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CBSE Class 12 Mathematics Solved Question Paper 2016

Short Answer Type

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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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?

For space unique space solution space vertical line straight A vertical line space not equal to 0 open vertical bar table row 1 1 1 row 2 1 cell negative 1 end cell row 3 2 straight k end table close vertical bar space not equal to space 0 straight C subscript 2 rightwards arrow straight C subscript 2 minus straight C subscript 1 semicolon space straight C subscript 3 rightwards arrow straight C subscript 3 minus straight C subscript 1 open vertical bar table row 1 0 0 row 2 cell negative 1 end cell cell negative 3 end cell row 3 cell negative 1 end cell cell straight k minus 3 end cell end table close vertical bar space not equal to space 0 expansion space along space straight R subscript 1 minus left parenthesis straight k minus 3 right parenthesis minus 3 space not equal to 0 minus straight k plus 3 minus 3 space not equal to 0 straight k not equal to 0

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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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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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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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