RS Aggarwal Solutions Class 10 Chapter 10 Quadratic Equations

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RS Aggarwal Solutions Class 10 Chapter 10

Question 1:

(i)   x²-x+3=0 is a quadratic polynomial.
∴  x²-x+3=0 is a quadratic equation.

(ii) 2x²+ [latex]\frac { 5 }{ 2 }   [/latex]x-√3=0
⇒ 4x²+5x-2√3=0
Clearly is 4x²+5x-2√3=0 a quadratic polynomial.
∴ 2x²+ [latex]\frac { 5 }{ 2 }   [/latex]x-√3=0 is a quadratic equation.

(iii) √2x²+7x+5√2=0 is a quadratic polynomial.
∴ √2x²+7x+5√2=0 is a quadratic equation.

(iv)[latex]\frac { 1 }{ 3 }   [/latex]x²+[latex]\frac { 1 }{ 5 }   [/latex]x-2=0
⇒ 5x²+3x-2=0
Clearly, 5x²+3x-2=0 is a quadratic equation.
[latex]\frac { 1 }{ 3 }   [/latex]x²+[latex]\frac { 1 }{ 5 }   [/latex] is a quadratic equation.

(v) x²-3x-√x+4=0 is not a quadratic polynomial since it contains √x, in which power 1/2 of x is not an integer.
∴ x²-3x-√x+4=0 is not a quadratic equation.

(vi) x-[latex]\frac { 6}{ x }   [/latex]=3
⇒ x²-3x-6 =0
And (x²-3x-6)Being a polynomial of degree 2, it is a quadratic polynomial.
Hence, x-[latex]\frac { 6}{ x }   [/latex]=3 is a quadratic equation.

(vii) x+[latex]\frac { 2}{ x }   [/latex]= x²
⇒ x³-x²-2 =0
And (x³-x²-2 =0) being a polynomial of degree 3, it is not a quadratic polynomial.
Hence, x+[latex]\frac { 2}{ x }   [/latex]= x² is not a quadratic equation.

(viii) [latex]{ x }^{ 2 }-\frac { 1 }{ { x }^{ 2 } } =5  [/latex] ⇒ x⁴ -1=5x²
⇒x⁴-5x²-1 =0
And (x⁴-5x²-1 =0) being a polynomial of degree 4.
Hence [latex]{ x }^{ 2 }-\frac { 1 }{ { x }^{ 2 } } =5  [/latex] is not a quadratic equation.

Question 2:
The given equation is 3x²+2x-1=0
(i) On substituting x = -1 in the equation, we get

(ii) On substituting [latex]x=\frac { 1 }{ 3 }   [/latex] in the equation, we get

(iii) On substituting [latex]x=-\frac { 1 }{ 2 }    [/latex] in the equation , we get

Question 3:
Since x = 1 is a solution of x²+kx+3=0 it must satisfy the equation.

Hence the required value of k = -4
Question 4:
Since [latex]x=\frac { 3 }{ 4 }   [/latex] is a root of  ax²+bx-6=0, we have


Again x = -2 being a root of ax²+bx-6=0, we have

Multiplying (2) by 4 adding the result from (1), we get
11a = 44 ⇒ a = 4
Putting a = 4 in (1), we get

Question 5:

Question 6:

Question 7:

Hence, 9 and -9 are the roots of the equation 3x²-243=0.
Question 8:

Hence, -5 and -7 are the roots of x²+12x+35=0.
Question 9:

Hence, 11 and 7 are the roots of equation x²=18x-77
Question 10:

Hence, [latex]x=-\frac { 1 }{ 3 }    [/latex] is the repeated root of the equation 9x²+6x+1=0
Question 11:

Hence,  is the repeated root of the equation
Question 12:

Hence, [latex]x=-\frac { 3 }{ 2 }   [/latex], [latex]x=-\frac { 1 }{ 2 }   [/latex]are the roots of 6x²+11x+3=0
Question 13:

Hence, [latex]x=\frac { 4 }{ 3 }   [/latex] and [latex]x=-\frac { 3 }{ 2 }   [/latex] are the roots of equation 6x²+x-12=0

Question 14:

Hence, [latex]x=-\frac { 1 }{ 3 }   [/latex] and 1 are the roots of the equation 3x²-2x-1=0.
Question 15:

Hence, [latex]x=\frac { 2 }{ 3 }   [/latex] and [latex]x=-\frac { 1 }{ 2 }   [/latex]are the roots of equation 6x²-x-2=0.
Question 16:

Hence, [latex]x=-\frac { 1 }{ 16 }   [/latex] and [latex]x=\frac { 2 }{ 3 }   [/latex] are the roots of 48x²-13x-1=0.
Question 17:

