JEE Main & Advanced Mathematics Equations and Inequalities Question Bank

done Self Evaluation Test - Linear Inequalities Total Questions - 50

• question_answer1)
  The shaded region shown in the figure is given by the in equations
  A)
  \[14x+5y\ge 70,\,\,\,y\le 14\] and \[x-y\ge 5\] done clear
  B)
  \[14x+5y\le 70,\,\,\,y\le 14\] and \[x-y\ge 5\] done clear
  C)
  \[14x+5y\ge 70,y\ge 14\] and \[x-y\ge 5\] done clear
  D)
  \[14x+5y\ge 70,y\le 14\] and \[x-y\le 5\] done clear
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• question_answer2)
  The number of real values of parameter k for which \[{{(lo{{g}_{16}}x)}^{2}}-{{\log }_{16}}x+{{\log }_{16}}k=0\] will have exactly one solution is
  A)
  0 done clear
  B)
  2 done clear
  C)
  1 done clear
  D)
  4 done clear
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• question_answer3)
  Solve for \[x,\,\,\,\frac{\left| x+3 \right|+x}{x+2}>1\]
  A)
  \[x\in (-5,-2)\cup (-1,\infty )\] done clear
  B)
  \[x\in (5,2)\cup (-1,\infty )\] done clear
  C)
  \[x\in (5,2)\] done clear
  D)
  \[x\in (-1,\infty )\] done clear
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• question_answer4)
  The solution set of the inequality \[37-(3x+5)\ge 9x-8(x-3)\] is
  A)
  \[(-\infty ,2)\] done clear
  B)
  \[(-\infty ,-2)\] done clear
  C)
  \[(-\infty ,2]\] done clear
  D)
  \[(-\infty ,-2]\]   done clear
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• question_answer5)
  Solution of \[2x-1=\left| x+7 \right|\] is
  A)
  -2 done clear
  B)
  8 done clear
  C)
  -2, 8      done clear
  D)
  None of these done clear
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• question_answer6)
  The inequality representing the following graph is
  A)
  \[\left| x \right|<3\] done clear
  B)
  \[\left| x \right|\le 3\] done clear
  C)
  \[\left| x \right|>3\] done clear
  D)
  \[\left| x \right|\ge 3\] done clear
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• question_answer7)
  If \[5\{x\}=x+[x]\] and \[[x]-\{x\}=\frac{1}{2}\] when \[\{x\}\] and \[[x]\] are fractional and integral part of x then x is
  A)
  \[\frac{1}{2}\] done clear
  B)
  \[\frac{3}{2}\] done clear
  C)
  \[\frac{5}{2}\] done clear
  D)
  \[\frac{7}{2}\] done clear
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• question_answer8)
  Ravi obtained 70 and 75 marks in first two unit tests. Then the minimum marks he should get in the third test to have an average of at least 60 marks, are
  A)
  45 done clear
  B)
  35 done clear
  C)
  25 done clear
  D)
  None of these done clear
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• question_answer9)
  If \[R\ge r>0\] and \[d>0,\] then \[0<\frac{{{d}^{2}}+{{R}^{2}}-{{r}^{2}}}{2dR}\le 1\]
  A)
  Is satisfied if \[\left| d-R \right|\le r\] done clear
  B)
  Is satisfied if \[\left| d-R \right|\le 2r\] done clear
  C)
  Is satisfied if \[\left| d-R \right|\ge r\] done clear
  D)
  Is not satisfied at all done clear
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• question_answer10)
  The length of a rectangle is three times the breadth. If the minimum perimeter of the rectangle is 160 cm, then what can you say about breadth?
  A)
  \[\operatorname{breadth} = 20\] done clear
  B)
  \[breadth\,\,\le 20\] done clear
  C)
  \[breadth\,\,\ge 20\] done clear
  D)
  \[breadth\,\,\ne 20\] done clear
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• question_answer11)
  Which of the following linear inequalities satisfy the shaded region of the given figure?
  A)
  \[2x+3y\ge 3\] done clear
  B)
  \[3x+4y\le 18\] done clear
  C)
  \[x-6y\le 3\] done clear
  D)
  All of these done clear
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• question_answer12)
  The solution set of \[{{(x)}^{2}}+{{(x+1)}^{2}}=25,\] where (x) is the least integer greater than or equal to x, is
  A)
  \[(2,4)\] done clear
  B)
  \[(-5,-4]\cup (2,3]\] done clear
  C)
  \[[-4,-3)\cup [3,4)\] done clear
  D)
  None of these done clear
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• question_answer13)
