#### 12th Standard English Medium Maths Reduced Syllabus Public Exam Model Question Paper - 2021

12th Standard

Reg.No. :
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Maths

Time : 02:40:00 Hrs
Total Marks : 90

Part I

Answer all the Questions.

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 |adj(adj A)| = |A|9, then the order of the square matrix A is

(a)

3

(b)

4

(c)

2

(d)

5

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

(a)

$\frac { -4 }{ 5 }$

(b)

$\frac { -3 }{ 5 }$

(c)

$\frac { 3 }{ 5 }$

(d)

$\frac { 4 }{ 5 }$

3. The complex number z which satisfies the condition $\left| \frac { 1+z }{ 1-z } \right|$ =1 lies on

(a)

circle x2+y2 =1

(b)

x-axis

(c)

y-axis

(d)

the lines x+y=1

4. 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

5. The domain of the function defined by f(x)=sin−1$\sqrt{x-1}$ is

(a)

[1,2]

(b)

[-1,1]

(c)

[0,1]

(d)

[-1,0]

6. The circle x2+y2=4x+8y+5intersects the line3x−4y=m at two distinct points if

(a)

15< m < 65

(b)

35< m <85

(c)

−85<m < −35

(d)

−35<m <15

7. The angle between the line $\vec { r } =(\hat { i } +2\hat { j } -3\hat { k } )+t(2\hat { i } +\hat { j } -2\hat { k } )$ and the plane $\vec { r } .(\hat { i } +\hat { j } )+4=0$ is

(a)

(b)

30°

(c)

45°

(d)

90°

8. The tangent to the curve y2 - xy + 9 = 0 is vertical when

(a)

y = 0

(b)

$\\ \\ y=\pm \sqrt { 3 }$

(c)

$y=\cfrac { 1 }{ 2 }$

(d)

$y=\pm 3$

9. The curve y= ax4 + bx2 with ab > 0

(a)

has, no horizontal tangent

(b)

is concave up

(c)

is concave down

(d)

has no points of inflection

10. The approximate change in the volume V of a cube of side x metres caused by increasing the side by 1% is

(a)

0.3xdx m3

(b)

0.03 xm3

(c)

0.03.x2 m3

(d)

0.03x3m3

11. The value of $\frac { (n+2) }{ (n) } =90$ then n is

(a)

10

(b)

5

(c)

8

(d)

9

12. For any value of n∈Z, $\int _{ 0 }^{ \pi }{ e{ cos }^{ 2x }{ cos }^{ 3 } } [(2n+1)x]$ is

(a)

$\frac{\pi}{2}$

(b)

$\pi$

(c)

0

(d)

2

13. The solution of the differential equation $\frac { dy }{ dx } =\frac { y }{ x } +\frac { \phi \left( \frac { y }{ x } \right) }{ \phi '\left( \frac { y }{ x } \right) }$is

(a)

$x\phi \left( \frac { y }{ x } \right) =k$

(b)

$\phi \left( \frac { y }{ x } \right) =kx$

(c)

$y\phi \left( \frac { y }{ x } \right) =k$

(d)

$\phi \left( \frac { y }{ x } \right) =ky$

14. Let X have a Bernoulli distribution with mean 0.4, then the variance of (2X - 3) is

(a)

0.24

(b)

0.48

(c)

0.6

(d)

0.96

15. Which one of the following is incorrect? For any two propositions p and q, we have

(a)

¬ (p∨q) ≡ ¬ p ∧ ¬q

(b)

¬ ( p∧ q)≡¬p ∨ ¬q

(c)

¬ (p ∨ q)≡¬p∨¬q

(d)

¬(¬p)≡ p

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

(a)

a≥0

(b)

a>0

(c)

a<0

(d)

a≤0

17. If ax2 + bx + c = 0, a, b, c E R has no real zeros, and if a + b + c < 0, then __________

(a)

c>0

(b)

c<0

(c)

c=0

(d)

c≥0

18. For real x, the equation $\left| \frac { x }{ x-1 } \right| +|x|=\frac { { x }^{ 2 } }{ |x-1| }$ has

(a)

one solution

(b)

two solution

(c)

at least two solution

(d)

no solution

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

(a)

0,2

(b)

0,1

(c)

0,3

(d)

0, √3

20. If |x|$\le$1, then 2tan-1 x-sin-1 $\frac{2x}{1+x^2}$ is equal to

(a)

tan-1x

(b)

sin-1x

(c)

0

(d)

$\pi$

1. Part II

Answer any seven questions. Question no. 23 is compulsory.

