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All Chapter 5 Marks

12th Standard EM

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Maths

Time : 03:00:00 Hrs
Total Marks : 240
48 x 5 = 240
1. Determine the values of λ for which the following system of equations (3λ − 8)x + 3y + 3z = 0, 3x + (3λ − 8)y + 3z = 0, 3x + 3y + (3λ − 8)z = 0. has a non-trivial solution.

2. Solve the following system of linear equations by matrix inversion method:
2x + 3y − z = 9, x + y + z = 9, 3x − y − z  = −1

3. Show that the equations -2x + y + z = a, x - 2y + z = b, x + y -2z = c are consistent only if a + b + c =0.

4. Using Gaussian Jordan method, find the values of λ and μ so that the system of equations 2x - 3y + 5z = 12, 3x + y + λz =μ, x - 7y + 8z = 17 has (i) unique solution (ii) infinite solutions and (iii) no solution.

5. Let z1,z2, and z3 be complex numbers such that $\left| { z }_{ 1 } \right\| =\left| { z }_{ 2 } \right| =\left| { z }_{ 3 } \right| =r>0$ and z1+z2+z3 $\neq$ 0 prove that $\left| \cfrac { { z }_{ 1 }{ z }_{ 2 }+{ z }_{ 2 }{ z }_{ 3 }+{ z }_{ 3 }{ z }_{ 1 } }{ { z }_{ 1 }+{ z }_{ 2 }+{ z }_{ 3 } } \right|$ =r

6. Find all cube roots of $\sqrt { 3 } +i$

7. Verify that 2 arg(-1) ≠ arg(-1)2

8. Verify that arg(1+i) + arg(1-i) = arg[(1+i) (1-i)]

9. Solve the equation (2x-)(6x-1)(3x-2)(x-12)-7=0

10. Find all zeros of the polynomial x6-3x5-5x4+22x3-39x2-39x+135, if it is known that 1+2i and $\sqrt{3}$ are two of its zeros.

11. If a, b, c, d and p are distinct non-zero real numbers such that (a2+b2+c2) p2-2 (ab+bc+cd) p+(b2+c2+d2)≤ 0 the n. Prove that a,b,c,d are in G.P and ad=bc

12. Solve: (2x2 - 3x + 1) (2x2 + 5x + 1) = 9x2.

13. If a1, a2, a3, ... an is an arithmetic progression with common difference d, prove that tan $x = {-b \pm \sqrt{b^2-4ac} \over 2a}\quad \left[ tan^{ -1 }\left( \frac { d }{ 1+{ a }_{ 1 }{ a }_{ 2 } } \right) +tan^{ -1 }\left( \frac { d }{ 1+{ a }_{ 2 }{ a }_{ 3 } } \right) +....tan^{ -1 }\left( \frac { d }{ 1+{ a }_{ n }{ a }_{ n-1 } } \right) \right] =\frac { { a }_{ n }-{ a }_{ 1 } }{ 1+{ a }_{ 1 }{ a }_{ n } }$

14. Solve $cos\left( sin^{ -1 }\left( \frac { x }{ \sqrt { 1+{ x }^{ 2 } } } \right) \right) =sin\left\{ cot^{ -1 }\left( \frac { 3 }{ 4 } \right) \right\}$

15. Write thefunction$f(x)={ tan }^{ -1 }\sqrt { \cfrac { a-x }{ a+x } } -a<x<a$ in the simplest form

16. Provethat ${ tan }^{ -1 }\left( \cfrac { 1-x }{ 1+x } \right) -{ tan }^{ -1 }\left( \cfrac { 1-y }{ 1+y } \right) ={ sin }^{ -1 }\left( \cfrac { y-x }{ \sqrt { 1+{ x }^{ 2 } } .\sqrt { 1+{ y }^{ 2 } } } \right) \\$

17. Show that the linex−y+4=0 is a tangent to the ellipse x2+3y2=12 . Also find the coordinates
of the point of contact.

18. Points A and B are 10km apart and it is determined from the sound of an explosion heard at those points at different times that the location of the explosion is 6 km closer to A than B . Show that the location of the explosion is restricted to a particular curve and find an equation of it.

19. 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.

20. The foci of a hyperbola coincides with the foci of the ellipse $\frac { { x }^{ 2 } }{ 25 } +\frac { y^{ 2 } }{ 9 } =1$. Find the equation of the hyperbola if its eccentricity is 2.

21. By vector method, prove that cos(α + β) = cos α cos β -  sin α sin β

22. Find the equation of a straight line passing through the point of intersection of the straight lines $\vec { r } =(\hat { i } +\hat { 3j } -\hat { k } )+t(2\hat { i } +3\hat { j } +2\hat { k } )\quad$ and $\frac { x-2 }{ 1 } =\frac { y-4 }{ 2 } =\frac { z+3 }{ 4 }$ and perpendicular to both straight lines.

