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CBSE

Subject

Mathematics

Class

JEE Class 12

JEE Mathematics 2009 Exam Questions

Multiple Choice Questions

1.

The remainder left out when 82n –(62)2n+1 is divided by 9 is 

  • 0

  • 2

  • 7

  • 8


B.

2

82n – (62)2n + 1
⇒ (9 – 1)2n – (63 – 1)2n + 1
⇒ (2nC0 92n2nC1 92n – 1 + ….. + 2nC2n)
– (2n + 1C0 632n + 1–2n + 1C1 632n + ….
2n +1C2n + 1
Clearly remainder is ‘2’.

264 Views

2.

Three distinct points A, B and C are given in the 2 – dimensional coordinate plane such that the ratio of the distance of any one of them from the point (1, 0) to the distance from the point ( - 1, 0) is equal to 1/3 . Then the circumcentre of the triangle ABC is at the point

  • (0,0)

  • (5/4, 0)

  • (5/2, 0)

  • (5/3,0)


B.

(5/4, 0)

P = (1,0);Q (-1,0)
Let A = (x,y)
AP over AQ space equals space BP over BQ space equals space CP over CQ space equals space 1 third space space.... space left parenthesis straight i right parenthesis

⇒ 3AP =AQ
⇒ 9AP2 = AQ2
⇒9 (x-1)2 +9y2
= (x+1)2 +y2
⇒ 9x2-18x +9 +9y2 = x2 +2x+ 1 +y2
⇒8x2-20x + 8y2 +8 = 0
⇒ x2+ y2-(5/2)+1 = 0 .. (2)
therefore A lies on the circle
Similarly B,C are also lies on the same circle
therefore, CIrcumcentre of ABC= Centre of circle (1) = (5/4, 0)

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3.

The projections of a vector on the three coordinate axis are 6, - 3, 2 respectively. The direction cosines of the vector are

  •  6, –3, 2 

  • 6/5, -3/5, 2/5

  • 6/7, -3/7, 2/7

  • -6/7, -3/7, 2/7


C.

6/7, -3/7, 2/7

Projection of a vector on coordinate axis are
x2-x1, y2-y1, z2-z1
x2-x1 = 6, y2-y1 = -3, z2-z1 = 2

square root of left parenthesis straight x subscript 2 minus straight x subscript 1 right parenthesis squared space plus left parenthesis straight y subscript 2 minus straight y subscript 1 right parenthesis squared space plus space left parenthesis straight z subscript 2 minus straight z subscript 1 right parenthesis squared end root space equals space square root of 36 plus 9 plus 4 end root space equals space 7
The space Direction space cosines space of space the space vector space are space 6 over 7 comma 3 over 7 comma 2 over 7
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4.

The sum to infinity of the series 1 space plus space 2 over 3 space plus space 6 over 3 squared space plus space 10 over 3 cubed space plus space 14 over 3 to the power of 4 space plus.... space is

  • 2

  • 3

  • 4

  • 6


B.

3

Let space straight S space equals space 1 space plus space 2 over 3 space plus space 6 over 3 squared space plus space 10 over 3 cubed space plus space 14 over 3 to the power of 4 space plus space.... infinity
straight S over 3 space equals space 1 third space plus 2 over 3 squared space plus 6 over 3 cubed space plus space 10 over 3 to the power of 4 plus...... infinity
_______________________________
fraction numerator 2 straight S over denominator 3 end fraction space equals space 1 space plus 1 third space plus 4 over 3 squared space plus 4 over 3 cubed space plus 4 over 3 to the power of 4 plus..... infinity
space equals space 4 over 3 space plus fraction numerator begin display style 4 over 3 squared end style over denominator 1 minus begin display style 1 third end style end fraction
space equals space 4 over 3 space plus 2 over 3 space equals space 2
fraction numerator 2 straight S over denominator 3 end fraction space equals space 2 space rightwards double arrow space straight S space equals 3
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5.

If the mean deviation of the numbers 1, 1 + d, 1+ 2d, ... , 1 + 100d from their mean is 255, then the d is equal to 

  • 10.0

  • 20.0

  • 10.1

  • 20.2


C.

