NAME : .....................................................................................................................................................
JEE (Main + Advanced) 2026
JEE (Main + Advanced) 2026
NURTURE COURSE
NURTURE COURSE
ASSIGNMENT # A (QUADRATIC EQUATION) MATHEMATICS
One or more than one correct :
1. If a , b , g are the roots of the equation, x3 + P0x2 + P1x + P2 = 0, then (1– a 2 ) (1– b 2 ) (1 – g 2 ) is equal
to
(A) (1 + P1)2– (P0 + P2)2 (B) (1 + P1)2 + (P0 + P2)2
(C) (1 – P1)2 – (P0 – P2)2 (D) none of these
2. The equations ax + bx + a = 0 and x – 2x + 2x – 1 = 0 have two roots in common. Then a + b must be
2 3 2
equal to
(A) 1 (B) –1 (C) 0 (D) none of these
3. The equations x + 5x + px + q = 0 and x + 7x + px + r = 0 have two roots in common. If the third root
3 2 3 2
of each equation is represented by x1 and x2 respectively, then the ordered pair (x1, x2) is
(A) (-5, -7) (B) (1, -1) (C) (-1, 1) (D) (5, 7)
4. If all roots of x4 – 8x3 + ax2 – bx + 16 = 0 are positive real numbers, then (a + b) is equal to -
(A) 32 (B) 24 (C) 56 (D) 8
5. Given a, b be the roots of quadratic equation, x – 2ax + 7 = 0 and g, d be the roots of the equation,
2
a2
x2 – 2bx + 1 = 0. If ad = bg, then is equal to -
b2
(A) 5 (B) 6 (C) 7 (D) 8
6. Let a, b be roots of x – x + b = 0 and Sk = a + b , then the value(s) of b such that S2, S3 and S4 are in
2 k k
arithmetic progression is -
(A) 0 (B) 1
1
(C) (D) no real value(s) of b exists
5
7. There is only one real value of l for which the quadratic equation lx2 – 4x + 9 = 0 has two positive
integral solutions. The sum of these two solutions is-
(A) 4 (B) 8 (C) 9 (D) 12
8. Let ƒ(x) is polynomial of two degree such that ƒ(1) = 6, ƒ(–1) = 30 and ƒ(3) > 0. If one root is twice of
the other root of the equation ƒ(x) = 0, then the value of ƒ(3) is -
(A) 54 (B) 78 (C) 96 (D) 110
9. The set of values of 'a' for which f(x) = ax2 + 2x(1 – a) – 4 is negative for exactly three integral values
of x, is -
(A) (0, 2) (B) (0, 1] (C) [1, 2) (D) [2, ¥)
ax2 + 7x - 2
10. The number of possible value of 'a' for which the expression y = has atleast one common
a + 7x - 2x2
linear factor in numerator and denominator, is -
(A) 0 (B) 1 (C) 2 (D) 3
11. If a, b are real numbers such that b > a & |a - b| < 2, then the equation (x – a)(x – b) + 1 = 0 has -
(A) both roots in (a, b) (B) both roots in (–¥, a) (C) both roots in (b, ¥) (D) no real roots
MATHEMATICS /ASSIGNMENT E-1/7
, JEE (Main + Advanced) 2026
NURTURE COURSE
12. If a, b, g are roots of equation 111x3 – 11x – 1 = 0, then (ab)–2 + (bg)–2 + (ga)–2 equals
(A) 2332 (B) 1331 (C) 1210 (D) 2442
9c
13. If a + b + c > and quadratic equation ax2 + 2bx – 5c = 0 has non-real roots, then -
4
(A) a > 0, c > 0 (B) a > 0, c < 0 (C) a < 0, c < 0 (D) a < 0, c > 0
14. Let ƒ(x) = x3 + x + 1 and P(x) be a cubic polynomial such that P(0) = –1 and the roots of P(x) = 0 are the
squares of the roots of ƒ(x) = 0, then value of P(9) is -
(A) 98 (B) 899 (C) 80 (D) 898
15. The set of values of 'a' for which the inequality x2 – (a + 2) x – (a + 3) < 0 is satisfied by atleast
one positive real x, is -
