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Quadratic Equations

10th Class - CBSE - Mathematics - 6109 Questions - 0 Concepts

Important Questions

Q1 Multiple Correct Hard
If the quadratic equation $$ax^{2}+2bx-4=0$$ ,where $$a,b,c\in R$$, does not have two real & distinct roots, then
  • A. $$a-b>1$$
  • B. $$2b-a\leq -4$$
  • C. $$2b-a\geq -4$$
  • D. $$a+b\leq 1$$

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Q2 Single Correct Hard
If $$p,q$$ are any two values of $$k$$ for which $$\dfrac{\displaystyle \sum _{ r=0 }^{ k-1 }{ { x }^{ 2r } } }{\displaystyle \sum _{ r=0 }^{ k-1 }{ { x }^{ r } } }$$ is a polynomial in $$x$$, then roots of equation $$3x^2 + px + 5q = 0 $$ cannot be
  • A. real
  • B. imaginary
  • C. irrational
  • D. rational

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Q3 Single Correct Hard
If, for a positive integer n, the quadratic equation, $$ x ( x + 1 ) + ( x + 1 ) ( x + 2 ) + \ldots + \left( x + \frac { 1 } { n - 1 } \right) ( x + n ) = 10 n $$ has two consecutive integral solutions, then n is equal to
  • A. 12
  • B. 9
  • C. 10
  • D. 11

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Q4 Single Correct Hard
A trader bought a number of articles for $$Rs\ 1200$$. Ten were damaged and he sold each of the remaining articles at $$Rs\ 2$$ more than what he paid for it, thus getting a profit of $$Rs\ 60$$ on the whole transaction. Taking no. of articles he bought as $$x$$, find $$x$$.
  • A. $$92$$
  • B. $$85$$
  • C. $$56$$
  • D. $$100$$

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Q5 One Word Hard
The hypotenuse of a right triangle is $$2$$ centimeters more than the longer of the side containing the right angle. The shorter of the side containing the right angle is $$7$$ centimeters less than the longer side containing the right angle. Find the length of the hypotenuse.

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Q6 Single Correct Hard
The sum of a number and its reciprocal is $$\dfrac {82}{9}$$. Find the number.
  • A. $$x=8$$ and $$x=\dfrac {1}{8}$$
  • B. $$x=10$$ and $$x=\dfrac {1}{10}$$
  • C. $$x=0.9$$ and $$x=\dfrac {1}{0.9}$$
  • D. $$x=9$$ and $$x=\dfrac {1}{9}$$

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Q7 Single Correct Hard
The values of $$\lambda $$ for which the equation 
$$(x^{2}+3x+5)^{2}-(\lambda -3)(x^{2}+3x+4)(x^{2}+3x+5)+(\lambda -4)(x^{2}+3x+4)^{2}=0$$
 has at least one real root are
  • A. $$\left ( 5,\dfrac{39}{7} \right )$$
  • B. $$\left ( 5,\dfrac{39}{7} \right ]$$
  • C. $$\left [ 5,\dfrac{39}{7} \right )$$
  • D. $$\left [ 5,\dfrac{39}{7} \right ]$$

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Q8 Single Correct Hard
If each pair among the equations $$x^{2} + qr =0$$, $$x^{2} +qx +rp=0$$ and $$x^{2} + rx+pq=0$$ have a common root then the product of these common roots is
  • A. $$-pq^{2}$$
  • B. $$pq (p+q)$$
  • C. $$-(pq^{2} +pq^{2})$$
  • D. $$2pqr$$

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Q9 Assertion & Reason Hard
ASSERTION

If the roots of the equations $${ x }^{ 2 }-bx+c=0$$ and $${ x }^{ 2 }-cx+b=0$$ differ by the same quantity, then $$b+c$$ is equal to $$-4$$

REASON

If $$\alpha, \beta$$ are the roots of the equation $$A{ x }^{ 2 }+Bx+C=0$$, then $$\displaystyle \alpha -\beta =\frac { \sqrt { { B }^{ 2 }-4AC }  }{ A } $$

  • A. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
  • B. Assertion is correct but Reason is incorrect
  • C. Assertion is incorrect but Reason is correct
  • D. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

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Q10 Single Correct Hard
If the quadratic equation $$ax^{2}+bx+a^{2}+b^{2}+c^{2}-ab-bc-ca=0$$, where $$a, b, c$$ are distinct real numbers, has imaginary roots, then
  • A. $$2\left ( a-b \right )+\left ( a-b \right )^{2}+\left ( b-c \right )^{2}+\left ( c-a \right )^{2}< 0$$
  • B. $$2\left ( a-b \right )+\left ( a-b \right )^{2}+\left ( b-c \right )^{2}+\left ( c-a \right )^{2}= 0$$
  • C. none of these
  • D. $$2\left ( a-b \right )+\left ( a-b \right )^{2}+\left ( b-c \right )^{2}+\left ( c-a \right )^{2}> 0$$

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Questions 122475
Subjects 10
Chapters 93
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