  A man wants to cut three lengths from a single piece of board of length 91 cm. The second length is to be 3 cm longer than the shortest and the third length is to be twice as long as the shortest. The possible length of the shortest board, if the third piece is to be at least 5 cm longer than the second, is
  A)
  Less than 8 cm done clear
  B)
  Greater than or equal to 8 cm but less then or equal to 22 cm done clear
  C)
  Less than 22 cm done clear
  D)
  Greater than 22 cm   done clear
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• question_answer14)
  The solution set of \[\frac{2x-1}{3}\ge \left( \frac{3x-2}{4} \right)-\left( \frac{2-x}{5} \right)\] is\[(-\infty ,a]\]. The value of ?a? is
  A)
  2 done clear
  B)
  3 done clear
  C)
  4 done clear
  D)
  5 done clear
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• question_answer15)
  For positive real numbers a, b, c such that \[a+b+c=p,\] which one does not hold?
  A)
  \[(p-a)(p-b)(p-c)\le \frac{8}{27}{{p}^{3}}\] done clear
  B)
  \[(p-a)(p-b)(p-c)\ge 8abc\] done clear
  C)
  \[\frac{bc}{a}+\frac{ca}{b}+\frac{ab}{c}\le p\] done clear
  D)
  None of these done clear
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• question_answer16)
  If x satisfies the inequalities \[x+7<2x+3\] and \[2x+4<5x+3,\] then x lies in the interval
  A)
  \[(-\infty ,3)\] done clear
  B)
  \[(1,3)\] done clear
  C)
  \[(4,\infty )\] done clear
  D)
         \[(-\infty ,-1)\] done clear
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• question_answer17)
  The solution set of the inequality \[{{5}^{x+2}}>{{\left( \frac{1}{25} \right)}^{1/x}}\]is
  A)
  \[(-2,0)\] done clear
  B)
  \[(-2,2)\] done clear
  C)
  \[(-5,5)\] done clear
  D)
  \[(0,\infty )\] done clear
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• question_answer18)
  The solution set of the inequality \[\left| x+2 \right|-\left| x-1 \right|<x-\frac{3}{2}\] is
  A)
  \[\left( \frac{9}{2},\infty  \right)\] done clear
  B)
  \[\left( -\infty ,\frac{3}{2} \right)\] done clear
  C)
  \[\left( -2,-\frac{3}{2} \right)\] done clear
  D)
  \[\left( -1,\frac{3}{2} \right)\] done clear
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• question_answer19)
  If \[\frac{3x-4}{2}\ge \frac{x+1}{4}-1,\] then \[x\in \]
  A)
  \[[1,\infty )\] done clear
  B)
  \[(1,\infty )\] done clear
  C)
  \[(-5,5)\] done clear
  D)
  \[[-5,5]\] done clear
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• question_answer20)
  Solution set of the inequality \[{{\log }_{3}}(x+2)(x+4)+lo{{g}_{1/3}}(x+2)<\frac{1}{2}\log \sqrt{3}\,\,7\,(1)\]  is
  A)
  \[(-2,-1)\] done clear
  B)
  \[(-2,3)\] done clear
  C)
  \[(-1,3)\] done clear
  D)
  \[(3,\infty )\] done clear
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• question_answer21)
  The area and perimeter of a rectangle are A and P respectively. Then P and A satisfy the inequality.
  A)
  \[P+A>PA\] done clear
  B)
  \[{{P}^{2}}\le A\] done clear
  C)
  \[A-P<2\] done clear
  D)
  \[{{P}^{2}}\ge 16A\] done clear
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• question_answer22)
  If \[\left| \frac{12x}{4{{x}^{2}}+9} \right|\ge 1\] for all real values of \[x,\] the inequality being satisfied only if         \[\left| x \right|\] is equal to
  A)
  \[\frac{3}{2}\] done clear
  B)
  \[\frac{2}{3}\] done clear
  C)
  \[\frac{1}{3}\] done clear
  D)
  \[\frac{1}{2}\] done clear
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• question_answer23)
  A manufacturer has 600 liters of a 12% solution of acid. How many liters of a 30% acid solution must be added to it so that acid content in the resulting mixture will be more than 15% but less than 18%?
  A)
  More than 120 liters but less than 300 liters done clear
  B)
  More than 140 liters but less than 600 liters done clear
  C)
  More than 100 liters but less than 280 liters done clear
  D)
  More than 160 liters but less than 500 liters done clear
