Joint Entrance Examination

Graduate Aptitude Test in Engineering

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1

MCQ (Single Correct Answer)

Let $$A = \left( {\matrix{
1 & 0 & 0 \cr
2 & 1 & 0 \cr
3 & 2 & 1 \cr
} } \right)$$. If $${u_1}$$ and $${u_2}$$ are column matrices such

that $$A{u_1} = \left( {\matrix{ 1 \cr 0 \cr 0 \cr } } \right)$$ and $$A{u_2} = \left( {\matrix{ 0 \cr 1 \cr 0 \cr } } \right),$$ then $${u_1} + {u_2}$$ is equal to :

that $$A{u_1} = \left( {\matrix{ 1 \cr 0 \cr 0 \cr } } \right)$$ and $$A{u_2} = \left( {\matrix{ 0 \cr 1 \cr 0 \cr } } \right),$$ then $${u_1} + {u_2}$$ is equal to :

A

$$\left( {\matrix{
-1 \cr
1 \cr
0 \cr
} } \right)$$

B

$$\left( {\matrix{
-1 \cr
1 \cr
-1 \cr
} } \right)$$

C

$$\left( {\matrix{
-1 \cr
-1 \cr
0 \cr
} } \right)$$

D

$$\left( {\matrix{
1 \cr
-1 \cr
-1 \cr
} } \right)$$

Let $$A{u_1} = \left( {\matrix{
1 \cr
0 \cr
0 \cr
} } \right)\,\,\,\,\,\,A{u_2} = \left( {\matrix{
0 \cr
1 \cr
0 \cr
} } \right)$$

Then, $$A{u_1} + A{u_2} = \left( {\matrix{ 1 \cr 0 \cr 0 \cr } } \right) + \left( {\matrix{ 0 \cr 1 \cr 0 \cr } } \right)$$

$$ \Rightarrow A\left( {{u_1} + {u_2}} \right) = \left( {\matrix{ 1 \cr 1 \cr 0 \cr } } \right)\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,...\left( 1 \right)$$

Also, $$A = \left( {\matrix{ 1 & 0 & 0 \cr 2 & 1 & 0 \cr 3 & 2 & 1 \cr } } \right)$$

$$ \Rightarrow \left| A \right| = 1\left( 1 \right) - 0\left( 2 \right) + 0\left( {4 - 3} \right) = 1$$

We know,

$${A^{ - 1}} = {1 \over {\left| A \right|}}\,adjA \Rightarrow {A^{ - 1}} = adj\left( A \right)$$

( as $$\left| A \right| = 1$$ )

Now, from equation $$(1)$$, we have

$${u_1} + {u_2} = {A^{ - 1}}\left( {\matrix{ 1 \cr 1 \cr 0 \cr } } \right)$$

$$ = \left[ {\matrix{ 1 & 0 & 0 \cr { - 2} & 1 & 0 \cr 1 & { - 2} & 1 \cr } } \right]\left( {\matrix{ 1 \cr 1 \cr 0 \cr } } \right)$$

$$ = \left[ {\matrix{ 1 \cr { - 1} \cr { - 1} \cr } } \right]$$

Then, $$A{u_1} + A{u_2} = \left( {\matrix{ 1 \cr 0 \cr 0 \cr } } \right) + \left( {\matrix{ 0 \cr 1 \cr 0 \cr } } \right)$$

$$ \Rightarrow A\left( {{u_1} + {u_2}} \right) = \left( {\matrix{ 1 \cr 1 \cr 0 \cr } } \right)\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,...\left( 1 \right)$$

Also, $$A = \left( {\matrix{ 1 & 0 & 0 \cr 2 & 1 & 0 \cr 3 & 2 & 1 \cr } } \right)$$

$$ \Rightarrow \left| A \right| = 1\left( 1 \right) - 0\left( 2 \right) + 0\left( {4 - 3} \right) = 1$$

We know,

$${A^{ - 1}} = {1 \over {\left| A \right|}}\,adjA \Rightarrow {A^{ - 1}} = adj\left( A \right)$$

( as $$\left| A \right| = 1$$ )

Now, from equation $$(1)$$, we have

$${u_1} + {u_2} = {A^{ - 1}}\left( {\matrix{ 1 \cr 1 \cr 0 \cr } } \right)$$

$$ = \left[ {\matrix{ 1 & 0 & 0 \cr { - 2} & 1 & 0 \cr 1 & { - 2} & 1 \cr } } \right]\left( {\matrix{ 1 \cr 1 \cr 0 \cr } } \right)$$

$$ = \left[ {\matrix{ 1 \cr { - 1} \cr { - 1} \cr } } \right]$$

2

MCQ (Single Correct Answer)

The number of values of $$k$$ for which the linear equations

$$4x + ky + 2z = 0,kx + 4y + z = 0$$ and $$2x+2y+z=0$$ possess a non-zero solution is

$$4x + ky + 2z = 0,kx + 4y + z = 0$$ and $$2x+2y+z=0$$ possess a non-zero solution is

