What will be the nature of roots of (2x^2+5x-12=0)?
Correct answer: A
Here (D=5^2-4\cdot2\cdot(-12)=121), so there are two distinct real roots. In exams, (D=121) is also a positive perfect square.
View question detailsMathematics
Nature of Roots
मूलों की प्रकृति
In Class 10 Mathematics, Nature of Roots explains how to determine the type and number of solutions of a quadratic equation. Students use the discriminant, b² − 4ac, to identify whether an equation has two distinct real roots, two equal real roots, or no real roots. The topic connects algebraic calculations with the graph of a quadratic function and helps learners interpret equations, compare cases, and solve related problems from the chapter Quadratic Equations.
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Hard · Level 39 · quadratic_equations,nature_of_roots,discriminant
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- Two distinct real roots
- Two equal real roots
- No real roots
- Two equal irrational roots
Hard · Level 39 · quadratic_equations,nature_of_roots,parameter
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- (h=\frac{3}{4})
- (h=\frac{9}{4})
- (h=3)
- (h=9)
Hard · Level 39 · quadratic_equations,nature_of_roots,no_real_roots
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- Two distinct real and irrational roots
- Two equal real roots
- No real roots
- Two rational roots
Hard · Level 39 · quadratic equations,nature of roots,discriminant,parameter
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- \(k=\pm9\)
- \(k=\pm18\)
- \(k=\pm3\)
- \(k=\pm27\)
Hard · Level 39 · quadratic_equations,nature_of_roots,fractions
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- Two equal real roots
- Two distinct real roots
- No real roots
- Two irrational roots
Hard · Level 39 · quadratic_equations,nature_of_roots,fractions
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- Two equal real roots
- Two distinct real roots
- No real roots
- Only one root is real
Hard · Level 39 · quadratic_equations,nature_of_roots,relation
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- (p^2=4q)
- (p^2>4q)
- (p^2<4q)
- (p=4q)
Hard · Level 39 · quadratic_equations,nature_of_roots,concept
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- Two equal real roots
- Two distinct real roots
- No real roots
- Two unequal imaginary roots
Hard · Level 39 · quadratic_equations,nature_of_roots,parameter_discriminant
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- Two distinct real roots for all real (k)
- Two equal real roots for all real (k)
- No real roots for any real (k)
- Equal roots only when (k=0)
Hard · Level 39 · quadratic_equations,nature_of_roots,real_roots_condition
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- (k\le1) and (k\neq0)
- (k<1) only
- (k>1)
- (k=0)
Hard · Level 39 · quadratic_equations,nature_of_roots,no_equal_roots
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- (m=2)
- (m=-2)
- No such real (m) exists
- All real (m)
Expert · Level 37 · quadratic-equations,nature-of-roots,expert
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- Two real rational and distinct
- Two real irrational and distinct
- Two real and equal
- No real roots
Expert · Level 37 · quadratic equations,nature of roots,discriminant,surd coefficients,equal roots
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- Two real and equal (\(D=0\))
- Two real, rational and distinct (\(D=4\))
- Two real, irrational and distinct (\(D=3\))
- No real roots (\(D<0\))
Expert · Level 37 · quadratic-equations,nature-of-roots,discriminant,surd-coefficients,no-real-roots
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- No real roots \(\Delta=-4\)
- Two real and equal roots \(\Delta=0\)
- Two real, rational and distinct roots \(\Delta=4\)
- Two real, irrational and distinct roots \(\Delta=5\)
Expert · Level 37 · quadratic-equations,nature-of-roots,discriminant,parameter-inequality,real-roots
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- \(k\geq\frac{3}{2}\)
- \(k<\frac{3}{2}\)
- Only \(k=\frac{3}{2}\)
- Every real \(k\)
Expert · Level 37 · quadratic-equations,discriminant,nature-of-roots,parameter-based-questions
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- \(m\ne 3\)
- \(m=3\)
- \(m>3\)
- \(m<3\)
Expert · Level 37 · quadratic-equations,nature-of-roots,discriminant,parameter-inequality,real-roots
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- \(p\leq 0,\ p\ne -1\)
- \(p<0,\ p\ne -1\)
- \(p\geq 0,\ p\ne -1\)
- All \(p\ne -1\)
Expert · Level 37 · quadratic-equations,nature-of-roots,equal-roots,discriminant,parameter-based-problems
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- \(k=4\)
- \(k=-4\)
- \(k=8\)
- \(k=12\)
Expert · Level 37 · quadratic-equations,nature-of-roots,discriminant,equal-roots,parameter
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- \(k=4\)
- \(k=12\)
- \(k=-4\)
- \(k=0\)
Expert · Level 37 · quadratic-equations,no-real-roots,parameter-interval
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- (0<k<3)
- (k\leq0) or (k\geq3)
- (k=0) or (k=3)
- Every (k)
If (x^2-3x+h=0) has equal roots, what is the value of (h)?
