If n is a positive integer, then `(sqrt(3)+1)^(2n)-(sqrt(3)-1)^(2n)` is (1) an irrational number (2) an odd positive integer (3) an even positive inte

Correct Answer - B `sqrt(log_(2)x)-0.5 = log_(2) sqrtx` ` or sqrt(log_(2)x)-0.5 = 0.5 log_(2)x` ` rArr y - 0.5 = 0.5y^(2)` `rArr y^(2) - 2y + 1 = 0` ` rArr...

Correct Answer - C `int_(-1)^(1//2)(e^(x)(2-x^(2))dx)/((1-x)sqrt(1-x^(2)))=int_(-1)^(1//2)(e^(x)(1-x^(2)+1))/((1-x)sqrt(1-x^(2)))` `=int_(-1)^(1//2) e^(x)[(sqrt(1+x))/(1-x)+1/((1-x)sqrt(1-x^(2)))]dx` `=e^(x)sqrt(1+x)/(1-x)|_(-1)^(1//2)=sqrt(3e)`

Correct Answer - D `(a)/(sqrt(bc))-2 = sqrt((b)/(c)) + sqrt((c)/(b))` `"or " a=b+c+2sqrt(bc)` `"or " a=(sqrt(b)+sqrt(c))^(2)` `"or " (sqrt(a)-sqrt(b)-sqrt(c))(sqrt(a)+sqrt(b)+sqrt(c))=0` `"or " sqrt(a)-sqrt(b)-sqrt(c)=0` `"since " sqrt(a)+sqrt(b)+sqrt(c)ne0 " " ("as "a,b,c gt 0).` `"Comparing...

Area of qudrilateral AbCD is maximum when area of ACD is maximum. Area of triangle ACD. `Delta_1=(1)/(2)||{:(3/sqrt2,,sqrt2,,),(3/sqrt2,,-sqrt2,,),(3sintheta,,2costheta,,),(3/sqrt2,,sqrt2,,):}||` `=|-3sqrt(2)costheta+3sqrt(2)sintheta|` `therefore Delta_1("max")=6`, when`theta=(7pi)/(4)` (as `theta epsilon(3pi//2,2pi`)) Maximum area is 12 sq.units (as...

Correct Answer - D According to the givn condition, `""^(n)C_(3)((1)/(2))^(n)=""^(n)C_(4)((1)/(2))^(n),` where n is the number of times die is thrown. `therefore""^(n)C_(3)=""^(n)C_(4)impliesn=7` Thus, the required probability is `=""^(7)C_(1)((1)/(2))^(7)=(7)/(2^(7))=(7)/(128)`

`(dy)/(dx)=(e^(sqrt(x)))/(2sqrt(x))-(e^(-sqrt(x)))/(2sqrt(x))=(e^(sqrt(x)-)e^(-sqrt(x)))/(2sqrt(x))` `=(sqrt((e^(sqrt(x))+e^(-sqrt(x)))^(2))-4)/(2sqrt(x))` `=(sqrt(y^(2)-4))/(2sqrt(x))`

Correct Answer - (i) 9.898 (ii) 17.32 (iii) 18.24 (iv) 6.928

Correct Answer - (i) 1 (ii) 1

Correct Answer - `-0.891`

Correct Answer - (i)`sqrt(2)` (ii)`(1)/(sqrt(3))` (iii)`(sqrt(3))/(2)` (iv)`3sqrt(2)` (v)`(1)/(2sqrt(5))` (vi)`2sqrt(2)`