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12th Maths Model Exam -2

St. Britto Hr. Sec. School - Madurai

12th Maths Model Exam -2-Aug 2020

Time : 2.30 Hours

Part - A

Multiple Choice Question - Answer All Question

If A = \(\left[ \begin{matrix} 3 & 5 \\ 1 & 2 \end{matrix} \right] \), B = adj A and C = 3A, then \(\frac { \left| adjB \right| }{ \left| C \right| } \) =

\(\frac { 1 }{ 3 } \)
\(\frac { 1 }{ 9 } \)
\(\frac { 1 }{ 4 } \)
1

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)| =

-40
-80
-60
-20

If A = \(\left[ \begin{matrix} 3 & 1 & -1 \\ 2 & -2 & 0 \\ 1 & 2 & -1 \end{matrix} \right] \) and A^-1 = \(\left[ \begin{matrix} { a }_{ 11 } & { a }_{ 12 } & { a }_{ 13 } \\ { a }_{ 21 } & { a }_{ 22 } & { a }_{ 23 } \\ { a }_{ 31 } & { a }_{ 32 } & { a }_{ 33 } \end{matrix} \right] \) then the value of a₂₃ is

0
-2
-3
-1

If A is a non-singular matrix such that A^-1 = \(\left[ \begin{matrix} 5 & 3 \\ -2 & -1 \end{matrix} \right] \), then (A^T)⁻¹ =

\(\left[ \begin{matrix} -5 & 3 \\ 2 & 1 \end{matrix} \right] \)
\(\left[ \begin{matrix} 5 & 3 \\ -2 & -1 \end{matrix} \right] \)
\(\left[ \begin{matrix} -1 & -3 \\ 2 & 5 \end{matrix} \right] \)
\(\left[ \begin{matrix} 5 & -2 \\ 3 & -1 \end{matrix} \right] \)

If A = \(\left[ \begin{matrix} \cos { \theta } & \sin { \theta } \\ -\sin { \theta } & \cos { \theta } \end{matrix} \right] \) and A(adj A) = \(\left[ \begin{matrix} k & 0 \\ 0 & k \end{matrix} \right] \) then adj (AB) is

0
sin θ
cos θ
1

The rank of the matrix \(\left[ \begin{matrix} 1 \\ \begin{matrix} 2 \\ -1 \end{matrix} \end{matrix}\begin{matrix} 2 \\ \begin{matrix} 4 \\ -2 \end{matrix} \end{matrix}\begin{matrix} 3 \\ \begin{matrix} 6 \\ -3 \end{matrix} \end{matrix}\begin{matrix} 4 \\ \begin{matrix} 8 \\ -4 \end{matrix} \end{matrix} \right] \) is

1
2
4
3

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

consistent and has a unique solution
consistent
consistent and has infinitely many solution
inconsistent

If A = \(\left[ \begin{matrix} 3 & -3 & 4 \\ 2 & -3 & 4 \\ 0 & -1 & 1 \end{matrix} \right] \), then adj(adj A) is

\(\left[ \begin{matrix} 3 & -3 & 4 \\ 2 & -3 & 4 \\ 0 & -1 & 1 \end{matrix} \right] \)
\(\left[ \begin{matrix} 6 & -6 & 8 \\ 4 & -6 & 8 \\ 0 & -2 & 2 \end{matrix} \right] \)
\(\left[ \begin{matrix} -3 & 3 & -4 \\ -2 & 3 & -4 \\ 0 & 1 & -1 \end{matrix} \right] \)
\(\left[ \begin{matrix} 3 & -3 & 4 \\ 0 & -1 & 1 \\ 2 & -3 & 4 \end{matrix} \right] \)

Let A be a 3 x 3 matrix and B its adjoint matrix If |B|=64, then |A|=

±2
±4
±8
±12

If A is a square matrix that IAI = 2, than for any positive integer n, |Aⁿ| =

0
2n
2ⁿ
n²

If the system of equations x + 2y - 3x = 2, (k + 3) z = 3, (2k + 1) y + z = 2. is inconsistent then k is

-3, -\(\frac{1}{2}\)
-\(\frac{1}{2}\)
1
2

If A is a matrix of order m x n, then \(\rho\)(A) is

m
n
≤ min (m,n)
≥ min (m,n)

Which of the following is not an elementary transformation?

R_i ↔️ R_j
R_i ⟶ 2R_i + R_j
C_j ⟶ C_j + C_i
R_i ⟶ R_i + C_j

In the non - homogeneous system of equations with 3 unknowns if \(\rho\)(A) = \(\rho\)([AIB]) = 2, then the system has _______

unique solution
one parameter family of solution
two parameter family of solutions
in consistent

In the system of equations with 3 unknowns, if Δ = 0, and one of Δ_x, Δ_y of Δ_z is non zero then the system is ______

Consistent
inconsistent
consistent with one parameter family of solutions
consistent with two parameter family of solutions

If A is a non-singular matrix then IA^-1|= ______

\(\left| \frac { 1 }{ { A }^{ 2 } } \right| \)
\(\frac { 1 }{ |A^{ 2 }| } \)
\(\left| \frac { 1 }{ A } \right| \)
\(\frac { 1 }{ |A| } \)

If α,β and γ are the roots of x³+px²+qx+r, then \(\Sigma \frac { 1 }{ \alpha } \) is

-\(\frac { q }{ r } \)
\(\frac { p }{ r } \)
\(\frac { q }{ r } \)
-\(\frac { q }{ p } \)

