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Mathematics

Algebra

Easy

Question

If the sum and product of the first three terms in AP are 33 and 1155, respectively, then a value of its 11^th term is

25
-36
-25
-35

The correct answer is: -25

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Let the sum of the first n terms of a non- constant AP $a subscript 1 comma a subscript 2 comma a subscript 3 horizontal ellipsis horizontal ellipsis$ be $50 n plus fraction numerator n left parenthesis n minus 7 right parenthesis over denominator 2 end fraction A comma$ where A is a constant, If d is the common difference of this AP, then the ordered pair $open parentheses d comma a subscript 50 close parentheses$ is equal to

Let the sum of the first n terms of a non- constant AP $a subscript 1 comma a subscript 2 comma a subscript 3 horizontal ellipsis horizontal ellipsis$ be $50 n plus fraction numerator n left parenthesis n minus 7 right parenthesis over denominator 2 end fraction A comma$ where A is a constant, If d is the common difference of this AP, then the ordered pair $open parentheses d comma a subscript 50 close parentheses$ is equal to

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If the length of the sides of a triangle are in AP and the greatest angle is double the smallest, then a ratio of lengths of the sides of this triangle is

If the length of the sides of a triangle are in AP and the greatest angle is double the smallest, then a ratio of lengths of the sides of this triangle is

MathematicsAlgebra

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

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

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

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

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

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

MathematicsAlgebra

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

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

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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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parallel

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

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

MathematicsAlgebra

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

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

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

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

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

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

MathematicsAlgebra

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

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

MathematicsAlgebra

parallel

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