Career Point DPP

DPPS For JEE Main & Advanced-2019
(LAKSHYA & SPECIAL BATCH)

SUBJECT: MATHEMATICS
Topic : Quadratic Equations
MULTIPLE CORRECT OPTION TYPE QUESTIONS

Q.1 If α and β are the roots of the equation ax2 + bx + c = 0 and αβ = 3, where a, b, c are in A.P., then α2 + β2 =
(A) 2 (B) – 2 (C) α4 + β4 (D) α + β
Q.2 If a, b, c ∈ R, then for which of the following graphs of the quadratic polynomial y = ax2 – 2bx + c (a ≠ 0) ;
the product (abc) is negative ?
y y y y
x
x
(A) (B) (C) x (D) x
Q.3 Which of the following quantities are irrational for the quadratic equation
(log108) x2 – (log105) x = 2 (log210)–1 – x ?
(A) sum of roots (B) products of roots (C) sum of coefficients (D) discriminant
Q.4 If p and q are chosen randomly from the set {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} with replacement, then the
probability that the roots of the equation x2 + px + q = 0
33 19
(A) are real is (B) are imaginary is
50 50
3 3
(C) equal is (D) real and distinct is
100 5
Q.5 If cos4θ + α, sin4θ + α are the roots of x2 + b(2x + 1) = 0 and cos2θ + β, sin2θ + β are the roots of the
equation x2 + 4x + 2 = 0, then b is equal to
(A) 1 (B) –1 (C) 2 (D) –2
Q.6 If quadratic equation x2 + 2(a + 2b) x + (2a + b – 1) = 0 has unequal real roots for all b ∈ R then the possible
values of a can be equal to
(A) 5 (B) –1 (C) –10 (D) 3
Q.7 The equation 2ax2 – 2x – a = 0, a ∈ N has rational roots, then

(A) a = 2 (B) a = 12 (C) a = 70 (D) a = 408
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Q.8 If exactly two integers lie between the roots of equation x2 + ax – 1 = 0. Then integral value(s) of 'a' is/are :
(A) –1 (B) –2 (C) 1 (D) 2
Q.9 Let f(x) = x2 + ax + b and g(x) = x2 + cx + d be two quadratic polynomials with real coefficients and satisfy
ac = 2(b + d). Then which of the following is (are) correct ?
(A) Exactly one of either f(x) = 0 or g (x) = 0 must have real roots
(B) Atleast one of either f(x) = 0 or g(x) = 0 must have real roots
(C) Both f(x) = 0 and g(x) = 0 must have real roots
(D) Both f(x) = 0 and g(x) = 0 must have imaginary roots
Q.10 If a, b, c are rational numbers (a > b > c > 0) and quadratic equation (a + b – 2c)x2 + (b + c – 2a)x + (c + a – 2b) = 0
has a root in the interval (–1, 0) then which of the following statement(s) is/are correct ?
(A) a + c < 2b (B) both roots are rational
2
(C) ax + 2bx + c = 0 have both roots negative (D) cx2 + 2bx + a = 0 have both roots negative
PASSAGE BASED QUESTIONS

Passage # 1 (Q. 11 & 12)
⎛ 7⎞ ⎛7 ⎞
For a, b ∈ R – {0}, let f(x) = ax2 + bx + a satisfies f ⎜ x + ⎟ = f ⎜ − x ⎟∀x ∈ R . Also the equation
⎝ 4⎠ ⎝4 ⎠
f(x) = 7x + a has only one real and distinct solution
Q.11 The value of (a + b) is equal to

(A) 4 (B) 5 (C) 6 (D) 7
⎡ 3⎤
Q.12 The minimum value of f(x) in ⎢0, ⎥ is equal to
⎣ 2⎦
− 33
(A) (B) 0 (C) 4 (D) –2
8
Passage # 2 (Q.13 to 15)

x 2 − 3x − 4
Consider a rational function f(x) = and a quadratic function g(x) = x2 – (b + 1)x + b – 1, where
x − 3x + 4
2
b is a parameter.
Q.13 The sum of integers in the range of f(x), is

(A) –5 (B) –6 (C) –9 (D) –10
Q.14 If both roots of the equation g(x) = 0 are greater than –1, then b lies in the interval
⎛ −1⎞ ⎛ −1 ⎞
(A) (–∞, –2) (B) ⎜ − ∞, ⎟ (C) (–2, ∞) (D) ⎜ ,∞⎟
⎝ 4 ⎠ ⎝ 2 ⎠
Q.15 The largest natural number b satisfying g(x) > – 2 ∀ x ∈ R, is

