JEE Advance - Mathematics (1979)

• 1

  Prove that the minimum value of $${{\left( {a + x} \right)\left( {b + x} \right)} \over {\left( {c + x} \right)}},$$
  $$a,b > c,x > - c$$ is $${\left( {\sqrt {a - c} + \sqrt {b - c} } \right)^2}$$.

  Одговорити

  (E)

  The minimum value is ${\left( {\sqrt {a - c} + \sqrt {b - c} } \right)^2}$
• 2

  Evaluate $$\int {{{{x^2}dx} \over {{{\left( {a + bx} \right)}^2}}}} $$

  Одговорити

  (A)

  ${1 \over {{b^3}}}\left[ {a + bx - 2a\log \left| {a + bx} \right| - {{{a^2}} \over {a + bx}}} \right] + C
• 3

  Two fair dice are tossed. Let $$x$$ be the event that the first die shows an even number and $$y$$ be the event that the second die shows an odd number. The two events $$x$$ and $$y$$ are:

  Одговорити

  (D)

  None of these.
• 4

  Six boys and six girls sit in a row randomly. Find the probability that
  (i) the six girls sit together
  (ii) the boys and girls sit alternately.

  Одговорити

  A

  B
• 5

  If $$\alpha + \beta + \gamma = 2\pi ,$$ then

  Одговорити

  (A)

  $$tan{\alpha \over 2} + \tan {\beta \over 2} + \tan {\gamma \over 2} = \tan {\alpha \over 2}\tan {\beta \over 2}\tan {\gamma \over 2}$$
• 6

  (b) Find the area of the smaller part of a disc of radius $$10$$ cm, cut off by a chord $$AB$$ which subtends an angle of at the circumference.

  Одговорити

  A

  B
• 7

  If $$\tan \theta = - {4 \over 3},then\sin \theta \,is\,$$

  Одговорити

  (B)

  $$ - {4 \over 5}\,or\,{4 \over 5}$$
• 8

  If the cube roots of unity are $$1,\,\omega ,\,{\omega ^2},$$ then the roots of the equation $${\left( {x - 1} \right)^3} + 8 = 0$$ are

  Одговорити

  (B)

  $$ - 1,1 - 2\omega ,\,1 - 2{\omega ^2}$$
• 9

  If x + iy = $$\sqrt {{{a + ib} \over {c + id}}} $$, prove that $${({x^2} + {y^2})^2} = {{{a^2} + {b^2}} \over {{c^2} + {d^2}}}$$.

  Одговорити

  (C)

  $$({x^2} + {y^2})^2 = {{{a^2} + {b^2}} over {{c^2} + {d^2}}}$$
• 10

  (b) If $$\cos \left( {\alpha + \beta } \right) = {4 \over 5},\,\,\sin \,\left( {\alpha - \beta } \right) = \,{5 \over {13}},$$ and $$\alpha ,\,\beta $$ lies between 0 and $${\pi \over 4}$$, find tan2$$\alpha $$.

  Одговорити

  (A)

  56/33
• 11

  deduce the condition that the equations have a common root.

  Одговорити

  (B)

  q(r - p)^2 - p(r - p)(s - q) + (s - q)^2; (q - s)^2 = (r - p)(ps - qr)
• 12

  The equation x + 2y + 2z = 1 and 2x + 4y + 4z = 9 have

  Одговорити

  (D)

  None of these.
• 13

  If x, y and z are real and different and $$\,u = {x^2} + 4{y^2} + 9{z^2} - 6yz - 3zx - 2xy$$, then u is always.

  Одговорити

  (A)

  non negative
• 14

  Let a > 0, b > 0 and c > 0. Then the roots of the equation $$a{x^2} + bx + c = 0$$

  Одговорити

  (C)

  both (a) and (b)
• 15

  If $$\ell $$, m, n are real, $$\ell \ne m$$, then the roots by the equation :
  $$(\ell - m)\,{x^2} - 5\,(\ell + m)\,x - 2\,(\ell - m) = 0$$ are

  Одговорити

  (C)

  Real and unequal
• 16

  Given that $${C_1} + 2{C_2}x + 3{C_3}{x^2} + ......... + 2n{C_{2n}}{x^{2n - 1}} = 2n{\left( {1 + x} \right)^{2n - 1}}$$
  where $${C_r} = {{\left( {2n} \right)\,!} \over {r!\left( {2n - r} \right)!}}\,\,\,\,\,r = 0,1,2,\,............,2n$$
  Prove that $${C_1}^2 - 2{C_2}^2 + 3{C_3}^2 - ............ - 2n{C_{2n}}^2 = {\left( { - 1} \right)^n}n{C_n}.$$

  Одговорити

  A

  B

  C

  D
• 17

  $${}^n{C_{r - 1}} = 36,{}^n{C_r} = 84\,\,and\,\,{}^n{C_{r + 1}} = 126$$, then r is :

  Одговорити

  (C)

  3
• 18

  The harmonic mean of two numbers is 4.Their arithmetic mean $$A$$ and the geometric mean $$G$$ satisfy the relation. $$2A + {G^2} = 27$$

  Одговорити

  (A)

  $$3$$ and $$6$$
• 19

  The points $$\left( { - a,\, - b} \right),\,\left( {0,\,0} \right),\,\left( {a,\,b} \right)$$ and $$\left( {{a^2},\,ab} \right)$$ are :

  Одговорити

  (A)

  Collinear
• 20

  (a) Two vertices of a triangle are $$(5, -1)$$ and $$(-2, 3).$$ If the orthocentre of the triangle is the origin, find the coordinates of the third point.
  (b) Find the equation of the line which bisects the obtuse angle between the lines $$x - 2y + 4 = 0$$ and $$4x - 3y + 2 = 0$$.

  Одговорити

  A

  D
• 21

  Find the derivative of $$$f\left( x \right) = \left\{ {\matrix{ {{{x - 1} \over {2{x^2} - 7x + 5}}} & {when\,\,x \ne 1} \cr { - {1 \over 3}} & {when\,\,x = 1} \cr } } \right.$$$
  at $$x=1$$

  Одговорити

  (B)

  -2/9
• 22

  If the bisector of the angle $$P$$ of a triangle $$PQR$$ meets $$QR$$ in $$S$$, then

  Одговорити

  (C)

  $$QS:SR=PQ:PR$$
• 23

  (b) If a triangle is inscribed in a circle, then the product of any two sides of the triangle is equal to the product of the diameter and the perpendicular distance of the third side from the opposite vertex. Prove the above statement.

  Одговорити

  A

  C