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 $$A = \left[ {\matrix{
{{e^t}} & {{e^{ - t}}\cos t} & {{e^{ - t}}\sin t} \cr
{{e^t}} & { - {e^{ - t}}\cos t - {e^{ - t}}\sin t} & { - {e^{ - t}}\sin t + {e^{ - t}}co{\mathop{\rm s}\nolimits} t} \cr
{{e^t}} & {2{e^{ - t}}\sin t} & { - 2{e^{ - t}}\cos t} \cr
} } \right]$$

then A is :

then A is :

A

invertible for all t$$ \in $$**R**.

B

invertible only if t $$=$$ $$\pi $$

C

not invertible for any t$$ \in $$**R**

D

invertible only if t $$=$$ $${\pi \over 2}$$.

$$A = \left[ {\matrix{
{{e^t}} & {{e^{ - t}}\cos t} & {{e^{ - t}}\sin t} \cr
{{e^t}} & { - {e^{ - t}}\cos t - {e^{ - t}}\sin t} & { - {e^{ - t}}\sin t + {e^{ - t}}co{\mathop{\rm s}\nolimits} t} \cr
{{e^t}} & {2{e^{ - t}}\sin t} & { - 2{e^{ - t}}\cos t} \cr
} } \right]$$

$$\left| A \right| = {e^t}.\,{e^{ - t}}.{e^{ - t}}\left| {\matrix{ 1 & {\cos t} & {\sin t} \cr 1 & { - \cos t - \sin t} & { - \sin t + \cos t} \cr 1 & {2\sin t} & { - 2\cos t} \cr } } \right|$$

Apply operations R_{2} < R_{2} $$-$$R_{1}, R_{3} < R_{3} $$-$$ R_{1}, R_{1} < R_{1}

$$\left| A \right| = {e^{ - t}}\left| {\matrix{ 1 & {\cos t} & {\sin t} \cr 0 & { - \sin t - 2\cos t} & { - 2\sin t + \cos t} \cr 0 & {2\sin t - \cos t} & { - 2\cos t - \sin t} \cr } } \right|$$

Open the determinant by R_{1}

$$\left| A \right| = 5{e^{ - t}}$$

Invertible for all t $$ \in $$ R

$$\left| A \right| = {e^t}.\,{e^{ - t}}.{e^{ - t}}\left| {\matrix{ 1 & {\cos t} & {\sin t} \cr 1 & { - \cos t - \sin t} & { - \sin t + \cos t} \cr 1 & {2\sin t} & { - 2\cos t} \cr } } \right|$$

Apply operations R

$$\left| A \right| = {e^{ - t}}\left| {\matrix{ 1 & {\cos t} & {\sin t} \cr 0 & { - \sin t - 2\cos t} & { - 2\sin t + \cos t} \cr 0 & {2\sin t - \cos t} & { - 2\cos t - \sin t} \cr } } \right|$$

Open the determinant by R

$$\left| A \right| = 5{e^{ - t}}$$

Invertible for all t $$ \in $$ R

2

If the system of equations

x + y + z = 5

x + 2y + 3z = 9

x + 3y + az = $$\beta $$

has infinitely many solutions, then $$\beta $$ $$-$$ $$\alpha $$ equals -

x + y + z = 5

x + 2y + 3z = 9

x + 3y + az = $$\beta $$

has infinitely many solutions, then $$\beta $$ $$-$$ $$\alpha $$ equals -

A

8

B

21

C

18

D

5

$$D = \left| {\matrix{
1 & 1 & 1 \cr
1 & 2 & 3 \cr
1 & 3 & \alpha \cr
} } \right| = \left| {\matrix{
1 & 1 & 1 \cr
0 & 1 & 2 \cr
0 & 2 & {\alpha - 1} \cr
} } \right|$$

$$ = \left( {\alpha - 1} \right) - 4 = \left( {\alpha - 5} \right)$$

for infinite solutions $$D = 0 \Rightarrow \alpha = 5$$

$${D_x} = 0 \Rightarrow \left| {\matrix{ 5 & 1 & 1 \cr 9 & 2 & 3 \cr \beta & 3 & 5 \cr } } \right| = 0$$

$$ \Rightarrow \left| {\matrix{ 0 & 0 & 1 \cr { - 1} & { - 1} & 3 \cr {\beta - 15} & { - 2} & 5 \cr } } \right| = 0$$

$$ \Rightarrow 2 + \beta - 15 = 0 \Rightarrow \beta - 13 = 0$$

on $$\beta = 13$$ we get $${D_y} = {D_z} = 0$$

$$\alpha = 5,\beta = 13$$

$$ = \left( {\alpha - 1} \right) - 4 = \left( {\alpha - 5} \right)$$

for infinite solutions $$D = 0 \Rightarrow \alpha = 5$$

$${D_x} = 0 \Rightarrow \left| {\matrix{ 5 & 1 & 1 \cr 9 & 2 & 3 \cr \beta & 3 & 5 \cr } } \right| = 0$$

$$ \Rightarrow \left| {\matrix{ 0 & 0 & 1 \cr { - 1} & { - 1} & 3 \cr {\beta - 15} & { - 2} & 5 \cr } } \right| = 0$$

