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Quadratic Equations - Past Year Questions

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Q1:

If a1,a2,...,a8a_1, a_2, ..., a_8 are the roots of the equation x8+x7+...+x+1=0x^8 + x^7 + ... + x + 1 = 0, then the value of a12025+a22025+...+a82025a_1^{2025} + a_2^{2025} + ... + a_8^{2025} is
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Q2:

If 8x2−2kx+k=08x^2 - 2kx + k = 0 is a quadratic equation in xx, such that one of its roots is pp times the other, and p,kp, k are positive real numbers, then kk equals
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Q3:

Let P(x)P(x) be a quadratic polynomial such that
∣P(0)P(1)P(0)P(2)∣=0\left|\begin{array}{ll} P(0) & P(1) \\ P(0) & P(2) \end{array}\right|=0
Let P(0)=2P(0)=2 and P(1)+P(2)+P(3)=14P(1)+P(2)+P(3)=14. Then P(4)P(4) equals
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Q4:

Let f(x)=a2x2+2bx+cf(x) = a^2x^2 + 2bx + c where, a≠0a \neq 0, b,cb, c are real numbers and xx is a real variable then
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Q5:

If the polynomial ax2+bx+5ax^2 + bx + 5 leaves a remainder 33 when divided by x−1x - 1, and a remainder 22 when divided by x+1x + 1, then 2b−4a2b - 4a equals
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Q6:

The difference between the maximum real root and the minimum real root of the equation (x2−5)4+(x2−7)4=16(x^2 - 5)^4 + (x^2 - 7)^4 = 16 is
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Q7:

The number of real solutions of the equation (x2−15x+55)x2−5x+6=1(x^2 - 15x + 55)^{x^2-5x+6} = 1 is:
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Q8:

If the harmonic mean of the roots of the equation (5+2)x2−bx+8+25=0(5 + \sqrt{2}) x ^ 2 - bx + 8 + 2\sqrt{5} = 0 is 44 then the value of bb is
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Q9:

Amit and Alok attempted to solve a quadratic equation. Amit made a mistake in writing down the constant term and ended up with roots (4,3)(4, 3). Alok made a mistake in writing down coefficient of xx to get roots (3,2)(3, 2). The correct roots of the equation are:
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Q10:

If α\alpha and β\beta are the roots of the equation ax2+bx+c=0a x^{2}+b x+c=0, then value of
1aα+b+1aβ+b\frac{1}{a \alpha+b}+\frac{1}{a \beta+b} is :
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Q11:

If highest common factor of x2−px−qx^{2}-p x-q and 5x2−3px−15q5 x^{2}-3 p x-15 q is (x−3)(x-3), then value of ( p,q)\left.p, q\right) will be :
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Q12:

If the roots of the equation x2−px+54=0x^{2}-p x+54=0 are in the ratio 2:32: 3, then value of pp is:
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Q13:

The factors of polynomial x3−6x2+11x−6x^{3}-6 x^{2}+11 x-6 are :
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Q14:

The graph of a polynomial y=f(x)y = f(x) is shown in figure below, then the number of its zeros is:
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Q15:

Given below are two statements:

Statement (I): (x2 + 3x + 1) = (x - 2)2 is not a quadratic equation.

Statement (II): The nature of roots of quadratic equations x2 + 2x√3 + 3 = 0 are real and equal.

In light of the above statements, choose the most appropriate answer from the options given below.

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Q16:

If sin⁡(α)\sin (\alpha) and cos⁡(α)\cos (\alpha) are the roots of the equation ax2+bx+c=0a x^{2}+b x+c=0, then b2b^{2} is
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Q17:

List I (Quadratic Equation)List II (Roots)A. 6x2+x−12=0I. (−6,4)B. 8x2+16x−10=202II. (9,36)C. x2+45x+324=0III. (3,−12)D. 2x2−5x−3=0IV. (−32,43)\begin{array}{|c|c|c|} \hline \textbf{List I (Quadratic Equation)} & \textbf{List II (Roots)} \\ \hline \text{A. } 6x^2 + x - 12 = 0 & \text{I. } (-6, 4) \\ \hline \text{B. } 8x^2 + 16x - 10 = 202 & \text{II. } (9, 36) \\ \hline \text{C. } x^2 + 45x + 324 = 0 & \text{III. } (3, -\frac{1}{2}) \\ \hline \text{D. } 2x^2 - 5x - 3 = 0 & \text{IV. } \left(-\frac{3}{2}, \frac{4}{3}\right) \\ \hline \end{array}
Match List I with List II. Choose the correct answer from the options given below:
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Q18:

If the minimum value of f(x)=x2+2bx+2c2f(x) = x^2 + 2bx + 2c^2 is greater than the maximum value of g(x)=−x2−2cx+b2g(x) = -x^2 - 2cx + b^2, then for real value of x.
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Q19:

The set of all real numbers x for which x2−∣x+2∣+x>0x^2 - |x + 2| + x > 0, is
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Quadratic Equations - Past Year Questions (Free PDF Download)

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