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 Multiple Choice QuestionsMultiple Choice Questions

1.

Two points A and B move from rest along a straight line with constant acceleration f and f′ respectively. If A takes m sec. more than B and describes ‘n’ units more than B in acquiring the same speed then

  • (f - f′)m2 = ff′n

  • (f + f′)m2 = ff′n

  • 1/2(f - f′)m = ff′n2

  • (f' -f) = 1/2ff'n2


D.

(f' -f) = 1/2ff'n2

v2 = 2f(d + n) = 2f′d
v = f′(t) = (m + t)f
eliminate d and m we get
(f'-f)n =1/2ff'm2

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2.

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.
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3.

The system of equations 
αx + y + z = α - 1, 
x + αy + z = α - 1, 
x + y + αz = α - 1 

has no solution, if α is

  • -2

  • either-2 or 1

  • not -2

  • 1


A.

-2

αx + y + z = α - 1, 
x + αy + z = α - 1, 
x + y + αz = α - 1 

increment space equals space open vertical bar table row straight alpha 1 1 row 1 straight alpha 1 row 1 1 straight alpha end table close vertical bar

= α(α2 – 1) – 1(α - 1) + 1(1 - α)
= α (α - 1) (α + 1) – 1(α - 1) – 1(α - 1)
⇒ (α - 1)[α2 + α - 1 – 1] = 0
⇒ (α - 1)[α2 + α - 2]= 0 [α2 + 2α - α - 2] = 0
(α - 1) [α(α + 2) – 1(α + 2)]= 0
(α - 1) = 0,
α + 2 = 0 ⇒ α = –2, 1; but α ≠ 1

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4.

If one of the lines given by 6x2 -xy +4cy2 = 0 is 3x + 4y = 0, then c equals

  • 1

  • -1

  • 3

  • -3


D.

-3

The pair of lines is 6x2-xy +4cy2 =0
On comparing with ax2 +2hxy by2 = 0
we get a = 6, 2h =-1, b= c
therefore
m1 +m2 = -2h/b = 1/4c and m1m2 = a/b = 6/4c
On line of given pair of lines is
3x+4y = 0
slope of line = -3/4 = m1
-3/4 +m2 = 1/4c
m2 = 1/4c + 3/4
left parenthesis negative 3 divided by 4 right parenthesis open parentheses fraction numerator 1 over denominator 4 straight c end fraction plus 3 divided by 4 close parentheses space equals space fraction numerator 6 over denominator 4 straight c end fraction
space equals space 3 over 4 space open parentheses fraction numerator 1 plus 3 straight c over denominator 4 straight c end fraction close parentheses space equals space fraction numerator 6 over denominator 4 straight c end fraction
space 1 plus 3 straight c space equals space fraction numerator negative 6 space straight x space 4 over denominator 4 end fraction
1 + 3c = -8 3c = -9
⇒ c = -3

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5.

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

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6.

If |z + 4| ≤ 3, then the maximum value of |z + 1| is

  • 4

  • 10

  • 6

  • 0


C.

6

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7.

The intercept on the line y = x by the circle x2 +y2 -2x = 0 is AB. Equation of the circle on AB as a diameter is

  • x2 +y2 -x-y =0

  • x2 -y2 -x-y =0

  • x2 +y2 +x-y =0

  • x2 +y2 -x+y =0


A.

x2 +y2 -x-y =0

The equation of line is y = x ........... (i)
and equation of circle is x2 + y2 - 2x = 0 ........... (ii)
On solving equation (i) and equation (ii),
we get x2 + x2 - 2x = 0
2x2 - 2x = 0
= 2x(x - 1) = 0
x = 0, x = 1
when x = 0, y = 0
when x = 1, y = 1
Let coordinate of A is (0, 0) and co-ordinate of B is (1, 1)

∴ Equation of circle (AB as a diameter)
(x - x1 )(x - x2 ) + (y - y1 )(y - y2 ) = 0
(x - 0)(x - 1) + (y - 0)(y - 1) = 0
x(x - 1) + y(y - 1) = 0
x2 - x + y2 - y = 0
x2 + y2 - x - y = 0

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8.

If the sum of the slopes of the lines given by x2 -2cxy -7y2 =0 is four times their product, then c has the value

  • -1

  • 2

  • -2

  • 1


B.

2

The given pair of line is x2 - 2cxy - 7y2 = 0
On comparing with ax2 + 2hxy + by2 = 0,
we get, a = 1, 2h = -2c, b = -7

straight m subscript 1 space plus space straight m subscript 2 equals space minus fraction numerator 2 straight h over denominator straight b end fraction
space equals space minus fraction numerator 2 straight c over denominator 7 end fraction
and space straight m subscript 1 straight m subscript 2 space equals space straight a over straight b space equals space fraction numerator negative 1 over denominator 7 end fraction


Given that, m1+m2 = 4m1m2
-2c/7 = - 4/7, c =4/2 =2

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9.

The equation of the straight line passing through the point (4, 3) and making intercepts on the co-ordinate axes whose sum is –1 is

  • straight x over 2 space plus straight y over 3 space equals space minus space 1 space and space fraction numerator straight x over denominator negative 2 end fraction space plus straight y over 1 space equals space minus space 1
  • straight x over 2 minus straight y over 3 space equals space minus space 1 space and space fraction numerator straight x over denominator negative 2 end fraction space plus straight y over 1 space equals space minus space 1
  • straight x over 2 space plus straight y over 3 space equals space 1 space and space straight x over 2 space plus straight y over 1 space equals space 1
  • straight x over 2 space minus straight y over 3 space equals space 1 space and space fraction numerator straight x over denominator negative 2 end fraction space plus straight y over 1 space equals 1

D.

straight x over 2 space minus straight y over 3 space equals space 1 space and space fraction numerator straight x over denominator negative 2 end fraction space plus straight y over 1 space equals 1

Let a and b be the intercepts on the co-ordinate axes. a + b = -1
⇒ b = - a - 1 = - (a + 1)

Equation of line is x/a + y/b = 1



rightwards double arrow space straight x over straight a space minus fraction numerator straight y over denominator straight a plus 1 end fraction space equals 1 space.... space left parenthesis straight i right parenthesis
Since space the space line space passes space through space left parenthesis 4 comma 3 right parenthesis
therefore 4 over straight a minus fraction numerator 3 over denominator straight a plus 1 end fraction space equals space 1
rightwards double arrow space fraction numerator 4 straight a space plus space 4 minus 3 straight a over denominator straight a left parenthesis straight a plus 1 right parenthesis end fraction space equals 1
straight a plus space 4 space equals space straight a squared space plus straight a
straight a squared space equals space 4
straight a equals space plus-or-minus 2
therefore space from space left parenthesis straight i right parenthesis
straight x over 2 minus straight y over 3 space equals 1 space or space fraction numerator straight x over denominator negative 2 end fraction space plus straight y over 1 space equals 1

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10.

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

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