JEE MAIN - Mathematics (2011)

• 1

  If the mean deviation about the median of the numbers a, 2a,........., 50a is 50, then |a| equals

  回答

  (A)

  4
• 2

  The value of $$p$$ and $$q$$ for which the function
  
  $$f\left( x \right) = \left\{ {\matrix{ {{{\sin (p + 1)x + \sin x} \over x}} & {,x < 0} \cr q & {,x = 0} \cr {{{\sqrt {x + {x^2}} - \sqrt x } \over {{x^{3/2}}}}} & {,x > 0} \cr } } \right.$$
  
  is continuous for all $$x$$ in R, are

  回答

  (B)

  $$p =$$ $$-{3 \over 2}$$, $$q = $$ $${1 \over 2}$$
• 3

  $$\mathop {\lim }\limits_{x \to 2} \left( {{{\sqrt {1 - \cos \left\{ {2(x - 2)} \right\}} } \over {x - 2}}} \right)$$

  回答

  (D)

  does not exist
• 4

  The domain of the function f(x) = $${1 \over {\sqrt {\left| x \right| - x} }}$$ is

  回答

  (B)

  $$\left( { - \infty ,0} \right)$$
• 5

  Let $$\alpha \,,\beta $$ be real and z be a complex number. If $${z^2} + \alpha z + \beta = 0$$ has two distinct roots on the line Re z = 1, then it is necessary that :

  回答

  (C)

  $$\beta \, \in (1,\infty )$$
• 6

  For $$x \in \left( {0,{{5\pi } \over 2}} \right),$$ define $$f\left( x \right) = \int\limits_0^x {\sqrt t \sin t\,dt.} $$ Then $$f$$ has

  回答

  (C)

  local maximum at $$\pi $$ and local minimum at $$2\pi $$
• 7

  The vectors $$\overrightarrow a $$ and $$\overrightarrow b $$ are not perpendicular and $$\overrightarrow c $$ and $$\overrightarrow d $$ are two vectors satisfying $$\overrightarrow b \times \overrightarrow c = \overrightarrow b \times \overrightarrow d $$ and $$\overrightarrow a .\overrightarrow d = 0\,\,.$$ Then the vector $$\overrightarrow d $$ is equal to :

  回答

  (C)

  $$\overrightarrow c - \left( {{{\overrightarrow a .\overrightarrow c } \over {\overrightarrow a .\overrightarrow b }}} \right)\overrightarrow b $$
• 8

  If $$C$$ and $$D$$ are two events such that $$C \subset D$$ and $$P\left( D \right) \ne 0,$$ then the correct statement among the following is :

  回答

  (A)

  $$P\left( {{C \over D}} \right)$$$$ \ge P\left( C \right)$$
• 9

  If $${{dy} \over {dx}} = y + 3 > 0\,\,$$ and $$y(0)=2,$$ then $$y\left( {\ln 2} \right)$$ is equal to :

  回答

  (D)

  $$7$$
• 10

  Let $$I$$ be the purchase value of an equipment and $$V(t)$$ be the value after it has been used for $$t$$ years. The value $$V(t)$$ depreciates at a rate given by differential equation $${{dv\left( t \right)} \over {dt}} = - k\left( {T - t} \right),$$ where $$k>0$$ is a constant and $$T$$ is the total life in years of the equipment. Then the scrap value $$V(T)$$ of the equipment is

  回答

  (A)

  $$I - {{k{T^2}} \over 2}$$
• 11

  The area of the region enclosed by the curves $$y = x,x = e,y = {1 \over x}$$ and the positive $$x$$-axis is :

  回答

  (B)

  $${3 \over 2}$$ square units
• 12

  The value of $$\int\limits_0^1 {{{8\log \left( {1 + x} \right)} \over {1 + {x^2}}}} dx$$ is

  回答

  (D)

  $$\pi \log 2$$
• 13

  The number of values of $$k$$ for which the linear equations
  $$4x + ky + 2z = 0,kx + 4y + z = 0$$ and $$2x+2y+z=0$$ possess a non-zero solution is :

  回答

  (A)

  $$2$$
• 14

  Let $$A$$ and $$B$$ be two symmetric matrices of order $$3$$.
  
