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

12th Standard

Reg.No. :
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Maths

Time : 02:40:00 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 = $\left[ \begin{matrix} 7 & 3 \\ 4 & 2 \end{matrix} \right]$, then 9I - A =

(a)

A-1

(b)

$\frac { { A }^{ -1 } }{ 2 }$

(c)

3A-1

(d)

2A-1

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. If AT is the transpose of a square matrix A, then

(a)

|A| ≠ |AT|

(b)

|A| = |AT|

(c)

|A| + |AT| =0

(d)

|A| = |AT| only

4. If $z=\cfrac { \left( \sqrt { 3 } +i \right) ^{ 3 }\left( 3i+4 \right) ^{ 2 } }{ \left( 8+6i \right) ^{ 2 } }$ , then |z| is equal to

(a)

0

(b)

1

(c)

2

(d)

3

5. The product of all four values of $\left( cos\cfrac { \pi }{ 3 } +isin\cfrac { \pi }{ 3 } \right) ^{ \frac { 3 }{ 4 } }$ is

(a)

-2

(b)

-1

(c)

1

(d)

2

6. If $\omega \neq 1$ is a cubic root of unity and $\left| \begin{matrix} 1 & 1 & 1 \\ 1 & { -\omega }^{ 2 } & { \omega }^{ 2 } \\ 1 & { \omega }^{ 2 } & { \omega }^{ 2 } \end{matrix} \right|$ =3k, then k is equal to

(a)

1

(b)

-1

(c)

$\sqrt { 3i }$

(d)

$-\sqrt { 3i }$

7. If cot-1 2 and cot-1 3 are two angles of a triangle, then the third angle is

(a)

$\frac{\pi}{4}$

(b)

$\frac{3\pi}{4}$

(c)

$\frac{\pi}{6}$

(d)

$\frac{\pi}{3}$

8. An ellipse hasOB as semi minor axes, F and F′ its foci and the angle FBF′ is a right angle. Then the eccentricity of the ellipse is

(a)

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

(b)

$\frac { 1 }{ 2 }$

(c)

$\frac { 1 }{ 4 }$

(d)

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

9. If $\vec { a } \times (\vec { b } \times \vec { c } )=(\vec { a } \times \vec { b } )\times \vec { c }$ where $\vec { a } ,\vec { b } ,\vec { c }$ are any three vectors such that $\vec { a } ,\vec { b }$ $\neq$ 0 and  $\vec { a } .\vec { b }$ $\neq$ 0 then $\vec { a }$ and $\vec { c }$ are

(a)

perpendicular

(b)

parallel

(c)

inclined at an angle $\frac{\pi}{3}$

(d)

inclined at an angle  $\frac{\pi}{6}$

10. A balloon rises straight up at 10 m/s. An observer is 40 m away from the spot where the balloon left the ground. Find the rate of change of the balloon's angle of elevation in radian per second when the balloon is 30 metres above the ground.

(a)

$\cfrac { 3 }{ 25 } radians/sec$

(b)

$\cfrac { 4 }{ 25 } radians/sec$

(c)

$\cfrac { 1 }{ 5 } radians/sec$

(d)

$\cfrac { 1 }{ 3 } radians/sec$

11. 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

12. The minimum value ofthe function |3-x|+9 is

(a)

0

(b)

3

(c)

6

(d)

9

13. If we measure the side of a cube to be 4 cm with an error of 0.1 cm, then the error in our calculation of the volume is

(a)

0.4 cu.cm

(b)

0.45 cu.cm

(c)

2 cu.cm

(d)

4.8 cu.cm

14. The change in the surface area S = 6x2 of a cube when the edge length varies from xo to xo+ dx is

(a)

12 xo+dx

(b)

12xo dx

(c)

6xo dx

(d)

6xo+ dx

15. 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

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

(a)

10

(b)

5

(c)

8

(d)

9

17. The order and degree of the differential equation $\frac { { d }^{ 2 }y }{ { dx }^{ 2 } } +{ \left( \frac { dy }{ dx } \right) }^{ 1/3 }+{ x }^{ 1/4 }=0$are respectively

(a)

2, 3

(b)

3, 3

(c)

2, 6

(d)

2, 4

18. The solution of $\frac { dy }{ dx } ={ 2 }^{ y-x }$is

(a)

2x+2y=C

(b)

2x-2y=C

(c)

$\frac { 1 }{ { 2 }^{ x } } -\frac { 1 }{ { 2 }^{ y } } =C$

(d)

x+y=C

19. 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

20.  p q (p ∧ q) ⟶ ¬q T T (a) T F (b) F T (c) F F (d)

Which one of the following is correct for the truth value of (p ∧ q)⟶ ¬p p?

