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#### Quarterly Model Question Paper

12th Standard EM

Reg.No. :
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Time : 02:45:00 Hrs
Total Marks : 90
20 x 1 = 20
1. If A=(1 2 3), then the rank of AAT is

(a)

0

(b)

2

(c)

3

(d)

1

2. if T= $_{ B }^{ A }\left( \begin{matrix} \overset { A }{ 0.7 } & \overset { B }{ 0.3 } \\ 0.6 & x \end{matrix} \right)$ is a transition probability matrix, then the value of x is

(a)

0.2

(b)

0.3

(c)

0.4

(d)

0.7

3. For what value of k, the matrix $A=\left( \begin{matrix} 2 & k \\ 3 & 5 \end{matrix} \right)$ has no inverse?

(a)

$\cfrac { 3 }{ 10 }$

(b)

$\cfrac { 10 }{ 3 }$

(c)

3

(d)

10

4. If A, B are two n x n non-singular matrices, then

(a)

AB is non-singular

(b)

AB is singular

(c)

(AB)-I = A-1 B-1

(d)

(AB)-1I does not exit

5. $\int _{ -1 }^{ 1 }{ { x }^{ 3 }{ e }^{ { x }^{ 4 } } }$ dx is

(a)

1

(b)

2$\int _{ -1 }^{ 1 }{ { x }^{ 3 }{ e }^{ { x }^{ 4 } } }$dx

(c)

0

(d)

${ e }^{ { x }^{ 4 } }$

6. If ∫ x sin x dx = - x cos x + α then α = __________ +c

(a)

sin x

(b)

cos x

(c)

C

(d)

none of these

7. ∫ sec2 (7-4x) dx = _____________ +c

(a)

$-\frac { 1 }{ 4 } tan(7-4x)$

(b)

$\frac { 1 }{ 7 } tan(7-4x)$

(c)

$\frac { 1 }{ 4 } tan(7-4x)$

(d)

$\frac { 1 }{ 7 } tan(7-4x)$

8. $\int { \frac { { e }^{ x }+1 }{ { e }^{ x }+x } }$ dx = ______________ +c

(a)

log |ex +1|

(b)

log |ex +x|

(c)

log |ex -1|

(d)

log |ex -x|

9. The demand function for the marginal function MR = 100 − 9x2 is

(a)

100 − 3x2

(b)

100x − 3x2

(c)

100x − 9x2

(d)

100 + 9x2

10. If MR and MC denote the marginal revenue and marginal cost and MR − MC = 36x − 3x2 − 81 , then the maximum profit at x is equal to

(a)

3

(b)

6

(c)

9

(d)

5

11. The area of the region bounded by the curve y2 = 2y - x and the y-axis _____ sq. units

(a)

$\frac{4}{3}$

(b)

$\frac{2}{3}$

(c)

4

(d)

$\frac{16}{3}$

12. The P.I of (3D+ D − 14)y = 13e2x is

(a)

$\frac {x}{2}$e2x

(b)

xe2x

(c)

$\frac {x^2}{2}$e2x

(d)

13xe2x

13. A homogeneous differential equation of the form $\frac { dy }{ dx }$ = f$\left( \frac { y }{ x } \right)$ can be solved by making substitution,

(a)

y = v x

(b)

v = y x

(c)

x = v y

(d)

x = v

14. The variable separable form of $\frac { dy }{ dx } =\frac { y(x-y) }{ x(x+y) }$ by taking y vx and $\frac { dy }{ dx } =v+x\frac { dv }{ dx }$

(a)

$\frac { 2{ v }^{ 2 } }{ 1+v } dv=\frac { dx }{ x }$

(b)

$\frac { 2{ v }^{ 2 } }{ 1+v } dv=-\frac { dx }{ x }$

(c)

$\frac { 2{ v }^{ 2 } }{ 1-v } dv=\frac { dx }{ x }$

(d)

$\frac { 1+v }{ 2{ v }^{ 2 } } dv=-\frac { dx }{ x }$

15. The differential equation obtained by eliminating a and b from y = a e3x + b e-3x is

(a)

$\frac { { d }^{ 2 }y }{ dx^{ 2 } }$+ay=0

(b)

$\frac { { d }^{ 2 }y }{ dx^{ 2 } }$-9y=0

(c)

$\frac { { d }^{ 2 }y }{ dx^{ 2 } } -9\frac { dy }{ dx }$

(d)

$\frac { { d }^{ 2 }y }{ dx^{ 2 } }$+9x=0

16. The particular integral of the differential equation $\frac { d^{ 2 }y }{ { dx }^{ 2 } } -5\frac { dy }{ dx }$+6y=e5x is _______

(a)

