Write the sum of intercepts cut off by the plane on the three a

Subject

Mathematics

Class

CBSE Class 12

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Sample Papers

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 Multiple Choice QuestionsShort Answer Type

11.

Show that the four points A(4,5,1), B(0,-1,-1), C(3,9,4) and D(-4,4,4)
are coplanar.

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 Multiple Choice QuestionsLong Answer Type

12.

Let A= R × R and * be a binary operation on A defined by

(a, b) * (c, d) = (a+c, b+d)

Show that * is commutative and associative. Find the identity element for *

on A. Also find the inverse of every element (a, b) ∈ A.

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13. Prove space that space straight y equals fraction numerator 4 space sin space straight theta over denominator 2 space plus cos space straight theta end fraction space minus straight theta space is space an space increasing space function space of space straight theta space on space open square brackets 0 comma straight pi over 2 close square brackets
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14.

Show that semi-vertical angle of a cone of maximum volume and given slant height is cos-1fraction numerator 1 over denominator square root of 3 end fraction.

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15.

Using properties of determinants, prove that

open vertical bar table row cell left parenthesis straight x plus straight y right parenthesis squared end cell zx zy row zx cell left parenthesis straight z plus straight y right parenthesis squared end cell xy row zy xy cell left parenthesis straight z plus straight x right parenthesis squared end cell end table close vertical bar space space equals space 2 xyz left parenthesis straight x plus straight y plus straight z right parenthesis cubed

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16. If space straight A space equals space open square brackets table row 1 0 2 row 0 2 1 row 2 0 3 end table close square brackets space and space straight A cubed minus 6 straight A squared plus 7 straight A space plus kI subscript 3 space equals space 0 space find space straight k.
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 Multiple Choice QuestionsShort Answer Type

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17.

Write the sum of intercepts cut off by the planestraight r with rightwards arrow on top. open parentheses 2 straight i with hat on top plus straight j with hat on top minus straight k with hat on top close parentheses minus 5 space equals 0 on the three axes.


Given,
straight r with rightwards arrow on top. open parentheses 2 straight i with hat on top plus straight j with hat on top minus straight k with hat on top close parentheses minus 5 equals 0
in space cartesian space form
2 straight x space plus straight y minus straight z minus 5 space equals 0
2 straight x plus straight y minus straight z equals 5
fraction numerator 2 straight x over denominator 5 end fraction plus straight y over 5 minus straight z over 5 equals 1
fraction numerator straight x over denominator 5 divided by 2 end fraction plus straight y over 5 plus fraction numerator straight z over denominator negative 5 end fraction equals 1
Intercept space cutt space of space on space the space axes space open parentheses 5 over 2 comma 5 comma negative 5 close parentheses
straight x over straight a plus straight y over straight b plus straight z over straight c equals 1
straight a equals 5 over 2 space straight b equals 5 space straight c equals negative 5
straight a plus straight b plus straight c equals 5 divided by 2

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18. find space straight lambda space and space straight mu space if

open parentheses straight i with hat on top plus space 3 space straight j with hat on top space plus space 9 space straight k with hat on top close parentheses space space straight x space space left parenthesis 3 space straight i with hat on top plus straight lambda space straight j with hat on top plus space straight mu space straight k with hat on top right parenthesis space equals space 0
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19. If space straight a with rightwards arrow on top equals 4 space straight i with hat on top space minus straight j with hat on top plus space straight k with hat on top space and space straight b with rightwards arrow on top space equals space 2 space straight i with hat on top space minus 2 straight j with hat on top plus space straight k with hat on top comma then space find space straight a space vector space parallel space to space the space vector space straight a with rightwards arrow on top plus straight b with rightwards arrow on top.
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20. space F i n d space colon integral fraction numerator left parenthesis 2 x minus 5 right parenthesis e to the power of 2 x end exponent over denominator left parenthesis 2 x minus 3 right parenthesis end fraction d x
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