(a)
 (a) (b) (c) (d) T T T T
(b)
 (a) (b) (c) (d) F T T T
(c)
 (a) (b) (c) (d) F F T T
(d)
 (a) (b) (c) (d) T T T F
1. Part II

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

7 x 2 = 14
21. Find the intervals of increasing and decreasing function for f(x) =x3 + 2x2 - 1.

22. Find the point on the parabola y2=18x at which the ordinate increases at twice the rate of the abscissa.

23. Use differentials to find $\sqrt{25.2}$

24. If of f(x, y) = x2 + y3 + 2xy2 find fxx, fyy, fxy and fyx.

25. If $u=log\sqrt { { x }^{ 2 }+{ y }^{ 2 } }$ then prove that $\left( \cfrac { \vartheta u }{ \vartheta x } \right) +\left( \cfrac { \vartheta u }{ \vartheta y } \right) =\cfrac { 1 }{ { x }^{ 2 }+{ y }^{ 2 } }$

26. If w=exy,x=at2,y=2at, find $\cfrac { dw }{ dt }$

27. If $w={ e }^{ { x }^{ 2 }+{ y }^{ 2 } }$ ,x=cosθ,y=sinθ, find $\cfrac { dw }{ d\theta }$

28. Find the volume of the solid y=x3,x=0,y=1 is revolved about the y-axis.

29. Solve : $\cfrac { dy }{ dx } =\cfrac { { e }^{ x }-{ e }^{ -x } }{ { e }^{ x }+{ e }^{ -x } }$

30. solve:xdy+ydx=xydx

1. Part III

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

7x 3 = 21
31. 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

32. If A = $\left[ \begin{matrix} 0 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 0 \end{matrix} \right]$, show that A-1 = $\frac {1}{2}$ (A2 - 3I).

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

34. The complex numbers u,v, and w are related by $\cfrac { 1 }{ u } =\cfrac { 1 }{ v } +\cfrac { 1 }{ w }$ If v=3−4i and w=4+3i, find u in rectangular form.

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

36. 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.

37. Find the domain of
f(x)=sin-1 $(\frac{|x|-2}{3})+cos^-1(\frac{1-|x|}{4})$

38. Find the domain of
g(x)=sin−1x+cos−1x

39. Find the equation of the ellipse with foci (±2,0) , vertices (±3,0) .

40. A room 34m long is constructed to be a whispering gallery. The room has an elliptical ceiling, as shown in Fig. 5.64. If the maximum height of the ceiling is 8m, determine where the foci are located.

1. Part IV

7x 5 = 35
1. If $\left| \overset { \rightarrow }{ A } \right| =\overset { \wedge }{ i } +\overset { \wedge }{ j } +\overset { \wedge }{ k }$ and $\overset { \wedge }{ i } =\overset { \wedge }{ j } -\overset { \wedge }{ k }$ are two given vector, then find a vector B satisfying the equations $\overset { \rightarrow }{ A } \times \overset { \rightarrow }{ B }$$\overset { \rightarrow }{ C }$ and $\overset { \rightarrow }{ A }$.$\overset { \rightarrow }{ B }$=3

2. 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

1. A manufacturer can sell x items at a price of rupees $\left( 5-\cfrac { x }{ 100 } \right)$ each. The cost price of x items is Rs.$\left( \cfrac { x }{ 5 } +500 \right)$ .Find the numbers of items he should sell to earn maximum profit.

2. If P=CLakB,c>0,$\alpha +\beta =1$, then prove that $k\cfrac { \partial P }{ \partial k } +L\cfrac { \partial P }{ \partial K } =P$ .(Without using Euler's theorem)

1. Show that the area under the curve y = sin x and y = sin 2x between x = 0 and x = $\frac { \pi }{ 3 }$ and x axis are as 2:3

2. Find the area bounded by x=at2,y=2at between the ordinates corresponding to t = 1 and t = 2.

1. Find the area bounded by the curve y=xex and y=xe-x and the line x=1.

2. The slope of the tangent at p(x,y) on the curve is -$\left( \frac { y+3 }{ x+2 } \right)$. If the curve passes through the origin, find the equation of the curve.

1. Solve: $\frac { dy }{ dx }$ = (3x+2y+1)2

2. Solve :x2dy+y(x+y)dx=0 given that y=1 when x=1.

1. Solve :$x\cfrac { dy }{ dx } sin\left( \cfrac { y }{ x } \right) +x-ysin\left( y\cfrac { y }{ x } \right) =,y(1)=\cfrac { \pi }{ 2 }$

2. Two cards are drawn simultaneously from a well shuffled pack of 52 cards. Find the mean and variance of the number of red cards.

1. 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.

2. Show that (Z7-[0],X7) satisfies closure, identity, inverse and commutative properties.