10th Class Mathematics Polynomials Question Bank

done Polynomial Total Questions - 40

• question_answer1)
  The zeros of the polynomials \[f(x)=ab{{x}^{2}}+\left( {{b}^{2}}+ac \right)x+bc\] is ____.
  A)
   \[-\left( {{b}^{2}}+ac \right),bc\]          done clear
  B)
   \[\left( -\frac{b}{a}\And \frac{c}{b} \right)\] done clear
  C)
   \[\left( \frac{{{b}^{2}}+ac}{a},\frac{bc}{b} \right)\]     done clear
  D)
   \[\left( -\frac{b}{c}\And \frac{c}{a} \right)\] done clear
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• question_answer2)
  2 and (\[-2\]) are two zeros of the polynomial\[{{x}^{4}}-5{{x}^{2}}+4\]. What are the other two zeros of the polynomial?
  A)
   \[1,-1\] done clear
  B)
  \[1,-2\]         done clear
  C)
   \[-1,2\]                          done clear
  D)
   \[1,0\] done clear
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• question_answer3)
  Find the zeroes of the polynomials \[f(y)={{y}^{3}}+{{y}^{2}}-9y-9\], if two zeroes are equal in magnitude but opposite in sign. \[\alpha ,-\alpha ,\beta \]
  A)
   \[(1,-1,-3)\]                    done clear
  B)
   \[(+3,-3,-1)\] done clear
  C)
   \[(-2+\sqrt{3},2-\sqrt{3},-1)\]      done clear
  D)
   \[\left( \frac{1}{2},\frac{1}{2},9 \right)\] done clear
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• question_answer4)
                    Which of the following is a cubic polynomial whose zeros are \[\left( -1 \right),\text{ }\left( -2 \right)\] and\[\left( -3 \right)\]?
  A)
   \[{{x}^{3}}-6\]                       done clear
  B)
   \[{{x}^{3}}-6{{x}^{2}}+6\]                       done clear
  C)
   \[{{x}^{3}}-6{{x}^{2}}+11x+6\]     done clear
  D)
   \[{{x}^{3}}-6{{x}^{2}}-11x-6\] done clear
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• question_answer5)
  If a and b are the roots of the equation\[3{{m}^{2}}+6m-11\], find a polynomial whose roots are \[2a+1\] and\[2b+1\].
  A)
   \[k\left( {{m}^{2}}+3m-\frac{53}{3} \right)\]    done clear
  B)
   \[k\left( {{m}^{2}}+9m-41 \right)\] done clear
  C)
   \[k\left( {{m}^{2}}+9m+41 \right)\]      done clear
  D)
   \[k\left( {{m}^{2}}-9m-41 \right)\] done clear
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• question_answer6)
  For what values of k is one zero of the polynomial\[\left( {{k}^{2}}-16 \right){{x}^{2}}+9x+6k,\] the reciprocal of the other?
  A)
    \[9,\,0\]                        done clear
  B)
   \[-3,-6\]         done clear
  C)
   \[-9,0\]             done clear
  D)
   \[8,-2\] done clear
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• question_answer7)
  If \[1\pm \sqrt{3}\] are the two zeroes of the polynomial\[f\left( x \right)={{x}^{4}}-4{{x}^{3}}+{{x}^{2}}+6x+2,\]then the other two zeroes of the polynomial f(x) are:
  A)
   \[\left( -5+\sqrt{5},-5-\sqrt{5} \right)\]      done clear
  B)
   \[(1,2)\] done clear
  C)
   \[\left( 3+\sqrt{2},3-\sqrt{2} \right)\]        done clear
  D)
   \[\left( 1+\sqrt{2},and1-\sqrt{2} \right)\] done clear
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• question_answer8)
  Which of the following is the division algorithm of polynomials, if dividend = f(x), divisor = p(x), quotient = q(x) and remainder = r(x)?
  A)
   \[f(x)=r(x)p(x)+q(x)\]     done clear
  B)
   \[f(x)=p(x)q(x)+r(x)\] done clear
  C)
   \[f(x)=p(x)q(x)-r(x)\] done clear
  D)
   \[f(x)=p(x)r(x)-q(x)\] done clear
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• question_answer9)
  Find the values of C for which the zeroes of the polynomial \[f\left( x \right)={{x}^{3}}+3{{x}^{2}}+6x+C\]are in A.P.
  A)
   \[-36\]               done clear
  B)
   \[22\] done clear
  C)
   \[-\sqrt{35}\]                 done clear
  D)
   \[\sqrt{21}\] done clear
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• question_answer10)
  \[\alpha ,\beta \]and\[\gamma \]are the zeros of a cubic polynomial. If\[\alpha +\beta +\gamma =3,\alpha \beta +\beta \gamma +\gamma \alpha =(-5)\]and\[\alpha \beta \gamma =(-24),\] find the polynomial.
