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The CORRECT option is (c) 7, 8 or 9

Explanation: If n is the number of terms and is EVEN, then the middle term is the \((\FRAC{n}{2} + 1)^{TH}\) term.

Else if n is the number of terms and is odd, there are two middle terms which are the \((\frac{n + 1}{2})^{th}\) term and the \((\frac{n + 3}{2})^{th}\) term.

Case 1: \(\frac{n}{2} + 1\) = 5 ; n = 8

Case 2: \(\frac{n + 1}{2}\) = 5 ; n = 9

Case 3: \(\frac{n + 3}{2}\) = 5 ; n = 7

Right option is (b) 9 \(\frac{2}{3}\)

Best EXPLANATION: Using BINOMIAL theorem we know that (a + b)^3 = a^3 + 3ab^2 + 3a^2b + b^3

Therefore, (7 + 2)^3 = 7^3 + 2^3 + (3 x 7 x 2^2) + (3 x 2 x 7^2)

(9)^3 = 7^3 + 2^3 + 84 + (3 x 2 x 7^2)

729 = 7^3 + 2^3 + 84 + 294

7^3 + 2^3 + 84 = 435

Also 7^2 – 2^2 = (7 – 2)(7 + 2)

7^2 – 2^2 = (5)(9)

7^2 – 2^2 = 45

So \(\frac{7^3 + 2^3 + 84}{7^2-2^2}\) = 435 / 45

= 9 \(\frac{30}{45}\)

= 9 \(\frac{2}{3}\).

The CORRECT answer is (C) (x + y) (x – y)^-1

The explanation: x^2 + y^2 + 2xy is the expansion of (x + y)^2

x^2 – y^2 can be written as (x – y)(x + y)

Substituting in the fraction we get, \(\frac{(x + y)^2}{(x – y)(x + y)}\).

After CANCELLING the terms we get, \(\frac{x^2+y^2+2xy}{x^2-y^2}\) = (x + y) (x – y)^-1.

The correct ANSWER is (b) \(\Sigma_{r = 0}^{r = 1000}\)(100Cr X^1000 – r y^r)

To ELABORATE: The expansion can be done using binomial theorem.

(x + y)^1000 = 1000C0 x^1000 y^0 + 1000C1 x^999 y^1 +….+ 1000C999 x^1 y^999 + 1000C1000 x^0 y^1000

This can also be WRITTEN as,

(x + y)^1000 = \(\Sigma_{r = 0}^{r = 1000}\)(100Cr x^1000 – r y^r).

The correct CHOICE is (d) 72

Easy EXPLANATION: Using BINOMIAL THEOREM (9 + 3I)^2 = 81 + 54i + 9i^2

We know that i^2 = –1

Therefore, (9 + 3i)^2 = 81 + 54i – 9

(9 + 3i)^2 = 72 + 54i

Real part = 72.

The correct choice is (C) 1552

To explain I would say: By binomial expansion,

(√3 + 1)^7 = \(\Sigma_{r = 0}^{r = 7}\)(7Cr √3^7 – r (1)^r)

Whenever, r is an EVEN number, 8 – r will also be even. Then √3 will also have an even power and thereby be integral.

Integral parts = 8C0 (√3)^0 + 8C2 (√3)^2 + 8C4 (√3)^4 + 8C6 (√3)^6 + 8C8 (√3)^8

Integral parts = 1 + 28 x 3 + 70 x 9 + 28 x 27 + 1 x 81

Integral part = 1552.

Correct choice is (B) ^80C40 (xyz)^40 (3)^40

Explanation: Since the POWER is even, there are odd NUMBER of terms.

The middle term is the \((\FRAC{n}{2} + 1)^{th}\) term.

= \((\frac{80}{2} + 1)^{th}\) term

= 41^st term

The 41^st term = ^80C40 (xyz)^40 (3)^40

Right option is (c) 1

The explanation is: 8^58 can be written as (8^2)^24.

8^48 = (64)^24

8^48 = (63 + 1)^24

We KNOW that (60 + 1)^24 = \(\Sigma_{r = 0}^{r = 24}\)(24Cr 63^24 – r 1^r)

= 24C0 63^24 4^0 + 24C1 63^23 4^1 +….+24C23 63^1 4^23 + 24C24 63^0 1^24

= 63 x k + 1

Therefore, the REMAINDER will be 1.

Right choice is (a) (x + y)^3

Easiest explanation: USING binomial expansions PROPERTIES, x^4 + 4x^3y + 6x^2y + 4xy^3 + y^4 can be written as

= 4C0x^4y^0 + 4C1x^3y^1 + 4C2x^2y^2 + 4C3x^1y^2 + 4C4x^0y^4

= (x + y)^4

When DIVIDED by (x + y), we GET (x + y)^3.

Right answer is (a) 125 ^96C3 x^93 y^3

The BEST I can explain: TR + 1 = ^nCr x^n – r y^r

Here first term is 4 and second term is 5y.

n = 96

r = 3

Therefore, Tr + 1 = ^96C3 x^96 – 3 (5y)^3

= 125 ^96C3 x^93 y^3

The CORRECT OPTION is (a) (n + 1)^th term

The best explanation: Clearly 2n is an EVEN number and the BINOMIAL has 2n + 1 terms.

