Nature of Roots

Correct answer: A. Two equal real roots

Explanation: 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.

Correct answer: A. Two distinct real roots

Explanation: Here (D=(2\sqrt{2})^2-4\cdot1\cdot1=4), so (D>0). In exams, (D=4) gives two distinct real roots.

Correct answer: A. \(m=3\) या \(m=-5\)

Explanation: 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 \(D=0\).

Correct answer: A. (r^2<\frac{25}{4}) and (r\neq0)

Explanation: For two distinct real roots, (D>0), so (25-4r^2>0). Also (r\neq0) is needed because the equation must remain quadratic.

Correct answer: A. (\lambda>1)

Explanation: For no real roots, (D<0), so (4-4\lambda<0) gives (\lambda>1). In exams, keep strict inequality separate from equality.

Correct answer: A. Two equal real roots

Explanation: 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.

Correct answer: C. It has no real roots

Explanation: 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).

Correct answer: A. (n<2)

Explanation: For real and distinct roots, (D>0), so (16-8n>0) gives (n<2). In exams, do not forget coefficient (a) in (4ac).

Correct answer: B. (s=1)

Explanation: 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).

Correct answer: C. No real roots

Explanation: (D=-5) is negative, so there will be no real roots. In exams, (D<0) may also be linked with complex roots.

Correct answer: A. (0) times

Explanation: 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.

Correct answer: B. (1) time

Explanation: Here (D=(-6)^2-4\cdot1\cdot9=0), so the graph touches the (x)-axis once. In exams, (D=0) indicates tangency.

Correct answer: C. (2) times

Explanation: 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.

Correct answer: A. True

Explanation: When (b^2-4ac<0), the discriminant is negative, so real roots do not exist. In exams, this is the standard rule for nature of roots.

Correct answer: B. False

Explanation: When (D>0), roots are not equal; they are distinct real roots. In exams, equal roots occur only when (D=0).

Correct answer: A. Two distinct real roots

Explanation: 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.

Correct answer: B. (h=\frac{9}{4})

Explanation: 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.

Correct answer: C. No real roots

Explanation: Here (D=(-8)^2-4\cdot8\cdot3=-32), so there are no real roots. In exams, the sign of (D) gives the final decision.

Correct answer: A. \(k=\pm9\)

Explanation: 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.

Correct answer: A. Two equal real roots

Explanation: 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.

Correct answer: A. Two equal real roots

Explanation: 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.

Correct answer: A. (p^2=4q)

Explanation: For equal roots, (D=0), so (p^2-4q=0) gives (p^2=4q). In exams, the formula becomes simpler when (a=1).

Correct answer: A. Two equal real roots

Explanation: 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.

Correct answer: A. Two distinct real roots for all real (k)

Explanation: 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.

Correct answer: A. (k\le1) and (k\neq0)

Explanation: For real roots, (D\ge0), so (36-36k\ge0) gives (k\le1). Also (k\neq0) is required because the equation must remain quadratic.