#### 12th Standard Maths English Medium Reduced Syllabus Model Question paper - 2021 Part - 2

12th Standard

Reg.No. :
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Maths

Time : 01:00:00 Hrs
Total Marks : 90

Part I

Choose the most suitable answer from the given four alternatives and write the option code with the corresponding answer.

20 x 1 = 20
1. If A = $\left[ \begin{matrix} 3 & 5 \\ 1 & 2 \end{matrix} \right]$, B = adj A and C = 3A, then $\frac { \left| adjB \right| }{ \left| C \right| }$

(a)

$\frac { 1 }{ 3 }$

(b)

$\frac { 1 }{ 9 }$

(c)

$\frac { 1 }{ 4 }$

(d)

1

2. 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

(a)

1

(b)

2

(c)

4

(d)

3

3. If xayb = em, xcyd = en, Δ1 = $\left| \begin{matrix} m & b \\ n & d \end{matrix} \right|$, Δ2 = $\left| \begin{matrix} a & m \\ c & n \end{matrix} \right|$, Δ3 = $\left| \begin{matrix} a & b \\ c & d \end{matrix} \right|$, then the values of x and y are respectively,

(a)

e21), e31)

(b)

log (Δ13), log (Δ23)

(c)

log (Δ21), log(Δ31)

(d)

e(Δ13),e(Δ23)

4. If |z-2+i|≤2, then the greatest value of |z| is

(a)

$\sqrt { 3 } -2$

(b)

$\sqrt { 3 } +2$

(c)

$\sqrt { 5 } -2$

(d)

$\sqrt { 5 } +2$

5. If $\left| z-\cfrac { 3 }{ z } \right| =2$ then the least value |z| is

(a)

1

(b)

2

(c)

3

(d)

5

6. The polynomial x3-kx2+9x has three real zeros if and only if, k satisfies

(a)

|k|≤6

(b)

k=0

(c)

|k|>6

(d)

|k|≥6

7. The equation of the circle passing through(1,5) and (4,1) and touching y -axis is x2+y2−5x−6y+9+(4x+3y−19)=0 whereλ is equal to

(a)

$0,-\frac { 40 }{ 9 }$

(b)

0

(c)

$\frac { 40 }{ 9 }$

(d)

$\frac { -40 }{ 9 }$

8. If $\vec { a } ,\vec { b } ,\vec { c }$ are three non-coplanar vectors such that $\vec { a } \times (\vec { b } \times \vec { c } )=\frac { \vec { b } +\vec { c } }{ \sqrt { 2 } }$, then the angle between

(a)

$\frac { \pi }{ 2 }$

(b)

$\frac { 3\pi }{ 6 }$

(c)

$\frac { \pi }{ 4 }$

(d)

${ \pi }$

9. If the planes $\vec { r } =(2\hat { i } -\lambda \hat { j } +\hat { k } )=3$ and $\vec { r } =(4+\hat { j } -\mu \hat { k } )=5$ are parallel, then the value of λ and μ are

(a)

$\frac { 1 }{ 2 } ,-2$

(b)

$-\frac { 1 }{ 2 } ,2$

(c)

$-\frac { 1 }{ 2 } ,-2$

(d)

$\frac { 1 }{ 2 } ,2$

10. If the length of the perpendicular from the origin to the plane 2x + 3y + λz =1, λ > 0 is $\frac{1}{5}$ then the value of λ is

(a)

$2\sqrt { 3 }$

(b)

$3\sqrt { 2 }$

(c)

0

(d)

1

11. The number given by the Rolle's theorem for the functlon x3-3x2, x∈[0,3] is

(a)

1

(b)

$\\ \\ \\ \sqrt { 2 }$

(c)

$\cfrac { 3 }{ 2 }$

(d)

2

12. The number given by the Mean value theorem for the function $\cfrac { 1 }{ x }$,x∈[1,9] is

(a)

2

(b)

2.5

(c)

3

(d)

3.5

13. If w (x, y, z) = x2 (v - z) + y2 (z - x) + z2(x - y), then $\frac { { \partial }w }{ \partial x } +\frac { \partial w }{ \partial y } +\frac { \partial w }{ \partial z }$ is

(a)

xy + yz + zx

(b)

x(y + z)

(c)

y(z + x)

(d)

0

14. If (x,y, z) = xy +yz +zx, then fx - fz is equal to

(a)

z - x

(b)

y - z

(c)

x - z

(d)

y - x

15. If f(x)$f(x)=\int _{ 1 }^{ x }{ \frac { { e }^{ { sin }^{ u } } }{ u } } du,x>1\quad and\quad \int _{ 1 }^{ 3 }{ \frac { { e }^{ { sinx }^{ 2 } } }{ x } } dx=\frac { 1 }{ 2 } [f(a)-f(1)]$, then one of the possible value of a is

