Joint Entrance Examination

Graduate Aptitude Test in Engineering

1

MCQ (Single Correct Answer)

All the values of $$m$$ for which both roots of the equation $${x^2} - 2mx + {m^2} - 1 = 0$$ are greater than $$ - 2$$ but less then 4, lie in the interval

A

$$ - 2 < m < 0$$

B

$$m > 3$$

C

$$ - 1 < m < 3$$

D

$$1 < m < 4$$

Equation $${x^2} - 2mx + {m^2} - 1 = 0$$

$${\left( {x - m} \right)^2} - 1 = 0$$

or $$\left( {x - m + 1} \right)\left( {x - m - 1} \right) = 0$$

$$x = m - 1,m + 1$$

$$m - 1 > - 2$$ and $$m + 1 < 4$$

$$ \Rightarrow m > - 1$$ and $$m<3$$

or $$\,\,\, - 1 < m < 3$$

$${\left( {x - m} \right)^2} - 1 = 0$$

or $$\left( {x - m + 1} \right)\left( {x - m - 1} \right) = 0$$

$$x = m - 1,m + 1$$

$$m - 1 > - 2$$ and $$m + 1 < 4$$

$$ \Rightarrow m > - 1$$ and $$m<3$$

or $$\,\,\, - 1 < m < 3$$

2

MCQ (Single Correct Answer)

If the roots of the quadratic equation $${x^2} + px + q = 0$$ are $$\tan {30^ \circ }$$ and $$\tan {15^ \circ }$$, respectively, then the value of $$2 + q - p$$ is

A

2

B

3

C

0

D

1

$${x^2} + px + q = 0$$

Sum of roots $$ = \tan {30^ \circ } + \tan {15^ \circ } = - p$$

Products of roots $$ = \tan {30^ \circ }.\tan {15^ \circ } = q$$

$$\tan {45^ \circ } = {{\tan {{30}^ \circ } + \tan {{15}^ \circ }} \over {1 - \tan {{30}^ \circ }.\tan {{15}^ \circ }}}$$

$$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = {{ - p} \over {1 - q}} = 1$$

$$ \Rightarrow - p = 1 - q \Rightarrow q - p = 1$$

$$\therefore$$ $$2 + q - p = 3$$

Sum of roots $$ = \tan {30^ \circ } + \tan {15^ \circ } = - p$$

Products of roots $$ = \tan {30^ \circ }.\tan {15^ \circ } = q$$

$$\tan {45^ \circ } = {{\tan {{30}^ \circ } + \tan {{15}^ \circ }} \over {1 - \tan {{30}^ \circ }.\tan {{15}^ \circ }}}$$

$$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = {{ - p} \over {1 - q}} = 1$$

$$ \Rightarrow - p = 1 - q \Rightarrow q - p = 1$$

$$\therefore$$ $$2 + q - p = 3$$

3

MCQ (Single Correct Answer)

If the roots of the equation $${x^2} - bx + c = 0$$ be two consecutive integers, then $${b^2} - 4c$$ equals

A

$$-2$$

B

$$3$$

C

$$2$$

D

$$1$$

Let n and (n + 1) be the roots of x^{2} $$-$$ bx + c = 0.

Then, n + (n + 1) = b and n(n + 1) = c

$$\therefore$$ b^{2} $$-$$ 4c = (2n + 1)^{2} $$-$$ 4n(n + 1)

= 4n^{2} + 4n + 1 $$-$$ 4n^{2} $$-$$ 4n = 1

4

MCQ (Single Correct Answer)

The value of $$a$$ for which the sum of the squares of the roots of the equation

$${x^2} - \left( {a - 2} \right)x - a - 1 = 0$$ assume the least value is

$${x^2} - \left( {a - 2} \right)x - a - 1 = 0$$ assume the least value is

A

$$1$$

B

$$0$$

C

$$3$$

D

$$2$$

Given quadratic equation,

$${x^2} - \left( {a - 2} \right)x - a - 1 = 0$$

Let $$\alpha $$ and $$\beta $$ are the roots of the equation.

$$ \therefore $$ $$\alpha $$ + $$\beta $$ = $$a - 2$$

and $$\alpha $$$$\beta $$ = $$ - a - 1$$

Now $${\alpha ^2} + {\beta ^2} = {\left( {\alpha + \beta } \right)^2} - 2\alpha \beta $$

$$ \Rightarrow $$ $${\alpha ^2} + {\beta ^2} = {\left( {a - 2} \right)^2} + 2\left( {a + 1} \right)$$

$$ \Rightarrow $$ $${\alpha ^2} + {\beta ^2} = {a^2} - 2a + 6$$

$$ \Rightarrow $$ $${\alpha ^2} + {\beta ^2} = {\left( {a - 1} \right)^2} + 5$$

$$ \Rightarrow $$ The value of $${\alpha ^2} + {\beta ^2}$$ will be minimum, when $${a - 1}$$ = 0

$$ \Rightarrow $$ $${a = 1}$$

$${x^2} - \left( {a - 2} \right)x - a - 1 = 0$$

Let $$\alpha $$ and $$\beta $$ are the roots of the equation.

$$ \therefore $$ $$\alpha $$ + $$\beta $$ = $$a - 2$$

and $$\alpha $$$$\beta $$ = $$ - a - 1$$

Now $${\alpha ^2} + {\beta ^2} = {\left( {\alpha + \beta } \right)^2} - 2\alpha \beta $$

$$ \Rightarrow $$ $${\alpha ^2} + {\beta ^2} = {\left( {a - 2} \right)^2} + 2\left( {a + 1} \right)$$

$$ \Rightarrow $$ $${\alpha ^2} + {\beta ^2} = {a^2} - 2a + 6$$

$$ \Rightarrow $$ $${\alpha ^2} + {\beta ^2} = {\left( {a - 1} \right)^2} + 5$$

$$ \Rightarrow $$ The value of $${\alpha ^2} + {\beta ^2}$$ will be minimum, when $${a - 1}$$ = 0

$$ \Rightarrow $$ $${a = 1}$$

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Complex Numbers

Quadratic Equation and Inequalities

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Sequences and Series

Matrices and Determinants

Vector Algebra and 3D Geometry

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Inverse Trigonometric Functions

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Circle

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Application of Derivatives

Indefinite Integrals

Definite Integrals and Applications of Integrals

Differential Equations