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• question_answer24)
  The number of integral roots of the equation\[\left| x-1 \right|+\left| x+2 \right|-\left| x-3 \right|=4\] is
  A)
  0 done clear
  B)
  1 done clear
  C)
  2 done clear
  D)
  4 done clear
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• question_answer25)
  Solution of \[\left| x-1 \right|\ge \left| x-3 \right|\] is
  A)
  \[x\le 2\] done clear
  B)
  \[x\ge 2\] done clear
  C)
  \[[1,3]\]  done clear
  D)
  None of these done clear
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• question_answer26)
  If, a, b, c, distinct positive real numbers then the expression \[\left( b+c-a \right)\text{ }\left( c+a-b \right)\text{ }\left( a+b-c \right)-abc\] is
  A)
  Positive done clear
  B)
  Negative done clear
  C)
  Non-positive done clear
  D)
  Non-negative done clear
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• question_answer27)
  The set of real values of x satisfying \[\left| x-1 \right|\le 3\] and \[\left| x-1 \right|\ge 1\] is
  A)
  \[[2,4]\] done clear
  B)
  \[(-\infty ,2]\cup [4,+\infty )\] done clear
  C)
  \[[-2,0]\cup [2,4]\] done clear
  D)
  None of these done clear
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• question_answer28)
  The set of real values of x satisfying \[\left| x-1 \right|\le 3\] And \[\left| x-1 \right|\ge 1\] is
  A)
  \[[2,4]\] done clear
  B)
  \[(-\infty ,2]\cup [4,+\infty )\] done clear
  C)
  \[[-2,0]\cup [2,4]\] done clear
  D)
  None of these done clear
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• question_answer29)
  Set of values of x satisfying the inequality \[\frac{{{x}^{2}}+6x-7}{\left| x+4 \right|}<0\] is/are
  A)
  \[(-\infty ,-7)\] done clear
  B)
  \[(-7,4)\] done clear
  C)
  \[(-4,1)\] done clear
  D)
  \[(1,\infty )\] done clear
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• question_answer30)
  A vertex of a feasible region by the linear constraints \[3x+4y\le 18,\,\,\,2x+3y\ge 3\] and \[x,y\ge 0\], is
  A)
  \[(0,2)\] done clear
  B)
  \[(4.8,0)\] done clear
  C)
  \[(0,3)\] done clear
  D)
  None of these done clear
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• question_answer31)
  For \[x\in R,\,\,\,\,\left\langle x \right\rangle \] is defined as follows: \[\left\langle x \right\rangle =\left\{ \begin{matrix}    x+1,  \\    \left| x-4 \right|,  \\ \end{matrix}\,\,\,\begin{matrix}    0\le x<2  \\    x\ge 2  \\ \end{matrix}\begin{matrix}    {}  \\    {}  \\ \end{matrix} \right.\] Then the solution set of the equation \[{{\left\langle x \right\rangle }^{2}}+x=\left\langle x \right\rangle +{{x}^{2}}\] is
  A)
  \[\{-1,1\}\] done clear
  B)
  \[[2,\infty )\] done clear
  C)
  \[[0,2)\] done clear
  D)
  \[\{0,2\}\] done clear
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• question_answer32)
  If \[\frac{\left| x+3 \right|+x}{x+2}>1,\] then \[x\in \]
  A)
  \[(-5,-2)\] done clear
  B)
  \[(-1,\infty )\] done clear
  C)
  \[(-5,-2)\cup (-1,\infty )\] done clear
  D)
  None of these done clear
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• question_answer33)
  The equation \[\left| |x-1|+a \right|=4\] can have real solution for x if 'a' belongs to the interval
  A)
  \[(-\infty ,+\infty )\] done clear
  B)
  \[(-\infty ,4]\] done clear
  C)
  \[(4,+\infty )\] done clear
  D)
  \[[-4,4]\] done clear
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• question_answer34)
  Number of integral values of x satisfying the inequality \[{{\left( \frac{3}{4} \right)}^{6x+10-{{x}^{2}}}}<\frac{27}{64}\] is
  A)
  5 done clear
  B)
  6 done clear
  C)
  7 done clear
  D)
  8 done clear
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• question_answer35)
  The graph of in equations \[x\le y\] and \[y\le x+3\] is located in
  A)
  II quadrant done clear
  B)
  I, II quadrants done clear
  C)
  I, II, III quadrants done clear
  D)
  II, III, IV quadrants done clear
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• question_answer36)
  Consider the following statements.