Hence, [latex]x=-\frac { 5 }{ 3 }   [/latex] and x=-2 are the roots of the equation 3x²+11x+10=0
Question 18:

Hence,[latex]x=\frac { 25 }{ 4 }   [/latex] and x=-4 are the roots of the equation 4x²-9x=100.
Question 19:

Hence,  [latex]x=\frac { 4 }{ 9 }   [/latex] and 2 are the roots of the equation 9x²-22+8=0
Question 20:

Hence,  [latex]x=\frac { 7 }{ 5 }   [/latex] and [latex]x=-\frac { 4 }{ 3 }   [/latex] are the roots of the given equation 15x²-28=x.
Question 21:

Hence, [latex]x=\frac { 1 }{ 3 }   [/latex] and -4 are the roots of given equation .
Question 22:

Hence, 1 and  √2 are the roots of the given equation
Question 23:

Question 24:

Question 25:

Hence, [latex]\frac { -\sqrt { 7 }  }{ 3 }      [/latex] and [latex]\frac { \sqrt { 7 }  }{ 7 }     [/latex] are the roots of given equation.
Question 26:

Hence, -√7 and [latex]\frac { 13\sqrt { 7 }  }{ 7 }     [/latex] are the roots of given equation.
Question 27:

Hence, [latex]\frac { 2\sqrt { 6 }  }{ 3 }     [/latex] and  [latex]\frac { -\sqrt { 6 }  }{ 8 }     [/latex]are the roots of given equation.
Question 28:

Hence, 5 and [latex]-\frac { 7 }{ 5 }    [/latex]are the roots of given equation
Question 29:

Hence, [latex]-\frac { 1 }{ 5 }    [/latex] and [latex]\frac { 1 }{ 2 }    [/latex]are the roots of given equation.
Question 30:

Hence, 2 and  [latex]\frac { 1 }{ 2 }    [/latex] are the roots of given equation.
Question 31:

Hence, [latex]-\frac { b }{ a }    [/latex] and [latex]\frac { c }{ b }    [/latex] are the roots of given equation.
Question 32:

Hence, [latex]\frac { -1 }{ { a }^{ 2 } }     [/latex] and [latex]\frac { 1 }{ { b }^{ 2 } }     [/latex]are the roots of given equation.
Question 33:

Hence, [latex]\frac { 3a }{ 4b }    [/latex] and [latex]\frac { -2b }{ 3a }    [/latex] are the roots of given equation.
Question 34:

Hence, [latex]\frac { { a }^{ 2 } }{ 2 }      [/latex] and [latex]\frac { { b }^{ 2 } }{ 2 }      [/latex]are the roots of given equation.
Question 35:

Hence, 2 and 1 are the roots of the given equation
Question 36:

Hence, -9 and 7 are the roots of the given equation
Question 37:

Hence, -4 and [latex]\frac { 9 }{ 4 }    [/latex] are the roots of the given equation
Question 38:

Hence  [latex]\frac { 40 }{ 13 }    [/latex] and 6 are the roots of the given equation
Question 39:

Hence, 4 and [latex]-\frac { 2 }{ 9 }    [/latex] are the roots of the given equation
Question 40:

Hence, 3 and [latex]\frac { 4 }{ 3 }    [/latex] are the roots of the given equation.
Question 41:

Hence, 5 and [latex]\frac { 1 }{ 2 }    [/latex] are the roots of the given equation.
Question 42:
Putting  the given equation become

Case I:

Case II:

Hence, [latex]-\frac { 3 }{ 2 }    [/latex] and -2 are the roots of the given equation
Question 43:

Putting  the given equation become

Case I:

Case II:

Hence, -1 and [latex]-\frac { 23 }{ 5 }    [/latex] are the roots of the given equation
Question 44:
On putting  the given equation become

Case I:

Case II:

Hence, -10 and [latex]-\frac { 1 }{ 5 }    [/latex] are the roots of the given equation.
Question 45:
Putting  the given equation become

Case I:

Case II:

Hence, -1 and [latex]\frac { 1 }{ 8 }    [/latex] are the roots of the given equation
Question 46:
The given equation

Hence, (a+b) and [latex]\frac { (a+b) }{ 2 }  [/latex] is the roots of the given equation
Question 47:

Hence, [latex]\frac { a+b }{ ab }   [/latex] and [latex]\frac { 2 }{ a+b }   [/latex] are the roots of the given equation
Question 48:

Hence, -2,0 are the roots of the given equation
Question 49:

Hence,  [latex]\frac { 1 }{ 2 }    [/latex] and [latex]\frac { 1 }{ 2 }    [/latex] are the roots of the given equation
Question 50:

Hence, 3 and 2 are the roots of the given equation.

Complete RS Aggarwal Solutions Class 10

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