  A)
   \[{{x}^{3}}-3{{x}^{2}}-5x+24\]       done clear
  B)
   \[{{m}^{3}}+3{{m}^{2}}-5m-24\] done clear
  C)
   \[{{n}^{3}}-2{{n}^{2}}+7n+12\] done clear
  D)
   \[{{y}^{3}}+6{{y}^{2}}-10y-48\] done clear
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• question_answer11)
  Find the value of a and b such that \[{{y}^{2}}+1\]is the factor of \[g\left( z \right)={{y}^{4}}+{{y}^{3}}+8{{y}^{2}}+ay+b\]
  A)
   \[(-1,-7)\]                       done clear
  B)
   \[(-1,-1)\] done clear
  C)
   \[(1,2)\]                         done clear
  D)
   \[(1,7)\] done clear
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• question_answer12)
  If a, b are the roots of the equation\[a{{x}^{2}}-2bx+c=0\] then \[{{\alpha }^{3}}{{\beta }^{3}}+{{\alpha }^{2}}{{\beta }^{3}}+{{\alpha }^{3}}{{\beta }^{2}}=\]
  A)
   \[\frac{{{c}^{2}}\left( 2b+c \right)}{{{a}^{3}}}\]        done clear
  B)
   \[\frac{b{{c}^{2}}}{{{a}^{3}}}\] done clear
  C)
   \[\frac{{{c}^{3}}}{{{a}^{3}}}\]                                done clear
  D)
   \[\frac{{{c}^{3}}(b+2c)}{{{a}^{3}}}\] done clear
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• question_answer13)
  Which of the following is the graph of a linear polynomial?
  A)
    done clear
  B)
    done clear
  C)
     done clear
  D)
    done clear
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• question_answer14)
  \[\alpha \] and \[\beta \]are the zeros of a polynomial, such that  \[\alpha +\beta =6\]and \[\alpha \beta =4\]. Identify the polynomial.
  A)
   \[{{x}^{2}}-6x+4\]     done clear
  B)
   \[{{x}^{2}}+8x+6\] done clear
  C)
   \[{{x}^{2}}+16\]                    done clear
  D)
   \[{{x}^{2}}-4\] done clear
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• question_answer15)
  Find the zeros of the polynomial \[7{{n}^{2}}+3\sqrt{3}n-1.\]
  A)
   \[\frac{3}{14}\left( \sqrt{5}-\sqrt{3} \right)\]and \[\frac{3}{14}\left( -\sqrt{3}-\sqrt{5} \right)\] done clear
  B)
   \[3\sqrt{3}\]and \[\sqrt{3}\] done clear
  C)
   \[\frac{3\sqrt{2}}{5},\frac{\sqrt{2}}{2}\]                        done clear
  D)
   \[-3\sqrt{5}\] and\[\sqrt{5}\] done clear
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• question_answer16)
  Identify the zeros of the cubic polynomial \[p\left( m \right)=\text{ }{{m}^{3}}-\text{ }8{{m}^{2}}+19m-12\text{ }.\]
  A)
   \[-1,-3,-4\]                     done clear
  B)
   \[0,-3,4\] done clear
  C)
   \[1,3,4\] done clear
  D)
   \[-1,1,12\] done clear
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• question_answer17)
  The zeros of a cubic polynomial\[{{y}^{3}}-6{{y}^{2}}+11y+8\] are \[m,m-a,\text{ }m+a\]. What is the value of m?
  A)
   3                                 done clear
  B)
   2   done clear
  C)
   1                                 done clear
  D)
   0 done clear
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• question_answer18)
  What is the quotient when \[\text{ }3{{m}^{2}}-2m+3\] is divided by \[\left( 1-m \right)\]leaving a remainder 9?
  A)
   \[-2m+3\]                     done clear
  B)
   \[-(2m-3)\] done clear
  C)
   \[2m+3\]                      done clear
  D)
   \[-(3m+1)\] done clear
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• question_answer19)
  Identify the quadratic polynomial with no zeros.
  A)
     done clear
  B)
    done clear
  C)
     done clear
  D)
    done clear
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• question_answer20)
  If a polynomial p(x) is divided by another polynomial g(x), with a quotient q(x) and remainder r(x), then p(x) = q(x) g(x) + r(x). What is the condition that r(x) must satisfy?