  I. Solution set of the inequality \[-15<\frac{3(x-2)}{5}\le 0\]is \[(-23,2]\]

  II. Solution set of the inequality

  \[7\le \frac{3x+11}{2}\le 11\] is \[\left[ 1,\frac{11}{3} \right]\]

  III. Solution set of the inequality

  \[-5\le \frac{2-3x}{4}\le 9\] is \[[-1,\,\,\,1]\cup [3,\,\,\,5]\]

  Choose the correct option

  A)
  Only I and II are true. done clear
  B)
  Only II and III are true. done clear
  C)
  Only I and III are true. done clear
  D)
  All are true.   done clear
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• question_answer37)
  The marks obtained by a student of class \[XI\]in first and second terminal examinations are 62 and 48, respectively. The minimum marks he should get in the annual examination to have an average of at least 60 marks, are
  A)
  70 done clear
  B)
  50 done clear
  C)
  74 done clear
  D)
  48 done clear
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• question_answer38)
  Solution set of the inequality \[\frac{1}{{{2}^{x}}-1}>\frac{1}{1-{{2}^{x-1}}}\] is
  A)
  \[(1,\infty )\] done clear
  B)
  \[(0,lo{{g}_{2}}(4/3))\] done clear
  C)
  \[(-1,\infty )\] done clear
  D)
  \[(0,lo{{g}_{2}}(4/3))\cup (1,\infty )\] done clear
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• question_answer39)
  If the equation \[{{2}^{x}}+{{4}^{y}}={{2}^{y}}+{{4}^{x}}\] is solved for y in terms of x where\[x<0\], then the sum of the solutions is
  A)
  \[x{{\log }_{2}}(1-{{2}^{x}})\] done clear
  B)
  \[x+{{\log }_{2}}(1-{{2}^{x}})\] done clear
  C)
  \[{{\log }_{2}}(1-{{2}^{x}})\] done clear
  D)
  \[x{{\log }_{2}}({{2}^{x}}+1)\] done clear
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• question_answer40)
  The set of real value of x satisfying \[\left| |x-1|-1 \right|\le 1\]is
  A)
  \[[-1,3]\] done clear
  B)
  \[[0,2]\] done clear
  C)
  \[[-1,1]\] done clear
  D)
  None of these   done clear
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• question_answer41)
  A company manufactures cassettes. Its cost and revenue functions are \[C(x)=26000+30x\] and \[R(x)=43x,\] respectively, where x is the number of cassettes produced and sold in a week. The number of cassettes must be sold by the company to realise some profit, is
  A)
  More than 2000 done clear
  B)
         Less than 2000 done clear
  C)
  More than 1000 done clear
  D)
         Less than 1000 done clear
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• question_answer42)
  If a, b and c are three positive real numbers such that \[a+b\ge c,\] then
  A)
  \[\frac{a}{1+a}+\frac{b}{1+b}\ge \frac{c}{1+c}\] done clear
  B)
