1
MCQ (Single Correct Answer)

JEE Main 2019 (Online) 9th January Evening Slot

If both the roots of the quadratic equation x2 $$-$$ mx + 4 = 0 are real and distinct and they lie in the interval [1, 5], then m lies in the interval :
A
($$-$$5, $$-$$4)
B
(4, 5)
C
(5, 6)
D
(3, 4)

Explanation

x2 $$-$$mx + 4 = 0

Case-I :

D > 0

m2 $$-$$ 16 > 0

$$ \Rightarrow $$  m $$ \in $$ ($$-$$ $$\infty $$, $$-$$ 4) $$ \cup $$ (4, $$\infty $$)

Case-II :

$$ \Rightarrow \,\,1 < {{ - b} \over {2a}} < 5$$

$$ \Rightarrow \,\,1 < {m \over 2} < 5 \Rightarrow \,m \in \left( {2,10} \right)$$

Case-III :

f(1) > 0   and f(5) > 0

1 $$-$$ m + 4 > 0   and 25 $$-$$ 5m + 4 > 0

m < 5   and m < $${{29} \over 5}$$

Case-IV :

Let one root is x = 1

1 $$-$$ m + 4 = 0

m = 5

Now equation x2 $$-$$ 5x + 4 = 0

(x $$-$$ 1) (x $$-$$ 4) = 0

x = 1 i.e. m = 5 is also included

hence m $$ \in $$ (4, 5]

So given option is (4, 5)
2
MCQ (Single Correct Answer)

JEE Main 2019 (Online) 9th January Evening Slot

The number of all possible positive integral values of $$\alpha $$  for which the roots of the quadratic equation, 6x2 $$-$$ 11x + $$\alpha $$ = 0 are rational numbers is :
A
3
B
2
C
4
D
5

Explanation

For rational D must be perfect square

D = 121 $$-$$ 24$$\alpha $$

for 121 $$-$$ 24$$\alpha $$ to be perfect square a must be 3, 4, 5

So, ans $$\alpha $$ = 3
3
MCQ (Single Correct Answer)

JEE Main 2019 (Online) 10th January Morning Slot

Consider the quadratic equation (c – 5)x2 – 2cx + (c – 4) = 0, c $$ \ne $$ 5. Let S be the set of all integral values of c for which one root of the equation lies in the interval (0, 2) and its other root lies in the interval (2, 3). Then the number of elements in S is -
A
12
B
18
C
10
D
11

Explanation



Let f(x) = (c $$-$$ 5)x2 $$-$$ 2cx + c $$-$$ 4

$$ \therefore $$  f(0)f(2) < 0      . . . . .(1)

& f(2)f(3) < 0      . . . . .(2)

from (1) and (2)

(c $$-$$ 4)(c $$-$$ 24) < 0

& (c $$-$$ 24)(4c $$-$$ 49) < 0

$$ \Rightarrow $$  $${{49} \over 4}$$ < c < 24

$$ \therefore $$  s = {113, 14, 15, . . . . . 23}

Number of elements in set S = 11
4
MCQ (Single Correct Answer)

JEE Main 2019 (Online) 10th January Evening Slot

The value of $$\lambda $$ such that sum of the squares of the roots of the quadratic equation, x2 + (3 – $$\lambda $$)x + 2 = $$\lambda $$ has the least value is -
A
1
B
2
C
$${{15} \over 8}$$
D
$${4 \over 9}$$

Explanation

$$\alpha $$ + $$\beta $$ = $$\lambda $$ $$-$$ 3

$$\alpha $$$$\beta $$ = 2 $$-$$ $$\lambda $$

$$\alpha $$2 + $$\beta $$2 = ($$\alpha $$ + $$\beta $$)2 $$-$$ 2$$\alpha $$$$\beta $$ = ($$\lambda $$ $$-$$ 3)2 $$-$$ 2$$\left( {2 - \lambda } \right)$$

= $$\lambda $$2 + 9 $$-$$ 6$$\lambda $$ $$-$$ 4 + 2$$\lambda $$

= $$\lambda $$2 $$-$$ 4$$\lambda $$ + 5

= ($$\lambda $$ $$-$$ 2)2 + 1

$$ \therefore $$  $$\lambda $$ = 2

Questions Asked from Quadratic Equation and Inequalities

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