lf $t_{0}, t_{1}, t_{2}, t_{n}$ are the consecutive terms in the expansion $(x + a i)^{n}$ , then

$(t_{0} - t_{2} + t_{4} - t_{6} + \dots .)^{2} + (t_{1} - t_{3} + t_{5} \dots .)^{2} =$

x^{2} + a^{2}

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

(x^{2} + a^{2})^{n}

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

x^{2} - a^{2}

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

(x^{2} - a^{2})^{n}

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

Open in App

Solution

The correct option is B $(x^{2} + a^{2})^{n}$
$(x + a i)^{n} =^{n} C_{0} x^{n} +^{n} C_{1} x^{n - 1} a i -^{n} C_{2} x^{n - 2} a^{2} -^{n} C_{3} x^{n - 3} a^{3} i +^{n} C_{4} x^{n - 4} a^{4} +^{n} C_{5} x^{n - 5} a^{5} i + . . . . .$
$(x + a i)^{n} = (^{n} C_{0} x^{n} -^{n} C_{2} x^{n - 2} a^{2} +^{n} C_{4} x^{n - 4} a^{4} + . . . .) + i (^{n} C_{1} x^{n - 1} a -^{n} C_{3} x^{n - 3} a^{3} +^{n} C_{5} x^{n - 5} a^{5})$
$\Rightarrow (x + a i)^{n} = (t_{0} - t_{2} + t_{4} + . . . . .) + i (t_{1} - t_{3} + t_{5} + . . . . .)$
$\Rightarrow | (x + a i)^{n} | = \sqrt{(t_{0} - t_{2} + t_{4} + . . . . .)^{2} + (t_{1} - t_{3} + t_{5} + . . . . .)^{2}}$ .....(1)
Now, $(x - a i)^{n} =^{n} C_{0} x^{n} -^{n} C_{1} x^{n - 1} a i -^{n} C_{2} x^{n - 2} a^{2} +^{n} C_{3} x^{n - 3} a^{3} i +^{n} C_{4} x^{n - 4} a^{4} -^{n} C_{5} x^{n - 5} a^{5} i + . . . . .$
$(x - a i)^{n} = (^{n} C_{0} x^{n} -^{n} C_{2} x^{n - 2} a^{2} +^{n} C_{4} x^{n - 4} a^{4} + . . . .) - i (^{n} C_{x}^{n - 1} a -^{n} C_{3} x^{n - 3} a^{3} +^{n} C_{5} x^{n - 5} a^{5})$
$\Rightarrow (x - a i)^{n} = (t_{0} - t_{2} + t_{4} + . . . . .) - i (t_{1} - t_{3} + t_{5} + . . . . .)$
$\Rightarrow | (x - a i)^{n} | = \sqrt{(t_{0} - t_{2} + t_{4} + . . . . .)^{2} + (t_{1} - t_{3} + t_{5} + . . . . .)^{2}}$ ....(2)
Multiplying eqn (1) and (2), we get
$| (x + a i)^{n} (x - a i)^{n} | = (t_{0} - t_{2} + t_{4} + . . . . .)^{2} + (t_{1} - t_{3} + t_{5} + . . . . .)^{2}$
$\Rightarrow (x^{2} + a^{2})^{n} = (t_{0} - t_{2} + t_{4} + . . . . .)^{2} + (t_{1} - t_{3} + t_{5} + . . . . .)^{2}$

Suggest Corrections

Similar questions

Q. If

T_{0}, T_{1}, T_{2}, \dots T_{n}

represent the terms in the expansion of

(x + a)^{n}

, then the value of

(T_{0} - T_{2} + T_{4} - T_{6} + \dots)^{2} + (T_{1} - T_{3} + T_{5} - \dots)^{2}

Q. If

T_{0}, T_{1}, T_{2}, . . . . T_{n}

represent the terms in the expansion of

{(x + a)}^{n}

, then the value of

{(T_{0} - T_{2} + T_{4} - T_{6} + . . . .)}_{0}^{2}

{(T_{1} - T_{3} + T_{5} - . . . .)}_{1}^{2}

If $T_{0}, T_{1}, T_{2}, . . . . . . . . . . . . . . . . . . T_{n}$ represent the terms in the expansion of $(x + a)^{n}$ , then the value of $(T_{0} - T_{2} + T_{4} - T_{6} + . . . . . . . . . . . . . . . . .)^{2}$ + $(T_{1} - T_{3} + T_{5} - . . . . . . . . . . . . . . . . .)^{2}$ is

Q. If

T_{0}, T_{1}, T_{2}, . . . . . T_{n}

represent the term in the expansion of

(x + a)^{n}

, then the value of

(T_{0} - T_{2} + T_{4} - T_{6} + . . . . .)^{2} + (T_{1} - T_{3} + T_{5} + . . . . .)^{2}

Q. If

T_{0}, T_{1}, T_{2}, . . ., T_{n}

represent in the terms in the expansion of

{(x + a)}^{n}

, then the value of

{(T_{0} - T_{2} + T_{4} - T_{6} + . . .)}_{0}^{2} + {(T_{1} - T_{3} + T_{5} + . . .)}_{1}^{2}

is equal to

Join BYJU'S Learning Program

lf t0,t1,t2,tn are the consecutive terms in the expansion (x+ai)n, then(t0−t2+t4−t6+….)2+(t1−t3+t5….)2=

lf $t_{0}, t_{1}, t_{2}, t_{n}$ are the consecutive terms in the expansion $(x + a i)^{n}$ , then

$(t_{0} - t_{2} + t_{4} - t_{6} + \dots .)^{2} + (t_{1} - t_{3} + t_{5} \dots .)^{2} =$