Subject

Mathematics

Class

CBSE Class 12

Pre Boards

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Sample Papers

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 Multiple Choice QuestionsShort Answer Type

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11.

Find the angle between the vector straight i with hat on top minus straight j with hat on top space and space straight j with hat on top space minus space straight k with hat on top.


 Let space straight a with rightwards arrow on top space equals space straight i with hat on top minus stack straight j semicolon with hat on top space space straight b with rightwards arrow on top space equals space straight j with hat on top space minus space straight k with hat on top
straight a with rightwards arrow on top. space straight b with rightwards arrow on top space equals space left parenthesis straight i with hat on top minus straight j with hat on top right parenthesis space left parenthesis straight j with hat on top space minus space straight k with hat on top right parenthesis space equals space 1 cross times 0 space plus left parenthesis negative 1 right parenthesis cross times 1 space plus 0 space cross times left parenthesis negative 1 right parenthesis space equals space minus 1
open vertical bar straight a with rightwards arrow on top close vertical bar space equals space square root of 1 squared plus left parenthesis negative 1 right parenthesis squared plus 0 squared end root space equals square root of 2
open vertical bar straight b with rightwards arrow on top close vertical bar space equals square root of 0 squared plus 1 squared plus left parenthesis negative 1 right parenthesis squared end root space equals square root of 2
We space know space that space straight a with rightwards arrow on top. straight b with rightwards arrow on top space equals open vertical bar straight a with rightwards arrow on top close vertical bar open vertical bar straight b with rightwards arrow on top close vertical bar cos space straight theta
Thus comma space cosθ space equals space fraction numerator straight a with rightwards arrow on top. straight b with rightwards arrow on top over denominator open vertical bar straight a with rightwards arrow on top close vertical bar. open vertical bar straight b with rightwards arrow on top close vertical bar end fraction space equals space fraction numerator negative 1 over denominator square root of 2 cross times square root of 2 end fraction equals fraction numerator negative 1 over denominator 2 end fraction
rightwards double arrow space space space cosθ space equals space cos 120 degree
rightwards double arrow space space space space straight theta space equals space 120 degree

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12.

Find the distance of a point (2, 5, −3) from the plane straight r with rightwards arrow on top. open parentheses 6 straight i with hat on top minus 3 straight j with hat on top plus 2 straight k with hat on top close parentheses space equals 4.

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13.

Find the differential equation of the family of lines passing through the origin. 

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14.

Find the integrating factor for the following differential equation:straight x space log space straight x dy over dx space plus space straight y space equals space 2 logx space

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15.

Evaluate: integral subscript 0 superscript straight pi divided by 2 end superscript fraction numerator sin squared straight x over denominator sinx plus cosx end fraction dx.

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16.

Evaluate integral subscript negative 1 end subscript superscript 2 left parenthesis straight e to the power of 3 straight x end exponent plus 7 straight x minus 5 right parenthesis dx

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17.

Evaluate:
integral fraction numerator straight x squared over denominator straight x to the power of 4 plus straight x squared minus 2 end fraction dx

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18.

In a set of 10 coins, 2 coins are with heads on both the sides. A coin is selected at random from this set and tossed five times. If all the five times, the result was heads, find the probability that the selected coin had heads on both the sides

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19.

How many times must a fair coin be tossed so that the probability of getting at least one head is more than 80%?

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20.

Find x such that the four points A(4, 1, 2), B(5, x, 6) , C(5, 1, -1) and D(7, 4, 0) are coplanar.

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