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

12th Standard

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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 P = $\left[ \begin{matrix} 1 & x & 0 \\ 1 & 3 & 0 \\ 2 & 4 & -2 \end{matrix} \right]$ is the adjoint of 3 × 3 matrix A and |A| = 4, then x is

(a)

15

(b)

12

(c)

14

(d)

11

2. If ATA−1 is symmetric, then A2 =

(a)

A-1

(b)

(AT)2

(c)

AT

(d)

(A-1)2

3. If $4{ cos }^{ -1 }x+{ sin }^{ -1 }x=\pi$ then x is

(a)

$\cfrac { 3 }{ 2 }$

(b)

$\cfrac { 1 }{ \sqrt { 2 } }$

(c)

$\cfrac { \sqrt { 3 } }{ 2 }$

(d)

$\cfrac { 2 }{ \sqrt { 3 } }$

4. The value of tan $\left( { cos }^{ -1 }\cfrac { 3 }{ 5 } +{ tan }^{ -1 }\cfrac { 1 }{ 4 } \right)$ is ______

(a)

$\cfrac { 19 }{ 8 }$

(b)

$\cfrac { 8 }{ 19 }$

(c)

$\cfrac { 19 }{ 12 }$

(d)

$\cfrac { 3 }{ 4 }$

5. Equation of tangent at (-4, -4) on x2 = -4y is

(a)

2x - y + 4 = 0

(b)

2x + y - 4 = 0

(c)

2x - y - 12 = 0

(d)

2x + y + 4 = 0

6. The distance between the foci of a hyperbola is 16 and e = $\sqrt { 2 }$ Its equation is

(a)

x2 - y2 = 32

(b)

y2 - x2 = 32

(c)

x2 - y2 = 16

(d)

y2 - x2 = 16

7. If the foci of the ellipse $\frac { { x }^{ 2 } }{ 16 } +\frac { { y }^{ 2 } }{ { b }^{ 2 } }$ = 1 and the hyperbola $\frac { { x }^{ 2 } }{ 144 } -\frac { { y }^{ 2 } }{ 81 } =\frac { 1 }{ 25 }$ coincide then b2 is

(a)

1

(b)

5

(c)

7

(d)

9

8. The length of major and minor axes of 4x2 + 3y2 = 12 are ____________

(a)

4, 2$\sqrt3$

(b)

2, $\sqrt3$

(c)

2$\sqrt3$, 4

(d)

$\sqrt3$, 2

9. The tangent at any point P on the ellipse $\frac { { x }^{ 2 } }{ 6 } +\frac { { y }^{ 2 } }{ 3 }$ = 1 whose centre C meets the major axis at T and PN is the perpendicular to the major axis; The CN CT = ______________

(a)

$\sqrt6$

(b)

3

(c)

$\sqrt3$

(d)

6

10. If $\lambda \overset { \wedge }{ i } +2\lambda \overset { \wedge }{ j } +2\lambda \overset { \wedge }{ k }$ is a unit vector, then the value of λ is

(a)

土 $\frac { 1 }{ 3 }$

(b)

土 $\frac { 1 }{ 4 }$

(c)

土 $\frac { 1 }{ 9 }$

(d)

$\frac { 1 }{ 2 }$

11. If the vectors $a\overset { \wedge }{ i } +\overset { \wedge }{ j } +\overset { \wedge }{ k }$$\overset { \wedge }{ i } +b\overset { \wedge }{ j } +\overset { \wedge }{ k }$ and $\overset { \wedge }{ i } +\overset { \wedge }{ j } +c\overset { \wedge }{ k }$ (a ≠ b ≠ c ≠ 1) are coplaner, then $\frac { 1 }{ 1-a } +\frac { 1 }{ 1-b } +\frac { 1 }{ 1-c } =$

(a)

0

(b)

1

(c)

2

(d)

$\frac { abc }{ (1-a)(1-b)(1-c) }$

12. The equation of the tangent to the curve y=x2-4x+2 at (4,2) is

(a)

x + 4y + 12 = 0

(b)

4x + y + 12 = 0

(c)

4x - y - 14 = 0

(d)

x + 4y - 12 = 0

13. If the radius of the sphere is measured as 9 em with an error of 0.03 cm, the approximate error in calculating its volume is

(a)

9.72 cm3

(b)

0.972 cm3

(c)

0.972π cm3

(d)

9.72π cm3

14. The value of $\int _{ -\pi }^{ \pi }{ { sin }^{ 3 }x \ { cos }^{ 3 }x \ } dx$ is

(a)

0

(b)

$\pi$

(c)

2$\pi$

(d)

4$\pi$

15. The I.F of y log y$\frac{dx}{dy}+x-log\ y=0$ is

(a)

log(log y)

(b)

log y

(c)

$\frac{1}{log\ y}$

(d)

$\frac{1}{log(log\ y)}$

16. If a random variable X has the p.d.f.$f(x)=\cfrac { k }{ { x }^{ 2 }+1 } ,0<x<\infty$ then k is

(a)

$\pi$

(b)

$\cfrac { 1 }{ \pi }$

(c)

1

(d)

$\cfrac { 2 }{ \pi }$

17. The sum of the mean and variance of a binomial distribution for 6 total is 2.16. Then the probability of success p=__________

(a)

0.4

(b)

0.6

(c)

0.8

(d)

0.2

18. The Identity element of $\left\{ \left( \begin{matrix} x & x \\ x & x \end{matrix} \right) \right\}$ |x$\in$R, x≠0} under matrix multiplication is

