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CBSE

Subject

Mathematics

Class

JEE Class 12

JEE Mathematics 2006 Exam Questions

Multiple Choice Questions

1.

All the values of m for which both roots of the equations x2 − 2mx + m2 − 1 = 0 are greater than −2 but less than 4, lie in the interval

  • −2 < m < 0

  • m > 3

  • −1 < m < 3 

  • 1 < m < 4 


C.

−1 < m < 3 

Equation x2 − 2mx + m2 − 1 = 0
(x − m)2 − 1 = 0
(x − m + 1) (x − m − 1) = 0
x = m − 1, m + 1 − 2 < m − 1 and m + 1 < 4

m > − 1 and m < 3 − 1 < m < 3.

152 Views

2.

The two lines x = ay + b, z = cy + d; and x = a′y + b′, z = c′y + d′ are perpendicular to each other if

  • aa′ + cc′ = −1

  • aa′ + cc′ = 1

  • fraction numerator straight a over denominator straight a apostrophe end fraction space plus fraction numerator straight c over denominator straight c apostrophe end fraction space equals 1
  • fraction numerator straight a over denominator straight a apostrophe end fraction space plus fraction numerator straight c over denominator straight c apostrophe end fraction space equals negative 1

A.

aa′ + cc′ = −1

Equation of lines 
fraction numerator straight x minus straight b over denominator straight a end fraction space equals space straight y space equals fraction numerator straight z minus straight d over denominator straight c end fraction
fraction numerator straight x minus straight b apostrophe over denominator straight a apostrophe end fraction space equals space straight y space equals space fraction numerator straight z minus straight d apostrophe over denominator straight c apostrophe end fraction
Lines space are space perpendicular
rightwards double arrow space aa apostrophe space plus space 1 space plus cc apostrophe space equals space 0

111 Views

3.

If the roots of the quadratic equation x2 + px + q = 0 are tan30° and tan15°, respectively then the value of 2 + q − p is

  • 2

  • 3

  • 0

  • 1


B.

3

x2 + px + q = 0
tan 30° + tan 15° = − p
tan 30° ⋅ tan 15° = q

tan space 45 to the power of straight o space equals space fraction numerator tan space 30 to the power of straight o space plus space tan space 15 to the power of straight o over denominator 1 minus tan space 30 to the power of straight o space tan space 15 to the power of straight o end fraction space
equals space fraction numerator negative straight p over denominator 1 minus straight q end fraction space equals 1

⇒ − p = 1 − q
⇒ q − p = 1
∴ 2 + q − p = 3

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4.

The value of sum from straight k equals 1 to 10 of space open parentheses sin space fraction numerator 2 kπ over denominator 11 end fraction plus space straight i space cos space fraction numerator 2 kπ over denominator 11 end fraction close parentheses space is

  • i

  • 1

  • -i

  • -1


C.

-i

sum from straight k space equals 1 to 10 of space open parentheses sin space fraction numerator 2 kπ over denominator 11 end fraction space plus space straight i space cos space fraction numerator 2 kπ over denominator 11 end fraction close parentheses
space equals space sum from straight k space equals 1 to 10 of space sin space fraction numerator 2 kπ over denominator 11 end fraction space plus space straight i space sum from straight k space equals 1 to 10 of space cos space fraction numerator 2 kπ over denominator 11 end fraction
space equals space 0 space plus space straight i space left parenthesis negative 1 right parenthesis space equals space minus space straight i
131 Views

5.

Let W denote the words in the English dictionary. Define the relation R by :
R = {(x, y) ∈ W × W | the words x and y have at least one letter in common}. Then R is

  • not reflexive, symmetric and transitive

  • reflexive, symmetric and not transitive

  • reflexive, symmetric and transitive

  • reflexive, not symmetric and transitive


B.

reflexive, symmetric and not transitive

Clearly (x, x) ∈ R ∀ x ∈ W. So, R is reflexive. Let (x, y) ∈ R,
then (y, x) ∈ R as x and y have at least one letter in common. So, R is symmetric. But R is not transitive for example

Let x = DELHI, y = DWARKA and z = PARK then
(x, y) ∈ R and (y, z) ∈ R but (x, z) ∉ R

141 Views

6.