(A) [–3, ¥) (B) (–3, ¥) (C) (–¥, –3) (D) (–¥, 3]
16. Set of the values of parameter 'm' for which every solution of the inequality x2 – 3x + 2 £ 0 is also
a solution of the inequality is 2x2 – (m + 1) x – (2m + 3) < 0,is -
æ3 ö æ 2 ö æ 2 3 ö
(A) çè 4 , ¥÷ø (B) çè - 3 , ¥÷ø (C) çè - 3 , 4 ÷ø (D) ( -¥, ¥)
17. Let ƒ(x) = x2 + ax + b such that ƒ(2) + ƒ(3) < 2, then equation ƒ(x) = 1 has (a, b Î R)
(A) both roots real and distinct (B) both roots real and equal
(C) non real roots (D) roots whose nature depends on value of a & b
18. If a, b, c are real numbers satisfying the condition a + b + c = 0 then the roots of the quadratic equation
3ax2 + 5bx + 7c = 0 are
(A) positive (B) negative (C) real and distinct (D) imaginary
( x - a )(x - b )
19. If x is real, the function, will assume all real values, provided
( x - c)
(A) a > b > c (B) a < b < c (C) a > c > b (D) none of these
20. Let p(x) = 0 be a polynomial equation of least possible degree, with rational coefficients, having 3
7 + 3 49
as one of its roots. Then the product of all the roots of p(x) = 0 is
(A) 7 (B) 49 (C) 56 (D) 63
21. The least value of the expression x2 + 4y2 + 3z2 – 2x – 12y – 6z + 14 is
(A) 0 (B) 1 (C) no least value (D) none of these
22. If the equation ax2 + bx + c = 0 has two positive and real root, then the equation ax2 + (b + 6a)x + (c +
3b) = 0 has
(A) no solution (B) atleast one positive solution
(C) atleast one negative solution (D) none of the above
23. The set of real value(s) of p for which the equation, |2x + 3| + |2x - 3| = px + 6 has more than two
solutions is
(A) [0, 4) (B) (-4, 4) (C) R - {4, -4, 0} (D) {0}
E-2/7 MATHEMATICS /ASSIGNMENT
JEE (Main + Advanced) 2026
JEE (Main + Advanced) 2026
NURTURE COURSE
NURTURE COURSE
ASSIGNMENT # A (QUADRATIC EQUATION) MATHEMATICS
One or more than one correct :
1. If a , b , g are the roots of the equation, x3 + P0x2 + P1x + P2 = 0, then (1– a 2 ) (1– b 2 ) (1 – g 2 ) is equal
to
(A) (1 + P1)2– (P0 + P2)2 (B) (1 + P1)2 + (P0 + P2)2
(C) (1 – P1)2 – (P0 – P2)2 (D) none of these
2. The equations ax + bx + a = 0 and x – 2x + 2x – 1 = 0 have two roots in common. Then a + b must be
2 3 2
equal to
(A) 1 (B) –1 (C) 0 (D) none of these
3. The equations x + 5x + px + q = 0 and x + 7x + px + r = 0 have two roots in common. If the third root
3 2 3 2
of each equation is represented by x1 and x2 respectively, then the ordered pair (x1, x2) is
(A) (-5, -7) (B) (1, -1) (C) (-1, 1) (D) (5, 7)
4. If all roots of x4 – 8x3 + ax2 – bx + 16 = 0 are positive real numbers, then (a + b) is equal to -
(A) 32 (B) 24 (C) 56 (D) 8
5. Given a, b be the roots of quadratic equation, x – 2ax + 7 = 0 and g, d be the roots of the equation,
2
a2
x2 – 2bx + 1 = 0. If ad = bg, then is equal to -
b2
(A) 5 (B) 6 (C) 7 (D) 8
6. Let a, b be roots of x – x + b = 0 and Sk = a + b , then the value(s) of b such that S2, S3 and S4 are in
2 k k
arithmetic progression is -
(A) 0 (B) 1
1
(C) (D) no real value(s) of b exists
5
7. There is only one real value of l for which the quadratic equation lx2 – 4x + 9 = 0 has two positive
integral solutions. The sum of these two solutions is-
(A) 4 (B) 8 (C) 9 (D) 12