Correct answer: B
For equal roots, (D=0), so (9-4h=0) and (h=\frac{9}{4}). In exams, the (D=0) method remains valid even with fractional answers.
View question detailsWhat is the nature of roots of (8x^2-8x+3=0)?
Correct answer: C
Here (D=(-8)^2-4\cdot8\cdot3=-32), so there are no real roots. In exams, the sign of (D) gives the final decision.
View question detailsIf the roots of the quadratic equation \(3x^2-2kx+27=0\) are equal, what are the possible values of \(k\)?
Correct answer: A
For equal roots, the discriminant must be zero: \(D=b^2-4ac=0\). Here, \(a=3\), \(b=-2k\), and \(c=27\), so \((-2k)^2-4(3)(27)=0\), giving \(4k^2=324\). Hence \(k^2=81\), and therefore \(k=\pm9\). Thus, option A is correct. Exam tip: For equal roots of a quadratic equation, set the discriminant equal to zero.
View question detailsWhat is the nature of roots of \(x^2+\frac{1}{2}x+\frac{1}{16}=0\)?
Correct answer: A
Here \(D=\left(\frac{1}{2}\right)^2-4\cdot1\cdot\frac{1}{16}=0\), so the roots are equal and real. In exams, square the denominator too while squaring a fraction.
View question detailsHow are the roots of \(4x^2+\frac{4}{3}x+\frac{1}{9}=0\)?
Correct answer: A
Here \(D=\left(\frac{4}{3}\right)^2-4\cdot4\cdot\frac{1}{9}=0\), so the roots are equal and real. In exams, calculate (4ac) carefully with fractions.
View question detailsIf (x^2+px+q=0) has two equal real roots, which relation is correct?
Correct answer: A
For equal roots, (D=0), so (p^2-4q=0) gives (p^2=4q). In exams, the formula becomes simpler when (a=1).
View question detailsIf (b^2=4ac) in a quadratic equation, what will be the nature of its roots?
Correct answer: A
When (b^2=4ac), (D=b^2-4ac=0), so the roots are equal and real. In exams, treat this as the discriminant zero condition.
View question detailsIf (x^2-2(k+3)x+(k^2+6k+5)=0), what will be the nature of its roots?
Correct answer: A
Here (D=[-2(k+3)]^2-4(k^2+6k+5)=16), so (D>0). In exams, the sign of (D) after simplification gives the final answer.
View question detailsIf (kx^2-6x+9=0) has real roots, what is the correct condition for (k)?
Correct answer: A
For real roots, (D\ge0), so (36-36k\ge0) gives (k\le1). Also (k\neq0) is required because the equation must remain quadratic.
View question detailsWhich conclusion is correct for equal real roots of ((m-2)x^2+2mx+(m+2)=0)?
Correct answer: C
Here (D=(2m)^2-4(m-2)(m+2)=16), so (D\neq0) always. In exams, (D=0) is necessary for equal roots.
View question detailsIf (a\neq0) and (b^2-4ac) is a positive perfect square in (ax^2+bx+c=0), what is the most accurate nature of the roots?
Correct answer: A
A positive perfect-square discriminant gives two distinct rational real roots. In exams check not only (D>0) but also whether (D) is a perfect square.
View question detailsWhat is the nature of the roots of the equation \(4x^2-4\sqrt{3}x+3=0\)?
Correct answer: A
Here, \(a=4\), \(b=-4\sqrt{3}\), and \(c=3\). Therefore, the discriminant is \(D=b^2-4ac=(-4\sqrt{3})^2-4(4)(3)=48-48=0\). When \(D=0\), the quadratic equation has two real and equal roots. In fact, the repeated root is \(x=\frac{\sqrt{3}}{2}\). Thus, option B is incorrect because the discriminant is not \(4\); it is \(0\). Exam tip: To determine the nature of the roots, first calculate \(D=b^2-4ac\).
