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

Let p, q and r be real numbers (p $$ \ne $$ q, r $$ \ne $$ 0), such that the roots of the equation $${1 \over {x + p}} + {1 \over {x + q}} = {1 \over r}$$ are equal in magnitude but opposite in sign, then the sum of squares of these roots is equal to :

A

$${{{p^2} + {q^2}} \over 2}$$

B

p^{2} + q^{2}

C

2(p^{2} + q^{2})

D

p^{2} + q^{2} + r^{2}

Given,

$${1 \over {x + p}} + {1 \over {x + q}} = {1 \over r}$$

$$ \Rightarrow $$$$\,\,\,$$ $${{x + p + x + q} \over {\left( {x + p} \right)\left( {x + q} \right)}} = {1 \over r}$$

$$ \Rightarrow $$$$\,\,\,$$ (2x + p + q) r = x^{2} + px + qx + pq

$$ \Rightarrow $$$$\,\,\,$$ x^{2} + (p + q $$-$$ 2r) x + pq $$-$$ pr $$-$$ qr = 0

Let $$\alpha $$ and $$\beta $$ are the roots,

$$\therefore\,\,\,$$ $$\alpha $$ + $$\beta $$ = $$-$$ (p + q $$-$$ 2r)

and $$\alpha $$ $$\beta $$ = pq $$-$$ pr $$-$$ qr

Given that, $$\alpha $$ = $$-$$ $$\beta $$ $$ \Rightarrow $$ $$\alpha $$ + $$\beta $$ = 0

$$\therefore\,\,\,$$ $$-$$ (p + q $$-$$ 2r) = 0

Now, $$\alpha $$^{2} + $$\beta $$^{2}

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

= ($$-$$ (p + q $$-$$ 2r))^{2} $$-$$ 2 (pq $$-$$ pr $$-$$ qr)

= p^{2} +q^{2} + 4r^{2} + 2pq $$-$$ 4pr $$-$$ 4qr $$-$$ 2pq + 2pr + 2qr

= p^{2} + q^{2} + 4r^{2} $$-$$ 2pr $$-$$ 2qr

= p^{2} + q^{2} $$-$$ 2r (p + q $$-$$ 2r)

= p^{2} + q^{2} $$-$$ 2r (0)

= p^{2} + q^{2}

$${1 \over {x + p}} + {1 \over {x + q}} = {1 \over r}$$

$$ \Rightarrow $$$$\,\,\,$$ $${{x + p + x + q} \over {\left( {x + p} \right)\left( {x + q} \right)}} = {1 \over r}$$

$$ \Rightarrow $$$$\,\,\,$$ (2x + p + q) r = x

$$ \Rightarrow $$$$\,\,\,$$ x

Let $$\alpha $$ and $$\beta $$ are the roots,

$$\therefore\,\,\,$$ $$\alpha $$ + $$\beta $$ = $$-$$ (p + q $$-$$ 2r)

and $$\alpha $$ $$\beta $$ = pq $$-$$ pr $$-$$ qr

Given that, $$\alpha $$ = $$-$$ $$\beta $$ $$ \Rightarrow $$ $$\alpha $$ + $$\beta $$ = 0

$$\therefore\,\,\,$$ $$-$$ (p + q $$-$$ 2r) = 0

Now, $$\alpha $$

= ($$\alpha $$ + $$\beta $$)

= ($$-$$ (p + q $$-$$ 2r))

= p

= p

= p

= p

= p

2

If an angle A of a $$\Delta $$ABC satiesfies 5 cosA + 3 = 0, then the roots of the quadratic equation, 9x^{2} + 27x + 20 = 0 are :

A

secA, cotA

B

sinA, secA

C

secA, tanA

D

tanA, cosA

Here, 9x^{2} + 27x + 20 = 0

$$\therefore\,\,\,$$ x = $${{ - b \pm \sqrt {{b^2} - 4ac} } \over {2a}}$$

$$ \Rightarrow $$$$\,\,\,$$ x = $${{ - 27 \pm \sqrt {{{27}^2} - 4 \times 9 \times 20} } \over {2 \times 9}}$$

$$ \Rightarrow $$$$\,\,\,$$ x = $$-$$ $${4 \over 3}$$, $$-$$ $${5 \over 3}$$

Given, cosA = $$-$$ $${3 \over 5}$$

$$\therefore\,\,\,$$ sec A = $${1 \over {\cos A}}$$ = $$-$$ $${5 \over 3}$$

Here, A is an obtuse angle.

$$\therefore\,\,\,$$ tan A = $$-$$ $$\sqrt {{{\sec }^2}A - 1} = - {4 \over 3}.$$

Hence, roots of the equation are sec A and tan A.

$$\therefore\,\,\,$$ x = $${{ - b \pm \sqrt {{b^2} - 4ac} } \over {2a}}$$

$$ \Rightarrow $$$$\,\,\,$$ x = $${{ - 27 \pm \sqrt {{{27}^2} - 4 \times 9 \times 20} } \over {2 \times 9}}$$

$$ \Rightarrow $$$$\,\,\,$$ x = $$-$$ $${4 \over 3}$$, $$-$$ $${5 \over 3}$$

Given, cosA = $$-$$ $${3 \over 5}$$

$$\therefore\,\,\,$$ sec A = $${1 \over {\cos A}}$$ = $$-$$ $${5 \over 3}$$

Here, A is an obtuse angle.

$$\therefore\,\,\,$$ tan A = $$-$$ $$\sqrt {{{\sec }^2}A - 1} = - {4 \over 3}.$$

Hence, roots of the equation are sec A and tan A.

3

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)

4

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

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*

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*

Sequences and Series *keyboard_arrow_right*

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*