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Mathematics

Algebra

Easy

Question

Suppose a, b, c, d and 2ac=b+d. Consider the quadratic equations $x squared plus 2 a x plus b equals 0$ and $x squared plus 2 c x plus d equals 0$ .Then

none of these equations have real roots
both the equations have real roots
exactly one of the equations has real roots
at least one of the equations has real roots

The correct answer is: at least one of the equations has real roots

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$text If end text a subscript 1 comma a subscript 2 comma a subscript 3 comma horizontal ellipsis.. text are in A.P. such that end text a subscript 1 plus a subscript 7 plus a subscript 16 equals 40 comma text then the sum of the first end text 15 text terms of this end text A text .P. is : end text$

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$text A value of end text theta element of open parentheses 0 comma pi over 3 close parentheses comma text for which end text open vertical bar table attributes columnalign center center center columnspacing 1em rowspacing 4 pt end attributes row cell 1 plus cos squared invisible function application theta end cell cell sin squared invisible function application theta end cell cell 4 cos invisible function application 6 theta end cell row cell cos squared invisible function application theta end cell cell 1 plus sin squared invisible function application theta end cell cell 4 cos invisible function application 6 theta end cell row cell cos squared invisible function application theta end cell cell sin squared invisible function application theta end cell cell 1 plus 4 cos invisible function application 6 theta end cell end table close vertical bar equals 0 comma text is end text$

$text A value of end text theta element of open parentheses 0 comma pi over 3 close parentheses comma text for which end text open vertical bar table attributes columnalign center center center columnspacing 1em rowspacing 4 pt end attributes row cell 1 plus cos squared invisible function application theta end cell cell sin squared invisible function application theta end cell cell 4 cos invisible function application 6 theta end cell row cell cos squared invisible function application theta end cell cell 1 plus sin squared invisible function application theta end cell cell 4 cos invisible function application 6 theta end cell row cell cos squared invisible function application theta end cell cell sin squared invisible function application theta end cell cell 1 plus 4 cos invisible function application 6 theta end cell end table close vertical bar equals 0 comma text is end text$

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$text Let end text a subscript 1 comma a subscript 2 comma a subscript 3 comma horizontal ellipsis. text be an A.P. with end text a subscript 6 equals 2. text Then the common difference of this A.P., which maximises the end text text product a end text subscript 1 text a end text subscript 4 text as, is : end text$

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$text Let end text a comma b text and end text c text be in G.P. with common ratio end text r comma text where end text a not equal to 0 text and end text 0 less than r less or equal than 1 half. text If end text 3 a comma 7 b text and end text 15 c text are the first three end text t e r m s space o f space a n space A. P. comma space t h e n space t h e space 4 t h space t e r m space o f space t h i s space A. P. space i s space colon$

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$I f space capital delta subscript 1 equals open vertical bar table row straight x cell sin theta end cell cell cos theta end cell row cell negative sin theta end cell cell negative straight x end cell 1 row cell cos theta end cell 1 straight x end table close vertical bar space a n d space capital delta subscript 2 equals open vertical bar table row straight x cell sin 2 theta end cell cell cos 2 theta end cell row cell negative sin 2 theta end cell cell negative straight x end cell 1 row cell cos 2 theta end cell 1 straight x end table close vertical bar comma space straight x not equal to 0 space t h e n space f o r space a l l space theta element of open parentheses 0 comma pi over 2 close parentheses colon$

$I f space capital delta subscript 1 equals open vertical bar table row straight x cell sin theta end cell cell cos theta end cell row cell negative sin theta end cell cell negative straight x end cell 1 row cell cos theta end cell 1 straight x end table close vertical bar space a n d space capital delta subscript 2 equals open vertical bar table row straight x cell sin 2 theta end cell cell cos 2 theta end cell row cell negative sin 2 theta end cell cell negative straight x end cell 1 row cell cos 2 theta end cell 1 straight x end table close vertical bar comma space straight x not equal to 0 space t h e n space f o r space a l l space theta element of open parentheses 0 comma pi over 2 close parentheses colon$

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$text If end text a subscript 1 comma a subscript 2 comma a subscript 3 comma horizontal ellipsis horizontal ellipsis horizontal ellipsis horizontal ellipsis horizontal ellipsis a subscript n text are in A.P. and end text a subscript 1 plus a subscript 4 plus a subscript 7 plus horizontal ellipsis horizontal ellipsis horizontal ellipsis.. plus a subscript 16 equals 114 comma text then end text a subscript 1 plus a subscript 6 plus a subscript 11 plus a subscript 16 text is equal to : end text$

$text If end text a subscript 1 comma a subscript 2 comma a subscript 3 comma horizontal ellipsis horizontal ellipsis horizontal ellipsis horizontal ellipsis horizontal ellipsis a subscript n text are in A.P. and end text a subscript 1 plus a subscript 4 plus a subscript 7 plus horizontal ellipsis horizontal ellipsis horizontal ellipsis.. plus a subscript 16 equals 114 comma text then end text a subscript 1 plus a subscript 6 plus a subscript 11 plus a subscript 16 text is equal to : end text$

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$I f space t h e space s y s t e m space o f space l i n e a r space e q u a t i o n s space x space plus space y space plus space z space equals space 5 space x space plus space 2 y space plus space 2 z space equals space 6 straight x plus 3 straight y plus lambda straight z equals mu comma left parenthesis lambda comma mu element of straight R right parenthesis comma text has infinitely many solutions, then the value of end text lambda plus mu text is : end text$

$I f space t h e space s y s t e m space o f space l i n e a r space e q u a t i o n s space x space plus space y space plus space z space equals space 5 space x space plus space 2 y space plus space 2 z space equals space 6 straight x plus 3 straight y plus lambda straight z equals mu comma left parenthesis lambda comma mu element of straight R right parenthesis comma text has infinitely many solutions, then the value of end text lambda plus mu text is : end text$

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$T h e space s u m space fraction numerator 3 cross times 1 cubed over denominator 1 squared end fraction plus fraction numerator 5 cross times open parentheses 1 cubed plus 2 cubed close parentheses over denominator 1 squared plus 2 squared end fraction plus fraction numerator 7 cross times open parentheses 1 cubed plus 2 cubed plus 3 cubed close parentheses over denominator 1 squared plus 2 squared plus 3 squared end fraction plus horizontal ellipsis horizontal ellipsis horizontal ellipsis text upto end text 10 to the power of th text term is end text$

$T h e space s u m space fraction numerator 3 cross times 1 cubed over denominator 1 squared end fraction plus fraction numerator 5 cross times open parentheses 1 cubed plus 2 cubed close parentheses over denominator 1 squared plus 2 squared end fraction plus fraction numerator 7 cross times open parentheses 1 cubed plus 2 cubed plus 3 cubed close parentheses over denominator 1 squared plus 2 squared plus 3 squared end fraction plus horizontal ellipsis horizontal ellipsis horizontal ellipsis text upto end text 10 to the power of th text term is end text$

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