Let f\left ( x \right ) be a polynomial function of second degree. If f\left (1 \right )= f\left (-1 \right ) and a,b,c are in A.P., then {f}'\left ( a \right ),{f}'\left ( b \right ),{f}'\left ( c \right ) are in 

  • Option 1)

    G.P.

  • Option 2)

    H.P.

  • Option 3)

    Arithmetic­Geometric Progression

  • Option 4)

    A.P.

 

Answers (1)

As we learnt in 

Properties of an A.P. -

If each term of an AP is multiplied by a fixed constant (or divided by a constant),then resultant is also an AP.

- wherein

If  a_{1},a_{2},a_{3}------is \: AP

Then Ka_{1},Ka_{2},Ka_{3}------is \: AP

and a_{1}/K,a_{2}/K,a_{3}/K------is \: also \: an \: AP

 

 Let the polynomial be

f\left ( x \right )= Ax^{2}+Bx+C

f\left ( 1 \right )= A+B+C

f\left ( -1 \right )= A-B+C

\therefore A+B+C= A-B+C

B=0

f\left ( x \right )= Ax^{2}+c

f'\left ( x \right )= 2Ax

f'\left ( a \right )= 2Aa

f'\left ( b \right )= 2Ab 

f'\left ( c \right )= 2Ac

Now, a, b, c in AP

So 2Aa, 2Ab, 2Ac in AP

So f'\left ( a \right ), f'\left ( b \right ), f'\left ( c \right ) in AP


Option 1)

G.P.

This option is incorrect.

Option 2)

H.P.

This option is incorrect.

Option 3)

Arithmetic­Geometric Progression

This option is incorrect.

Option 4)

A.P.

This option is correct.

Most Viewed Questions

Preparation Products

Knockout JEE Main April 2021 (One Month)

Personalized AI Tutor and Adaptive Time Table, Self Study Material, Weekend Live Classes, Mentorship from our Experts, Unlimited Mock Tests and Personalized Analysis Reports, 24x7 Doubt Chat Support,.

₹ 14000/- ₹ 4999/-
Buy Now
Knockout JEE Main May 2021

Personalized AI Tutor and Adaptive Time Table, Self Study Material, Weekend Live Classes, Mentorship from our Experts, Unlimited Mock Tests and Personalized Analysis Reports, 24x7 Doubt Chat Support,.

₹ 22999/- ₹ 9999/-
Buy Now
Test Series JEE Main May 2021

Unlimited Chapter Wise Tests, Unlimited Subject Wise Tests, Unlimited Full Mock Tests, Get Personalized Performance Analysis Report,.

₹ 6999/- ₹ 2999/-
Buy Now
Knockout JEE Main May 2022

Personalized AI Tutor and Adaptive Time Table, Self Study Material, Weekend Live Classes, Mentorship from our Experts, Unlimited Mock Tests and Personalized Analysis Reports, 24x7 Doubt Chat Support,.

₹ 34999/- ₹ 14999/-
Buy Now
JEE Main Rank Booster 2021

Booster and Kadha Video Lectures, Unlimited Full Mock Test, Adaptive Time Table, 24x7 Doubt Chat Support,.

₹ 13999/- ₹ 6999/-
Buy Now
Boost your Preparation for JEE Main 2021 with Personlized Coaching
 
Exams
Articles
Questions