  \[\frac{a}{1+a}+\frac{b}{1+b}<\frac{c}{1+c}\] done clear
  C)
  \[\frac{a}{1+a}+\frac{b}{1+b}>\frac{c}{1+c}\] done clear
  D)
  None of these done clear
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• question_answer43)
  The least integer a, for which \[1+{{\log }_{5}}({{x}^{2}}+1)\le lo{{g}_{5}}(a{{x}^{2}}+4x+a)\] is true for all \[x\in R\] is
  A)
  6 done clear
  B)
  7 done clear
  C)
  10 done clear
  D)
  1 done clear
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• question_answer44)
  Solution of \[\left| 2x-3 \right|<\left| x+2 \right|\] is
  A)
  \[\left( -\infty ,\frac{1}{3} \right)\] done clear
  B)
  \[\left( \frac{1}{3},5 \right)\] done clear
  C)
  \[(5,\infty )\] done clear
  D)
  \[\left( -\infty ,\frac{1}{3} \right)\cup (5,\infty )\] done clear
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• question_answer45)
  The solution set of the inequalities \[3x-7>2(x-6)\] and \[6-x>11-2x,\] is
  A)
  \[(-5,\infty )\] done clear
  B)
  \[[5,\infty )\] done clear
  C)
  \[(5,\infty )\] done clear
  D)
  \[[-5,\infty )\] done clear
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• question_answer46)
  If \[{{(\sqrt{2})}^{x}}+{{(\sqrt{3})}^{x}}={{(\sqrt{13})}^{x/2}},\] then the number of values of x is
  A)
  2 done clear
  B)
  4 done clear
  C)
  1 done clear
  D)
  None of these done clear
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• question_answer47)
  Number of real roots of the equation \[\sqrt{x}+\sqrt{x-\sqrt{1-x}}=1\] is
  A)
  0 done clear
  B)
  1 done clear
  C)
  2 done clear
  D)
  3 done clear
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• question_answer48)
  The number of real roots of the equation \[\left| 2-\left| 1-\left| x \right| \right| \right|=1\] is
  A)
  1 done clear
  B)
  3 done clear
  C)
  5 done clear
  D)
  6 done clear
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• question_answer49)
  The set of points (x, y) satisfying the inequalities \[x+y\le 1,-x-y\le 1\] lies in the region bounded by the two straight lines passing through the respective pair of points
  A)
  \[\{(1,0)(0,1)\}\] and \[\{(-1,0),(0,-1)\}\] done clear
  B)
  \[\{(1,0),(1,1)\}\] and \[\{(-1,0),(0,-1)\}\] done clear
  C)
  \[\{(-1,0),(0,-1)\}\] and \[\{(1,0),(-1,1)\}\] done clear
  D)
  \[\{(1,0),(0,-1)\}\] and \[\{(-1,0),(0,1)\}\]   done clear
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• question_answer50)
  The equation \[{{\log }_{3}}({{3}^{x}}-8)=2-x\] has the solution
  A)
  \[x=1\] done clear
  B)
  \[x=2\] done clear
  C)
  \[x=3\] done clear
  D)
  \[x=4\] done clear
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