10.1

Mean space left parenthesis straight x with minus on top right parenthesis space equals space fraction numerator sum space of space quantities over denominator straight n end fraction
space equals space fraction numerator begin display style straight n over 2 end style space left parenthesis straight a plus 1 right parenthesis over denominator straight n end fraction space equals space 1 half left square bracket space 1 plus space 1 plus 100 straight d right square bracket space equals space 1 plus 50 space straight d
straight M. straight D. space equals space 1 over straight n space begin inline style stack sum space with space below and space on top end style vertical line straight x subscript straight i space minus straight x with minus on top vertical line
rightwards double arrow space 255 space equals space 1 over 101 left square bracket 50 straight d space plus 49 straight d space plus 48 straight d space plus..... space straight d plus space 0 space plus straight d plus space.... plus 50 straight d right square bracket
space equals space fraction numerator 2 straight d over denominator 101 end fraction open square brackets fraction numerator 50 space straight x 51 over denominator 2 end fraction close square brackets
rightwards double arrow space straight d space equals space fraction numerator 255 space straight x space 101 over denominator 50 space straight x space 51 end fraction space equals space 10.1
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6.

Let A and B denote the statements
A: cos α + cosβ + cosγ = 0
B : sinα + sinβ + sinγ = 0
If cos(β – γ) + cos(γ – α) + cos(α – β) = – 3/2, then

  •  A is true and B is false

  • A is false and B is true

  • both A and B are true

  • both A and B are false


C.

both A and B are true

cos(β–γ) + cos(γ – α) + cos(α – β) = –3/2
⇒ 2cos (β – γ) + 2cos(γ– α) + 2cos(α – β) = –3
⇒ Σ(cosβcosγ + 2Σsin α sinβ + 3 = 0
⇒ (cosα+cosβ + cosγ)2+ (sinα + sinβ +sinγ)2=0
⇒ cosα + cosβ + cosγ = 0
sinα + sinα + sinγ = 0

127 Views

7.

The ellipse x2+ 4y2= 4 is inscribed in a rectangle aligned with the coordinate axes, which in turn is inscribed in another ellipse that passes through the point (4, 0). Then the equation of the ellipse is

  • x2+ 16y2= 16 

  • x2+ 12y2= 16 

  • 4x2+ 48y2= 48 

  • 4x2+ 64y2= 48


B.

x2+ 12y2= 16 



straight x squared over 16 space plus straight y squared over straight b squared space equals space 1
It space passes space through space left parenthesis 2 comma 1 right parenthesis
So space 4 over 16 space plus 1 over straight b squared space equals space 1
1 over straight b squared space equals space 3 over 4
straight b squared space equals space 4 over 3
rightwards double arrow space straight x squared space plus space 12 straight y squared space equals space 16
371 Views

8.

If the roots of the equation bx2+ cx + a = 0 be imaginary, then for all real values of x, the expression 3b2x2 + 6bcx + 2c2 is

  • greater than 4ab

  • less than 4ab

  • greater than -4ab

  • less than 4ab


C.

greater than -4ab

As, bx2 + cx + a = 0 has imaginary roots
So, c2< 4ab
Now, 3b2x2 + 6bcx + 2c2
= 3(bx + c)2– c2≥ – c2≥ – 4ab

134 Views

9.

If A, B and C are three sets such that A ∩ B = A∩ C and A ∪ B = A ∪ C, then

  • A = B

  • A = C

  • B = C

  • A ∩ B = φ


C.

B = C

A ∪ B = A ∪ C
⇒ n (A ∪ B) = n(A ∪ C)
⇒ n(A) + n(B) – n(A ∩ B)
= n(A) + n(C) – n(A ∩C)
n(B) = n(C)

105 Views

10.

The differential equation which represents the family of curves y=c1ec2xe, where c1 and c2 are arbitrary constants, is

  • y' =y2

  •  y″ = y′ y

  • yy″ = y′

  • yy″ = (y′)


D.

yy″ = (y′)

y c1ec2x = …..(i)
y' = c1c2ec2x
y' = c2y.....(from (i) ....(ii)
y" = c2y' ....... (iii)
from (ii) & (iii)
fraction numerator straight y apostrophe over denominator straight y end fraction space equals space fraction numerator straight y " over denominator straight y apostrophe end fraction
rightwards double arrow space yy " space equals space left parenthesis straight y apostrophe right parenthesis squared

120 Views

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