$\frac { e^{ 5x } }{ 6 }$

(b)

$\frac { xe^{ 5x } }{ 21 }$

(c)

6e5x

(d)

$\frac { { e }^{ 5x } }{ 25 }$

17. For the given points (x0, y0) and (x1,y1) the Lagrange’s formula is

(a)

$y(x)=\frac { x-{ x }_{ 1 } }{ { x }_{ 0 }-{ x }_{ 1 } } { y }_{ 0 }+\frac { x-{ x }_{ 0 } }{ { x }_{ 1 }-{ x }_{ 0 } } { y }_{ 1 }$

(b)

$y(x)=\frac { { x }_{ 1 }-{ x }_{ 0 } }{ { x }_{ 0 }-{ x }_{ 1 } } { y }_{ 0 }+\frac { { x }_{ 1 }-{ x }_{ 0 } }{ { x }_{ 1 }-{ x }_{ 0 } } { y }_{ 1 }$

(c)

$y(x)=\frac { x-{ x }_{ 1 } }{ { x }_{ 0 }-{ x }_{ 1 } } { y }_{ 1 }+\frac { x-{ x }_{ 0 } }{ { x }_{ 1 }-{ x }_{ 0 } } { y }_{ 0 }$

(d)

$y(x)=\frac { { x }_{ 1 }-{ x } }{ { x }_{ 0 }-{ x }_{ 1 } } { y }_{ 1 }+\frac { x-{ x }_{ 0 } }{ { x }_{ 1 }-{ x }_{ 0 } } { y }_{ 0 }$

18. If f (x)=x+ 2x + 2 and the interval of differencing is unity then Δf (x)

(a)

2x −3

(b)

2x +3

(c)

x + 3

(d)

x − 3

19. E [f(x0)] is

(a)

f(xo + h)

(b)

f(xo - h)

(c)

f(xo) + h

(d)

f(xo) - h

20. If y is to be estimated for the value of x between two extreme points in a set of values, it is called ___________

(a)

Interpolation

(b)

extrapolation

(c)

Forward interpolation

(d)

backward interpolation

21. 7 x 2 = 14
22. For what values of the parameterl , will the following equations fail to have unique solution: 3x−y+λz=1,2x+y+z=2,x+2y−lz = −1 by rank method.

23. Integrate the following with respect to x.
sin3 x

24. Evaluate the following integrals:
$\int _{ -1 }^{ 1 }{ { x }^{ 2 }{ e }^{ -2x } } dx$

25. If $\int _{ 0 }^{ a }{ { 3x }^{ 2 } } dx=8$ find the value of a

26. Solve the following:
$\frac { dy }{ dx } +ycosx=sinxcosx$.

27. Write down the order and degree of the following differential equations.
$\sqrt { 1+\left( \frac { dy }{ dx } \right) ^{ 2 } }$=4x

28. When h = 1, find Δ (x3).

29. 7 x 3 = 21
30. Find the rank of the matrix $\left( \begin{matrix} 5 & 3 & 0 \\ 1 & 2 & -4 \\ -2 & -4 & 8 \end{matrix} \right)$

31. Show that the equations x- 3y + 4z = 3, 2x - 5y + 7z = 6, 3x - 8y + 11z = 1 are inconsistent

32. Evaluate ഽ$\sqrt { { x }^{ 2 }-16 }$dx

33. Solve 9y'' − 12y' + 4y = 0

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

35. From the following table find the missing value

 x 2 3 4 5 6 f(x) 45 49.2 54.1 - 67.4
36. Using graphic method, find the value of y when x=27.

 x 10 15 20 25 30 y 35 32 29 26 23
37. 7 x 5 = 35
38. The total cost of 11 pencils and 3 erasers is Rs 64 and the total cost of 8 pencils and 3 erasers is Rs 49. Find the cost of each pencil and each eraser by Cramer’s rule.

39. If $\int _{ a }^{ b }{ dx } =1$ and $\int _{ a }^{ b }{ xdx } =1$, then find a and b

40. Evaluate ഽ x3 sin (x4) dx

41. The price of a machine is 6,40,000 if the rate of cost saving is represented by the function f(t) = 20,000 t. Find out the number of years required to recoup the cost of the function.

42. The sum of Rs. 2,000 is compounded continuously, the nominal rate of interest being 5% per annum. In how many years will the amount be double the original principal? (loge2 = 0.6931)

43. Equipment maintenance and operating costs (are related to the overhaul interval x by the equation ${ x }^{ 2 }\frac { dc }{ dx } -10xc=-10$ with c=c0 and x=x0. Find c as a function of x.

44. Using Lagrange's formula find the value of y when x = 4 from the following table.

 x 0 3 5 6 8 y 276 460 414 343 110