7x 2 = 14
21. Expand the polynomial f(x)=x2-3x+2 in power of (x-2)

22. Without using any kind of computational aid use linear approximation to find the value of e0.1

23. Find the volume of the solid generated when the region enclosed by $y=\sqrt { x } ,y=2$  and x = 0 is revolved about y-axis.

24. Form the differential equation satisfied by are the straight lines in my-plane.

25. A coin is tossed twice. If X is a random variable defined as the number of heads minus the number of tails, then obtain its probability distribution.

26. Write the equivalent forms of p➝q and (~p)➝q.

27. If α, β, γ  and $\delta$ are the roots of the polynomial equation 2x4+5x3−7x2+8=0 , find a quadratic equation with integer coefficients whose roots are α + β + γ + $\delta$ and αβ૪$\delta$.

28. Find value of a for which the sum of the squares of the equation x2 - (a- 2) x - a-1=0 assumes the least value.

29. Part III

Answer any seven questions. Question no. 32 is compulsory.

7 x 3 = 21
30. Obtain the Cartesian form of the locus of z in
|z|=|z-i|

31. Find the value of
$tan\left( { sin }^{ -1 }\frac { 3 }{ 5 } +{ cot }^{ -1 }\frac { 3 }{ 2 } \right)$

32. Find the equation of the parabola with vertex (-1,-2) , axis parallel to y -axis and passing through (3,6)

33. Identify the type of conic and find centre, foci, vertices, and directrices of each of the following:
$\frac { { x }^{ 2 } }{ 16 } +\frac { { y }^{ 2 } }{ 9 } =1$

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

35. The vector equation in parametric form of a line is $\vec { r } =(3\hat { i } -2\hat { j } +6\hat { k } )+t(2\hat { i } -\hat { j } +3\hat { k } )$. Find
(i) the direction cosines of the straight line
(ii) vector equation in non-parametric form of the line
(iii) Cartesian equations of the line.

36. Find the domain of cos-1$(\frac{2+sinx}{3})$

37. Find the real solutions of the equation
${ tan }^{ -1 }\sqrt { x(x+1) } +{ sin }^{ -1 }\sqrt { { x }^{ 2 }+x+1 } =\cfrac { \pi }{ 2 }$

1. Part IV

Answer all the questions.

7x 5 = 35
1. The length x of a rectangle is decreasing at the rate of 3cm/min and the width is increasing at the rate of 2cm/min. When x = 10cm, y = 6cm, find the rate of change of (i) the perimeter (ii) the area of the rectan

2. missle fired from ground level rises x metres vertically upwards in t seconds and $x=100t-\cfrac { 25 }{ 2 } { t }^{ 2 }$. Find the
(i) initial velocity of the missile
(ii) the time when the height of the missile is maximum
(iii) the maximum height reached
(iv) the velocity which the missile strikes the ground.

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

2. Find the area of the loop of the curve 3ay2=x(x-a)2

1. A population grows at the rate of 2% per year. How long does it take for the population to double?

2. A thermometer reading 80°

1. The p.d.f of a continuous random variable X is $f(x)=\begin{cases} a+b{ x }^{ 2 },0\le x\le 1 \\ 0,otherwise \end{cases}$ where a and b are some constants. Find
(a) a and b if $E(x)=\cfrac { 3 }{ 5 }$
(b) Var (X)

2. 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.

1. Verify (p ∧ -p) ∧ (~q ∧ p) is a tautlogy, contradiction or contingency.

2. Prove by using truth table $\sim (pV(qVr)\equiv \left( \sim p \right) \wedge \left( \sim q\wedge \sim r \right)$

1. If U(x, y, z) = log (x3 + y3 + z3), find $\frac { \partial U }{ \partial x } +\frac { \partial U }{ \partial y } +\frac { \partial U }{ \partial z }$

2. If V(x,y) = ex(x cos y - y siny), then prove that $\frac { { \partial }^{ 2 }V }{ \partial { x }^{ 2 } } =\frac { { \partial }^{ 2 }V }{ \partial { y }^{ 2 } }$ = 0

1. W(x, y, z) = xy + yz + zx, x =u -v, y = uv, z = u + v, u ∈ R. Find $\frac { \partial W }{ \partial u } ,\frac { \partial W }{ \partial v }$, and evaluate them at $\left( \frac { 1 }{ 2 } ,1 \right)$

2. Prove that f(x, y) = x3 - 2x2y + 3xy2 + y3 is homogeneous; what is the degree? Verify Euler's Theorem for f.