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Quadratic Equation — JEE Main & Advanced IOQM PYQs

38 previous year questions from Quadratic Equation with answers and solutions. Numbered list, year tags, and one-tap solutions — built for serious JEE / NEET practice.

38 questionsIOQMSolutions on every page
1

Suppose the prime numbers p and q satisfy q^2+3 p=197 p^2+q . Write q p as l+ m n , where l, m, n are positive integers, m < n and GCD (m, n)=1 . Find the maximum value of l+m+n .

Olympiad workbook Solution View → 2

Let X= -5,-4,-3,-2,-1,0,1,2,3,4,5 and S= (a, b) X X: x^2+a x+b . and x^3+b x+a have at least a common real zero . How many elements are there in S ?

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If x= 2 + 3 + 6 is a root of x^4+a x^3+b x^2+c x+d=0 where a, b, c, d are integers, what is the value of | a + b + c + d | ?

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Suppose that P is the polynomial of least degree with integer coefficients such that P ( 7 + 5 )=2( 7 - 5 ) . Find P (2) .

Olympiad workbook Solution View → 5

Let A= m: m . an integer and the roots of x^2+m x + 2020 = 0 are positive integers) and B= (n: n . an integer and the roots of x^2+2020 x+ n=0 are negative integers). Suppose a is

Olympiad workbook Solution View → 6

Let x, y, z be complex numbers such that aligned & x y+z + y z+x + z x+y =9 & x^2 y+z + y^2 z+x + z^2 x+y =64 & x^3 y+z + y^3 z+x + z^3 x+y =488 aligned If x y z + y z x + z x y =

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Suppose a, b are positive real numbers such that a a +b b =183, a b +b a =182 . Find 9 5 (a+b) .

Olympiad workbook Solution View → 8

Let f(x)=x^2+a x+b . If for all nonzero real xf (x+ 1 x )=f(x)+f ( 1 x ) and the roots of f(x)=0 are integers, what is the value of a^2+b^2 ?

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Let t be the area of a regular pentagon with each side equal to 1 . Let P(x)=0 be the polynomial equation with least degree, having integer coefficients, satisfied by x=t and the g

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Determine the sum of all possible positive integers n , the product of whose digits equals n^2-15 n- 27.

Olympiad workbook Solution View → 11

Three positive integers a, b, c with a>c satisfy the following equations: a c+b+c=b c+a+66, a+b+c=32 . Find the value of a .

Olympiad workbook Solution View → 12

Suppose a , b are integers and a + b is a root of x ^2+ ax + b =0 . What is the maximum possible values of b^2 ?

Olympiad workbook Solution View → 13

Suppose an integer r , a natural number n and a prime number p satisfy the equation 7 x^2-44 x+12=p^n . Find the largest value of p .

Olympiad workbook Solution View → 14

A positive integer m has the property that m^2 is expressible in the form 4 n^2-5 n+16 where n is an integer (of any sign). Find the maximum possible value of |m-n| .

Olympiad workbook Solution View → 15

Suppose 1,2,3 are the roots of the equation x^4+a x^2+b x=c . Find the value of c .

Olympiad workbook Solution View → 16

Let a, b be integers such that all the roots of the equation (x^2+a x+20 ) (x^2+17 x+b )=0 are negative integers. What is the smallest possible value of a + b ?

Olympiad workbook Solution View → 17

Let a, b, c be reals satisfying 3 a b+2=6 b, 3 b c+2=5 c, 3 c a+2=4 a Let Q denote the set of all rational numbers. Given that the product a b c can take two values r s Q and t u Q

Olympiad workbook Solution View → 18

Let a, b be positive integers satisfying a^3-b^3-a b=25 . Find the largest possible value of a^2+b^3 .