(a)

3

(b)

6

(c)

9

(d)

5

16. The value of $\int _{ 0 }^{ 1 }{ { ({ sin }^{ -1 }x) }^{ 2 } } dx$

(a)

$\frac { { \pi }^{ 2 } }{ 4 } -1$

(b)

$\frac { { \pi }^{ 2 } }{ 4 } +2$

(c)

$\frac { { \pi }^{ 2 } }{ 4 } +1$

(d)

$\frac { { \pi }^{ 2 } }{ 4 } -2$

17. The number of arbitrary constants in the general solutions of order n and n +1are respectively

(a)

n-1,n

(b)

n,n+1

(c)

n+1,n+2

(d)

n+1,n

18. The number of arbitrary constants in the particular solution of a differential equation of third order is

(a)

3

(b)

2

(c)

1

(d)

0

19. Subtraction is not a binary operation in

(a)

R

(b)

Z

(c)

N

(d)

Q

20. 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

1. Part II

Answer any 7 questions. Question no. 27 is compulsory.

7 x 2 = 14
21. Find the rank of each of the following matrices:
$\left[ \begin{matrix} 4 & 3 \\ -3 & -1 \\ 6 & 7 \end{matrix}\begin{matrix} 1 & -2 \\ -2 & 4 \\ -1 & 2 \end{matrix} \right]$

22. Write $\cfrac { 3+4i }{ 5-12i }$ in the x+iy form, hence find its real and imaginary parts.

23. Find the modulus of the following complex numbers
$\cfrac { 2i }{ 3+4i }$

24. If $\omega \neq 1$ is a cube root of unity, then the show that $\cfrac { a+b\omega +c{ \omega }^{ 2 } }{ b+c\omega +{ a\omega }^{ 2 } } +\cfrac { a+b\omega +{ c\omega }^{ 2 } }{ c+a\omega +b{ \omega }^{ 2 } } =-1$

25. Evaluate the following if z=5−2i and w= −1+3i
2z+3w

26. Find the following $\left| \overline { (1+i) } (2+3i)(4i-3 \right|$

27. Show that the equation x9-5x5+4x4+2x2+1=0 has atleast 6 imaginary solutions.

1. Part III

Answer any 7 questions. Question no. 34 is compulsory.

7 x 3 = 21
28. If w (x,y, z) =x2 +y2 +y2, x= et, y = esin t, z = et cos t, find $\frac{dw}{dt}$

29. Let z(x, y) = x3 - 3x2y3, where x = set, y = se-t, s, t ∈ R. Find $\frac { \partial z }{ \partial s }$ and $\frac { \partial z }{ \partial t }$

30. Evaluate the following
$\int _{ 0 }^{ \pi /4 }{ { sin}^{ 6}x\quad dx }$

31. Solve the differential equation:
x cos y dy = ex(x log x + 1)dx

32. The probability density function of X is

Find P(0.2≤X<0.6)

33. If the probability mass function f (x) of a random variable X isx

 x 1 2 3 4 f (x) $\cfrac { 1 }{ 12 }$ $\cfrac { 5 }{ 12 }$ $\cfrac { 5 }{ 12 }$ $\cfrac { 1 }{ 12 }$

find (i) its cumulative distribution function, hence find
(ii) P(X ≤ 3) and,
(iii) P(X ≥ 2)

34. 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.

1. Part IV

7 x 5 = 35
35. Verify Euler’s theorem for the function $f(x,y)=\cfrac { 1 }{ \sqrt { { x }^{ 2 }+{ y }^{ 2 } } }$

36. Show that the ratio of the area under the curve y=sinx and y=sin2x between x=0 and $x=\cfrac { \pi }{ 3 }$ and x- axis are as 2 : 3.

37. Solve : $\left( 1+{ x }^{ 2 } \right) \cfrac { dy }{ dx } -x={ 2tan }^{ -1 }x$

39. The probability distribution of a random variable X is given by

 X 0 1 2 3 P(X) 0.1 0.3 0.5 0.1

If Y=X2+3X, find the mean and the variance of Y.

40. Show that (2018)2017+(2020)2017≡0(mod2019).

41. Prove without using the truth table $\sim \left( pV\left( qVr \right) \right) \equiv \left( \sim pV\sim q \right) V\left( -r \right)$