Joint Entrance Examination

Graduate Aptitude Test in Engineering

Geomatics Engineering Or Surveying

Engineering Mechanics

Hydrology

Transportation Engineering

Strength of Materials Or Solid Mechanics

Reinforced Cement Concrete

Steel Structures

Irrigation

Environmental Engineering

Engineering Mathematics

Structural Analysis

Geotechnical Engineering

Fluid Mechanics and Hydraulic Machines

General Aptitude

1

If both the roots of the quadratic equation x^{2} $$-$$ mx + 4 = 0 are real and distinct and they lie in the interval [1, 5], then m lies in the interval :

A

($$-$$5, $$-$$4)

B

(4, 5)

C

(5, 6)

D

(3, 4)

x^{2} $$-$$mx + 4 = 0

**Case-I :**

D > 0

m^{2} $$-$$ 16 > 0

$$ \Rightarrow $$ m $$ \in $$ ($$-$$ $$\infty $$, $$-$$ 4) $$ \cup $$ (4, $$\infty $$)

**Case-II :**

$$ \Rightarrow \,\,1 < {{ - b} \over {2a}} < 5$$

$$ \Rightarrow \,\,1 < {m \over 2} < 5 \Rightarrow \,m \in \left( {2,10} \right)$$

**Case-III :**

f(1) > 0 and f(5) > 0

1 $$-$$ m + 4 > 0 and 25 $$-$$ 5m + 4 > 0

m < 5 and m < $${{29} \over 5}$$

**Case-IV :**

Let one root is x = 1

1 $$-$$ m + 4 = 0

m = 5

Now equation x^{2} $$-$$ 5x + 4 = 0

(x $$-$$ 1) (x $$-$$ 4) = 0

x = 1 i.e. m = 5 is also included

hence m $$ \in $$ (4, 5]

So given option is (4, 5)

D > 0

m

$$ \Rightarrow $$ m $$ \in $$ ($$-$$ $$\infty $$, $$-$$ 4) $$ \cup $$ (4, $$\infty $$)

$$ \Rightarrow \,\,1 < {{ - b} \over {2a}} < 5$$

$$ \Rightarrow \,\,1 < {m \over 2} < 5 \Rightarrow \,m \in \left( {2,10} \right)$$

f(1) > 0 and f(5) > 0

1 $$-$$ m + 4 > 0 and 25 $$-$$ 5m + 4 > 0

m < 5 and m < $${{29} \over 5}$$

Let one root is x = 1

1 $$-$$ m + 4 = 0

m = 5

Now equation x

(x $$-$$ 1) (x $$-$$ 4) = 0

x = 1 i.e. m = 5 is also included

hence m $$ \in $$ (4, 5]

So given option is (4, 5)

2

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

3

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

4

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

Number in Brackets after Paper Name Indicates No of Questions

AIEEE 2002 (5) *keyboard_arrow_right*

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Trigonometric Functions & Equations *keyboard_arrow_right*

Properties of Triangle *keyboard_arrow_right*

Inverse Trigonometric Functions *keyboard_arrow_right*

Complex Numbers *keyboard_arrow_right*

Quadratic Equation and Inequalities *keyboard_arrow_right*

Permutations and Combinations *keyboard_arrow_right*

Mathematical Induction and Binomial Theorem *keyboard_arrow_right*

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Matrices and Determinants *keyboard_arrow_right*

Vector Algebra and 3D Geometry *keyboard_arrow_right*

Probability *keyboard_arrow_right*

Statistics *keyboard_arrow_right*

Mathematical Reasoning *keyboard_arrow_right*

Functions *keyboard_arrow_right*

Limits, Continuity and Differentiability *keyboard_arrow_right*

Differentiation *keyboard_arrow_right*

Application of Derivatives *keyboard_arrow_right*

Indefinite Integrals *keyboard_arrow_right*

Definite Integrals and Applications of Integrals *keyboard_arrow_right*

Differential Equations *keyboard_arrow_right*

Straight Lines and Pair of Straight Lines *keyboard_arrow_right*

Circle *keyboard_arrow_right*

Conic Sections *keyboard_arrow_right*