If x³+12x²+10ax+1999 definitely has a positive root, if and only if

a≥0
a>0
a<0
a≤0

Ifj(x) = 0 has n roots, thenf'(x) = 0 has __________ roots

n
n -1
n+1
(n-r)

If (2+√3)x²-2x+1+(2-√3)x²-2x-1=\(\frac { 2 }{ 2-\sqrt { 3 } } \) then x=

0,2
0,1
0,3
0, √3

Part - B

Generic 2 - Answer All Question

Find the rank of the following matrices by minor method:
\(\left[ \begin{matrix} 2 & -4 \\ -1 & 2 \end{matrix} \right] \)

Find the adjoint of the following:
\(\left[ \begin{matrix} 2 & 3 & 1 \\ 3 & 4 & 1 \\ 3 & 7 & 2 \end{matrix} \right] \)

Find the rank of the following matrices by minor method:
\(\left[ \begin{matrix} -1 & 3 \\ 4 & -7 \\ 3 & -4 \end{matrix} \right] \)

Find the rank of the following matrices by row reduction method:
\(\left[ \begin{matrix} 3 \\ \begin{matrix} 2 \\ -1 \end{matrix} \end{matrix}\begin{matrix} -8 \\ \begin{matrix} -5 \\ 2 \end{matrix} \end{matrix}\begin{matrix} 5 \\ \begin{matrix} 1 \\ 3 \end{matrix} \end{matrix}\begin{matrix} 2 \\ \begin{matrix} 4 \\ -2 \end{matrix} \end{matrix} \right] \)

Test for consistency and if possible, solve the following systems of equations by rank method.
2x + 2y + z = 5, x - y + z = 1, 3x + y + 2z = 4

Construct a cubic equation with roots 1,2, and 3

For any vector \(\vec { a } \), prove that \(\hat { i } \times (\vec { a } \times \hat { i } )+\hat { j } \times (\vec { a } \times \hat { j } )+\hat { k } \times (\vec { a } \times \hat { k }) =2\vec { a } \).

Evaluate the following definite integrals : \(\overset { 1 }{ \underset { 0 }{ʃ } }\) \(\sqrt{\frac{1-x}{1+x}}\)dx

Define a random variable

Consider p→q : If today is Monday, then 4 + 4 = 8.

Part - C

Generic 3 - Answer All Question

Solve: 2x + 3y = 10, x + 6y = 4 using Cramer's rule.

Solve: 2x+2^x-1+2^x-2=7x+7^x-1+7^x-2

If \(\overset { \rightarrow }{ a } =\overset { \wedge }{ i } -\overset { \wedge }{ j } ,\overset { \rightarrow }{ b } =\overset { \wedge }{ j } -\overset { \wedge }{ k } ,\overset { \rightarrow }{ c } =\overset { \wedge }{ k } -\overset { \wedge }{ i } \) then find \(\left[ \overset { \rightarrow }{ a } -\overset { \rightarrow }{ b } ,\overset { \rightarrow }{ b } -\overset { \rightarrow }{ c } ,\overset { \rightarrow }{ c } -\overset { \rightarrow }{ a } \right] \)

A random variable X has the following probability distribution

X

0

1

2

3

4

5

6

7

P(x)

0

K

2K

2K

3K

K²

2K²

7K²+K

Evaluate (i) k

(ii) P(X ≥ 6)

(iii) P(0< X< 3)

Three fair coins are tossed simultaneously. Find the probability mass function for number of heads
occurred

Verify the
(i) closure property,
(ii) commutative property,
(iii) associative property
(iv) existence of identity and
(v) existence of inverse for the arithmetic operation - on Z.

Define an operationon Q as follows: ab=\(\left(\frac{a+b}{2} \right)\) ; a,b ∈Q. Examine the closure, commutative, and
associative properties satisfied by*on Q.

If A = \(\left[ \begin{matrix} 3 & 2 \\ 7 & 5 \end{matrix} \right] \) and B = \(\left[ \begin{matrix} -1 & -3 \\ 5 & 2 \end{matrix} \right] \), verify that (AB)^-1 = B^-1A^-1

Solve the equation 2x³+11x²−9x−18=0.

Prove that \([\vec { a } \times \vec { b } ,\vec { b } \times \vec { c } ,\vec { c } \times \vec { a } ]\) = \([{ \vec { a } ,\vec { b } ,\vec { c } }]^{ 2 }\)

Part - D

Generic 5 - Answer All Question

Solve: \(\frac { 2 }{ x } +\frac { 3 }{ y } +\frac { 10 }{ z } =4,\frac { 4 }{ x } -\frac { 6 }{ y } +\frac { 5 }{ z } =1,\frac { 6 }{ x } +\frac { 9 }{ y } -\frac { 20 }{ z } \)=2

A box contains 4 red and 5 black marbles. Find the probability distribution of the red marbles in a
random draw of three marbles. Also find the mean and standard deviation of the distribution

Using the equivalence property, show that p ↔ q ≡ ( p ∧ q) v (ㄱp ∧ ㄱq)

Solve the equation (x-2)(x-7)(x-3)(x+2)+19=0

By vector method, prove that cos(α + β) = cos α cos β - sin α sin β

Find the point of intersection of the lines \(\frac { x-1 }{ 2 } =\frac { y-2 }{ 3 } =\frac { z-3 }{ 4 } \) and \(\frac { x-4 }{ 5 } =\frac { y-1 }{ 2 } =z\)

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