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CBSE Class 12 Mathematics Solved Question Paper 2014

Short Answer Type

If A is a square matrix such that $straight A squared equals straight A comma$ then write the value of $7 straight A minus left parenthesis straight I plus straight A right parenthesis cubed comma$ where I is an identity matrix.

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If space open square brackets table row cell straight x minus straight y end cell cell space straight z end cell row cell 2 straight x minus straight y end cell cell space straight w end cell end table close square brackets space equals space open square brackets table row cell negative 1 end cell cell space 4 end cell row 0 cell space 5 end cell end table close square brackets comma space find space the space value space of space straight x plus straight y.

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If space tan to the power of negative 1 end exponent straight x plus tan to the power of negative 1 end exponent straight y space equals space straight pi over 4 comma space xy less than 1 comma space then space write space the space value space of space straight x plus straight y plus xy.

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If $open vertical bar table row cell 3 straight x end cell cell space 7 end cell row cell negative 2 end cell cell space 4 end cell end table close vertical bar space equals open vertical bar table row 8 cell space 7 end cell row 6 cell space 4 end cell end table close vertical bar comma$ find the value of x.

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If $straight R equals open curly brackets open parentheses straight x comma space straight y close parentheses colon straight x plus 2 straight y space equals space 8 close curly brackets$ is a relation on N, write the range of R.

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Find the value(s) of x for which y = $open square brackets straight x left parenthesis straight x minus 2 right parenthesis close square brackets squared$ is an increasing function.

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7.
Find the equations of the tangent and normal to the curve $straight x squared over straight a squared minus straight y squared over straight b squared equals 1 space at space the space point space open parentheses square root of 2 straight a comma space straight b close parentheses.$

Let $straight x squared over straight a squared minus straight y squared over straight b squared equals 1$ be the equation of the curve.
Rewriting the above equation as,

$straight y squared over straight b squared equals straight x squared over straight a squared minus 1 space rightwards double arrow space space space straight y squared space equals space straight b squared over straight a squared straight x squared minus straight b squared$
Differentiating the above function with respect to x, we get,
$2 straight y dy over dx equals straight b squared over straight a squared 2 straight x rightwards double arrow space space dy over dx equals straight b squared over straight a squared straight x over straight y rightwards double arrow space space open square brackets dy over dx close square brackets subscript left parenthesis square root of 2 straight a comma space straight b right parenthesis end subscript space equals space straight b squared over straight a squared fraction numerator square root of 2 straight a over denominator straight b end fraction equals fraction numerator square root of 2 straight b over denominator straight a end fraction$

Slope of the tangent is $straight m equals fraction numerator square root of 2 straight b over denominator straight a end fraction$
Equation of the tangent is
$left parenthesis straight y minus straight y subscript 1 right parenthesis space equals space straight m left parenthesis straight x minus straight x subscript 1 right parenthesis rightwards double arrow space space space left parenthesis straight y minus straight b right parenthesis equals fraction numerator square root of 2 straight b over denominator straight a end fraction open parentheses straight x minus square root of 2 straight a close parentheses rightwards double arrow space space straight a left parenthesis straight y minus straight b right parenthesis equals square root of 2 straight b left parenthesis straight x minus square root of 2 straight a right parenthesis rightwards double arrow space square root of 2 bx minus ay plus ab minus 2 ab equals 0 rightwards double arrow square root of 2 bx minus ay minus ab space equals space 0$
Slope of the normal is $negative fraction numerator 1 over denominator begin display style fraction numerator square root of 2 straight b over denominator straight a end fraction end style end fraction$
Equation of the normal is
$open parentheses straight y minus straight y subscript 1 close parentheses space equals space straight m left parenthesis straight x minus straight x subscript 1 right parenthesis rightwards double arrow space space left parenthesis straight y minus straight b right parenthesis space equals space fraction numerator negative straight a over denominator square root of 2 straight b end fraction left parenthesis straight x minus square root of 2 straight a right parenthesis rightwards double arrow square root of 2 straight b left parenthesis straight y minus straight b right parenthesis equals negative straight a left parenthesis straight x minus square root of 2 straight a right parenthesis rightwards double arrow ax plus square root of 2 by minus square root of 2 straight b squared plus square root of 2 straight a squared equals 0 rightwards double arrow ax plus square root of 2 by plus square root of 2 left parenthesis straight a squared minus straight b squared right parenthesis space equals space 0$

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If the function f: R $rightwards arrow$ R be given by $straight f left parenthesis straight x right parenthesis space equals space straight x squared plus 2 space space and space straight g space colon thin space straight R rightwards arrow space straight R$ be given by $straight g left parenthesis straight x right parenthesis space equals fraction numerator straight x over denominator straight x minus 1 end fraction comma space straight x not equal to 1 comma$ find fog and gof and hence find fog (2) and gof ( −3).

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Prove that
$tan to the power of negative 1 end exponent open square brackets fraction numerator square root of 1 plus straight x end root minus square root of 1 minus straight x end root over denominator square root of 1 plus straight x end root plus square root of 1 minus straight x end root end fraction close square brackets space equals space straight pi over 4 minus 1 half cos to the power of negative 1 end exponent straight x comma space space fraction numerator negative 1 over denominator square root of 2 end fraction less or equal than straight x less or equal than 1$

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10.

Prove that
If $tan to the power of negative 1 end exponent open parentheses fraction numerator straight x minus 2 over denominator straight x minus 4 end fraction close parentheses plus tan to the power of negative 1 end exponent open parentheses fraction numerator straight x plus 2 over denominator straight x plus 4 end fraction close parentheses equals straight pi over 4 comma$ find the value of x.