For 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 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.
TOPIC PRACTICE
Quiz this set
Up to 20 questions from this page. Select your focus, then start.
20 questions
Choose questions
Hard · Level 39 · quadratic_equations,nature_of_roots,parameter
View options
- (q=\pm6)
- (q=\pm12)
- (q=\pm18)
- (q=\pm24)
Hard · Level 39 · quadratic_equations,nature_of_roots,rational_roots
View options
- Two distinct real and rational roots
- Two equal real roots
- No real roots
- Two distinct irrational roots
Hard · Level 39 · quadratic_equations,nature_of_roots,no_real_roots
View options
- Two distinct real roots
- Two equal real roots
- No real roots
- Two rational roots
Hard · Level 39 · quadratic equations,nature of roots,discriminant,equal roots,parameter
View options
- \(k=4\) या \(k=-8\)
- \(k=8\) या \(k=-4\)
- \(k=6\) या \(k=-6\)
- \(k=3\) या \(k=-9\)
Hard · Level 39 · quadratic_equations,nature_of_roots,parameter
View options
- (t=10)
- (t=20)
- (t=25)
- (t=100)
Hard · Level 39 · quadratic_equations,nature_of_roots,discriminant
View options
- Two real and distinct roots
- Two real and equal roots
- No real roots
- Two irrational roots
Hard · Level 39 · quadratic_equations,nature_of_roots,parameter
View options
- (k=0) or (k=1)
- (k=1) or (k=2)
- (k=0) or (k=4)
- (k=-1) or (k=1)
Hard · Level 39 · quadratic_equations,nature_of_roots,surd_coefficient
View options
- Two equal real roots
- Two distinct real roots
- No real roots
- Two rational roots
Hard · Level 39 · quadratic_equations,nature_of_roots,surd_coefficient
View options
- Two distinct real roots
- Two equal real roots
- No real roots
- Two zero roots
Hard · Level 39 · quadratic equations,nature of roots,discriminant,parameter
View options
- \(m=3\) या \(m=-5\)
- \(m=4\) या \(m=-4\)
- \(m=5\) या \(m=-3\)
- \(m=2\) या \(m=-6\)
Hard · Level 39 · quadratic_equations,nature_of_roots,parameter_inequality
View options
- (r^2<\frac{25}{4}) and (r\neq0)
- (r^2=\frac{25}{4})
- (r^2>\frac{25}{4})
- (r=0)
Hard · Level 39 · quadratic_equations,nature_of_roots,inequality
View options
- (\lambda>1)
- (\lambda=1)
- (\lambda<1)
- (\lambda\le1)
Hard · Level 39 · quadratic_equations,nature_of_roots,common_factor
View options
- Two equal real roots
- Two distinct real roots
- No real roots
- Two irrational roots
Hard · Level 39 · quadratic_equations,nature_of_roots,no_real_roots
View options
- It has two distinct real roots
- It has two equal real roots
- It has no real roots
- It has two rational roots
Hard · Level 39 · quadratic_equations,nature_of_roots,inequality
View options
- (n<2)
- (n=2)
- (n>2)
- (n\le2)
Hard · Level 39 · quadratic_equations,nature_of_roots,quadratic_condition
View options
- (s=0)
- (s=1)
- (s=2)
- (s=4)
Hard · Level 39 · quadratic_equations,nature_of_roots,concept
View options
- Two distinct real roots
- Two equal real roots
- No real roots
- Two rational roots
Hard · Level 39 · quadratic_equations,nature_of_roots,graphical
View options
- (0) times
- (1) time
- (2) times
- (4) times
Hard · Level 39 · quadratic_equations,nature_of_roots,graphical
View options
- (0) times
- (1) time
- (2) times
- (3) times
Hard · Level 39 · quadratic_equations,nature_of_roots,graphical
View options
- (0) times
- (1) time
- (2) times
- (5) times
What 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.
View question detailsWhat type of roots will (3x^2-4x+2=0) have?
Correct answer: C
Here (D=(-4)^2-4\cdot3\cdot2=-8), so there are no real roots. In exams, negative (D) means no intersection on the real number line.
View question detailsIf the two roots of the equation \(x^2+(k+2)x+9=0\) are equal, what are the possible values of \(k\)?
Correct answer: A
For a quadratic equation to have equal roots, its discriminant must be zero. Here, \(a=1\), \(b=k+2\), and \(c=9\), so \(D=(k+2)^2-4(1)(9)=0\). Therefore, \((k+2)^2=36\), giving \(k+2=\pm6\) and hence \(k=4\) or \(k=-8\). Option B results from reversing the signs incorrectly. Exam tip: For equal roots, begin by setting the discriminant \(D\) equal to zero.