View question detailsChoose the correct conclusion about the nature of the roots of \(x^2-2\sqrt{5}x+6=0\).
Correct answer: A
For the given quadratic equation, \(a=1\), \(b=-2\sqrt{5}\), and \(c=6\). Thus, the discriminant is \(\Delta=b^2-4ac=(-2\sqrt{5})^2-4(1)(6)=20-24=-4\). Since \(\Delta<0\), the equation has no real roots, so option A is correct. Exam tip: a negative discriminant means the roots are non-real; equal real roots occur only when \(\Delta=0\).
View question detailsIf \(k\) is a real number and the roots of the equation \(x^2-2(k+1)x+(k^2+4)=0\) are real, what is the correct condition on \(k\)?
Correct answer: A
For a quadratic equation to have real roots, its discriminant must satisfy \(D\geq0\). Here, \(a=1\), \(b=-2(k+1)\), and \(c=k^2+4\), so \(D=b^2-4ac=4(k+1)^2-4(k^2+4)=8k-12\). Thus, \(8k-12\geq0\), giving \(k\geq\frac{3}{2}\). At \(k=\frac{3}{2}\), the roots are equal and real, so option C is too restrictive. Exam tip: use \(D\geq0\) for real roots and \(D>0\) only for distinct real roots.
View question detailsWhat condition on \(m\) is necessary for the equation \(x^2-(m+3)x+3m=0\) to have two real and distinct roots?
Correct answer: A
A quadratic equation has two real and distinct roots only when its discriminant satisfies \(D>0\). Here, \(D=(m+3)^2-4(1)(3m)=m^2-6m+9=(m-3)^2\). This is zero when \(m=3\) and positive for every other real value of \(m\). Therefore, the required condition is \(m\ne3\). In option B, the roots are real but equal, not distinct. Exam tip: use \(D>0\) for distinct real roots, \(D=0\) for equal roots, and \(D<0\) for non-real roots.
View question detailsFor the equation \((p+1)x^2-2(p+2)x+(p+4)=0\), where \(p\ne -1\), what condition on \(p\) ensures that the roots are real?
Correct answer: A
Here, \(a=p+1\), \(b=-2(p+2)\), and \(c=p+4\). Therefore, the discriminant is \(D=b^2-4ac=4(p+2)^2-4(p+1)(p+4)=-4p\). For real roots, \(D\geq 0\), which gives \(p\leq 0\). Since \(p=-1\) is excluded, the complete condition is \(p\leq 0,\ p\ne -1\). Option B is incorrect because at \(p=0\), \(D=0\), giving two equal real roots. Exam tip: For real roots, apply \(D\geq 0\) and also verify that the coefficient of \(x^2\) is non-zero.
View question detailsFor what value of \(k\) will the quadratic equation \(2x^2-(k+4)x+2k=0\) have equal roots?
Correct answer: A
For equal roots, the discriminant \(D=b^2-4ac\) must be zero. Here, \(a=2\), \(b=-(k+4)\), and \(c=2k\). Thus, \(D=(k+4)^2-16k=k^2-8k+16=(k-4)^2\). Setting this equal to zero gives \(k=4\). The other values do not make the discriminant zero. Exam tip: equal roots in a quadratic equation always require \(D=0\).
View question detailsFor which value of \(k\) will the two roots of the quadratic equation \(2x^2-(k+4)x+2k=0\) be equal?
Correct answer: A
Equal roots require the discriminant to be zero. Here \(a=2\), \(b=-(k+4)\), and \(c=2k\), so \(D=b^2-4ac=(k+4)^2-16k=(k-4)^2\). Thus, \((k-4)^2=0\) gives \(k=4\). For example, when \(k=-4\), \(D=64\), so the roots are not equal. Exam tip: For equal roots of a quadratic equation, set \(b^2-4ac=0\).
View question detailsIf (x^2+2(k-1)x+(k+5)=0) has no real roots, in which interval will (k) lie?
Correct answer: A
Here (D=4(k-1)^2-4(k+5)=4(k^2-3k-4)). Use (D<0) and factor carefully before choosing the interval.
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