  A)
   \[r\left( x \right)=0\] done clear
  B)
   \[r\left( x \right)=0\]or deg of \[r\left( x \right)>\]deg \[g\left( x \right)\] done clear
  C)
   Either \[r\left( x \right)=0\]or deg \[r\left( x \right)<\]deg \[(g(x))\] done clear
  D)
  \[r\left( x \right)=g\left( x \right)\] done clear
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• question_answer21)
  Find the sum of zeros of the polynomial \[7{{x}^{2}}-17.\]
  A)
   0     done clear
  B)
   1            done clear
  C)
   -1                                done clear
  D)
   2 done clear
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• question_answer22)
  If \[\alpha \] and \[\beta \]are the zeros of \[a{{x}^{2}}-{{a}^{2}}x+{{a}^{3}}\]where a < 1, and a > 0 then, which of the following is correct?
  A)
   \[\alpha +\beta <\alpha \beta \]              done clear
  B)
   \[\alpha +\beta >\alpha \beta \] done clear
  C)
   \[\beta -\alpha >\alpha \beta \]                done clear
  D)
   \[\alpha +\beta =\alpha \beta \] done clear
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• question_answer23)
  If \[\alpha \] and \[\beta \] are two zeros of the quadratic polynomial \[p\left( x \right)=3{{x}^{2}}-6x+5,\] find the value of \[{{\alpha }^{3}}+{{\beta }^{3}}\].
  A)
   \[\frac{30}{13}\]                                   done clear
  B)
   \[\frac{35}{6}\] done clear
  C)
   \[-2\]    done clear
  D)
   \[2\] done clear
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• question_answer24)
  What is the nature of the zeros of the quadratic polynomial \[2{{x}^{2}}+63x+\text{ }63\].
  A)
   Both positive                      done clear
  B)
   Both negative done clear
  C)
   One positive, one negative          done clear
  D)
   Cannot be said done clear
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• question_answer25)
  What is the relation between the zeros and the coefficients of the polynomial\[4\sqrt{3}{{x}^{2}}+5x-2\sqrt{3}\]?
  A)
   \[\text{Sum}\,\,\text{of}\,\,\text{zeros}=\frac{-(Coefficient\,\,of\,\,x)}{(coefficient\,\,of\,\,{{x}^{2}})}\] done clear
  B)
   \[\text{Sum}\,\,\text{of}\,\,\text{zeros}=\frac{(Coefficient\,\,of\,\,x)}{(coefficient\,\,of\,\,{{x}^{2}})}\] done clear
  C)
   \[\text{Sum}\,\,\text{of}\,\,\text{zeros}=\frac{(Coefficient\,\,of\,\,{{x}^{2}})}{(coefficient\,\,of\,\,x)}\] done clear
  D)
   \[\text{Product}\,\,\text{of}\,\,\text{zeros}=\frac{(Coefficient\,\,of\,\,{{x}^{2}})}{\text{constant}}\] done clear
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• question_answer26)
  Choose the zeros of the polynomial whose graph is given.
  A)
   0, 2, 3              done clear
  B)
   - 3, 2, 5 done clear
  C)
   0, 0, 0              done clear
  D)
   0 done clear
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• question_answer27)
  Find the quadratic polynomial, one of whose zeros is \[\frac{-\sqrt{3}}{\sqrt{2}}\] and the product of zeros is 1.
  A)
   \[3\sqrt{2}{{x}^{2}}+7x+3\]  done clear
  B)
   \[\sqrt{6}{{x}^{2}}+5x+\sqrt{6}\] done clear
  C)
   \[\sqrt{6}{{x}^{2}}+5x+\sqrt{6}\]      done clear
  D)
   \[3{{x}^{2}}+\frac{5}{2\sqrt{3}}x-\frac{1}{6}\] done clear
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• question_answer28)
  Find the remainder when \[8{{p}^{3}}-10{{p}^{2}}+11p-24\] is divided by \[(1-3p+{{p}^{2}})\].
  A)
   \[53p-24\]         done clear
  B)
   \[-10p+2\] done clear
  C)
   \[p-10\]             done clear
  D)
   \[6p-7\] done clear
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• question_answer29)
  A quadratic polynomial\[f(x)={{x}^{2}}-(m+n)x+mn\] has two zeros. Find the value of\[{{\alpha }^{2}}{{\beta }^{2}}\].
  A)
   \[\frac{1}{4}\left( {{m}^{2}}+{{n}^{2}} \right)\]                    done clear
  B)
   \[\frac{1}{6}\left( {{m}^{2}}+4{{n}^{3}} \right)\] done clear
  C)
   \[{{m}^{2}}+{{n}^{2}}\] done clear
  D)
   \[\frac{1}{8}\left( {{m}^{2}}+4{{n}^{2}} \right)\] done clear
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• question_answer30)
  The polynomial \[{{a}^{3}}+{{a}^{2}}+a+1\]is divided by a polynomial g. The quotient and remainder obtained are \[\left( a+1 \right)\] and \[\left( -2a-3 \right)\] respectively, Find g.
