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 z = 1 + ai be a complex number, a > 0, such that z^{3} is a real number.

Then the sum 1 + z + z^{2} + . . . . .+ z^{11} is equal to :

Then the sum 1 + z + z

A

$$ - 1250\,\sqrt 3 \,i$$

B

$$ 1250\,\sqrt 3 \,i$$

C

$$1365\,\sqrt 3 i$$

D

$$-$$ $$1365\,\sqrt 3 i$$

z = 1 + ai

z^{2} = 1 $$-$$ a^{2} + 2ai

z^{2} . z = {(1 $$-$$ a^{2}) + 2ai} {1 + ai}

= (1 $$-$$ a^{2}) + 2ai + (1 $$-$$ a^{2}) ai $$-$$ 2a^{2}

$$ \because $$ z^{3} is real $$ \Rightarrow $$ 2a + (1 $$-$$ a^{2}) a = 0

a (3 $$-$$ a^{2}) = 0 $$ \Rightarrow $$ a = $$\sqrt 3 $$ (a > 0)

1 + z + z^{2} . . . . . . . z^{11} = $${{{z^{12}} - 1} \over {z - 1}} = {{{{\left( {1 + \sqrt 3 i} \right)}^{12}} - 1} \over {1 + \sqrt 3 i - 1}}$$

= $${{{{\left( {1 + \sqrt 3 i} \right)}^{12}} - 1} \over {\sqrt 3 i}}$$

(1 + $${\sqrt 3 i}$$)^{12} = 2^{12} $${\left( {{1 \over 2} + {{\sqrt 3 } \over 2}i} \right)^{12}}$$

= 2^{12} (cos$${\pi \over 3}$$ + isin$${\pi \over 3}$$)^{12} = 2^{12} (cos4$$\pi $$ + isin4$$\pi $$) = 2^{12}

$$ \Rightarrow $$ $${{{2^{12}} - 1} \over {\sqrt 3 i}} = {{4095} \over {\sqrt 3 i}} = - {{4095} \over 3}\sqrt 3 i = - 1365\sqrt 3 i$$

z

z

= (1 $$-$$ a

$$ \because $$ z

a (3 $$-$$ a

1 + z + z

= $${{{{\left( {1 + \sqrt 3 i} \right)}^{12}} - 1} \over {\sqrt 3 i}}$$

(1 + $${\sqrt 3 i}$$)

= 2

$$ \Rightarrow $$ $${{{2^{12}} - 1} \over {\sqrt 3 i}} = {{4095} \over {\sqrt 3 i}} = - {{4095} \over 3}\sqrt 3 i = - 1365\sqrt 3 i$$

2

Let a_{1}, a_{2}, a_{3}, . . . . . . . , a_{n}, . . . . . be in A.P.

If a_{3} + a_{7} + a_{11} + a_{15} = 72,

then the sum of its first 17 terms is equal to :

If a

then the sum of its first 17 terms is equal to :

A

306

B

153

C

612

D

204

As a_{1} a_{2} . . . . . a_{n} . . . . . are in A.P.

$$ \therefore $$ a_{3} + a_{15} = a_{7} + a_{11} = a_{1} + a_{17}

Given,

a_{3} + a_{7} + a_{11} + a_{15} + a_{15} = 72

$$ \Rightarrow $$ (a_{3} + a_{15}) + (a_{7} + a_{11}) = 72

$$ \Rightarrow $$ 2(a_{1} + a_{17}) = 72

$$ \Rightarrow $$ (a_{1} + a_{17}) = 36

$$ \therefore $$ Sum of first 17 terms

= $${{17} \over 2}$$ (a_{1} + a_{17})

= $${{17} \over 2}$$ $$ \times $$ 36

= 306

$$ \therefore $$ a

Given,

a

$$ \Rightarrow $$ (a

$$ \Rightarrow $$ 2(a

$$ \Rightarrow $$ (a

$$ \therefore $$ Sum of first 17 terms

= $${{17} \over 2}$$ (a

= $${{17} \over 2}$$ $$ \times $$ 36

= 306

3

If A > 0, B > 0 and A + B = $${\pi \over 6}$$,

then the minimum value of tanA + tanB is :

then the minimum value of tanA + tanB is :

A

$$\sqrt 3 - \sqrt 2 $$

B

$$2 - \sqrt 3 $$

C

$$4 - 2\sqrt 3 $$

D

$${2 \over {\sqrt 3 }}$$

Given,

A + B = $${\pi \over 6}$$

$$ \therefore $$ tan(A + B) = tan$$\left( {{\pi \over 6}} \right)$$ = $${1 \over {\sqrt 3 }}$$

