The number of common tangent to the circles x2+y2-4x-6y-12=0 and

Subject

Mathematics

Class

JEE Class 12

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 Multiple Choice QuestionsMultiple Choice Questions

11.

If the function g(x) =open curly brackets table attributes columnalign left end attributes row cell straight k square root of straight x plus 1 end root comma space space 0 space less or equal than straight x space less or equal than 3 end cell row cell mx plus 2 space comma space 3 space less than straight x less or equal than 5 end cell end table close is differntiable, then the value of k+m is

  • 2

  • 16/5

  • 10/3

  • 10/3

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12.

The normal to the curve x2 + 2xy-3y2 =0 at (1,1)

  • does not meet the curve again

  • meets the curve again in the second quadrant

  • meets the curve again in the third quadrant

  • meets the curve again in the third quadrant

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13.

The number of common tangent to the circles x2+y2-4x-6y-12=0 and x2+y2+6x+18y+26 = 0 is

  • 1

  • 2

  • 3

  • 3


C.

3

Number of common tangents depend on the position of the circle with respect to the each other.
(i) If circles touch externally ⇒C1C2  = r1+ r2,3 common tangents
(ii) If circles touch internally ⇒ C1C2 = r2-r1, 1 common tangents
(iii) If circles do not touch each other, 4 common tangents

Given  equations of circles are
x2 +y2-4x-6y-12 = 0 .. (i) 
x2+y2+6x+18y+26 =0 ... (ii)
Centre of circle (i) is C1 (2,3) and radius
=square root of 4 plus 9 plus 12 end root space equals space 5 space left parenthesis straight r subscript 1 right parenthesis
Centre of circle (ii) is C2(-3,-9) and radius
square root of 9 plus 81 minus 26 end root equals space 8 space left parenthesis straight r subscript 2 right parenthesis
Now comma space straight C subscript 1 straight C subscript 2 space equals space square root of left parenthesis 2 plus 3 right parenthesis squared plus left parenthesis 3 plus 9 right parenthesis squared end root
rightwards double arrow space straight C subscript 1 straight C subscript 2 space equals space square root of 5 squared plus 12 squared end root
straight C subscript 1 straight C subscript 2 space equals space square root of 25 plus 144 end root space equals space 13
straight r subscript 1 plus straight r subscript 2 space equals space 5 plus 8 space equals space 13
Also comma space straight C subscript 1 straight C subscript 2 space equals space straight r subscript 1 plus straight r subscript 2
Thus, both circles touch each other externally. Hence, there are three common tangents.

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14.

A = open square brackets table row 1 2 2 row 2 1 cell negative 2 end cell row straight a 2 straight b end table close square brackets is a matrix satisfying the equation AAT = 9I, Where I is 3 x 3 identity matrix, then the ordered pair (a,b) is equal to

  • (2,-1)

  • (-2,1)

  • (2,1)

  • (2,1)

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15.

The set of all values of λ for which the system of linear equations 

2x1-2x2+x3 = λx1
2x1- 3x2 + 2x3 = λx2
-x1 + 2x2 = λx3
a non- trivial solution.

  • is an empty set

  • is a singleton set

  • contains two elements

  • contains two elements

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16. straight l im with straight x rightwards arrow 0 below space fraction numerator left parenthesis 1 minus cos 2 straight x right parenthesis left parenthesis 3 space plus cos space straight x right parenthesis over denominator straight x space tan space 4 straight x end fraction space is equal to 
  • 4

  • 3

  • 2

  • 2

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17.

Let tan to the power of negative 1 end exponent space straight y space equals tan to the power of negative 1 end exponent space straight x space plus space tan to the power of negative 1 end exponent space open parentheses fraction numerator 2 straight x over denominator 1 minus straight x squared end fraction close parentheses comma where vertical line straight x vertical line space less than space fraction numerator 1 over denominator square root of 3 space end fraction space.Then, a value of y is

  • fraction numerator 3 straight x squared minus straight x squared over denominator 1 minus 3 straight x squared end fraction
  • fraction numerator 3 straight x space plus space straight x cubed over denominator 1 minus 3 straight x squared end fraction
  • fraction numerator 3 straight x minus straight x cubed over denominator 1 plus 3 straight x squared end fraction
  • fraction numerator 3 straight x minus straight x cubed over denominator 1 plus 3 straight x squared end fraction
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18.

Let f (x) be a polynomial of degree four having extreme values at x =1 an x =2. If limit as straight x rightwards arrow 0 of open square brackets 1 plus fraction numerator straight f left parenthesis straight x right parenthesis over denominator straight x squared end fraction close square brackets space equals space 3 comma then f(2) is equal to 

  • -8

  • -4

  • 0

  • 0

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19.

If 12 identical balls are to be placed in 3 identical boxes, then the probability that  one of the boxes contains exactly 3 balls, is

  • 55 over 33 open parentheses 2 over 3 close parentheses to the power of 11
  • 55 space open parentheses 2 over 3 close parentheses to the power of 10
  • 220 space open parentheses 1 third close parentheses to the power of 12
  • 220 space open parentheses 1 third close parentheses to the power of 12
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20.

The integral integral fraction numerator dx over denominator straight x squared space left parenthesis straight x to the power of 4 plus 1 right parenthesis to the power of begin display style 3 over 4 end style end exponent end fraction space equals

  • open parentheses fraction numerator straight x to the power of 4 plus 1 over denominator straight x to the power of 4 end fraction close parentheses to the power of 1 fourth end exponent space plus straight C
  • left parenthesis straight x to the power of 4 plus 1 right parenthesis to the power of 1 fourth end exponent space plus straight C
  • negative left parenthesis straight x to the power of 4 plus 1 right parenthesis to the power of 1 fourth end exponent space plus straight C
  • negative left parenthesis straight x to the power of 4 plus 1 right parenthesis to the power of 1 fourth end exponent space plus straight C
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