8. Let ƒ(x) is polynomial of two degree such that ƒ(1) = 6, ƒ(–1) = 30 and ƒ(3) > 0. If one root is twice of
the other root of the equation ƒ(x) = 0, then the value of ƒ(3) is -
(A) 54 (B) 78 (C) 96 (D) 110
9. The set of values of 'a' for which f(x) = ax2 + 2x(1 – a) – 4 is negative for exactly three integral values
of x, is -
(A) (0, 2) (B) (0, 1] (C) [1, 2) (D) [2, ¥)
ax2 + 7x - 2
10. The number of possible value of 'a' for which the expression y = has atleast one common
a + 7x - 2x2
linear factor in numerator and denominator, is -
(A) 0 (B) 1 (C) 2 (D) 3
11. If a, b are real numbers such that b > a & |a - b| < 2, then the equation (x – a)(x – b) + 1 = 0 has -
(A) both roots in (a, b) (B) both roots in (–¥, a) (C) both roots in (b, ¥) (D) no real roots
MATHEMATICS /ASSIGNMENT E-1/7
, JEE (Main + Advanced) 2026
NURTURE COURSE
12. If a, b, g are roots of equation 111x3 – 11x – 1 = 0, then (ab)–2 + (bg)–2 + (ga)–2 equals
(A) 2332 (B) 1331 (C) 1210 (D) 2442
9c
13. If a + b + c > and quadratic equation ax2 + 2bx – 5c = 0 has non-real roots, then -
4
(A) a > 0, c > 0 (B) a > 0, c < 0 (C) a < 0, c < 0 (D) a < 0, c > 0
14. Let ƒ(x) = x3 + x + 1 and P(x) be a cubic polynomial such that P(0) = –1 and the roots of P(x) = 0 are the
squares of the roots of ƒ(x) = 0, then value of P(9) is -
(A) 98 (B) 899 (C) 80 (D) 898
15. The set of values of 'a' for which the inequality x2 – (a + 2) x – (a + 3) < 0 is satisfied by atleast
one positive real x, is -
(A) [–3, ¥) (B) (–3, ¥) (C) (–¥, –3) (D) (–¥, 3]
16. Set of the values of parameter 'm' for which every solution of the inequality x2 – 3x + 2 £ 0 is also
a solution of the inequality is 2x2 – (m + 1) x – (2m + 3) < 0,is -
æ3 ö æ 2 ö æ 2 3 ö
(A) çè 4 , ¥÷ø (B) çè - 3 , ¥÷ø (C) çè - 3 , 4 ÷ø (D) ( -¥, ¥)
17. Let ƒ(x) = x2 + ax + b such that ƒ(2) + ƒ(3) < 2, then equation ƒ(x) = 1 has (a, b Î R)
(A) both roots real and distinct (B) both roots real and equal
(C) non real roots (D) roots whose nature depends on value of a & b
18. If a, b, c are real numbers satisfying the condition a + b + c = 0 then the roots of the quadratic equation
3ax2 + 5bx + 7c = 0 are
(A) positive (B) negative (C) real and distinct (D) imaginary
( x - a )(x - b )
19. If x is real, the function, will assume all real values, provided
( x - c)
(A) a > b > c (B) a < b < c (C) a > c > b (D) none of these
20. Let p(x) = 0 be a polynomial equation of least possible degree, with rational coefficients, having 3
7 + 3 49
as one of its roots. Then the product of all the roots of p(x) = 0 is
(A) 7 (B) 49 (C) 56 (D) 63
21. The least value of the expression x2 + 4y2 + 3z2 – 2x – 12y – 6z + 14 is
(A) 0 (B) 1 (C) no least value (D) none of these
22. If the equation ax2 + bx + c = 0 has two positive and real root, then the equation ax2 + (b + 6a)x + (c +
3b) = 0 has
(A) no solution (B) atleast one positive solution
(C) atleast one negative solution (D) none of the above
23. The set of real value(s) of p for which the equation, |2x + 3| + |2x - 3| = px + 6 has more than two
solutions is
(A) [0, 4) (B) (-4, 4) (C) R - {4, -4, 0} (D) {0}
E-2/7 MATHEMATICS /ASSIGNMENT