(a)

$\left( \begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix} \right)$

(b)

$\left( \begin{matrix} \frac { 1 }{ 4x } & \frac { 1 }{ 4x } \\ \frac { 1 }{ 4x } & \frac { 1 }{ 4x } \end{matrix} \right)$

(c)

$\left( \begin{matrix} \frac { 1 }{ 2 } & \frac { 1 }{ 2 } \\ \frac { 1 }{ 2 } & \frac { 1 }{ 2 } \end{matrix} \right)$

(d)

$\left( \begin{matrix} \frac { 1 }{ 2x } & \frac { 1 }{ 2x } \\ \frac { 1 }{ 2x } & \frac { 1 }{ 2x } \end{matrix} \right)$

19. Define * on Z by a*b = a+b+1 ∀ a,b $\in$ Z. Then the identity element of z is

(a)

1

(b)

0

(c)

1

(d)

-1

20. Which of the following is a statement?

(a)

7+2<10

(b)

Wish you all success

(c)

All the best

(d)

How old are you?

21. Part II

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

7 x 2 = 14
22. If $\overset { \rightarrow }{ a } =\overset { \wedge }{ i } +2\overset { \wedge }{ j } +3\overset { \wedge }{ k }$$\overset { \rightarrow }{ b } =-\overset { \wedge }{ i } +2\overset { \wedge }{ j } +\overset { \wedge }{ k }$ and $\overset { \rightarrow }{ c } =3\overset { \wedge }{ i } +\overset { \wedge }{ j }$ find $\frac { \lambda }{ c }$ such that $\overset { \rightarrow }{ a } +\lambda \overset { \rightarrow }{ b }$ is perpendicular to $\overset { \rightarrow }{ c }$

()

-a

23.

Compute the limit

$\underset{x\rightarrow 1}{lim}(\frac{x^{2}-3x+2}{x^{2}-4x+3})$.

24. Find df for f(x) = x2 + x 3 and evaluate it for
x = 3 and dx = 0.02

25. If w=log(x2+y2),x=cosθ,y=sinθ, find $\cfrac { dw }{ d\theta }$

26. Evaluate: $\int _{ 0 }^{ 3 }{ (3{ x }^{ 2 }-4x+5) } dx$

27. A differential equation, determine its order, degree (if exists)
${ \left( \frac { dy }{ dx } \right) }^{ 3 }=\sqrt { 1+\left( \frac { dy }{ dx } \right) }$

28. Write the converse, inverse, and contrapositive of each of the following implication.
(i) If x and y are numbers such that x = y, then x2 = y2
(ii) If a quadrilateral is a square then it is a rectangle.

1. Part III

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

7 x 3 = 21
29. Find,the rank of the matrix math $\left[ \begin{matrix} 4 \\ -2 \\ 1 \end{matrix}\begin{matrix} 4 \\ 3 \\ 4 \end{matrix}\begin{matrix} 0 \\ -1 \\ 8 \end{matrix}\begin{matrix} 3 \\ 5 \\ 7 \end{matrix} \right]$.

30. Prove that ${ tan }^{ -1 }\left( \cfrac { m }{ n } \right) -{ tan }^{ -1 }\left( \cfrac { m-n }{ m+n } \right) =\cfrac { \pi }{ 4 }$

31. Show that the lines $\frac { x-1 }{ 3 } =\frac { y+1 }{ 2 } =\frac { z-1 }{ 5 }$ and $\frac { x+2 }{ 4 } =\frac { y-1 }{ 3 } =\frac { z+1 }{ -2 }$ do not intersect

()

a, b, c

32. A ball is thrown vertically upwards, moves according to the law s = 13.8 t - 4.9 t2 where s
is in metres and t is in seconds.
(i) Find the acceleration at t = 1
(ii) Find velocity at t = 1
(iii) Find the maximum height reached by the ball?

33. If v=log(tanx+tany+tanz), then prove that $\Sigma sin2x\cfrac { \partial v }{ \partial x } =2$

34. Solve : ydx+(x-y2)dy=0

35. A random variable X has the following probability distribution

 x 0 1 2 3 4 5 6 7 P(X) 0 k 2k 2k 3k k2 2k2 7k2+k

Evaluate (i) k
(ii) $\\ \\ P(X\ge 6)$
(iii) P(0<X<3)

1. Part IV

7 x 5 = 35
36. Solve: (2x2 - 3x + 1) (2x2 + 5x + 1) = 9x2.

37. The guides of a railway bridge is a parabola with its vertex at the highest point 15 m above the ends. If the span is 120 m, find the height of the bridge at 24 m from the middle point.

38. Gas is escaping from a spherical balloon at the rate of 900 cm3/sec. How fast is the surface area and radius of the balloon shrinking when the radius of the balloon is 30 cm?

39. Find the intervals for which the function f(x)=2x2-9x2-12x+1 is increasing or decfreasing and find the local extermems.

40. Find the local maximum and local minimum values for f(x)=12x2-2x2-x4.

41. Let X be a random variable denoting the life time of an electrical equipment having probability density function
$f(x)=\begin{cases} \begin{matrix} { ke }^{ -2x } & forx>0 \end{matrix} \\ \begin{matrix} 0 & forx\le 0 \end{matrix} \end{cases}$
Find
(i) the value of k
(ii) Distribution function
(iii) P(X < 2)
(iv) calculate the probability that X is at least for four unit of time
(v) P(X = 3)

42. Prove that p➝(¬q V r) ≡ ¬pV(¬qVr) using truth table.