If A = \(\left[ \begin{matrix} 2 & 0 \\ 1 & 5 \end{matrix} \right] \) and B = \(\left[ \begin{matrix} 1 & 4 \\ 2 & 0 \end{matrix} \right] \) then |adj (AB)| = 

(a)

-40

(b)

-80

(c)

-60

(d)

-20

If \(\rho\) (A) = \(\rho\)([A| B]), then the system AX = B of linear equations is

(a)

consistent and has a unique solution

(b)

consistent

(c)

consistent and has infinitely many solution

(d)

inconsistent

If \(\left| z-\frac { 3 }{ z } \right| =2\), then the least value |z| is

(a)

1

(b)

2

(c)

3

(d)

5

If \(\omega \neq 1\) is a cubic root of unity and \(\left( 1+\omega \right) ^{ 7 }=A+B\omega \), then (A, B) equals

(a)

(1, 0)

(b)

(−1, 1)

(c)

(0, 1)

(d)

(1, 1)

A polynomial equation in x of degree n always has

(a)

n distinct roots

(b)

n real roots

(c)

n complex roots

(d)

at most one root

\(\sin ^{-1}(\cos x)=\frac{\pi}{2}-x\) is valid for

(a)

\(-\pi \le x\le 0\)

(b)

\(0 \le x\le \pi\)

(c)

\(-\frac { \pi }{ 2 } \le x\le \frac { \pi }{ 2 } \)

(d)

\(-\frac { \pi }{ 4 } \le x\le \frac { 3\pi }{ 4 } \)

If \(\sin ^{-1} \frac{x}{5}+\operatorname{cosec}^{-1} \frac{5}{4}=\frac{\pi}{2}\), then the value of x is

(a)

4

(b)

5

(c)

2

(d)

3

If the normals of the parabola y² = 4x drawn at the end points of its latus rectum are tangents to the circle (x − 3)² + (y + 2)² = r² , then the value of r² is

(a)

2

(b)

3

(c)

1

(d)

4

If \(\vec { a } ,\vec { b } ,\vec { c } \) are three unit vectors such that \(\vec { a } \) is perpendicular to \(\vec { b } \) and is parallel to \(\vec { c } \) then \(\vec { a } \times (\vec { b } \times \vec { c } )\) is equal to

(a)

\(\vec { a } \)

(b)

\(\vec { b} \)

(c)

\(\vec { c } \)

(d)

\(\vec { 0 } \)

Distance from the origin to the plane 3x − 6y + 2z + 7 = 0 is

(a)

0

(b)

1

(c)

2

(d)

3

The slope of the line normal to the curve f(x) = 2cos 4x at \(x=\cfrac { \pi }{ 12 } \) is

(a)

\(-4\sqrt { 3 } \)

(b)

-4

(c)

\(\cfrac { \sqrt { 3 } }{ 12 } \)

(d)

\(4\sqrt { 3 } \)

The point of inflection of the curve y = (x - 1)³ is

(a)

(0, 0)

(b)

(0, 1)

(c)

(1, 0)

(d)

(1, 1)

The change in the surface area S = 6x² of a cube when the edge length varies from x_o to x_o+ dx is

(a)

12 x_o+dx

(b)

12x_o dx

(c)

6x_o dx

(d)

6x_o+ dx

The value of \(\int _{ 0 }^{ \frac { \pi }{ 6 } }{ { cos }^{ 3 }3x\ dx }\ is\)

(a)

\(\frac{2}{3}\)

(b)

\(\frac{2}{9}\)

(c)

\(\frac{1}{9}\)

(d)

\(\frac{1}{3}\)

The order and degree of the differential equation \(\frac { { d }^{ 2 }y }{ { dx }^{ 2 } } +{ \left( \frac { dy }{ dx } \right) }^{ 1/3 }+{ x }^{ 1/4 }=0\) are respectively

(a)

2, 3

(b)

3, 3

(c)

2, 6

(d)

2, 4

The solution of \(\frac { dy }{ dx } ={ 2 }^{ y-x }\) is

(a)

2^x + 2^y = C

(b)

2^x - 2^y = C

(c)

\(\frac { 1 }{ { 2 }^{ x } } -\frac { 1 }{ { 2 }^{ y } } =C\)

(d)

x + y = C

A pair of dice numbered 1, 2, 3, 4, 5, 6 of a six-sided die and 1, 2, 3, 4 of a four-sided die is rolled and the sum is determined. Let the random variable X denote this sum. Then the number of elements in the inverse image of 7 is

(a)

1

(b)

2

(c)

3

(d)

4

Suppose that X takes on one of the values 0, 1, and 2. If for some constant k, \(P(X=i)=k P(X=i-1) \text { for } i=1,2 \text { and } P(X=0)=\frac{1}{7}\) , then the value of k is

(a)

1

(b)

2

(c)

3

(d)

4

If a compound statement involves 3 simple statements, then the number of rows in the truth table is

(a)

9

(b)

8

(c)

6

(d)

3

Which one of the following is not true?

(a)

Negation of a negation of a statement is the statement itself

(b)

If the last column of the truth table contains only T then it is a tautology.

(c)

If the last column of its truth table contains only F then it is a contradiction

(d)

If p and q are any two statements then p↔️q is a tautology.