12th Standard Maths English Medium Reduced Syllabus Important Questions with Answer key - 2021 Part - 2

12th Standard

Reg.No. :
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Maths

Time : 02:45:00 Hrs
Total Marks : 165

Multiple Choice Questions

15 x 1 = 15
1. If A = $\left[ \begin{matrix} 2 & 0 \\ 1 & 5 \end{matrix} \right]$ and B = $\left[ \begin{matrix} 1 & 4 \\ 2 & 0 \end{matrix} \right]$ then |adj (AB)| =

(a)

-40

(b)

-80

(c)

-60

(d)

-20

2. If ATA−1 is symmetric, then A2 =

(a)

A-1

(b)

(AT)2

(c)

AT

(d)

(A-1)2

3. Which of the following is/are correct?
(i) Adjoint of a symmetric matrix is also a symmetric matrix.
(ii) Adjoint of a diagonal matrix is also a diagonal matrix.
(iii) If A is a square matrix of order n and λ is a scalar, then adj(λA) = λn adj(A).

(a)

Only (i)

(b)

(ii) and (iii)

(c)

(iii) and (iv)

(d)

(i), (ii) and (iv)

4. The area of the triangle formed by the complex numbers z,iz, and z+iz in the Argand’s diagram is

(a)

$\cfrac { 1 }{ 2 } \left| z \right| ^{ 2 }$

(b)

|z|2

(c)

$\cfrac { 3 }{ 2 } \left| z \right| ^{ 2 }$

(d)

2|z|2

5. The conjugate of a complex number is $\cfrac { 1 }{ i-2 }$/Then the complex number is

(a)

$\cfrac { 1 }{ i+2 }$

(b)

$\cfrac { -1 }{ i+2 }$

(c)

$\cfrac { -1 }{ i-2 }$

(d)

$\cfrac { 1 }{ i-2 }$

6. If z is a non zero complex number, such that 2iz2=$\bar { z }$ then |z| is

(a)

$\cfrac { 1 }{ 2 }$

(b)

1

(c)

2

(d)

3

7. According to the rational root theorem, which number is not possible rational root of 4x7+2x4-10x3-5?

(a)

-1

(b)

$\frac { 5 }{ 4 }$

(c)

$\frac { 4 }{ 5 }$

(d)

5

8. If a, b, c ∈ Q and p +√q (p,q ∈ Q) is an irrational root of ax2+bx+c=0 then the other root is

(a)

-p+√q

(b)

p-iq

(c)

p-√q

(d)

-p-√q

9. Let a > 0, b > 0, c >0. h n both th root of th quatlon ax2+b+C= 0 are

(a)

real and negative

(b)

real and positive

(c)

rational numb rs

(d)

none

10. A stone is thrown up vertically. The height it reaches at time t seconds is given by x = 80t -16t2. The stone reaches the maximum height in time t seconds is given by

(a)

2

(b)

2.5

(c)

3

(d)

3.5

11. The maximum value of the function x2 e-2x,

(a)

$\cfrac { 1 }{ e }$

(b)

$\cfrac { 1 }{ 2e }$

(c)

$\cfrac { 1 }{ { e }^{ 2 } }$

(d)

$\cfrac { 4 }{ { e }^{ 4 } }$

12. If f (x, y) = exy then $\frac { { \partial }^{ 2 }f }{ \partial x\partial y }$ is equal to

(a)

xyexy

(b)

(1 +xy)exy

(c)

(1 +y)exy

(d)

(1 + x)exy

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. If p and q are the order and degree of the differential equation $y=\frac { dy }{ dx } +{ x }^{ 3 }\left( \frac { { d }^{ 2 }y }{ { dx }^{ 2 } } \right) +xy=cosx,$When

(a)

p < q

(b)

p = q

(c)

p>q

(d)

p exists and q does not exist

15. A binary operation * is defined on the set of positive rational numbers Q+ by a*b = $\frac { ab }{ 4 }$. Then 3 * $\left( \frac { 1 }{ 5 } *\frac { 1 }{ 2 } \right)$ is

(a)

$\frac { 3 }{ 160 }$

(b)

$\frac { 5 }{ 160 }$

(c)

$\frac { 3 }{ 10 }$

(d)

$\frac { 3 }{ 40 }$

1. 2 Marks

15 x 2 = 30
16. Find the modulus of the following complex numbers
$\cfrac { 2i }{ 3+4i }$

17. Find the modulus of the following complex numbers
(1-i)10

18. Find the modulus and principal argument of the following complex numbers.
$-\sqrt { 3 } +i$

19. Find the modulus of the complex number i25.

20. 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$\delta$.