In an ellipse, the distance between its foci is 6 and minor axis is 8. Then its eccentricity is

  • 3/5

  • 1/5

  • 2/5

  • 4/5


A.

3/5

2ae = 6 ⇒ ae = 3
2b = 8 ⇒ b = 4
b2= a2(1 − e2)
16 = a2 − a2e2
a2= 16 + 9 = 25
a = 5
∴e = 3/a = 3/5

216 Views

7.

Suppose a population A has 100 observations 101, 102, … , 200, and another population B has 100 observations 151, 152, … , 250. If VA and VB represent the variances of the two populations, respectively, then VA/VB is 

  • 1

  • 9/4

  • 4/9

  • 2/3


A.

1

straight sigma subscript straight x superscript 2 space equals space fraction numerator sum from space to space of straight d subscript straight i superscript 2 over denominator straight n end fraction(Here deviations are taken from the mean) Since A and B both have 100 consecutive integers, therefore both have same standard deviation and hence the variance. 
therefore straight V subscript straight A over straight V subscript straight B space equals space 1 space left parenthesis As space begin inline style sum from space to space of end style space straight d subscript straight i superscript 2 space is space same space in space both space the space cases right parenthesis
387 Views

8.

A body falling from rest under gravity passes a certain point P. It was at a distance of 400 m from P, 4s prior to passing through P. If g = 10 m/s2 , then the height above the point P from where the body began to fall is

  • 720 m

  • 900 m

  • 320 m

  • 680 m


A.

720 m


straight h space equals space 1 half gt squared space and space straight h space plus space 400 space equals space 1 half space straight g space left parenthesis straight t plus 4 right parenthesis squared
Subtracting we get 400 = 8g + 4gt
⇒ t = 8 sec
∴ straight h space equals space 1 half space straight x space 10 space straight x space 64 space equals space 320 space straight m
∴ Desired height = 320 + 400 = 720 m.
130 Views

9.

The locus of the vertices of the family of parabolas straight y space equals fraction numerator straight a cubed straight x squared over denominator 3 end fraction space plus fraction numerator straight a squared straight x over denominator 2 end fraction minus 2 straight a space is

  • xy space equals space 105 over 64
  • xy space equals space 5 over 4
  • xy space equals space 35 over 16
  • xy space equals space 64 over 105

A.

xy space equals space 105 over 64
Parabola colon space straight y space equals fraction numerator straight a cubed straight x squared over denominator 3 end fraction plus fraction numerator straight a squared straight x over denominator 2 end fraction minus 2 straight a
Vertex colon thin space left parenthesis straight alpha comma space straight beta right parenthesis
straight alpha space equals space fraction numerator negative straight a squared divided by 2 over denominator 2 straight a cubed divided by 3 end fraction space equals space minus space fraction numerator 3 over denominator 4 straight a end fraction comma
straight beta space equals space fraction numerator open parentheses begin display style straight a to the power of 4 over 4 end style plus 4. begin display style straight a cubed over 3 end style.2 straight a close parentheses over denominator 4 begin display style straight a cubed over 3 end style end fraction space space equals space fraction numerator open parentheses negative begin display style 1 fourth end style plus begin display style 8 over 3 end style close parentheses straight a to the power of 4 over denominator begin display style 4 over 3 end style straight a cubed end fraction
αβ space equals space minus space fraction numerator 3 over denominator 4 straight a end fraction open parentheses negative 35 over 16 close parentheses straight a space equals space 105 over 64
393 Views

10.

A straight line through the point A(3, 4) is such that its intercept between the axes is bisected at A. Its equation is

  • x + y = 7

  • 3x − 4y + 7 = 0

  • 4x + 3y = 24

  • 3x + 4y = 25


C.

4x + 3y = 24

The equation of axes is xy = 0 
⇒ the equation of the line is

fraction numerator straight x.4 space plus straight y.3 over denominator 2 end fraction space equals space 12
rightwards double arrow space 4 straight x space plus 3 straight y space equals space 24

133 Views

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