Joint Entrance Examination

Graduate Aptitude Test in Engineering

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1

MCQ (Single Correct Answer)

Let S_{n} = 1 + q + q^{2} + . . . . . + q^{n} and T_{n} = 1 + $$\left( {{{q + 1} \over 2}} \right) + {\left( {{{q + 1} \over 2}} \right)^2}$$ + . . . . . .+ $${\left( {{{q + 1} \over 2}} \right)^n}$$ where q is a real number and q $$ \ne $$ 1. If ^{101}C_{1} + ^{101}C_{2} . S_{1} + .... + ^{101}C_{101} . S_{100} = $$\alpha $$T_{100} then $$\alpha $$ is equal to

A

202

B

200

C

2^{100}

D

2^{99}

$$=$$ $$\alpha $$T

. . . . . .+

$$ = 2\alpha {{\left( {1 - {{\left( {{{1 + q} \over 2}} \right)}^{101}}} \right)} \over {\left( {1 - q} \right)}}$$

$$ \Rightarrow $$

. . . . . . +

$$ = 2\alpha \left( {1 - {{\left( {{{1 + q} \over 2}} \right)}^{101}}} \right)$$

$$ \Rightarrow $$ (2

$$ = 2\alpha \left( {1 - {{\left( {{{1 + q} \over 2}} \right)}^{101}}} \right)$$

$$ \Rightarrow $$ $${2^{101}}\left( {1 - {{\left( {{{1 + q} \over 2}} \right)}^{101}}} \right) = 2\alpha \left( {1 - {{\left( {{{1 + q} \over 2}} \right)}^{101}}} \right)$$

$$ \Rightarrow $$ $$\alpha = {2^{100}}$$

2

MCQ (Single Correct Answer)

Let (x + 10)^{50} + (x $$-$$ 10)^{50} = a_{0} + a_{1}x + a_{2}x^{2} + . . . . + a_{50}x^{50}, for all x $$ \in $$ R; then $${{{a_2}} \over {{a_0}}}$$ is equal to

A

12.25

B

12.75

C

12.00

D

12.50

(10 + x)^{50} + (10 $$-$$ x)^{50}

$$ \Rightarrow $$ a_{2} = 2.^{50}C_{2} 10^{48}, a_{0} = 2.10^{50}

$${{{a_2}} \over {{a_0}}} = {{^{50}{C_2}} \over {{{10}^2}}} = 12.25$$

$$ \Rightarrow $$ a

$${{{a_2}} \over {{a_0}}} = {{^{50}{C_2}} \over {{{10}^2}}} = 12.25$$

3

MCQ (Single Correct Answer)

The sum of the real values of x for which the middle term in the binomial expansion of $${\left( {{{{x^3}} \over 3} + {3 \over x}} \right)^8}$$ equals 5670 is :

A

0

B

8

C

6

D

4

$${T_5} = {}^8{C_4}{{{x^{12}}} \over {81}} \times {{81} \over {{x^4}}} = 5670$$

$$ \Rightarrow 70{x^8} = 5670$$

$$ \Rightarrow x = \pm \sqrt 3 $$

$$ \Rightarrow 70{x^8} = 5670$$

$$ \Rightarrow x = \pm \sqrt 3 $$

4

MCQ (Single Correct Answer)

The value of r for which ^{20}C_{r} ^{20}C_{0} + ^{20}C_{r$$-$$1} ^{20}C_{1} + ^{20}C_{r$$-$$2} ^{20}C_{2} + . . . . .+ ^{20}C_{0} ^{20}C_{r} is maximum, is

A

20

B

15

C

10

D

11

Given sum = coefficient of x^{r} in the expansion of

(1 + x)^{20}(1 + x)^{20},

Which is equal to^{40}C_{r}

It is maximum when r = 20

(1 + x)

Which is equal to

It is maximum when r = 20

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