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

Download the PDF Sample Papers Free for off line practice and view the Solutions online.
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 Multiple Choice QuestionsShort Answer Type

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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.


Given  four points A(4,5,1), B(0,-1,-1), C(3,9,4) and D(-4,4,4).
therefore,

AB with rightwards arrow on top space equals space left parenthesis negative 4 i with hat on top minus 6 j with hat on top minus 2 k with hat on top right parenthesis

AC with rightwards arrow on top space equals space left parenthesis negative i with hat on top plus 4 j with hat on top plus 3 k with hat on top right parenthesis

AD with rightwards arrow on top space equals space left parenthesis negative 8 i with hat on top plus j with hat on top plus 3 k with hat on top right parenthesis

therefore space space vertical line AB with rightwards arrow on top space space AC with rightwards arrow on top space space AD with rightwards arrow on top vertical line space equals space open vertical bar table row cell negative 4 end cell cell negative 6 end cell cell negative 2 end cell row cell negative 1 end cell 4 3 row cell negative 8 end cell 1 3 end table close vertical bar

equals space minus 4 left parenthesis 12 space plus space 3 right parenthesis space plus space 6 left parenthesis negative 3 space plus space 24 right parenthesis space minus space 2 left parenthesis 1 space plus space 32 right parenthesis

equals space minus 60 space plus space 126 space minus space 66

equals 0
Hence comma space Four space points space straight A comma space straight B comma space straight C comma straight D space are space 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
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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.
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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