  A)
   \[{{a}^{2}}+2\]                     done clear
  B)
   \[{{a}^{2}}+4\] done clear
  C)
   \[2{{a}^{2}}-a\]                     done clear
  D)
   \[{{a}^{2}}+a+4\] done clear
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• question_answer31)
  When is a real number 'a' called the zero of the polynomial f(x)?
  A)
   f (0) = a           done clear
  B)
   f  = a          done clear
  C)
   f  = 0      done clear
  D)
   f  = f (0) done clear
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• question_answer32)
  If A and B are the zeros of the polynomial \[a{{x}^{2}}+bx+c,\]what is the value of\[{{A}^{2}}+{{B}^{2}}\]?
  A)
   \[\frac{{{b}^{2}}+2ac}{{{a}^{2}}}\]                        done clear
  B)
   \[\frac{{{a}^{2}}+2ac}{{{b}^{2}}}\] done clear
  C)
   \[\frac{{{a}^{2}}-2ac}{{{b}^{2}}}\]             done clear
  D)
   \[\frac{{{b}^{2}}-2ac}{{{a}^{2}}}\] done clear
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• question_answer33)
  If a and b can take values 1, 2, 3, 4, then the number of the equations of the form \[a{{x}^{2}}+bx+1=0\] having real roots is
  A)
   10                                done clear
  B)
   7         done clear
  C)
   6                                 done clear
  D)
   12 done clear
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• question_answer34)
  If one zero of the quadratic polynomial \[2{{x}^{2}}+(6m+5)x-3=0\] is negative of the other, find the value of 'm'.
  A)
   \[-\frac{35}{6}\]                        done clear
  B)
   \[2\] done clear
  C)
   \[\frac{9}{4}\]                          done clear
  D)
   \[-\frac{5}{6}\] done clear
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• question_answer35)
  The roots of \[x+\frac{1}{x}=3,\] when \[x\ne 0\] are:
  A)
   \[3,-3\]              done clear
  B)
   \[\frac{3+\sqrt{5}}{2},\frac{3-\sqrt{5}}{2}\] done clear
  C)
   \[\frac{5+\sqrt{3}}{2},\frac{5-\sqrt{3}}{2}\]      done clear
  D)
   \[\frac{5+\sqrt{6}}{2},\frac{5-\sqrt{6}}{2}\] done clear
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• question_answer36)
  Choose the graph of a quadratic polynomial.
  A)
             done clear
  B)
    done clear
  C)
             done clear
  D)
    done clear
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• question_answer37)
  If \[\alpha \] and \[\beta \] are the roots of the given equation \[2\sqrt{3}{{x}^{2}}+4x-3\sqrt{3},\] then the value of\[\frac{1}{{{\alpha }^{3}}}+\frac{1}{{{\beta }^{3}}}\]is _______.
  A)
   0                                 done clear
  B)
   \[-\frac{280\sqrt{3}}{243}\] done clear
  C)
   \[+\frac{280\sqrt{3}}{243}\]     done clear
  D)
   \[\frac{280}{243}\] done clear
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• question_answer38)
  If the equation \[(sin\theta -1){{x}^{2}}+(sin\theta )x+cos\theta =0\] has real roots, then \[\theta \] =
  A)
   \[[0,\pi ]\]                      done clear
  B)
   \[\left[ 0,\frac{3\pi }{2} \right]\] done clear
  C)
   \[\left[ -\frac{\pi }{2},\frac{\pi }{2} \right]\]          done clear
  D)
   \[[0,2\pi ]\] done clear
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• question_answer39)
  If \[\alpha ,\beta \] and \[\gamma \] are the zeros of p(x), which of the following is true?
  A)
   \[p(x)=(x-\alpha )(x-\beta )(x-\gamma )\] done clear
  B)
   \[p(x)=(x+\alpha )(x-\beta )(x-\gamma )\] done clear
  C)
   \[p(x)=(x+\alpha )(x+\beta )(x-\gamma )\]           done clear
  D)
   \[p(x)=(x+\alpha )(x+\beta )(x+\gamma )\] done clear
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• question_answer40)
  Which of the following is not correct?
  A)
   The degree of a zero polynomial is zero. done clear
  B)
   The polynomial \[{{x}^{5}}-{{x}^{3}}-{{x}^{2}}+1\] has utmost five zeros (three real and two imaginary). done clear
  C)
   The degree of a cubic polynomial is 3. done clear
  D)
   A quadratic polynomial has a maximum of two zeros. done clear
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