CAREER POINT Fresher Course for IIT JEE (Main & Advanced)–2017 DAILY PRACTICE PROBLEM SHEET Course : Fresher(XL) Batch Subject : Mathematics
DPPS 2
Topic : Quadratic Equation Q.1
Q.2
2
2
2
If min. (2x – ax + 2) > max. (b – 1 + 2x – x ) then roots of the equation 2x + ax + (2 – b) = 0, are (A) positive and distinct
(B) negative and distinct
(C) opposite in sign
(D) imaginary
The number of integral values of
α for which the inequality x 2 – 2(4α – 1)x + 15α2 > 2α + 7 is true for every
x ∈ R, is (A) 0 Q.3
(B) 1
(C) 2 2
If roots of the quadratic equation bx – 2ax + a = 0 are real and distinct, where a, b ∈ R and b ≠ 0, then (A) atleast one roots lies in the interval (0, 1) (C) atleast one root lies in the interval (–1, 0)
Q.4
(B) no roots lies in the interval (0, 1) (D) None of the above 2
2
Let a, b, c ∈ R 0 and 1 be a root of the equation ax + bx + c = 0, then the equation 4ax + 3bx + 2c = 0 has (A) imaginary roots
Q.5
(D) 3
(B) real and equal roots
(C) real and unequal roots (D) rational roots 2
If p and q are the roots of the quadratic equation x – (α – 2)x – α = 1 (α ∈ R), then the minimum value of 2
2
(p + q ) is equal to (A) 2 Q.6
(B) 3
(C) 5
(D) 6 2
Number of integral values of a for which every solution of the inequality x – 3x + 4 > 0 is also the solution 2
of the inequality (a – 1)x – (a + |a – 1| + 2) x + 1 ≥ 0, is (A) 0 Q.7
(B) 1
(C) 2
(D) 3
2
If α and β are the roots of equation x – a(x + 1) – b = 0 where a, b
1
+
1
α − aα β − aβ 2
(A)
4 a + b
2
−
2 a + b (B)
∈ R–{0} and a + b ≠ 0 then the value of
is equal to
2
(C) 0
a + b
(D)
1 a + b
Passage for Q. No. 8 & 9
For a, b ∈ R – {0}, let f(x) = ax + bx + a satisfies f x + 2
7
7 = f − x ∀x ∈ R . Also the equation 4 4
f(x) = 7x + a has only one real and distinct solution Q.8
Q.9
The value of (a + b) is equal to (A) 4 (B) 5
The minimum value of f(x) in 0, (A)
− 33 8
(B) 0
3 2
(C) 6
(D) 7
(C) 4
(D) –2
is equal to
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Passage for Q. No. 10 to 12
Consider a rational function f(x) =
x2 x
2
− 3x − 4 2 and a quadratic function g(x) = x – (b + 1)x + b – 1, where − 3x + 4
b is a parameter. Q.10
The sum of integers in the range of f(x), is (A) –5
Q.11
(B) –6
(D) –10
If both roots of the equation g(x) = 0 are greater than –1, then b lies in the interval (A) (– ∞, –2)
Q.12
(C) –9
− ∞, − 1 4
(B)
(C) (–2, ∞)
(D)
− 1 , ∞ 2
The largest natural number b satisfying g(x) > – 2 ∀ x ∈ R, is (A) 1
(B) 2
(C) 3
(D) 4
Passage for Q. No. 13 to 15
Consider a function f(x) =
Q.13
+3
(B) 2
3 2
.
(C) 3
(D) 4
The minimum value of f(x) is equal to
− π 3
(A) tan Q.15
x2
which has greatest value equal to
The value of the constant number a is equal to (A) 1
Q.14
3x + a
(B) sin
− π 6
(C) cos
− π 3
(D) cot
π 2
If the equation f(x) = b has two distinct real roots then the number of integral values of b is equal to (A) 0 (B) 1 (C) 2 (D) 3
Passage for Q. No. 16 to 18 2
2
2
2
Consider two quadratic trinomials f(x) = x – 2ax + a – 1 and g(x) = (4b – b – 5) x – (2b – 1) x + 3b, where a, b ∈ R. Q.16
The values of a for which both roots of the equation f(x) = 0 are greater than – 2 but less than 4, lie in the interval (A) – ∞ < a < – 3
Q.17
(D) 5 < a < ∞
(C) – 1 < a < 3
If roots of the quadratic equation g(x) = 0 lie on either side of unity, then number of integral values of b is equal to (A) 1
Q.18
(B) – 2 < a < 0
(B) 2
(C) 3
(D) 4
(C) 0 < a < 1
(D) a > 3
If f(x) < 0 ∀ x ∈ [0, 1], then a lie in the interval (A) – 1 < a < 1
(B) 0 < a < 2
ANSWERS :
1. (D)
2. (B)
3. (A)
4. (C)
5. (C)
6. (A)
7. (C)
12. (B)
13. (C)
14. (B)
15. (B)
16. (C)
17. (B)
18. (C)
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8. (B)
9. (D)
10. (B)
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11. (D)
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