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Graduate Aptitude Test in Engineering

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Engineering Mathematics

General Aptitude

1

The number of all possible positive integral values of $$\alpha $$ for which the roots of the quadratic equation, 6x^{2} $$-$$ 11x + $$\alpha $$ = 0 are rational numbers is :

A

3

B

2

C

4

D

5

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

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

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

So, ans $$\alpha $$ = 3

2

Consider the quadratic equation (c – 5)x^{2} – 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

Let f(x) = (c $$-$$ 5)x

$$ \therefore $$ 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

3

The value of $$\lambda $$ such that sum of the squares of the roots of the quadratic equation, x^{2} + (3 – $$\lambda $$)x + 2 = $$\lambda $$ has the least value is -

A

1

B

2

C

$${{15} \over 8}$$

D

$${4 \over 9}$$

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

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

$$\alpha $$

= $$\lambda $$

= $$\lambda $$

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

$$ \therefore $$ $$\lambda $$ = 2

4

If one real root of the quadratic equation 81x^{2} + kx + 256 = 0 is cube of the other root, then a value of k is

A

$$-$$ 81

B

$$-$$ 300

C

100

D

144

81x^{2} + kx + 256 = 0 ; x = $$\alpha $$, $$\alpha $$^{3}

$$ \Rightarrow $$ $$\alpha $$^{4} = $${{256} \over {81}}$$ $$ \Rightarrow $$ $$\alpha $$ = $$ \pm $$ $${{4} \over {3}}$$

Now $$-$$ $${k \over {81}}$$ = $$\alpha $$ + $$\alpha $$^{3} = $$ \pm $$ $${{100} \over {27}}$$

$$ \Rightarrow $$ k = $$ \pm $$300

$$ \Rightarrow $$ $$\alpha $$

Now $$-$$ $${k \over {81}}$$ = $$\alpha $$ + $$\alpha $$

$$ \Rightarrow $$ k = $$ \pm $$300

Number in Brackets after Paper Name Indicates No of Questions

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Trigonometric Functions & Equations *keyboard_arrow_right*

Properties of Triangle *keyboard_arrow_right*

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Complex Numbers *keyboard_arrow_right*

Quadratic Equation and Inequalities *keyboard_arrow_right*

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Circle *keyboard_arrow_right*

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