A

$$2$$

B

$$1$$

C

zero

D

$$3$$

$$\Delta = 0 \Rightarrow \left| {\matrix{
4 & k & 2 \cr
k & 4 & 1 \cr
2 & 2 & 1 \cr
} } \right| = 0$$

$$ \Rightarrow 4\left( {4 - 2} \right) - k\left( {k - 2} \right) + $$

$$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,2\left( {2k - 8} \right) = 0$$

$$ \Rightarrow 8 - {k^2} + 2k + 4k - 16 = 0$$

$$ \Rightarrow {k^2} - 6k + 8 = 0$$

$$ \Rightarrow \left( {k - 4} \right)\left( {k - 2} \right) = 0,k = 4,2$$

$$ \Rightarrow 4\left( {4 - 2} \right) - k\left( {k - 2} \right) + $$

$$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,2\left( {2k - 8} \right) = 0$$

$$ \Rightarrow 8 - {k^2} + 2k + 4k - 16 = 0$$

$$ \Rightarrow {k^2} - 6k + 8 = 0$$

$$ \Rightarrow \left( {k - 4} \right)\left( {k - 2} \right) = 0,k = 4,2$$

3

MCQ (Single Correct Answer)

Let $$A$$ and $$B$$ be two symmetric matrices of order $$3$$.

**Statement - 1:** $$A(BA)$$ and $$(AB)$$$$A$$ are symmetric matrices.

**Statement - 2:** $$AB$$ is symmetric matrix if matrix multiplication of $$A$$ with $$B$$ is commutative.

A

statement - 1 is true, statement - 2 is true; statement - 2 is **not** a correct explanation for statement - 1.

B

statement - 1 is true, statement - 2 is false.

C

statement - 1 is false, statement -2 is true

D

statement -1 is true, statement - 2 is true; statement - 2 is a correct explanation for statement - 1.

$$\therefore$$ $$\,\,\,\,\,A' = A,B' = B$$

Now $$\,\,\,\left( {A\left( {BA} \right)} \right)' = \left( {BA} \right)'A'$$

$$ = \left( {A'B'} \right)A' = \left( {AB} \right)A = A\left( {BA} \right)$$

Similarly $$\left( {\left( {AB} \right)A} \right)' = \left( {AB} \right)A$$

So, $$A\left( {BA} \right)\,\,\,\,$$ and $$A\left( {BA} \right)\,\,\,\,$$ are symmetric matrices.

Again $$\left( {AB} \right)' = B'A' = BA$$

Now if $$BA=AB$$, then $$AB$$ is symmetric matrix.

Now $$\,\,\,\left( {A\left( {BA} \right)} \right)' = \left( {BA} \right)'A'$$

$$ = \left( {A'B'} \right)A' = \left( {AB} \right)A = A\left( {BA} \right)$$

Similarly $$\left( {\left( {AB} \right)A} \right)' = \left( {AB} \right)A$$

So, $$A\left( {BA} \right)\,\,\,\,$$ and $$A\left( {BA} \right)\,\,\,\,$$ are symmetric matrices.

Again $$\left( {AB} \right)' = B'A' = BA$$

Now if $$BA=AB$$, then $$AB$$ is symmetric matrix.

4

MCQ (Single Correct Answer)

The number of $$3 \times 3$$ non-singular matrices, with four entries as $$1$$ and all other entries as $$0$$, is

A

$$5$$

B

$$6$$

C

at least $$7$$

D

less than $$4$$

$$\left[ {\matrix{
1 & {...} & {...} \cr
{...} & 1 & {...} \cr
{...} & {...} & 1 \cr
} } \right]\,\,$$ are $$6$$ non-singular matrices because $$6$$

blanks will be filled by $$5$$ zeros and $$1$$ one.

Similarly, $$\left[ {\matrix{ {...} & {...} & 1 \cr {...} & 1 & {...} \cr 1 & {...} & {...} \cr } } \right]\,\,$$ are $$6$$ non-singular matrices.

So, required cases are more than $$7,$$ non-singular $$3 \times 3$$ matrices.

blanks will be filled by $$5$$ zeros and $$1$$ one.

Similarly, $$\left[ {\matrix{ {...} & {...} & 1 \cr {...} & 1 & {...} \cr 1 & {...} & {...} \cr } } \right]\,\,$$ are $$6$$ non-singular matrices.

So, required cases are more than $$7,$$ non-singular $$3 \times 3$$ matrices.

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Complex Numbers

Quadratic Equation and Inequalities

Permutations and Combinations

Mathematical Induction and Binomial Theorem

Sequences and Series

Matrices and Determinants

Vector Algebra and 3D Geometry

Probability

Statistics

Mathematical Reasoning

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Properties of Triangle

Inverse Trigonometric Functions

Straight Lines and Pair of Straight Lines

Circle

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Application of Derivatives

Indefinite Integrals

Definite Integrals and Applications of Integrals

Differential Equations