Evaluate: from Class 12 CBSE Previous Year Board Papers | Mathe

Subject

Mathematics

Class

CBSE Class 12

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Sample Papers

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 Multiple Choice QuestionsShort Answer Type

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.

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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


integral fraction numerator straight x squared over denominator straight x to the power of 4 plus straight x squared minus 2 end fraction dx
equals integral fraction numerator straight x squared over denominator left parenthesis straight x squared minus 1 right parenthesis thin space left parenthesis straight x squared plus 2 right parenthesis end fraction dx
equals integral fraction numerator straight x squared over denominator left parenthesis straight x minus 1 right parenthesis thin space left parenthesis straight x plus 1 right parenthesis thin space left parenthesis straight x squared plus 2 right parenthesis end fraction dx
Using space partial space fraction comma
space space fraction numerator straight x squared over denominator left parenthesis straight x minus 1 right parenthesis left parenthesis straight x plus 1 right parenthesis thin space left parenthesis straight x squared plus 2 right parenthesis end fraction space equals space fraction numerator straight A over denominator left parenthesis straight x minus 1 right parenthesis end fraction plus fraction numerator straight B over denominator left parenthesis straight x plus 1 right parenthesis end fraction plus fraction numerator Cx plus straight D over denominator left parenthesis straight x squared plus 2 right parenthesis end fraction
fraction numerator straight x squared over denominator left parenthesis straight x minus 1 right parenthesis left parenthesis straight x plus 1 right parenthesis left parenthesis straight x squared plus 2 right parenthesis end fraction equals fraction numerator straight A left parenthesis straight x plus 1 right parenthesis left parenthesis straight x squared plus 2 right parenthesis plus straight B left parenthesis straight x squared plus 2 right parenthesis left parenthesis straight x minus 1 right parenthesis plus left parenthesis Cx plus straight D right parenthesis left parenthesis straight x minus 1 right parenthesis left parenthesis straight x plus 1 right parenthesis over denominator left parenthesis straight x minus 1 right parenthesis left parenthesis straight x plus 1 right parenthesis left parenthesis straight x squared plus 2 right parenthesis end fraction

Equating the coefficients from both the numerators we get,
 A+B+C = 0 ....(1)
  A-B+D = 1 ....(2)
  2A+2B-C = 0 ...(3)
  2A-2B-D = 0 .....(4)
Solving the above equations we get,
 straight A equals 1 over 6 comma space space straight B space equals space minus 1 over 6 comma space straight C equals space space 0 comma space straight D space equals space 2 over 3
Our space Intergral space becomes comma
integral fraction numerator straight x squared over denominator left parenthesis straight x minus 1 right parenthesis thin space left parenthesis straight x plus 1 right parenthesis space left parenthesis straight x squared plus 2 right parenthesis end fraction dx space equals space integral fraction numerator 1 over denominator 6 left parenthesis straight x minus 1 right parenthesis end fraction minus fraction numerator 1 over denominator 6 left parenthesis straight x plus 1 right parenthesis end fraction plus fraction numerator 2 over denominator 3 left parenthesis straight x squared plus 2 right parenthesis end fraction dx
equals space 1 over 6 log left parenthesis straight x minus 1 right parenthesis minus 1 over 6 log left parenthesis straight x plus 1 right parenthesis plus 2 over 3 cross times fraction numerator 1 over denominator square root of 2 end fraction tan to the power of negative 1 end exponent open parentheses fraction numerator straight x over denominator square root of 2 end fraction close parentheses plus straight C
equals 1 over 6 open square brackets log left parenthesis straight x minus 1 right parenthesis minus log left parenthesis straight x plus 1 right parenthesis plus 2 square root of 2 tan to the power of negative 1 end exponent open parentheses fraction numerator straight x over denominator square root of 2 end fraction close parentheses close square brackets plus straight C
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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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