You visited us 1 times! Enjoying our articles? Unlock Full Access!

Definition of Sets

Let S=a∈ N,...

Question

Let $S = {a \in N, a \leq 100}$ if the equation $[{tan}^{2} x] - tan x - a = 0$ has real roots, then the number of elements $S$ is (when $[.]$ is greatest integer function)

10

Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses

1

No worries! We‘ve got your back. Try BYJU‘S free classes today!

9

No worries! We‘ve got your back. Try BYJU‘S free classes today!

0

No worries! We‘ve got your back. Try BYJU‘S free classes today!

Open in App

Solution

The correct option is A $10$
Given equation is,
$[t a n^{2} x] - t a n x - a = 0$
$∴ D = b^{2} - 4 a c = (- 1)^{2} - 4 (1) (- a)$

$= 1 + 4 a$
So D must be a odd perfect square

$⟹ \sqrt{1 + 4 a} = 2 λ + 1$

$⟹ 1 + 4 a = 4 λ^{2} + 1 + 4 λ$

$⟹ a = λ (λ + 1)$

For different values og=f $λ$ we get
$a = 0, 2, 6, 12, 20, 30, 42, 56, 72, 90$

Hence there are $10$ values of $a$ .

Suggest Corrections

Similar questions

Let $S = {a \in N, a \leq 100}$ . If the equation $[T a n^{2} x] - T a n x - a = 0$ has real roots then number of elements in S is (where [] is step functin).

Q. The number of integral values of a,

a \in (6, 100)

for which the equation

[t a n x]^{2} + t a n x - a = 0

has real roots (where [.] denotes greatest integer function) is:

Q. If the equation

{a x}^{2} + 2 b x + c = 0

has real roots,

a, b, c

being real numbers and if

m

and

n

are real number such that

m^{2} > n > 0

then show that the equation

{a x}^{2} + 2 m b x + n c = 0

has real roots.

Q. Let

a, b

and

c

be three distinct real roots of the cubic

x^{3} + 2 x^{2} - 4 x - 4 = 0

. If the equation

x^{3} + q x^{2} + r x + s = 0

has roots

\frac{1}{a}, \frac{1}{b}

and

\frac{1}{c}

, then the value of

(q + r + s)

is equal to

Q. Quadratic equation

x^{2} + [a^{2} - 5 a + b + 4] x + b = 0

has roots

- 5

and

1

, then number of integral values of

a

are
Note:

[.]

denotes the greatest integer function.

Join BYJU'S Learning Program

Let S={a∈N,a≤100} if the equation [tan2x]−tanx−a=0 has real roots, then the number of elements S is (when [.] is greatest integer function)

The correct option is A 10Given equation is,[tan2x]−tanx−a=0∴D=b2−4ac=(−1)2−4(1)(−a)=1+4aSo D must be a odd perfect square⟹√1+4a=2λ+1⟹1+4a=4λ2+1+4λ⟹a=λ(λ+1)For different values og=f λ we geta=0,2,6,12,20,30,42,56,72,90Hence there are 10 values of a.

Let $S = {a \in N, a \leq 100}$ if the equation $[{tan}^{2} x] - tan x - a = 0$ has real roots, then the number of elements $S$ is (when $[.]$ is greatest integer function)