We know,

tan(A + B) = $${{\tan A + \tan B} \over {1 - \tan A\tan B}}$$

$$ \Rightarrow $$ $${1 \over {\sqrt 3 }}$$ = $${y \over {1 - \tan A\tan B}}$$

where y = tan A + tan B

$$ \Rightarrow $$ tanA tanB = 1 $$-$$ $$\sqrt 3 $$ y

Also AM $$ \ge $$ GM

$$ \Rightarrow $$ $${{\tan A + \tan B} \over 2} \ge \sqrt {\tan A\tan B} $$

$$ \Rightarrow $$ y $$ \ge $$ 2$$\sqrt {1 - \sqrt 3 y} $$

$$ \Rightarrow $$ y^{2} $$ \ge $$ 4 $$-$$ 4$${\sqrt 3 y}$$

$$ \Rightarrow $$ y^{2} + 4$${\sqrt 3 y}$$ $$-$$ 4 $$ \ge $$ 0

$$ \Rightarrow $$ y $$ \le $$ $$-$$ 2$$\sqrt 3 $$ $$-$$ 4

or y $$ \ge $$ $$-$$ 2$$\sqrt 3 $$ + 4

(y $$ \le $$ $$-$$ 2$$\sqrt 3 $$ $$-$$ 4 is not possible as tan B > 0)

A + B = $${\pi \over 6}$$

$$ \therefore $$ tan(A + B) = tan$$\left( {{\pi \over 6}} \right)$$ = $${1 \over {\sqrt 3 }}$$

We know,

tan(A + B) = $${{\tan A + \tan B} \over {1 - \tan A\tan B}}$$

$$ \Rightarrow $$ $${1 \over {\sqrt 3 }}$$ = $${y \over {1 - \tan A\tan B}}$$

where y = tan A + tan B

$$ \Rightarrow $$ tanA tanB = 1 $$-$$ $$\sqrt 3 $$ y

Also AM $$ \ge $$ GM

$$ \Rightarrow $$ $${{\tan A + \tan B} \over 2} \ge \sqrt {\tan A\tan B} $$

$$ \Rightarrow $$ y $$ \ge $$ 2$$\sqrt {1 - \sqrt 3 y} $$

$$ \Rightarrow $$ y

$$ \Rightarrow $$ y

$$ \Rightarrow $$ y $$ \le $$ $$-$$ 2$$\sqrt 3 $$ $$-$$ 4

or y $$ \ge $$ $$-$$ 2$$\sqrt 3 $$ + 4

(y $$ \le $$ $$-$$ 2$$\sqrt 3 $$ $$-$$ 4 is not possible as tan B > 0)

4

Let $$a$$, b, c $$ \in R$$. If $$f$$(x) = ax^{2} + bx + c is such that

$$a$$ + b + c = 3 and $$f$$(x + y) = $$f$$(x) + $$f$$(y) + xy, $$\forall x,y \in R,$$

then $$\sum\limits_{n = 1}^{10} {f(n)} $$ is equal to

$$a$$ + b + c = 3 and $$f$$(x + y) = $$f$$(x) + $$f$$(y) + xy, $$\forall x,y \in R,$$

then $$\sum\limits_{n = 1}^{10} {f(n)} $$ is equal to

A

165

B

190

C

255

D

330

f(x) = ax^{2} + bx + c

f(1) = a + b + c = 3 $$ \Rightarrow $$ f (1) = 3

Now f(x + y) = f(x) + f(y) + xy ...(1)

Put x = y = 1 in eqn (1)

f(2) = f(1) + f(1) + 1

= 2f(1) + 1

$$ \Rightarrow $$ f(2) = 7

Similarly f(3) = 12

f(4) = 18

$$\sum\limits_{n = 1}^{10} {f(n)} $$ = 3 + 7 + 12 + 18 + 25 + 33 + 42 + 52 + 63 + 75 = 330

f(1) = a + b + c = 3 $$ \Rightarrow $$ f (1) = 3

Now f(x + y) = f(x) + f(y) + xy ...(1)

Put x = y = 1 in eqn (1)

f(2) = f(1) + f(1) + 1

= 2f(1) + 1

$$ \Rightarrow $$ f(2) = 7

Similarly f(3) = 12

f(4) = 18

$$\sum\limits_{n = 1}^{10} {f(n)} $$ = 3 + 7 + 12 + 18 + 25 + 33 + 42 + 52 + 63 + 75 = 330

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*