What will be the nature of roots of (7x^2-5x-3=0)?
Correct answer: A
Here (D=(-5)^2-4(7)(-3)=109). (109) is positive but not a perfect square, so the roots are irrational and distinct.
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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Medium · Level 39 · quadratic-equations,irrational-roots,numerical
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- Two real irrational and distinct ((D=109))
- Two real rational and distinct ((D=25))
- Two real and equal ((D=0))
- No real roots ((D<0))
Hard · Level 39 · quadratic_equations,nature_of_roots,discriminant,real_roots
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- Two distinct real roots
- Two equal real roots
- No real roots
- One root is zero
Hard · Level 39 · quadratic_equations,nature_of_roots,equal_roots
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- Two distinct real roots
- Two equal real roots
- Imaginary roots
- Not real and unequal
Medium · Level 39 · quadratic-equations,discriminant,nature-of-roots,Nature of Roots,Quadratic Equations,Mathematics,Class 10 MCQ
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- Two real and equal roots
- Two real and distinct roots
- No real roots
- One real root
Hard · Level 39 · quadratic_equations,nature_of_roots,parameter
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- (2)
- (4)
- (8)
- (16)
Hard · Level 39 · quadratic_equations,nature_of_roots,parameter
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- (k=\pm3)
- (k=\pm6)
- (k=3) only
- (k=-3) only
Hard · Level 39 · quadratic_equations,nature_of_roots,concept
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- (b^2-4ac>0)
- (b^2-4ac=0)
- (b^2-4ac<0)
- (b^2+4ac=0)
Hard · Level 39 · quadratic_equations,nature_of_roots,concept
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- Two distinct real roots
- Two equal real roots
- No real roots
- Two negative roots
Easy · Level 39 · quadratic-equations,discriminant,sign-of-D,Nature of Roots,Quadratic Equations,Mathematics,Class 10 MCQ
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- D>0
- D=0
- D<0
- D=1
Hard · Level 39 · quadratic_equations,nature_of_roots,parameter
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- (m=\pm4)
- (m=\pm8)
- (m=16)
- (m=-16)
Hard · Level 39 · quadratic_equations,nature_of_roots,inequality
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- (k<\frac{25}{4})
- (k=\frac{25}{4})
- (k>\frac{25}{4})
- (k<-\frac{25}{4})
Hard · Level 39 · quadratic_equations,nature_of_roots,inequality
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- (k^2<64)
- (k^2=64)
- (k^2>64)
- (k^2<16)
Hard · Level 39 · quadratic_equations,nature_of_roots,signs
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- Two distinct real roots
- Two equal real roots
- No real roots
- Both roots are zero
Hard · Level 39 · quadratic_equations,nature_of_roots,perfect_square
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- Roots are real and equal
- Roots are real and distinct
- Roots are not real
- Roots are irrational
Medium · Level 39 · quadratic-equations,discriminant,equal-roots,Nature of Roots,Quadratic Equations,Mathematics,Class 10 MCQ
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- p=1
- p=2
- p=3
- p=4
Hard · Level 39 · quadratic_equations,nature_of_roots,rational_roots
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- Two distinct real and rational roots
- Two equal real roots
- No real roots
- Two irrational roots
Hard · Level 39 · quadratic_equations,nature_of_roots,irrational_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,irrational_roots
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- Two distinct real and irrational roots
- Two equal real roots
- No real roots
- Two distinct rational roots
Hard · Level 39 · quadratic_equations,nature_of_roots,parameter
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- (q=\pm6)
- (q=\pm12)
- (q=\pm18)
- (q=\pm24)
Hard · Level 39 · quadratic_equations,nature_of_roots,rational_roots
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- Two distinct real and rational roots
- Two equal real roots
- No real roots
- Two distinct irrational roots
What is the nature of the roots of the quadratic equation \(3x^2-5x+2=0\)?
Correct answer: A
Here, \(a=3\), \(b=-5\), and \(c=2\). The discriminant is \(\Delta=b^2-4ac=(-5)^2-4(3)(2)=25-24=1\). Since \(\Delta>0\), the equation has two distinct real roots. Two equal real roots occur only when \(\Delta=0\), so option B is incorrect. Exam tip: calculate the discriminant first to determine the nature of the roots.
View question detailsHow will the roots of the quadratic equation (x^2+6x+9=0) be?
Correct answer: B
Here (D=6^2-4\cdot1\cdot9=0), so the roots are equal real roots. In exams, (D=0) directly indicates equal roots.
View question detailsFind the nature of the roots of the quadratic equation 2x²+3x+5=0.
Correct answer: C
For a quadratic equation ax²+bx+c=0, the discriminant D=b²−4ac determines the nature of its roots. Here a=2, b=3 and c=5, so D=3²−4(2)(5)=9−40=−31. Because D is negative, the equation has no real roots; its two roots are complex conjugates. Therefore option C is correct. Equal real roots would require D=0, while two distinct real roots would require D>0. Option D is also incorrect because a non-degenerate quadratic equation does not have exactly one real root under the usual discriminant classification; the negative discriminant gives no real root at all.