$$ \Rightarrow 2 + \beta - 15 = 0 \Rightarrow \beta - 13 = 0$$

on $$\beta = 13$$ we get $${D_y} = {D_z} = 0$$

$$\alpha = 5,\beta = 13$$

3

Let d $$ \in $$ R, and

$$A = \left[ {\matrix{ { - 2} & {4 + d} & {\left( {\sin \theta } \right) - 2} \cr 1 & {\left( {\sin \theta } \right) + 2} & d \cr 5 & {\left( {2\sin \theta } \right) - d} & {\left( { - \sin \theta } \right) + 2 + 2d} \cr } } \right],$$

$$\theta \in \left[ {0,2\pi } \right]$$ If the minimum value of det(A) is 8, then a value of d is -

$$A = \left[ {\matrix{ { - 2} & {4 + d} & {\left( {\sin \theta } \right) - 2} \cr 1 & {\left( {\sin \theta } \right) + 2} & d \cr 5 & {\left( {2\sin \theta } \right) - d} & {\left( { - \sin \theta } \right) + 2 + 2d} \cr } } \right],$$

$$\theta \in \left[ {0,2\pi } \right]$$ If the minimum value of det(A) is 8, then a value of d is -

A

$$-$$ 7

B

$$2\left( {\sqrt 2 + 2} \right)$$

C

$$-$$ 5

D

$$2\left( {\sqrt 2 + 1} \right)$$

$$\det A = \left| {\matrix{
{ - 2} & {4 + d} & {\sin \theta - 2} \cr
1 & {\sin \theta + 2} & d \cr
5 & {2\sin \theta - d} & { - \sin \theta + 2 + 2d} \cr
} } \right|$$

(R_{1} $$ \to $$ R_{1} + R_{3} $$-$$ 2R_{2})

$$ = \left| {\matrix{ 1 & 0 & 0 \cr 1 & {\sin \theta + 2} & d \cr 5 & {2\sin \theta - d} & {2 + 2d - \sin \theta } \cr } } \right|$$

= (2 + sin $$\theta $$) ( 2 + 2d $$-$$ sin$$\theta $$) $$-$$ d(2sin$$\theta $$ $$-$$ d)

= 4 + 4d $$-$$ 2sin$$\theta $$ + 2sin$$\theta $$ + 2dsin$$\theta $$ $$-$$ sin^{2}$$\theta $$ $$-$$ 2dsin$$\theta $$ + d^{2}

d^{2} + 4d + 4 $$-$$ sin^{2}$$\theta $$

= (d + 2)^{2} $$-$$ sin^{2}$$\theta $$

For a given d, minimum value of

det(A) = (d + 2)^{2} $$-$$ 1 = 8

$$ \Rightarrow $$ d = 1 or $$-$$ 5

(R

$$ = \left| {\matrix{ 1 & 0 & 0 \cr 1 & {\sin \theta + 2} & d \cr 5 & {2\sin \theta - d} & {2 + 2d - \sin \theta } \cr } } \right|$$

= (2 + sin $$\theta $$) ( 2 + 2d $$-$$ sin$$\theta $$) $$-$$ d(2sin$$\theta $$ $$-$$ d)

= 4 + 4d $$-$$ 2sin$$\theta $$ + 2sin$$\theta $$ + 2dsin$$\theta $$ $$-$$ sin

d

= (d + 2)

For a given d, minimum value of

det(A) = (d + 2)

$$ \Rightarrow $$ d = 1 or $$-$$ 5

4

Let A = $$\left[ {\matrix{
2 & b & 1 \cr
b & {{b^2} + 1} & b \cr
1 & b & 2 \cr
} } \right]$$ where b > 0.

Then the minimum value of $${{\det \left( A \right)} \over b}$$ is -

Then the minimum value of $${{\det \left( A \right)} \over b}$$ is -

A

$$\sqrt 3 $$

B

$$-$$ $$2\sqrt 3 $$

C

$$ - \sqrt 3 $$

D

$$2\sqrt 3 $$

A = $$\left[ {\matrix{
2 & b & 1 \cr
b & {{b^2} + 1} & b \cr
1 & b & 2 \cr
} } \right]$$ (b > 0)

$$\left| A \right|$$ = 2(2b^{2} + 2 $$-$$ b^{2}) $$-$$ b(2b $$-$$ b) + 1(b_{2} $$-$$ b_{2} $$-$$ 1)

$$\left| A \right|$$ = 2(b^{2} + 2) $$-$$ b^{2} $$-$$ 1

$$\left| A \right|$$ = b^{2} + 3

$${{\left| A \right|} \over b} = b + {3 \over b} \Rightarrow {{b + {3 \over b}} \over 2} \ge \sqrt 3 $$

$$b + {3 \over b} \ge 2\sqrt 3 $$

$$\left| A \right|$$ = 2(2b

$$\left| A \right|$$ = 2(b

$$\left| A \right|$$ = b

$${{\left| A \right|} \over b} = b + {3 \over b} \Rightarrow {{b + {3 \over b}} \over 2} \ge \sqrt 3 $$

$$b + {3 \over b} \ge 2\sqrt 3 $$

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*