The middle term for a binomial with even power, is the term equal to (n/2 + 1) where n is number of terms.

In this case,(2n/2 + 1) = n + 1.

Correct choice is (a) 3x^4

Easy explanation: SINCE the power is odd, there will be even NUMBER of terms and TWO middle terms.

r = \(\FRAC{n + 1}{2}\) and r = \(\frac{n – 1}{2}\)

Here n = 3

Therefore, r = 2 and r = 1.

When r = 2, ^3C2 (x^2)^3 – 2(x)^2 = 3x^4

When r = 1, ^3C1 (x^2)^3 – 1(x)^1 = 3x^5

Correct option is (a) ^7C3

The explanation is: SINCE the MIDDLE term is the FOURTH term

Considering 3x to be EVEN, \((\frac{3x + 1}{2})^{th}\) term = 4

x = 7/3

Therefore, the fourth term coefficients are ^7C3

The correct choice is (d) 1/50

To explain I would say: The numerator when simplified is of the form (101 – 99)^3

The DENOMINATOR can be simplified as (101 – 99)(101 + 99)

When we SUBSTITUTE in the numerator and denominator we get (2 X 2 x 2) / (2 x 200)

= 1/50.

Correct option is (a) 4

Easy explanation: The powers of FOUR follow the GIVEN order:

4^1 = 4

4^2 = 16

4^3 = 64

4^4 = 256

4^5 = 1024 and so on.

Odd powers of 4, have the number 4 in the units PLACE. When 5 divides the nearest ten, 4 will be obtained as the remainder each TIME.

Correct CHOICE is (d) x^2y^2 + 4xy + 4

To explain I would say: (a + b)^2 can be expanded using binomial theorem to get:

(a + b)^2 = a^2 + 2AB + b^2

Here, a = xy and b = 2

Therefore, (xy + 2)^2 = (xy)^2 + 2(xy)(2) + (2)^2

(xy + 2)^2 = x^2y^2 + 4xy + 4.

The correct answer is (b) 3

Easiest explanation: Coefficient of the second TERM of (x – y)^3 is ^3C1 and the coefficient of the third term of the EXPANSION (x + y)^n is ^nC2.

^3C1 = ^nC2

3 = \(\frac{n!}{2!(n – 2)!}\)

6 = \(\frac{n(n-1)(n-2)!}{(n – 2)!}\)

6 = n^2 – n

n^2 – n – 6 = 0

n^2 – 3n + 2n – 6 = 0

(n – 3) (n + 2) = 0

n = 3, – 2

Since n cannot be negative, n = 3.

Correct option is (a) 12

The best explanation: If 2N – 3 is EVEN, then the middle term is the \((\frac{2n-3}{2}+1)^{th}\) term.

Else if 2n – 3 is ODD, there are two middle terms which are the \((\frac{2n-3+1}{2})^{th}\) term and the \((\frac{2n-3+3}{2})^{th}\) term.

Case 1: \(\frac{2n-3}{2}\) + 1= 11 ; N = 11.5

Case 2: \(\frac{2n-2}{2}\) = 11 ; n = 12

Case 3: \(\frac{2n}{2}\) = 11 ; n = 11

The correct option is (C) 1328

Easy EXPLANATION: (11 + i)^3 = 11^3 + 3.11^2.i +3.i^2.11 +i^3

= 1331 + 363i – 3 – i

= 1328 + 365i.

Correct choice is (c) (x – 1)^91

Easiest explanation: The general term of an expansion is ^nCr x^N – r y^r.

Clearly here n is 91 and the first term is x raised to the power 89.

The second term is raised to power 2.

y^2 = 1

y = +1 or -1

Therefore the expansion can EITHER be (x + 1)^91 or (x – 1)^91.

Right option is (d) 16

To EXPLAIN: 4^103 = 4 x 4^102

4^103 = 4 x (4^2)^51

4^103 = 4 x (16)^51

4^103 = 4 x (17 – 1)^51

4^103 = 4 x \(\Sigma_{r = 0}^{r = 51}\)(51Cr 17^24 – r (-1)^r

4^103 = 4 x [51C0 17^51 (-1)^0 + 51C1 17^51 (-1)^1 +….+ 51C50 17^1 (-1)^50 + 51C51 17^0 (-1)^51]

4^103 = 4 x 17 x K – 1

The remainder = 17 – 1

Remainder = 16.

Right option is (b) 10240 x^3

To explain: The MIDDLE TERM will be the 4^th term

4^th term = ^6C3 (4)^6 – 3(2X)^3

= 20 (64) (8x^3)

= 10240 x^3

The correct choice is (c) ^xyCr (X^xy – R . -y^r)

For explanation I would SAY: The general term of a binomial series is given by ^nCr a^N – r b^r.

Here a = x, b = -y and n = xy

Therefore the general term is given by ^xyCr (x^xy – r . -y^r).