23. ABCD is a quadrilateral with $\overset { \rightarrow }{ AB } =\overset { \rightarrow }{ \alpha }$ and $\overset { \rightarrow }{ AD } =\overset { \rightarrow }{ \beta }$ and $\overset { \rightarrow }{ AC } =2\overset { \rightarrow }{ \alpha } +3\overset { \rightarrow }{ \beta }$. If. the area of the quadrilateral is λ times the area of the parallelogram with $\overset { \rightarrow }{ AB }$ and $\overset { \rightarrow }{ AD }$ as adjacent sides, then prove that $\lambda =\frac { 5 }{ 2 }$

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24. 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

25. A ladder 17 metre long is leaning against the wall. The base of the ladder is pulled away from the wall at a rate of 5 m/s. When the base of the ladder is 8 metres from the wall.
How fast is the top of the ladder moving down the wall?

26. Find the local extrema for the following function using second derivative test:
f(x) = x2 e-2x

27. Sand is pouring from a pipe at the rate of 12 cm3/sec. The falling sand forms a cone on the ground in such a way that the height of the cone is always one-sixth of the radius of the base. How fast is the height of the sand cone increasing, when the height is 4 cm?

28. Find the intervals of concavity and points of inflexion for f(x)=x3-15x2+75x-50.

29. Let U(x, y) = ex sin y, where x = st2, y = s2 t, s, t ∈ R. Find $\frac { \partial U }{ \partial s } ,\frac { \partial U }{ \partial t }$ and evaluate them at s = t = 1.

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

31. If V = log r and r2 = x2 +y2 + z2, then prove that $\frac { { \partial }^{ 2 }V }{ \partial { x }^{ 2 } } +\frac { { \partial }^{ 2 }V }{ \partial { y }^{ 2 } } +\frac { { \partial }^{ 2 } }{ \partial { z }^{ 2 } } =\frac { 1 }{ { r }^{ 2 } }$

32. Find $\cfrac { \partial w }{ \partial u } ,\cfrac { \partial w }{ \partial v }$ if w=sin-1(x,y) where x=u+v,y=u-v

33. Evaluate the following integrals using properties of integration:
$\int _{ 0 }^{ \pi }{ x\left[ { sin }^{ 2 }(sinx)+{ cos }^{ 2 }(cosx) \right] } dx$

34. Find the area of the region bounded between the curves y = sin x and y = cos x and the lines x = 0 and x =$\pi$

35. Find the value of ‘c’ for which the area bounded by the curve y=8x2-x5,the lines x=1,x=c and x-axis $\cfrac { 16 }{ 3 }$

36. Find volume of the solid generated by the revolution of the loop of the curve x=t2,$y=t-\cfrac { { t }^{ 3 } }{ 3 }$ about the x-axis.

37. The velocity v , of a parachute falling vertically satisfies the equation $\\ \\ \\ \\ \\ \\ \\ v\frac { dv }{ dx } =g\left( 1-\frac { { v }^{ 2 } }{ { k }^{ 2 } } \right) \\ \\$, where g and k are constants. If v and x are both initially zero, find v in terms of x.

38. A pot of boiling water at 100oC is removed from a stove at time t = 0 and left to cool in the kitchen. After 5 minutes, the water temperature has decreased to 80oC , and another 5 minutes later it has dropped to 65oC . Determine the temperature of the kitchen.

39. Solve : $\cfrac { dy }{ dx } =\left( { sin }^{ 2 }x{ cos }^{ 2 }x+{ xe }^{ x } \right) dx$

40. Solve : (x-siny)dy+tanydx=0,y(0)=0

41. 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.

42. Suppose that f (x) given below represents a probability mass function

 x 1 2 3 4 5 6 f(x) c2 2c2 3c2 4c2 c 2c

Find
(i) the value of c
(ii) Mean and variance.

43. In a business venture a man can make a profit of Rs.2,000 with a probability of 0.4 or have a loss of Rs.1,000 with a probability of 0.6. What is his expectation, variance and S.D of profit?

44. The probability that an engineering college student will graduate is 0.3. Find the probability that out of 6 students (i) none (ii) one (iii) at least one will graduate.

45. Establish the equivalence property connecting the bi-conditional with conditional: p ↔️ q ≡ (p ➝ q) ∧ (q⟶ p)

46. Let A be Q\{1}. Define ∗ on A by x*y = x + y − xy . Is ∗ binary on A? If so, examine the commutative and associative properties satisfied by ∗ on A.

47. Let Q, be the set of all nonzero rational numbers and k is a nonzero fixed rational number and * be a binary operation defined as a*b=kab. Show that (Q,*) satisfies closure, associative, inverse and commutative properties.

48. Prove that (2019)10+(2020)10≡1025(mod 2018)