JEE Advance - Mathematics (1981)

• 1

  Show that : $$\mathop {\lim }\limits_{n \to \infty } \left( {{1 \over {n + 1}} + {1 \over {n + 2}} + .... + {1 \over {6n}}} \right) = \log 6$$

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  (A)

  The limit can be evaluated by recognizing it as a Riemann sum approximation of the integral of 1/x from 1 to 6.
• 2

  For a biased die the probabilities for the different faces to turn up are given below :

  This die tossed and you are told that either face $$1$$ or face $$2$$ has turned up. Then the probability that it is face $$1$$ is ...............

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  (A)

  5/21
• 3

  An anti-aircraft gun can take a maximum of four shots at an enemy plane moving away from it. The probabilities of hitting the plane at the first, second, third and fourth shot are $$0.4, 0.3, 0.2$$ and $$0.1$$ respectively. What is the probability that the gun hits the plane?

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  (C)

  0.69
• 4

  Let $$\overrightarrow A ,\overrightarrow B ,\overrightarrow C $$ be vectors of length $$3, 4, 5$$ respectively. Let $$\overrightarrow A $$ be perpendicular to $$\overrightarrow B + \overrightarrow C ,\overrightarrow B $$ to $$\overrightarrow C + \overrightarrow A $$ to $$\overrightarrow A + \overrightarrow B .$$ Then the length of vector $$\overrightarrow A + \overrightarrow B + \overrightarrow C $$ is ..........

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  (B)

  5√2
• 5

  Let $$\overrightarrow A ,\overrightarrow B $$ and $${\overrightarrow C }$$ be unit vectors suppose that $$\overrightarrow A .\overrightarrow B = \overrightarrow A .\overrightarrow C = 0,$$ and thatthe angle between $${\overrightarrow B }$$ and $${\overrightarrow C }$$ is $$\pi /6.$$ Then $$\overrightarrow A = \pm 2\left( {\overrightarrow B \times \overrightarrow C } \right).$$

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  (B)

  FALSE
• 6

  The scalar $$\overrightarrow A .\left( {\overrightarrow B + \overrightarrow C } \right) \times \left( {\overrightarrow A + \overrightarrow B + \overrightarrow C } \right)$$ equals :

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  (A)

  $$0$$
• 7

  Find the area bounded by the curve $${x^2} = 4y$$ and the straight

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  (C)

  9/8 sq. units
• 8

  Suppose that the normals drawn at three different points on the parabola $${y^2} = 4x$$ pass through the point $$(h, k)$$. Show that $$h>2$$.

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  (D)

  The statement is false for h < 2.
• 9

  The general solution of the trigonometric equation sin x+cos x=1 is given by:

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  (C)

  $$x = n\pi + {\left( { - 1} \right)^n}\,\,\,\,\,\,\,{\pi \over 4} - {\pi \over 4}$$ ; $$n = 0,\, \pm 1,\, \pm 2..$$
• 10

  For complex number $${z_1} = {x_1} + i{y_1}$$ and $${z_2} = {x_2} + i{y_2},$$ we write $${z_1} \cap {z_2},\,\,if\,\,{x_1} \le {x_2}\,\,and\,\,{y_1} \le {y_2}.$$
  Then for all complex numbers $$z\,\,with\,\,1 \cap z,$$ we have $${{1 - z} \over {1 + z}} \cap 0.$$

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  (B)

  FALSE
• 11

  The complex numbers $$z = x + iy$$ which satisfy the equation $$\,\left| {{{z - 5i} \over {z + 5i}}} \right| = 1$$ lie on

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  (A)

  the x-axis
• 12

  Let the complex number $${{z_1}}$$, $${{z_2}}$$ and $${{z_3}}$$ be the vertices of an equilateral triangle. Let $${{z_0}}$$ be the circumcentre of the triangle. Then prove that $$z_1^2 + z_2^2 + z_3^2 = 3z_0^2$$.

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  (A)

  The given equation holds true if and only if the triangle is equilateral and $$z_0$$ is its circumcenter.
• 13

  For every integer n > 1, the inequality $${(n!)^{1/n}} < {{n + 1} \over 2}$$ holds.

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  (B)

  FALSE
• 14

  Five balls of different colours are to be placed in there boxes of different size. Each box can hold all five. In how many different ways can be place the balls so that no box remains emply?

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  (B)

  300
• 15

  The area enclosed within the curve $$\left| x \right| + \left| y \right| = 1$$ is .................

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  (B)

  2 sq. units
• 16

  Let A be the centre of the circle $${x^2}\, + \,{y^2}\, - \,2x\,\, - 4y\, - 20 = 0\,$$. Suppose that the tangents at the points B (1, 7) and D (4. - 2) on the circle meet at the point C. Find the area of the quadrilateral ABCD.