View question detailsFor which (t) will (x^2-10x+t=0) have real and equal roots?
Correct answer: C
For equal roots, (D=0), so (100-4t=0) gives (t=25). In exams, (b^2=4ac) is a quick sign of equal roots.
View question detailsWhat is the nature of roots of (2x^2-7x+6=0)?
Correct answer: A
Here (D=(-7)^2-4\cdot2\cdot6=1), so the roots are two distinct real roots. In exams, (D=1) is also a positive perfect square.
View question detailsIf (4x^2+4kx+k=0) has equal roots, what will be the values of (k)?
Correct answer: A
From (D=0), ((4k)^2-4\cdot4\cdot k=0), that is (16k(k-1)=0). So (k=0) or (k=1), and (a=4\neq0).
View question detailsWhat is the nature of roots of (x^2+2\sqrt{3}x+3=0)?
Correct answer: A
Here (D=(2\sqrt{3})^2-4\cdot1\cdot3=0), so the roots are equal and real. In exams, be careful while squaring a surd coefficient.
View question detailsWhat will be the nature of roots of (x^2+2\sqrt{2}x+1=0)?
Correct answer: A
Here (D=(2\sqrt{2})^2-4\cdot1\cdot1=4), so (D>0). In exams, (D=4) gives two distinct real roots.
View question detailsIf the two roots of the equation \(x^2-2(m+1)x+16=0\) are equal, what are the possible values of \(m\)?
Correct answer: A
For equal roots, the discriminant must be zero. Here, \(a=1\), \(b=-2(m+1)\), and \(c=16\). Thus, \(D=b^2-4ac=[-2(m+1)]^2-64=0\), giving \((m+1)^2=16\). Hence, \(m+1=\pm4\), so \(m=3\) or \(m=-5\). Exam tip: In questions involving equal roots of a quadratic equation, begin by using \(D=0\).
View question detailsFor which (r) will (rx^2-5x+r=0) have two distinct real roots?
Correct answer: A
For two distinct real roots, (D>0), so (25-4r^2>0). Also (r\neq0) is needed because the equation must remain quadratic.
View question detailsIf (x^2+2x+\lambda=0) has no real roots, what is the correct condition for (\lambda)?
Correct answer: A
For no real roots, (D<0), so (4-4\lambda<0) gives (\lambda>1). In exams, keep strict inequality separate from equality.
View question detailsState the nature of roots of (5x^2+20x+20=0).
Correct answer: A
Here (D=20^2-4\cdot5\cdot20=0), so the roots are equal and real. In exams, removing a common factor does not change the nature.
View question detailsWhich conclusion is correct for (10x^2-3x+2=0)?
Correct answer: C
Here (D=(-3)^2-4\cdot10\cdot2=-71), so it has no real roots. In exams, small (b^2) and large (4ac) can make (D<0).
View question detailsIf (2x^2-4x+n=0) has real and distinct roots, which condition is correct for (n)?
Correct answer: A
For real and distinct roots, (D>0), so (16-8n>0) gives (n<2). In exams, do not forget coefficient (a) in (4ac).
View question detailsFor which (s) will (s x^2+2sx+1=0) have equal roots?
Correct answer: B
For equal roots, (D=0), so ((2s)^2-4s=0) gives (4s(s-1)=0). (s=0) does not make a quadratic, so (s=1).
View question detailsIf (D=-5), which statement about the roots of a quadratic equation is correct?
Correct answer: C
(D=-5) is negative, so there will be no real roots. In exams, (D<0) may also be linked with complex roots.
View question detailsHow many times will the graph of (x^2+4x+8=0) cut the (x)-axis?
Correct answer: A
Here (D=4^2-4\cdot1\cdot8=-16), so there are no real roots. In exams, (D<0) means no cut with the (x)-axis.
View question detailsHow many times will the graph of (x^2-6x+9=0) touch the (x)-axis?
Correct answer: B
Here (D=(-6)^2-4\cdot1\cdot9=0), so the graph touches the (x)-axis once. In exams, (D=0) indicates tangency.
View question detailsHow many times will the graph of (x^2-5x+4=0) cut the (x)-axis?
Correct answer: C
Here (D=(-5)^2-4\cdot1\cdot4=9), so there are two distinct real roots. In exams, (D>0) means two cuts with the (x)-axis.
View question detailsQUIZ COMPLETE