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 $${\left( {\sqrt {a - c} + \sqrt {b - c} } \right)^2}$$.

  отговор

  (E)

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

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

  отговор

  (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:

  отговор

  (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.

  отговор

  A

  B
• 5

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

  отговор

  (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.

  отговор

  A

  B
• 7

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

  отговор

  (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

  отговор

  (B)

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

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

  отговор

  (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$$\alpha $$.

  отговор

  (A)

  56/33
• 11

  deduce the condition that the equations have a common root.

  отговор

  (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

  отговор

  (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.

  отговор

  (A)

  non negative
• 14

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

  отговор

  (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

  отговор

  (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 $${C_1}^2 - 2{C_2}^2 + 3{C_3}^2 - ............ - 2n{C_{2n}}^2 = {\left( { - 1} \right)^n}n{C_n}.$$

  отговор

  A

  B

  C

  D
• 17

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

  отговор

  (C)

  3
• 18

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

  отговор

  (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 :

  отговор

  (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 $$4x - 3y + 2 = 0$$.

  отговор

  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 $$x=1$$

  отговор

  (B)

  -2/9
• 22

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

  отговор

  (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.

  отговор

  A

  C