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    Mathematics

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    CBSE Class 12

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    CBSE Class 12 Mathematics Solved Question Paper 2016

    Multiple Choice QuestionsShort Answer Type

    1. If space straight A space equals space open parentheses table row cell cos space straight alpha end cell cell sin space straight alpha end cell row cell negative sin space straight alpha end cell cell cos space straight alpha end cell end table close parentheses comma space find space straight alpha space satisfying space 0 space less than space straight alpha space less than space straight pi over 2 space when space straight A space plus space straight A to the power of straight T space space equals space square root of 2 straight I subscript 2 semicolon space Where space straight A to the power of straight T space is space transpose space of space straight A.
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    2.

    If A is a 3 x 3 matrix |3A| = k|A|, then write the value of k.

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    3.

    For what values of k, the system of linear equations

    x + y + z = 2
    2x + y - z = 3
    3x + 2y + kz = 4

    has a unique solution?

    1308 Views
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    4. Solve space for space straight x space colon space tan to the power of negative 1 end exponent space left parenthesis straight x space plus 1 right parenthesis space plus space tan to the power of negative 1 end exponent straight x space plus tan to the power of negative 1 end exponent space left parenthesis straight x plus 1 right parenthesis space equals space tan to the power of negative 1 end exponent 3 straight x.
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    5. Prove space that space tan to the power of negative 1 end exponent space open parentheses fraction numerator 6 straight x minus 8 straight x cubed over denominator 1 minus 12 straight x squared end fraction close parentheses minus tan to the power of negative 1 end exponent space open parentheses fraction numerator 4 straight x over denominator 1 minus 4 straight x squared end fraction close parentheses space equals space tan to the power of negative 1 end exponent space 2 straight x semicolon space vertical line 2 straight x vertical line less than fraction numerator 1 over denominator square root of 3 end fraction
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    6.

    A typist charges Rs. 145 for typing 10 English and 3 Hindi pages, while charges for typing 3 English and 10 Hindi pages are Rs. 180. Using matrices, find the charges of typing one English and one Hindi page separately. However, typist charged only Rs. 2 per page from a poor student Shyam for 5 Hindi pages. How much less was charged from this poor boy? Which values are reflected in this problem?

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    7. If space space straight f open parentheses straight x close parentheses equals open curly brackets table attributes columnalign left end attributes row cell fraction numerator sin open parentheses straight a plus 1 close parentheses plus 2 sinx over denominator straight x end fraction comma space straight x less than 0 end cell row cell 2 space space space space space space space space space space space space space space space space space space space space space space comma space x equals 0 end cell row cell fraction numerator square root of 1 plus b x end root minus 1 over denominator straight x end fraction space space space space space space comma space straight x greater than 0 end cell end table close
    is continuous at x = 0, then find the values of a and b.
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    8. if space cos space left parenthesis straight a plus straight y right parenthesis space equals space cos space straight y space then space prove space that space dy over dx space equals fraction numerator cos squared left parenthesis straight a plus straight y right parenthesis over denominator sin space straight a end fraction Hence space show space that space sin space straight a space fraction numerator straight d squared straight y over denominator dx squared end fraction plus sin space 2 space left parenthesis straight a plus straight y right parenthesis dy over dx equals 0

    Given space that comma straight x space cos space left parenthesis straight a plus straight y right parenthesis space equals space cos space straight y space.... space left parenthesis straight i right parenthesis rightwards double arrow space straight x space equals space fraction numerator cos space straight y over denominator cos space left parenthesis straight a space plus space straight y right parenthesis end fraction space... space left parenthesis ii right parenthesis Differentiating space both space sides space of space the space equation space left parenthesis straight i right parenthesis comma space we space have comma straight x space left parenthesis negative sin space left parenthesis straight a space plus space straight y right parenthesis right parenthesis space dy over dx equals space minus space cos space left parenthesis straight a plus straight y right parenthesis space equals negative sin space straight y space dy over dx rightwards double arrow space space open square brackets sin space straight y minus space straight x space sin space left parenthesis straight a space plus space straight y right parenthesis close square brackets dy over dx space equals space minus cos space left parenthesis straight a space plus space straight y right parenthesis rightwards double arrow space open square brackets sin space straight y space minus fraction numerator cos space straight y over denominator cos space left parenthesis straight a plus straight y right parenthesis end fraction sin space left parenthesis straight a space plus space straight y right parenthesis close square brackets dy over dx space space equals negative cos space left parenthesis straight a space plus space straight y right parenthesis rightwards double arrow space open square brackets fraction numerator cos space left parenthesis straight a space plus space straight y right parenthesis space space straight x space space sin space straight y space minus space cos space straight y space sin space left parenthesis straight a space plus space straight y right parenthesis over denominator cos space left parenthesis straight a plus straight y right parenthesis end fraction close square brackets dy over dx space equals space minus space cos left parenthesis straight a space plus space straight y right parenthesis space rightwards double arrow space open square brackets cos space left parenthesis straight a space plus space straight y right parenthesis space straight x space sin space straight y space minus space cosy space space straight x space space sin space left parenthesis straight a space plus space straight y right parenthesis close square brackets dy over dx equals negative cos left parenthesis straight a space plus space straight y right parenthesis space space straight x space space cos space left parenthesis straight a space plus space straight y right parenthesis rightwards double arrow space space open square brackets sin space left parenthesis straight a space plus space straight y space minus space straight y right parenthesis close square brackets dy over dx space space equals space minus cos squared left parenthesis straight a space plus space straight y right parenthesis space space space space space open square brackets sin left parenthesis straight A minus straight B right parenthesis space equals space sinA space cos space straight B space minus space cosA space sinB close square brackets rightwards double arrow left square bracket sina space right square bracket space dy over dx equals negative cos squared left parenthesis straight a space plus space straight y right parenthesis rightwards double arrow space dy over dx space equals fraction numerator negative cos squared left parenthesis straight a space plus space straight y right parenthesis over denominator sin space straight a space end fraction space space space space.. space left parenthesis iii right parenthesis differentiating space once space again space with space respect space to space straight x space comma we space have comma sin space straight a space fraction numerator straight d squared straight y over denominator dx squared end fraction space plus space 2 cos left parenthesis straight a space plus space straight y right parenthesis space sin space left parenthesis straight a space plus space straight y right parenthesis dy over dx rightwards double arrow space sin space straight a space fraction numerator straight d squared straight y over denominator dx squared end fraction space plus space 2 space cos space left parenthesis straight a space plus space straight y right parenthesis space sin space left parenthesis straight a space plus space straight y right parenthesis dy over dx equals 0 rightwards double arrow space sin space straight a space fraction numerator straight d squared straight y over denominator dx squared end fraction space plus space sin space 2 space left parenthesis straight a space plus space straight y right parenthesis dy over dx equals 0 Hence space proved.
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    9. Find space dy over dx space if space straight y equals sin to the power of negative 1 end exponent open square brackets fraction numerator 6 straight x minus 4 square root of 1 minus 4 straight x squared end root over denominator 5 end fraction close square brackets
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    10.

    Find the equation of tangents to the curve y= x3+2x-4, which are perpendicular to line x+14y+3 =0.

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