#### 12th Standard Maths English Medium Reduced Syllabus Model Question paper with answer key - 2021 Part - 1

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 is a 3 × 3 non-singular matrix such that AAT = ATA and B = A-1AT, then BBT =

(a)

A

(b)

B

(c)

I

(d)

BT

2. If (AB)-1 = $\left[ \begin{matrix} 12 & -17 \\ -19 & 27 \end{matrix} \right]$ and A-1 = $\left[ \begin{matrix} 1 & -1 \\ -2 & 3 \end{matrix} \right]$, then B-1 =

(a)

$\left[ \begin{matrix} 2 & -5 \\ -3 & 8 \end{matrix} \right]$

(b)

$\left[ \begin{matrix} 8 & 5 \\ 3 & 2 \end{matrix} \right]$

(c)

$\left[ \begin{matrix} 3 & 1 \\ 2 & 1 \end{matrix} \right]$

(d)

$\left[ \begin{matrix} 8 & -5 \\ -3 & 2 \end{matrix} \right]$

3. If ATA−1 is symmetric, then A2 =

(a)

A-1

(b)

(AT)2

(c)

AT

(d)

(A-1)2

4. If adj A = $\left[ \begin{matrix} 2 & 3 \\ 4 & -1 \end{matrix} \right]$ and adj B = $\left[ \begin{matrix} 1 & -2 \\ -3 & 1 \end{matrix} \right]$ then adj (AB) is

(a)

$\left[ \begin{matrix} -7 & -1 \\ 7 & -9 \end{matrix} \right]$

(b)

$\left[ \begin{matrix} -6 & 5 \\ -2 & -10 \end{matrix} \right]$

(c)

$\left[ \begin{matrix} -7 & 7 \\ -1 & -9 \end{matrix} \right]$

(d)

$\left[ \begin{matrix} -6 & -2 \\ 5 & -10 \end{matrix} \right]$

5. Let A = $\left[ \begin{matrix} 2 & -1 & 1 \\ -1 & 2 & -1 \\ 1 & -1 & 2 \end{matrix} \right]$ and 4B = $\left[ \begin{matrix} 3 & 1 & -1 \\ 1 & 3 & x \\ -1 & 1 & 3 \end{matrix} \right]$. If B is the inverse of A, then the value of x is

(a)

2

(b)

4

(c)

3

(d)

1

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

(a)

$\left[ \begin{matrix} 3 & -3 & 4 \\ 2 & -3 & 4 \\ 0 & -1 & 1 \end{matrix} \right]$

(b)

$\left[ \begin{matrix} 6 & -6 & 8 \\ 4 & -6 & 8 \\ 0 & -2 & 2 \end{matrix} \right]$

(c)

$\left[ \begin{matrix} -3 & 3 & -4 \\ -2 & 3 & -4 \\ 0 & 1 & -1 \end{matrix} \right]$

(d)

$\left[ \begin{matrix} 3 & -3 & 4 \\ 0 & -1 & 1 \\ 2 & -3 & 4 \end{matrix} \right]$

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

(a)

1

(b)

2

(c)

3

(d)

5

8. If |z|=1, then the value of $\cfrac { 1+z }{ 1+\overline { z } }$ is

(a)

z

(b)

$\bar { z }$

(c)

$\cfrac { 1 }{ z }$

(d)

1

9. 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)

10. The principal argument of the complex number $\cfrac { \left( 1+i\sqrt { 3 } \right) ^{ 2 } }{ 4i\left( 1-i\sqrt { 3 } \right) }$ is

(a)

$\cfrac { 2\pi }{ 3 }$

(b)

$\cfrac { \pi }{ 6 }$

(c)

$\cfrac { 5\pi }{ 6 }$

(d)

$\cfrac { \pi }{ 2 }$

11. ${ sin }^{ -1 }\frac { 3 }{ 5 } -{ cos }^{ -1 }\frac { 12 }{ 13 } +{ sec }^{ -1 }\frac { 5 }{ 3 } { -cosec }^{ 1- }\frac { 13 }{ 2 }$is equal to

(a)

2$\pi$

(b)

$\pi$

(c)

0

(d)

tan-1$\frac{12}{65}$

12. The equation of the normal to the circle x2+y2−2x−2y+1=0 which is parallel to the line
2x+4y=3 is

(a)

x+2y=3

(b)

x+2y+3= 0

(c)

2x+4y+3=0

(d)

x−2y+3= 0

13. If P(x, y) be any point on 16x2+25y2=400 with foci F1 (3,0) and F2 (-3,0) then PF1 PF2 +
is

(a)

8

(b)

6

(c)

10

(d)

12

14. If $\vec { a } ,\vec { b } ,\vec { c }$ are non-coplanar, non-zero vectors such that $[\vec { a } ,\vec { b } ,\vec { c } ]$ = 3, then ${ \{ [\vec { a } \times \vec { b } ,\vec { b } \times \vec { c } ,\vec { c } \times \vec { a } }]\} ^{ 2 }$ is equal to

(a)

81

(b)

9

(c)

27

(d)

18

15. 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)

${ \pi }$

16. If the distance of the point (1,1,1) from the origin is half of its distance from the plane x + y + z + k =0, then the values of k are

(a)

$\pm 3$

(b)

$\pm 6$

(c)

-3, 9

(d)

