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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.

4254 Views

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

920 Views

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

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.

1117 Views

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

676 Views

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?

2328 Views

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.

1266 Views

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

1206 Views

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

783 Views

10.
Find the equation of tangents to the curve y= x³+2x-4, which are perpendicular to line x+14y+3 =0.

Taking the given equation,
y = x³+2x-4
Differentiating the above function with respect to x, we have,
$dy over dx equals space 3 straight x squared plus 2 rightwards double arrow space straight m subscript 1 equals space 3 straight x squared plus 2$

Given that the tangents to the given curve are perpendicular to the line x+ 14y + 3 = 0
Slope of this line, m₂=-1/14
Since the given line and the tangents to the curve are perpendicular, we have,

m₁ x m₂ =-1

$rightwards double arrow space left parenthesis 3 straight x squared plus 2 right parenthesis open parentheses fraction numerator negative 1 over denominator 14 end fraction close parentheses space equals negative 1 rightwards double arrow space 3 straight x squared space plus 2 space equals space 14 rightwards double arrow space 3 straight x squared space equals space 12 rightwards double arrow space straight x squared space equals 4 rightwards double arrow space straight x space equals plus-or-minus 2 If space straight x equals 2 comma space straight y equals straight x cubed space plus 2 straight x minus 4 rightwards double arrow space straight y equals left parenthesis negative 2 right parenthesis cubed plus 2 straight x space left parenthesis negative 2 right parenthesis minus 4 rightwards double arrow straight y equals negative 16 Equation space of space the space tangent space having space slope space straight m space at space the space point space left parenthesis straight x subscript 1 comma straight y subscript 1 right parenthesis space is left parenthesis straight y minus straight y subscript 1 right parenthesis space equals straight m left parenthesis straight x minus straight x subscript 1 right parenthesis Equation space of space the space tangent space at space straight P space left parenthesis 2 comma 8 right parenthesis space with space slope space 14 left parenthesis straight y minus 8 right parenthesis equals 14 left parenthesis straight x minus 2 right parenthesis rightwards double arrow space straight y minus 8 space equals space 14 space straight x minus 28 rightwards double arrow 14 straight x minus straight y equals 20 Equation space of space the space tangent space at space straight P left parenthesis negative 2 comma negative 16 right parenthesis space with space slope space 14 left parenthesis straight y minus 8 right parenthesis equals 14 left parenthesis straight x minus 2 right parenthesis rightwards double arrow space straight y minus 8 space equals space 14 space straight x minus 28 rightwards double arrow 14 space straight x minus straight y space equals 20 Equation space of space the space tangent space at space straight P left parenthesis negative 2 comma negative 16 right parenthesis space with space slope space 1 left parenthesis straight y plus 16 right parenthesis equals 14 left parenthesis straight x plus 2 right parenthesis rightwards double arrow straight y plus 16 space equals space 14 straight x plus 28 rightwards double arrow 14 space straight x minus straight y space equals space minus 12$
thus equation of the tangent is 14 x- y =-12