Olympiad workbook Solution View → 19

Find the number of ordered triples (x, y, z) of real numbers that satisfy the system of equation : x+y+z=7 ; x^2+y^2+z^2=27 ; x y z=5

Olympiad workbook Solution View → 20

Let X= -5,-4,-3,-2,-1,0,1,2,3,4,5 and S= (a, b) X X: x^2+a x+b . and x^3+b x+a have at least a common real zero . How many elements are there in S ?

Olympiad workbook Solution View → 21

Let A= m: m . an integer and the roots of x^2+m x + 2020 = 0 are positive integers) and B= (n: n . an integer and the roots of x^2+2020 x+ n=0 are negative integers). Suppose a is

Olympiad workbook Solution View → 22

Find the number of ordered triples (x, y, z) of real numbers that satisfy the system of equation : x+y+z=7 ; x^2+y^2+z^2=27 ; x y z=5

Olympiad workbook Solution View → 23

Let a, b, c be reals satisfying 3 a b+2=6 b, 3 b c+2=5 c, 3 c a+2=4 a Let Q denote the set of all rational numbers. Given that the product a b c can take two values r s Q and t u Q

Olympiad workbook Solution View → 24

Let x, y, z be complex numbers such that aligned & x y+z + y z+x + z x+y =9 & x^2 y+z + y^2 z+x + z^2 x+y =64 & x^3 y+z + y^3 z+x + z^3 x+y =488 aligned If x y z + y z x + z x y =

Olympiad workbook Solution View → 25

Let a, b be positive integers satisfying a^3-b^3-a b=25 . Find the largest possible value of a^2+b^3 .

Olympiad workbook Solution View → 26

Suppose that P is the polynomial of least degree with integer coefficients such that P ( 7 + 5 )=2( 7 - 5 ) . Find P (2) .

Olympiad workbook Solution View → 27

Suppose the prime numbers p and q satisfy q^2+3 p=197 p^2+q . Write q p as l+ m n , where l, m, n are positive integers, m < n and GCD (m, n)=1 . Find the maximum value of l+m+n .

Olympiad workbook Solution View → 28

A positive integer m has the property that m^2 is expressible in the form 4 n^2-5 n+16 where n is an integer (of any sign). Find the maximum possible value of |m-n| .

Olympiad workbook Solution View → 29

Suppose 1,2,3 are the roots of the equation x^4+a x^2+b x=c . Find the value of c .

Olympiad workbook Solution View → 30

Suppose a, b are positive real numbers such that a a +b b =183, a b +b a =182 . Find 9 5 (a+b) .

Olympiad workbook Solution View → 31

Let a, b be integers such that all the roots of the equation (x^2+a x+20 ) (x^2+17 x+b )=0 are negative integers. What is the smallest possible value of a + b ?

Olympiad workbook Solution View → 32

Three positive integers a, b, c with a>c satisfy the following equations: a c+b+c=b c+a+66, a+b+c=32 . Find the value of a .

Olympiad workbook Solution View → 33

If x= 2 + 3 + 6 is a root of x^4+a x^3+b x^2+c x+d=0 where a, b, c, d are integers, what is the value of | a + b + c + d | ?

Olympiad workbook Solution View → 34

Let f(x)=x^2+a x+b . If for all nonzero real xf (x+ 1 x )=f(x)+f ( 1 x ) and the roots of f(x)=0 are integers, what is the value of a^2+b^2 ?

Olympiad workbook Solution View → 35

Suppose an integer r , a natural number n and a prime number p satisfy the equation 7 x^2-44 x+12=p^n . Find the largest value of p .

Olympiad workbook Solution View → 36

Suppose a , b are integers and a + b is a root of x ^2+ ax + b =0 . What is the maximum possible values of b^2 ?

Olympiad workbook Solution View → 37

Let t be the area of a regular pentagon with each side equal to 1 . Let P(x)=0 be the polynomial equation with least degree, having integer coefficients, satisfied by x=t and the g

Olympiad workbook Solution View → 38

Determine the sum of all possible positive integers n , the product of whose digits equals n^2-15 n- 27.

Olympiad workbook Solution View →
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