View question detailsIf the roots of (kx^2+4x+1=0) are equal, what is the value of (k)?
Correct answer: B
For equal roots, (D=0), so (4^2-4\cdot k\cdot1=0) gives (k=4). In exams, use (D=0) when equal roots are given.
View question detailsIf the roots of (x^2-2kx+9=0) are equal, what are the possible values of (k)?
Correct answer: A
For equal roots, (D=0), so ((-2k)^2-4\cdot1\cdot9=0) gives (k=\pm3). In exams, when (k^2) appears, check both signs.
View question detailsUnder which condition will (ax^2+bx+c=0) have two distinct real roots?
Correct answer: A
For two distinct real roots, the discriminant (D=b^2-4ac) is positive. In exams, first calculate (D) to decide the nature.
View question detailsIf (D) is the discriminant of a quadratic equation and (D=0), what is the nature of roots?
Correct answer: B
When (D=0), both roots are equal and real. In exams, these may also be called repeated roots.
View question detailsWhat is the sign of D for the quadratic equation 5x²−2x+3=0?
Correct answer: C
For ax²+bx+c=0, the discriminant is D=b²−4ac. In this equation, a=5, b=−2 and c=3. Substitution gives D=(−2)²−4(5)(3)=4−60=−56. Since −56 is less than zero, the sign of D is negative, so option C is correct. The question asks only for the sign, not for the roots themselves; nevertheless, a negative discriminant also tells us that the equation has no real roots. Option A would correspond to a positive value, option B to exactly zero, and option D is not the result of the discriminant calculation.
View question detailsFor which (m) will (x^2+mx+16=0) have equal roots?
Correct answer: B
For equal roots, (D=0), so (m^2-64=0) and (m=\pm8). In exams, remember (m^2=a^2) gives (m=\pm a).
View question detailsIf (x^2+5x+k=0) has two distinct real roots, what is the correct condition for (k)?
Correct answer: A
For two distinct real roots, (D>0), so (25-4k>0) gives (k<\frac{25}{4}). In exams, keep the inequality sign correct while solving.
View question detailsIf (2x^2+kx+8=0) has no real roots, which condition is correct for (k)?
Correct answer: A
For no real roots, (D<0), so (k^2-64<0), that is (k^2<64). In exams, connect (D<0) with no real roots.
View question detailsWhat will be the nature of roots of (7x^2+2x-3=0)?
Correct answer: A
Here (D=2^2-4\cdot7\cdot(-3)=88), so (D>0). In exams, when (c) is negative, (D) can often become positive.
View question detailsIf (9x^2+12x+4=0), which statement about the nature of roots is correct?
Correct answer: A
Here (D=12^2-4\cdot9\cdot4=0), so the roots are real and equal. In exams, (9x^2+12x+4) can also be recognized as ((3x+2)^2).
View question detailsFor which value of p will px²+6x+3=0 have equal roots?
Correct answer: C
A quadratic equation has two equal real roots precisely when its discriminant is zero, provided the coefficient of x² is nonzero. For px²+6x+3=0, a=p, b=6 and c=3. Thus D=b²−4ac=6²−4(p)(3)=36−12p. Set D=0: 36−12p=0, so 12p=36 and p=3. This value also satisfies p≠0, so the equation remains genuinely quadratic. Therefore option C is correct. The other proposed values give D=24, 12 and −12 respectively, so they produce distinct real roots for p=1 or 2 and no real roots for p=4, not equal roots.
View question detailsWhat is the nature and type of roots of (x^2-7x+10=0)?
Correct answer: A
Here (D=(-7)^2-4\cdot1\cdot10=9), which is a positive perfect square. In exams, positive perfect-square (D) gives rational and distinct roots.
View question detailsChoose the correct nature of roots of (x^2-4x+1=0).
Correct answer: A
Here (D=(-4)^2-4\cdot1\cdot1=12), which is positive but not a perfect square. In exams, such (D) gives irrational distinct real roots.
View question detailsIf (D=18) is the discriminant of a quadratic equation, what is the correct statement about its roots?
Correct answer: A
(D=18) is positive but not a perfect square, so the roots are real, distinct, and irrational. In exams, also check whether (D) is a perfect square.
View question detailsFor which (q) will (3x^2+qx+12=0) have equal roots?
Correct answer: B
For equal roots, (D=0), so (q^2-144=0) and (q=\pm12). In exams, both signs of (b) may be possible.
View question detailsWhat is the nature of roots of (6x^2-11x+3=0)?
Correct answer: A
Here (D=(-11)^2-4\cdot6\cdot3=49), a positive perfect square. In exams, roots are rational in such cases.
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