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    Mathematics
    Algebra
    Easy

    Question

    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

    1. 120
    2. 150
    3. 280
    4. 200

    The correct answer is: 200

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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

    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

    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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    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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    T h e space s u m stack space space sum with k equals 1 below and 20 on top k 1 over 2 to the power of k space i s space e q u a l space t o

    T h e space s u m stack space space sum with k equals 1 below and 20 on top k 1 over 2 to the power of k space i s space e q u a l space t o

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    I f space t h e space s u m space a n d space p r o d u c t space o f space t h e space f i r s t space t h r e e space t e r m s space i n space a n space A. P. space a r e space 33 space a n d space 1155 comma space r e s p e c t i v e l y comma space t h e n space a space v a l u e space o f space i t s space 11 t h space t e r m space i s space colon

    I f space t h e space s u m space a n d space p r o d u c t space o f space t h e space f i r s t space t h r e e space t e r m s space i n space a n space A. P. space a r e space 33 space a n d space 1155 comma space r e s p e c t i v e l y comma space t h e n space a space v a l u e space o f space i t s space 11 t h space t e r m space i s space colon

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    text  Some identical balls are arranged in rows to form an equilateral triangle. The first row consists of one ball,  end text text  the second row consists of two balls and so on. If 99  more identical balls are added to the total number of  end text text  balls used in forming the equilateral triangle, then all these balls can be arranged in a square whose each side  end text text  contains exactly  end text 2 text  balls less than the number of balls each side of the triangle contains. Then the number of  end text text  balls used to form the equilateral triangle is:  end text

    text  Some identical balls are arranged in rows to form an equilateral triangle. The first row consists of one ball,  end text text  the second row consists of two balls and so on. If 99  more identical balls are added to the total number of  end text text  balls used in forming the equilateral triangle, then all these balls can be arranged in a square whose each side  end text text  contains exactly  end text 2 text  balls less than the number of balls each side of the triangle contains. Then the number of  end text text  balls used to form the equilateral triangle 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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    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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    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

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    L e t space t h e space n u m b e r space 2 comma space b comma c space b e space i n space a n space A. P. space a n d space A equals open square brackets table attributes columnalign left left left columnspacing 1em rowspacing 4 pt end attributes row cell 1      1      1 end cell row cell 2      b      c end cell row cell 4      b squared      c squared end cell end table close square brackets. space I f space det invisible function application left parenthesis straight A right parenthesis element of left square bracket 2 comma 16 right square bracket space t h e n space c space l i e s space i n space t h e space i n t e r v a l space minus space

    L e t space t h e space n u m b e r space 2 comma space b comma c space b e space i n space a n space A. P. space a n d space A equals open square brackets table attributes columnalign left left left columnspacing 1em rowspacing 4 pt end attributes row cell 1      1      1 end cell row cell 2      b      c end cell row cell 4      b squared      c squared end cell end table close square brackets. space I f space det invisible function application left parenthesis straight A right parenthesis element of left square bracket 2 comma 16 right square bracket space t h e n space c space l i e s space i n space t h e space i n t e r v a l space minus space

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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 – space 2 y space plus space k z space equals space 1 space 2 x space plus space y space plus space z space equals space 2 space 3 x space – space y space – space k z space equals space 3 h a s space a space s o l u t i o n space left parenthesis x comma space y comma space z right parenthesis comma space space z not equal to 0 comma space t h e n space left parenthesis x comma space y right parenthesis space l i e s space o n space t h e space s t r a i g h t space l i n e space w h o s e space e q u a t i o n space i s space minus

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    text  The sum of all natural numbers 'n' such that  end text 100 less than n less than 200 space text  and H.C.F.  end text left parenthesis 91 comma n right parenthesis greater than 1 text  is  end text minus

    text  The sum of all natural numbers 'n' such that  end text 100 less than n less than 200 space text  and H.C.F.  end text left parenthesis 91 comma n right parenthesis greater than 1 text  is  end text minus

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    text  Let  end text A equals open parentheses table attributes columnalign center center columnspacing 1em rowspacing 4 pt end attributes row cell cos invisible function application alpha minus sin invisible function application alpha end cell blank row cell sin invisible function application alpha end cell cell cos invisible function application alpha end cell end table close parentheses comma left parenthesis a element of R right parenthesis text  such that  end text A to the power of 32 equals open parentheses table attributes columnalign center center columnspacing 1em rowspacing 4 pt end attributes row 0 cell negative 1 end cell row 1 0 end table close parentheses. text  Then a value of  end text alpha text  is -  end text

    text  Let  end text A equals open parentheses table attributes columnalign center center columnspacing 1em rowspacing 4 pt end attributes row cell cos invisible function application alpha minus sin invisible function application alpha end cell blank row cell sin invisible function application alpha end cell cell cos invisible function application alpha end cell end table close parentheses comma left parenthesis a element of R right parenthesis text  such that  end text A to the power of 32 equals open parentheses table attributes columnalign center center columnspacing 1em rowspacing 4 pt end attributes row 0 cell negative 1 end cell row 1 0 end table close parentheses. text  Then a value of  end text alpha text  is -  end text

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    T h e space g r e a t e s t space v a l u e space o f space c space element of space R space f o r space w h i c h 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 x space – space c y space – space c z space equals space 0 space c x space – space y space plus space c z space equals space 0 space c x space plus space c y space – space z space equals space 0 space h a s space a space n o n minus t r i v i a l space s o l u t i o n comma space i s space minus

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