21. Find a polynomial equation of minimum degree with rational coefficients, having 2-$\sqrt{3}$i as a root.

22. Find a polynomial equation of minimum degree with rational coefficients, having 2i+3 as a root.

23. Examine for the rational roots of x8-3x+1=0

24. Find the eccentricity of the hyperbola. with foci on the x-axis if the length of its conjugate axis is ${ \left( \frac { 3 }{ 4 } \right) }^{ th }$ of the length of its tranverse axis.

25. Solve :$\cfrac { dy }{ dx } =\cfrac { 2x }{ { x }^{ 2 }+1 }$

26. Define Distribution function.

27. Define Probability Density function

28. A coin is tossed until a head appears or the tail appears 4 times in succession. Find the probability distribution of the number of tosses.

29. The probability distribution of a random variable X is given under :

Find (i) k
(ii) E(X)

30. Check whether dot product is defined on the set of vectors. Explain?

1. 3 Marks

15 x 3 = 45
31. Given the complex number z=2+3i, represent the complex numbers in Argand diagram
z, iz , and z+iz

32. Given the complex number z=2+3i, represent the complex numbers in Argand diagram
z, −iz , and z−iz

33. For the hyperbola 3x2 - 6y2 = -18, find the length of transverse and conjugate axes and eccentricity.

34. Find the equation of the plane passing through the line of intersection of the planes $\vec { r } .(2\hat { i } -7\hat { j } +4\hat { k } )=3$ and 3x - 5y + 11 = 0, and the point (-2, 1, 3)

35. Find the equation of the plane which passes through the point (3, 4, -1) and is parallel to the plane 2x - 3y + 5z = 0. Also, find the distance between the two planes.

36. Find the equation of the plane passing through the intersection of the planes $\vec { r } .(\hat { i } +\hat { j } +\hat { k } )+1=0$ and $\vec { r } .(2\hat { i } -3\hat { j } +5\hat { k } )=2$ and the point (-1, 2, 1)

37. Find the absolute extrema of the following function on the given closed interval
f(x) = x2 -12x + 10; [1,2]

38. A right circular cylinder has radius r =10 cm. and height h = 20 cm. Suppose that the radius of the cylinder is increased from 10 cm to 10. 1 cm and the height does not change. Estimate the change in the volume of the cylinder. Also, calculate the relative error and percentage error.

39. Assuming log10e = 0.4343, find an approximate value of log10 1003

40. Find the partial derivatives of the following functions at the indicated point
h (x, y, z) = x sin (xy) + z2x, $\left( 2,\frac { \pi }{ 4 }, 1\right)$

41. A firm produces two types of calculators each week, x number of type A and y number of type B. The weekly revenue and cost functions (in rupees) are R(x, y) = 80x + 90y + 0.04xy − 0.05x2 − 0.05y2 and C(x, y) = 8x + 6y + 2000 respectively
Find the profit function P(x, y)

42. Let U(x, y, z) = xyz, x = e-t, y = e-t cos t, z = sin t, t ∈ R. Find $\frac{dU}{dt}$

43. Evaluate the following definite integrals:
$\int _{ 0 }^{ 1 }{ \sqrt { \frac { 1-x }{ 1+x } } } dx$

44. On the set Q of rational numbers, an operation * is defined as a*b=k(a+b) where k is a given non zero number. Is it associative

45. If on the set Q of rational numbers, a binary operation * is defined as a*b=λ(a+b) were λ is a nonzero fixed number and its given that * is associative, then the value of λ and what can we say about the operation*?

1. 5 Marks

15 x 5 = 75
46. For what value of λ, the system of equations x+y+z=1, x+2y+4z=λ, x+4y+10z=λ2 is consistent.

47. Find a polynomial equation of minimum degree with rational coefficients, having $\sqrt{5}$$\sqrt{3}$ as a root.

48. Find the domain of the following functions
(i) f(x) = sin-1(2x - 3)
(ii) f(x) = sin-1x + cos x

49. Show that the equation of the normal to the curve x=cos3θ,y=asin3θ at 'θ' is xcosθ-ysinθ=acos2θ.

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

51. Find the ratio of the area between the curves y=cosx and y=cos2x and x- axis from x=0 to $x=\cfrac { \pi }{ 3 }$

52. Find the area bounded by the curve y2(2a-x)=x2 and the line x=2a.

53. Solve : ${ e }^{ \frac { dy }{ dx } }=x+1,y(0)=5$

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

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

56. Four bad oranges are accidentally mixed with sixteen good ones. Find the probability distribution of bad oranges in a draw of two oranges. Also find the mean, variance and standard deviation of the distribution.

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

58. 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.

59. Show that (2018)2017+(2020)2017≡0(mod2019).

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