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  (B)

  72 sq units
• 17

  Find the equations of the circle passing through (- 4, 3) and touching the lines x + y = 2 and x - y = 2.

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  A

  B
• 18

  The equation $${{{x^2}} \over {1 - r}} - {{{y^2}} \over {1 + r}} = 1,\,\,\,\,r > 1$$ represents

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  (D)

  none of these
• 19

  Each of the four inequalties given below defines a region in the $$xy$$ plane. One of these four regions does not have the following property. For any two points $$\left( {{x_1},{y_1}} \right)$$ and $$\left( {{x_2},{y_2}} \right)$$ in the region, the point $$\left( {{{{x_1} + {x_2}} \over 2},{{{y_1} + {y_2}} \over 2}} \right)$$ is also in the region. The inequality defining this region is

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  (C)

  $${x^2} - {y^2} \le 1$$
• 20

  Suppose $${\sin ^3}\,x\sin 3x = \sum\limits_{m = 0}^n {{C_m}\cos \,mx} $$ is an identity in x, where C₀, C₁ ,....C_n are constants, and $${C_n} \ne 0$$ , then the value of n is _____.

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  (C)

  6
• 21

  Let $$y = {e^{x\,\sin \,{x^3}}} + {\left( {\tan x} \right)^x}$$. Find $${{dy} \over {dx}}$$

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  (C)

  ${e^{x\,\sin {x^3}}}\left[ {\sin {x^3} + 3{x^3}\cos {x^3}} \right] + {\left( {\tan x} \right)^x}\left[ {{{x\sec^2 x} \over {\tan x}} + \log \,\tan x} \right]$
• 22

  Let the angles $$A, B, C$$ of a triangle $$ABC$$ be in A.P. and let $$b:c = \sqrt 3 :\sqrt 2 $$. Find the angle $$A$$.

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  (D)

  75°
• 23

  Let $$a, b, c$$ be positive real numbers Let
  $$\theta = {\tan ^{ - 1}}\sqrt {{{a\left( {a + b + c} \right)} \over {bc}}} + {\tan ^{ - 1}}\sqrt {{{b\left( {a + b + c} \right)} \over {ca}}} $$ $$ + {\,\,\tan ^{ - 1}}\sqrt {{{c\left( {a + b + c} \right)} \over {ab}}} $$

  Then $$\tan \theta = $$ ____________

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  (A)

  0
• 24

  Find the value of : $$\cos \left( {2{{\cos }^{ - 1}}x + {{\sin }^{ - 1}}x} \right)$$ at $$x = {1 \over 5}$$, where
  $$0 \le {\cos ^{ - 1}}x \le \pi $$ and $$ - \pi /2 \le {\sin ^{ - 1}}x \le \pi /2$$.

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  (A)

  $$\frac{-2\sqrt{6}}{5}$$
• 25

  For all $$x$$ in $$\left[ {0,1} \right]$$, let the second derivative $$f''(x)$$ of a function $$f(x)$$ exist and satisfy $$\left| {f''\left( x \right)} \right| < 1.$$ If $$f(0)=f(1)$$, then show that $$\left| {f\left( x \right)} \right| < 1$$ for all $$x$$ in $$\left[ {0,1} \right]$$.

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  (B)

  The problem statement is true, and the result follows from Taylor's theorem with remainder term.
• 26

  Let $$x$$ and $$y$$ be two real variables such that $$x>0$$ and $$xy=1$$. Find the minimum value of $$x+y$$.

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  (C)

  2
• 27

  Use the function $$f\left( x \right) = {x^{1/x}},x > 0$$. to determine the bigger of the two numbers $${e^\pi }$$ and $${\pi ^e}$$

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  (A)

  $$e^\pi$$ is bigger
• 28

  Evaluate $$\int {\left( {{e^{\log x}} + \sin x} \right)\cos x\,\,dx.} $$

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  (C)

  x \sin x + \cos x - \frac{1}{4}\cos 2x + C
• 29

  The value of the definite integral $$\int\limits_0^1 {\left( {1 + {e^{ - {x^2}}}} \right)} \,\,dx$$

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  (D)

  none of these
• 30

  Let $$a, b, c$$ be non-zero real numbers such that
  $$\int\limits_0^1 {\left( {1 + {{\cos }^8}x} \right)\left( {a{x^2} + bx + c} \right)dx = \int\limits_0^2 {\left( {1 + {{\cos }^8}x} \right)\left( {a{x^2} + bx + c} \right)dx.} } $$
  Then the quadratic equation $$a{x^2} + bx + c = 0$$ has

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  (B)

  at least one root in $$(0, 2)$$