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

  إجابة

  (E)

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

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

  إجابة

  (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:

  إجابة

  (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.

  إجابة

  A

  B
• 5

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

  إجابة

  (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.

  إجابة

  A

  B
• 7

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

  إجابة

  (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

  إجابة

  (B)

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

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

  إجابة

  (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$$\alpha $$.

  إجابة

  (A)

  56/33
• 11

  deduce the condition that the equations have a common root.

  إجابة

  (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

  إجابة

  (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.

  إجابة

  (A)

  non negative
• 14

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

  إجابة

  (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

  إجابة

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

  إجابة

  A

  B

  C

  D
• 17

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

  إجابة

  (C)

  3
• 18

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

  إجابة

  (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 :

  إجابة

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

  إجابة

  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 $$x=1$$

  إجابة

  (B)

  -2/9
• 22

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

  إجابة

  (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.

  إجابة

  A

  C