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

  Răspuns

  (E)

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

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

  Răspuns

  (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:

  Răspuns

  (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.

  Răspuns

  A

  B
• 5

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

  Răspuns

  (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.

  Răspuns

  A

  B
• 7

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

  Răspuns

  (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

  Răspuns

  (B)

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

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

  Răspuns

  (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$$\alpha $$.

  Răspuns

  (A)

  56/33
• 11

  deduce the condition that the equations have a common root.

  Răspuns

  (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

  Răspuns

  (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.

  Răspuns

  (A)

  non negative
• 14

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

  Răspuns

  (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

  Răspuns

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

  Răspuns

  A

  B

  C

  D
• 17

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

  Răspuns

  (C)

  3
• 18

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

  Răspuns

  (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 :

  Răspuns

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

  Răspuns

  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 $$x=1$$

  Răspuns

  (B)

  -2/9
• 22

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

  Răspuns

  (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.

  Răspuns

  A

  C