3, 9

17. The function sin4 x + cos4X is increasing in the interval

(a)

$\left[ \cfrac { 5\pi }{ 8 } ,\cfrac { 3\pi }{ 4 } \right]$

(b)

$\left[ \cfrac { \pi }{ 2 } ,\cfrac { 5\pi }{ 8 } \right]$

(c)

$\left[ \cfrac { \pi }{ 4 } ,\cfrac { \pi }{ 2 } \right]$

(d)

$\left[ 0,\cfrac { \pi }{ 4 } \right]$

18. 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

19. If u(x, y) = x2+ 3xy + y - 2019, then $\frac { \partial u }{ \partial x }$(4, -5) is equal to

(a)

-4

(b)

-3

(c)

-7

(d)

13

20. Linear approximation for g(x) = cos x at x=$\frac{-\pi}{2}$ is

(a)

x + $\frac{-\pi}{2}$

(b)

- x + $\frac{\pi}{2}$

(c)

x - $\frac{\pi}{2}$

(d)

- x + $\frac{\pi}{2}$

1. Part II

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

7 x 2 = 14
21. Show that (1+x)n=1+nx approximately if x is close to zero.

22. Evaluate $\int _{ 0 }^{ \infty }{ \left( { a }^{ -x }-{ b }^{ -x } \right) } dx$

23. Find the area bounded by the curve y=sin2x between the ordinates x=0.x=π and x-axis.

24. Find the order and degree of $\\ y+\cfrac { dy }{ dx } =\cfrac { 1 }{ 4 } \int { ydx }$

25. Form the D.E of family of parabolas having vertex at the origin and axis along positive y-axis.

26. Find the variance of the binomial distribution with parameters 8 and $\cfrac { 1 }{ 4 }$

27. If a≡b(mod n) and b≡c(mod n),check whether a≡c(mod n).

1. Part III

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

7 x 3 = 21
28. Show that $\left( \cfrac { 19-7i }{ 9+i } \right) ^{ 12 }+\left( \cfrac { 20-5i }{ 7-6i } \right) ^{ 12 }$ is real

29. If z=2−2i, find the rotation of z by θ radians in the counter clockwise direction about the origin when $\theta =\cfrac { 3\pi }{ 2 }$.

30. Obtain the Cartesian form of the locus of z in
|2z-3-i|=3

31. If p and q are the roots of the equation lx2+nx+n = 0, show that $\sqrt { \frac { p }{ q } } +\sqrt { \frac { q }{ p } } +\sqrt { \frac { n }{ l } }$=0.

32. How many rows are needed for following statement formulae?
p ∨ ¬ t ( p ∨ ¬s)

33. How many rows are needed for following statement formulae?
(( p ∧ q) ∨ (¬r ∨¬s)) ∧ (¬ t ∧ v))

34. Let $A=\left( \begin{matrix} 1 & 0 \\ 0 & 1 \\ 1 & 0 \end{matrix}\begin{matrix} 1 & 0 \\ 0 & 1 \\ 0 & 1 \end{matrix} \right) ,B=\left( \begin{matrix} 0 & 1 \\ 1 & 0 \\ 1 & 0 \end{matrix}\begin{matrix} 0 & 1 \\ 1 & 0 \\ 0 & 1 \end{matrix} \right) ,C=\left( \begin{matrix} 1 & 1 \\ 0 & 1 \\ 1 & 1 \end{matrix}\begin{matrix} 0 & 1 \\ 1 & 0 \\ 1 & 1 \end{matrix} \right)$be any three boolean matrices of the same type.
Find AΛB

1. Part IV

7 x 5 = 35
35. Solve the Linear differential equation:
$(x+a)\frac { dy }{ dx } -2y={ (x+a) }^{ 4 }$

36. The probability density function of X is given
$f(x)=\begin{cases} \begin{matrix} { Ke }^{ \frac { -x }{ 3 } } & \begin{matrix} for & x>0 \end{matrix} \end{matrix} \\ \begin{matrix} 0 & \begin{matrix} for & x\le 0 \end{matrix} \end{matrix} \end{cases}$
Find
(i) the value of k
(ii) the distribution function.
(iii) P(X <3)
(iv) P(5 ≤X)
(v) P(X ≤ 4)

37. A retailer purchases a certain kind of' electronic device from a manufacturer. The manufacturer ,indicates that the defective rate of the device is 5%. The inspector of the retailer randomly picks 10 items from a shipment. What is the probability that there will be
(i) at least one defective item
(ii) exactly two defective items.

38. Find the constant C such that the function $f(x)=\begin{cases} \begin{matrix} { Cx }^{ 2 } & 1 is a density function, and compute (i) P(1.5 < X < 3.5) (ii) P(X ≤2) (iii) P(3 < X ) . 39. Verify (i) closure property, (ii) commutative property, (iii) associative property, (iv) existence of identity, and (v) existence of inverse for the operation +5 on Z5 using table corresponding to addition modulo 5. 40. Let M=\(\left\{ \left( \begin{matrix} x & x \\ x & x \end{matrix} \right) :x\in R-\{ 0\} \right\}$ and let * be the matrix multiplication. Determine whether M is closed under ∗. If so, examine the commutative and associative properties satisfied by ∗ on M.

41. Using truth table check whether the statements ¬(p V q) V (¬p ∧ q) and ¬p are logically equivalent.