Using properties of determinants, prove that from Class 12 CBSE

Subject

Mathematics

Class

CBSE Class 12

Pre Boards

Practice to excel and get familiar with the paper pattern and the type of questions. Check you answers with answer keys provided.

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.

1283 Views

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
1368 Views

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


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 equals space 2 xyz left parenthesis straight x plus straight y plus straight z right parenthesis cubed
straight L. straight H. straight S comma space Multiplying space straight R subscript 1 comma space straight R subscript 2 space and space straight R subscript 3 space by space straight z comma space straight x space comma space straight y space respectively

equals space 1 over xyz space open vertical bar table row cell straight z left parenthesis straight x plus straight y right parenthesis squared end cell zx cell straight z squared straight y end cell row cell zx squared end cell cell straight x left parenthesis straight z plus straight y right parenthesis squared end cell cell straight x squared straight y end cell row cell zy squared end cell cell xy squared end cell cell straight y left parenthesis straight z plus straight x right parenthesis squared end cell end table close vertical bar
taking space common space straight z comma space straight x comma space straight y comma space from space straight C subscript 1 comma end subscript straight C subscript 2 comma end subscript and space straight C subscript 3

equals space xyz over xyz space open vertical bar table row cell left parenthesis straight x plus straight y right parenthesis squared end cell straight z cell straight z squared end cell row cell straight x squared end cell cell left parenthesis straight z plus straight y right parenthesis squared end cell cell straight x squared end cell row cell straight y squared end cell cell straight y squared end cell cell left parenthesis straight z plus straight x right parenthesis squared end cell end table close vertical bar
straight C subscript 1 rightwards arrow straight C subscript 2 minus straight C subscript 3 space and space straight C subscript 2 rightwards arrow straight C subscript 2 minus straight C subscript 3

taking space common space straight x plus straight y plus straight z space from space straight C subscript 1 space and space straight C subscript 2
left parenthesis straight x plus straight y plus straight z right parenthesis squared space open vertical bar table row cell straight x plus straight y minus straight z end cell 0 cell straight z squared end cell row 0 cell left parenthesis straight z plus straight y right parenthesis squared end cell cell straight x squared end cell row cell straight y minus straight z minus straight x end cell cell straight y minus straight z minus straight x end cell cell left parenthesis straight z plus straight x right parenthesis squared end cell end table close vertical bar
straight R subscript 3 rightwards arrow straight R subscript 3 rightwards arrow left parenthesis straight R subscript 1 plus straight R subscript 2 right parenthesis

equals space left parenthesis straight x plus straight y plus straight z right parenthesis squared space open vertical bar table row cell straight x plus straight y minus straight z end cell 0 cell straight z squared end cell row 0 cell straight z plus straight y minus straight x end cell cell straight x squared end cell row cell negative 2 straight x end cell cell negative 2 zx end cell cell 2 xz end cell end table close vertical bar
straight C subscript 1 rightwards arrow zC subscript 1 comma space straight C subscript 2 space rightwards arrow xC subscript 3

equals space fraction numerator left parenthesis straight x plus straight y plus straight z right parenthesis squared over denominator xz end fraction space open vertical bar table row cell straight z left parenthesis straight x plus straight y minus straight z right parenthesis end cell 0 cell straight z squared end cell row 0 cell straight x left parenthesis straight z plus straight y minus straight x right parenthesis end cell cell straight x squared end cell row cell negative 2 xz end cell cell negative 2 zx end cell cell 2 xz end cell end table close vertical bar
straight C subscript 1 rightwards arrow straight C subscript 1 plus straight C subscript 3 comma space space straight C subscript 2 space rightwards arrow straight C subscript 2 plus straight C subscript 3

equals space fraction numerator left parenthesis straight x plus straight y plus straight z right parenthesis squared over denominator xz end fraction space open vertical bar table row cell straight z left parenthesis straight x plus straight y right parenthesis end cell cell straight z squared end cell cell straight z squared end cell row cell straight x squared end cell cell straight x left parenthesis straight z plus straight y right parenthesis end cell cell straight x squared end cell row 0 0 cell 2 xz end cell end table close vertical bar
taking space straight z space and space straight x space common space from space straight R subscript 1 space and space straight R subscript 2

expansion space along space space straight R subscript 3
equals space left parenthesis straight x plus straight y plus straight z right parenthesis squared space straight x space 2 xz space left parenthesis left parenthesis straight x plus straight y right parenthesis left parenthesis straight z plus straight y right parenthesis minus xz right parenthesis

equals space left parenthesis straight x plus straight y plus straight z right parenthesis squared space straight x space 2 xz left parenthesis xz plus xy plus yz plus straight y squared minus xz right parenthesis

equals space left parenthesis straight x plus straight y plus straight z right parenthesis squared space straight x space 2 xz left parenthesis xy space plus yz space plus straight y squared right parenthesis

equals space 2 xyz space 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.
860 Views

 Multiple Choice QuestionsShort Answer Type

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.

1312 Views

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.
1199 Views

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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