#### First Mid Term Model Questions

12th Standard EM

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Maths

Time : 01:00:00 Hrs
Total Marks : 50
10 x 1 = 10
1. If A$\left[ \begin{matrix} 1 & -2 \\ 1 & 4 \end{matrix} \right] =\left[ \begin{matrix} 6 & 0 \\ 0 & 6 \end{matrix} \right]$, then A =

(a)

$\left[ \begin{matrix} 1 & -2 \\ 1 & 4 \end{matrix} \right]$

(b)

$\left[ \begin{matrix} 1 & 2 \\ -1 & 4 \end{matrix} \right]$

(c)

$\left[ \begin{matrix} 4 & 2 \\ -1 & 1 \end{matrix} \right]$

(d)

$\left[ \begin{matrix} 4 & -1 \\ 2 & 1 \end{matrix} \right]$

2. If A, B and C are invertible matrices of some order, then which one of the following is not true?

(a)

(b)

(c)

det A-1 = (det A)-1

(d)

(ABC)-1 = C-1B-1A-1

3. The system of linear equations x + y + z  = 6, x + 2y + 3z =14 and 2x + 5y + λz =μ (λ, μ $\in$ R) is consistent with unique solution if

(a)

λ = 8

(b)

λ = 8, μ ≠ 36

(c)

λ ≠ 8

(d)

none

4. The value of (1+i)4 + (1-i)4 is

(a)

8

(b)

4

(c)

-8

(d)

-4

5. The value of $\frac { (cos{ 45 }^{ 0 }+isin{ 45 }^{ 0 })^{ 2 }(cos{ 30 }^{ 0 }-isin{ 30 }^{ 0 }) }{ cos{ 30 }^{ 0 }+isin{ 30 }^{ 0 } }$ is

(a)

$\frac { 1 }{ 2 } +i\frac { \sqrt { 3 } }{ 2 }$

(b)

$\frac { 1 }{ 2 } -i\frac { \sqrt { 3 } }{ 2 }$

(c)

$-\frac { \sqrt { 3 } }{ 2 } +\frac { 1 }{ 2 }$

(d)

$\frac { \sqrt { 3 } }{ 2 } +\frac { 1 }{ 2 }$

6. According to the rational root theorem, which number is not possible rational root of 4x7+2x4-10x3-5?

(a)

-1

(b)

$\frac { 5 }{ 4 }$

(c)

$\frac { 4 }{ 5 }$

(d)

5

7. The quadratic equation whose roots are ∝ and β is

(a)

(x - ∝)(x -β) =0

(b)

(x - ∝)(x + β) =0

(c)

∝+β=$\frac{b}{a}$

(d)

∝.β=$\frac{-c}{a}$

8. If p(x) = ax2 + bx + c and Q(x) = -ax2 + dx + c where ac ≠ 0 then p(x). Q(x) = 0 has at least _______ real roots.

(a)

no

(b)

1

(c)

2

(d)

infinite

9. ${ tan }^{ -1 }\left( \frac { 1 }{ 4 } \right) +{ tan }^{ -1 }\left( \frac { 2 }{ 3 } \right)$is equal to

(a)

$\frac { 1 }{ 2 } { cos }^{ -1 }\left( \frac { 3 }{ 5 } \right)$

(b)

$\frac { 1 }{ 2 } { sin }^{ -1 }\left( \frac { 3 }{ 5 } \right)$

(c)

$\frac { 1 }{ 2 } {tan }^{ -1 }\left( \frac { 3 }{ 5 } \right)$

(d)

${ tan}^{ -1 }\left( \frac { 1}{ 2 } \right)$

10. ${ tan }^{ -1 }\left( tan\cfrac { 9\pi }{ 8 } \right)$

(a)

$\cfrac { 9\pi }{ 8 }$

(b)

$\cfrac { 9\pi }{ 8 }$

(c)

$\cfrac { \pi }{ 8 }$

(d)

$\cfrac { -\pi }{ 8 }$

11. 5 x 2 = 10
12. Find the adjoint of the following:
$\left[ \begin{matrix} -3 & 4 \\ 6 & 2 \end{matrix} \right]$

13. If A is a square matrix such that A3 = I, then prove that A is non-singular.

14. Find z−1, if z=(2+3i)(1− i).

15. Solve the equations:
6x4-35x3+62x2-35x+6=0

16. Evaluate $sin\left( { cos }^{ -1 }\left( \cfrac { 1 }{ 2 } \right) \right)$

17. 5 x 3 = 15
18. If A = $\frac { 1 }{ 9 } \left[ \begin{matrix} -8 & 1 & 4 \\ 4 & 4 & 7 \\ 1 & -8 & 4 \end{matrix} \right]$, prove that A−1 = AT.

19. Find the inverse of the non-singular matrix A =  $\left[ \begin{matrix} 0 & 5 \\ -1 & 6 \end{matrix} \right]$, by Gauss-Jordan method.

20. Find,the rank of the matrix math $\left[ \begin{matrix} 4 \\ -2 \\ 1 \end{matrix}\begin{matrix} 4 \\ 3 \\ 4 \end{matrix}\begin{matrix} 0 \\ -1 \\ 8 \end{matrix}\begin{matrix} 3 \\ 5 \\ 7 \end{matrix} \right]$.

21. Show that the complex numbers 3 + 2i, 5i, -3 + 2i and -i form a square.

22. Find all real numbers satisfying 4x-3(2x+2)+25=0

23. 3 x 5 = 15
24. If F($\alpha$) = $\left[ \begin{matrix} \cos { \alpha } & 0 & \sin { \alpha } \\ 0 & 1 & 0 \\ -\sin { \alpha } & 0 & \cos { \alpha } \end{matrix} \right]$, show that [F($\alpha$)]-1 = F(-$\alpha$).

25. Simplify: (1+i)18

26. Provethat ${ tan }^{ -1 }\left( \cfrac { 1-x }{ 1+x } \right) -{ tan }^{ -1 }\left( \cfrac { 1-y }{ 1+y } \right) ={ sin }^{ -1 }\left( \cfrac { y-x }{ \sqrt { 1+{ x }^{ 2 } } .\sqrt { 1+{ y }^{ 2 } } } \right) \\$