+^(25)C_(1)+^(25)C_(2)*2+...+^(25)C_(25)*2^(24)])/(2)` `=2[13+^(25)C_(2)+...+^(25)C_(25)*2^(23)]`