Sign in
12thPass logo
New chatPYQ LibraryDoubtsRank report
Sign in to see Recents

Your guest activity stays on this device

Sign in to save progress →
Sign in

Quadratic Equation and Inequalities question

2022 · 26 Jul · Shift 2 · Q20
Guest · filters and generic practice availableBrowsing as a guest · PYQ filters and generic practice are available. Sign in only for personalised features and saved progress.
  1. PYQ Library
  2. /JEE Main
  3. /Mathematics
  4. /Quadratic Equation and Inequalities
  5. /2022 · 26 Jul · Shift 2 · Q20

Quadratic Equation and Inequalities question

2022 · 26 Jul · Shift 2 · Q20

JEE MainMathematicsQuadratic Equation and InequalitiesMCQ+4 / −1
The minimum value of the sum of the squares of the roots of x2+(3−a)x+1=2ax^{2}+(3-a) x+1=2 ax2+(3−a)x+1=2a is:
  1. A
    4
  2. B
    5
  3. C
    6
  4. D
    8
View written solutionFree

Correct answer: C

  1. Write the quadratic in standard form

Given: x2+(3−a)x+1=2ax^2+(3-a)x+1=2ax2+(3−a)x+1=2a

Bring all terms to one side: x2+(3−a)x+(1−2a)=0x^2+(3-a)x+(1-2a)=0x2+(3−a)x+(1−2a)=0

Let the roots be α\alphaα and β\betaβ.

  1. Use relations between roots and coefficients

For x2+(3−a)x+(1−2a)=0,x^2+(3-a)x+(1-2a)=0,x2+(3−a)x+(1−2a)=0, we have α+β=−(3−a)=a−3\alpha+\beta=-(3-a)=a-3α+β=−(3−a)=a−3 and αβ=1−2a.\alpha\beta=1-2a.αβ=1−2a.

  1. Form the sum of squares of the roots

We need α2+β2.\alpha^2+\beta^2.α2+β2.

Using α2+β2=(α+β)2−2αβ,\alpha^2+\beta^2=(\alpha+\beta)^2-2\alpha\beta,α2+β2=(α+β)2−2αβ, we get α2+β2=(a−3)2−2(1−2a).\alpha^2+\beta^2=(a-3)^2-2(1-2a).α2+β2=(a−3)2−2(1−2a).

Now simplify: =(a2−6a+9)−2+4a=(a^2-6a+9)-2+4a=(a2−6a+9)−2+4a =a2−2a+7.=a^2-2a+7.=a2−2a+7.

  1. Find its minimum value

So we must minimize f(a)=a2−2a+7.f(a)=a^2-2a+7.f(a)=a2−2a+7.

Complete the square: f(a)=(a−1)2+6.f(a)=(a-1)^2+6.f(a)=(a−1)2+6.

Since (a−1)2≥0(a-1)^2\ge 0(a−1)2≥0, the minimum value is 6,6,6, attained at a=1a=1a=1.

  1. Check the options
  • A: 444 ✗
  • B: 555 ✗
  • C: 666 ✓
  • D: 888 ✗

Therefore, the correct option is C.

PreviousNext

More from Quadratic Equation and Inequalities

  • The sum of the cubes of all the roots of the equation x4−3x3−2x2+3x+1=0 is ​.2022 · Numerical
  • Let p and q be two real numbers such that p + q = 3 and p4 + q4 = 369. Then (p1​+q1​)−2 is equal to ​.2022 · Numerical
  • If α,β are the roots of the equation x2−(5+3log3​5​−5log5​3​)x+3(3(log3​5)31​−5(log5​3)32​−1)=0, then the…2022 · MCQ
  • The number of distinct real roots of x4 − 4x + 1 = 0 is :2022 · MCQ
  • If the sum of all the roots of the equation e2x−11ex−45e−x+281​=0 is loge​p, then p is equal to ​.2022 · Numerical
  • Let α, β be the roots of the equation x2−4λx+5=0 and α, γ be the roots of the equation x2−(32​+23​)x+7+3λ3​=0, λ> 0. If β+γ=32​…2022 · Numerical
  • The sum of all real values of x for which x2+3x+103x2−9x+17​=3x2+5x+125x2−7x+19​ is equal to ​.2022 · Numerical
  •  Let S={x∈[−6,3]−{−2,2}:∣x∣−2∣x+3∣−1​≥0} and T={x∈Z:x2−7∣x∣+9≤0}.  Then the number of elements in S∩T is :2022 · MCQ