  Statement - 1 : $$A(BA)$$ and $$(AB)$$$$A$$ are symmetric matrices.
  
  Statement - 2 : $$AB$$ is symmetric matrix if matrix multiplication of $$A$$ with $$B$$ is commutative.

  回答

  (A)

  statement - 1 is true, statement - 2 is true; statement - 2 is not a correct explanation for statement - 1.
• 15

  $${{{d^2}x} \over {d{y^2}}}$$ equals:

  回答

  (C)

  $$ - \left( {{{{d^2}y} \over {d{x^2}}}} \right){\left( {{{dy} \over {dx}}} \right)^{ - 3}}$$
• 16

  Equation of the ellipse whose axes of coordinates and which passes through the point $$(-3,1)$$ and has eccentricity $$\sqrt {{2 \over 5}} $$ is :

  回答

  (D)

  $$3{x^2} + 5{y^2} - 32 = 0$$
• 17

  A man saves ₹ 200 in each of the first three months of his service. In each of the subsequent months his saving increases by ₹ 40 more than the saving of immediately previous month. His total saving from the start of service will be ₹ 11040 after

  回答

  (C)

  21 months
• 18

  
  Statement - 1: The number of ways of distributing 10 identical balls in 4 distinct boxes such that no box is emply is $${}^9{C_3}$$.
  Statement - 2: The number of ways of choosing any 3 places from 9 different places is $${}^9{C_3}$$.

  回答

  (A)

  Statement - 1 is true, Statement - 2 is true, Statement - 2 is not a correct explanation for Statement - 1.
• 19

  These are 10 points in a plane, out of these 6 are collinear, if N is the number of triangles formed by joining these points. then:

  回答

  (A)

  $$N \le 100$$
• 20

  If $$\omega ( \ne 1)$$ is a cube root of unity, and $${(1 + \omega )^7} = A + B\omega \,$$. Then $$(A,B)$$ equals :

  回答

  (A)

  (1 ,1)
• 21

  If $$A = {\sin ^2}x + {\cos ^4}x,$$ then for all real $$x$$:

  回答

  (D)

  $${{3} \over {4}} \le A \le 1$$
• 22

  Statement - 1 : The point $$A(1,0,7)$$ is the mirror image of the point
  
  $$B(1,6,3)$$ in the line : $${x \over 1} = {{y - 1} \over 2} = {{z - 2} \over 3}$$
  
  Statement - 2 : The line $${x \over 1} = {{y - 1} \over 2} = {{z - 2} \over 3}$$ bisects the line
  
  segment joining $$A(1,0,7)$$ and $$B(1, 6, 3)$$

  回答

  (A)

  Statement -1 is true, Statement -2 is true; Statement -2 is not a correct explanation for Statement -1.
• 23

  Let $R$ be the set of real numbers.
  
  Statement I : $A=\{(x, y) \in R \times R: y-x$ is an integer $\}$ is an equivalence relation on $R$.
  
  Statement II : $ B=\{(x, y) \in R \times R: x=\alpha y$ for some rational number $\alpha\}$ is an equivalence relation on $R$.

  回答

  (B)

  Statement I is true, Statement II is false.
• 24

  Let $$\overrightarrow a $$, $$\overrightarrow b $$, $$\overrightarrow c $$ be three non-zero vectors which are pairwise non-collinear. If $\overrightarrow a+3 \overrightarrow b$ is collinear with $\overrightarrow c$ and $\overrightarrow b+2 \overrightarrow c$ is collinear with $\overrightarrow a$, then $\overrightarrow a+\overrightarrow b+6 \overrightarrow c$ is :

  回答

  (D)

  $\overrightarrow 0$