(A) 1 (B) 2 (C) 3 (D) 4
Passage # 3 (Q.16 to 18)
Let f(x) be a polynomial of degree 5 with leading coefficient unity, such that f(1) = 5, f(2) = 4, f(3) = 3,
f(4) = 2, and f(5) = 1, then :
Q.16 f(6) is equal to :
(A) 120 (B) –120 (C) 0 (D) 6
Q.17 Sum of the roots of f(x) is equal to :

(A) 15 (B) –15 (C) 21 (D) can't be determine
Q.18 Product of the roots of f(x) is equal to :

(A) 120 (B) –120 (C) 114 (D) –114
COLUMN MATCHING TYPE QUESTIONS

Q.19 Match the following :
Column-I Column-II
2x − 1 4
(P) The least positive integer x, for which is positive, is (1)
2 x + 3x 2 + x
3
3
equal to
(Q) If the quadratic equation 3x2 + 2(a2 + 1)x + (a2 – 3a + 2) = 0 possesses (2) 3
roots of opposite sign then a can be equal to 4
(R) The roots of equation x + 3 − 4 x − 1 + x + 8 − 6 x − 1 = 1 belongs (3) 1
to [a, b], then (a + b) is equal to
(S) If the roots of equation x4 – 8x3 + bx2 – cx + 16 = 0 are all positive then (4) 15
b
is equal to
c
Codes :
P Q R S
(A) 4 1 2 3
(B) 3 1 4 2
(C) 1 3 2 4
(D) 2 3 3 1
NUMERIC RESPONSE TYPE QUESTIONS

2 3 4 8 ⎛ 8 ⎞
Q.20 Let x1 = 97, x2 = , x3 = , x4 = ,……, x8 = then log 3 ⎜ ∏ x i − 60 ⎟ =
2 ⎜ ⎟
x1 x2 x3 x7 ⎝ i =1 ⎠
Q.21 Let α, β are two real roots of equation x2 + px + q = 0, p, q ∈ R, q ≠ 0. If the quadratic equation g(x) = 0 has
1 1
two roots α + ,β+ such that sum of its roots is equal to product of roots, then the number of integral
α β
values q can attain is :
Q.22 Let M be the minimum value of f(θ) = (3 cos2θ + sin2θ) (sec2θ + 3 cosec2θ), for permissible real values of θ
( x − 1)(50 − 10 x )
and P denotes the product of all real solutions of the equation = x2 – 8x + 7. Find (P M)
x 2 − 5x
x –8 n
Q.23 The number of all values of n, (where n is a whole number) for which the equation = has no
n – 10 x
solution.
Q.24 Find the smallest value of k for which both the roots of equation x2 – 8kx + 16(k2 – k + 1) = 0 are real,
distinct and have values atleast 4.
Q.25 If sin θ and cos θ are the roots of the quadratic equation ax2 + bx + c = 0 (ac ≠ 0). Then find the value of
b2 – a 2
.
ac
Answer key
MULTIPLE CORRECT OPTION TYPE QUESTIONS

1. (B,D) 2. (A,C,D) 3. (C,D) 4. (B,C) 5. (B,C) 6. (B,C) 7. (A,B,C,D)
8. (A,C) 9. (B) 10. (A,B,C,D)
PASSAGE BASED QUESTIONS

11. (B) 12. (D) 13. (B) 14. (D) 15. (B) 16. (A) 17. (A)
18. (C)
COLUMN MATCHING TYPE QUESTIONS

19. (B)
NUMERIC RESPONSE TYPE QUESTIONS
20. 4 21. 3 22. 24 23. 6 24. 2 25. 2

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DPPS For JEE Main & Advanced-2019

(LAKSHYA & SPECIAL BATCH)

MULTIPLE CORRECT OPTION TYPE QUESTIONS

Q.7 The equation 2ax2 – 2x – a = 0, a ∈ N has rational roots, then

PASSAGE BASED QUESTIONS

Q.11 The value of (a + b) is equal to

Passage # 2 (Q.13 to 15)

Q.13 The sum of integers in the range of f(x), is

Q.15 The largest natural number b satisfying g(x) > – 2 ∀ x ∈ R, is

Q.17 Sum of the roots of f(x) is equal to :

Q.18 Product of the roots of f(x) is equal to :

COLUMN MATCHING TYPE QUESTIONS

NUMERIC RESPONSE TYPE QUESTIONS

MULTIPLE CORRECT OPTION TYPE QUESTIONS

PASSAGE BASED QUESTIONS

COLUMN MATCHING TYPE QUESTIONS

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