Match the following :
`{:("Column I" , " Column II"),(" (A) The total number or real solutions of the equation" (x^(2)-7x+1)^((x^(2)-11x+30)) =1 "is", "(p)" 2), ("(B) Total number of values of a so that " x^(2) -x-a=0 " has distinct integral roots, where a" in and 6 ge a ge 100 " is equal to " , "(q)" 4),("(C) The least value of n such that " (n-2)x^(2) + 8x + n +4 gt 0 AA x in R " where " n in N "is" , "(r)" 5 ),("(D) Total number of integral values of a such that " x^(2) +ax +a +1=0 " has integral roots is equal to " , "(t)" 8),(,"(t)"9):}`

Answer

A(q),B(s) , C (r ),D(p)

Explanation

`(A) (x^(2)-7x +1)^((x^(2)-11x+30))=1`
` Rightarrow x^(2) -11x + 30 = 0 or x^(2) -7x + 1 =1`
` Rightarrow (x -5) (x-6) = 0 or x (x-7) =0`
x = 5 or 6 OR x = 0 or 7
So there are four such possible values of x.
(B) Let `alpha and beta` the distinct integral roots
` Rightarrow alpha +beta =1 and alpha beta = -a`
So , ` alpha and beta` are integers such that their sum is unity and their product a negative integer less than or equal to 6 and greater than or equal to 100 there will be such 8 pairs.
`3xx (-2), 4 xx (-3), 5 xx (-4), 6 xx (-5) , 7 xx (-6), 8 xx (-7), 9xx (-8), 10 xx (-9)`
(c) The given quadratic equation
`(n-2)x^(2) + 8x + (n+4) gt 0 AAx in R`
` Rightarrow n -2 gt 0 Rightarrow n gt2`
` D lt 0`
` Rightarrow 64 - 4( n-2) (n+ 4) lt 0`
` Rightarrow 16 -n^(2)-2n + 8 lt 0`
` Rightarrow n^(2) + 2n - 24 lt 0`
` Rightarrow (n+6) (n-4) lt 0`
` n gt -6 , n lt 4, ` but n is a natural number ,so the value of n is 5.
Let` alpha and beta ` be the two integral roots of the equation ` x^(2) + ax +a+1=0` ,then
` alpha + beta = -a, beta xx alpha = a+1`
` Rightarrow -betan (alpha +beta) = a+1`
` Rightarrow beta^(2) + abeta = - (a+1)`
` Rightarrow (beta^(2) +1) = - a(1 + beta) `
` Rightarrow a = - (beta^(2) +1)/(beta+1) `
` - beta + 1 - 2/(beta +1)`
` = ( 1-beta)- 2/(beta +1)`
Since `beta` is an integer a is an integer, so ` beta ` must be either 0 or 1 or -3. The corresponding value of a is -1 , -1 and 5. So, there are only 2 distinct value of ` beta`.