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

  پاسخ دهید

  (E)

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

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

  پاسخ دهید

  (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:

  پاسخ دهید

  (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.

  پاسخ دهید

  A

  B
• 5

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

  پاسخ دهید

  (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.

  پاسخ دهید

  A

  B
• 7

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

  پاسخ دهید

  (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

  پاسخ دهید

  (B)

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

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

  پاسخ دهید

  (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$$\alpha $$.

  پاسخ دهید

  (A)

  56/33
• 11

  deduce the condition that the equations have a common root.

  پاسخ دهید

  (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

  پاسخ دهید

  (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.

  پاسخ دهید

  (A)

  non negative
• 14

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

  پاسخ دهید

  (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

  پاسخ دهید

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

  پاسخ دهید

  A

  B

  C

  D
• 17

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

  پاسخ دهید

  (C)

  3
• 18

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

  پاسخ دهید

  (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 :

  پاسخ دهید

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

  پاسخ دهید

  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 $$x=1$$

  پاسخ دهید

  (B)

  -2/9
• 22

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

  پاسخ دهید

  (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.

  پاسخ دهید

  A

  C