+1 Half Yearly Model Question

11th Standard

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Maths

Time : 02:30:00 Hrs
Total Marks : 90
    I. Choose the best suitable answer:
    20 x 1 = 20
  1. Let R be a relation on the set N given by R={(a,b):a=b-2, b>6}. Then

    (a)

    (2,4)∈R

    (b)

    (3,8)∈R

    (c)

    (6,8)∈R

    (d)

    (8,7)∈R

  2. If n(A) = 2 and n(B ∪ C) = 3, then n[(A x B) ∪ (A x C)] is

    (a)

    23

    (b)

    32

    (c)

    6

    (d)

    5

  3. If a and b are the roots of the equation x2-kx+c = 0 then the distance between the points (a, 0) and (b, 0)

    (a)

    \(\sqrt { { 4k }^{ 2 }-c } \)

    (b)

    \(\sqrt { { k }^{ 2 }-4c } \)

    (c)

    \(\sqrt { 4c-{ k }^{ 2 } } \)

    (d)

    \(\sqrt { k-8c } \)

  4. \(\left( 1+cos\frac { \pi }{ 8 } \right) \left( 1+cos\frac { 3\pi }{ 8 } \right) \left( 1+cos\frac { 5\pi }{ 8 } \right) \left( 1+cos\frac { 7\pi }{ 8 } \right) \)=

    (a)

    \(\frac { 1 }{ 8 } \)

    (b)

    \(\frac { 1 }{ 2 } \)

    (c)

    \(\frac { 1 }{ \sqrt { 3 } } \)

    (d)

    \(\frac { 1 }{ \sqrt { 2 } } \)

  5. Which of the following is incorrect?

    (a)

    sinx=\(\frac { -1 }{ 5 } \)

    (b)

    cosx=1

    (c)

    secx=\(\frac { 1 }{ 2 } \)

    (d)

    tanx=20

  6. The number of rectangles that a chessboard has

    (a)

    81

    (b)

    99

    (c)

    1296

    (d)

    6561

  7. 5c1+5c2+5c3+5c4+5c5 is equal to

    (a)

    30

    (b)

    31

    (c)

    32

    (d)

    33

  8. The value of 2 + 4 + 6 + + 2n is

    (a)

    \(\frac { n\left( n-1 \right) }{ 2 } \)

    (b)

    \(\frac { n\left( n+1 \right) }{ 2 } \)

    (c)

    \(\frac { 2n\left( 2n-1 \right) }{ 2 } \)

    (d)

    n(n + 1)

  9. Find the nearest point on the line 3x + y = 10 from the origin is:

    (a)

    (2, 1)

    (b)

    (1, 2)

    (c)

    (3, 1)

    (d)

    (1,3)

  10. The points (a,0),(0,b) and (1,1) will be collinear if

    (a)

    a+b=1

    (b)

    a+b=2

    (c)

    \(\frac{1}{a}+\frac{1}{b}=1\)

    (d)

    a+b=0

  11. The matrix A satisfying the equation \(\begin{bmatrix} 1 & 3 \\ 0 & 1 \end{bmatrix}\)A=\(\begin{bmatrix} 1 & 1 \\ 0 & -1 \end{bmatrix}\)is

    (a)

    \(\begin{bmatrix} 1 & 4 \\ -1 & 0 \end{bmatrix}\)

    (b)

    \(\begin{bmatrix} 1 & -4 \\ 1 & 0 \end{bmatrix}\)

    (c)

    \(\begin{bmatrix} 1 & 4 \\ 0 & -1 \end{bmatrix}\)

    (d)

    \(\begin{bmatrix} 1 & -4 \\ 1 & 1 \end{bmatrix}\)

  12. A matrix which is not a square matrix is called a_________matrix.

    (a)

    singular

    (b)

    non-singular

    (c)

    non-square

    (d)

    rectangular

  13. The value of the expression \({ \left| \overrightarrow { a } \times \overrightarrow { b } \right| }^{ 2 }+{ (\overrightarrow { a } .\overrightarrow { b } ) }^{ 2 }\) is

    (a)

    cos2 \(\theta\)

    (b)

    sin2 \(\theta\)

    (c)

    \({ |\overrightarrow { a } | }^{ 2 }|{ \overrightarrow { b } | }^{ 2 }\)

    (d)

    \({ (|\overrightarrow { a } |+|\overrightarrow { b } |) }^{ 2 }\)

  14. \(lim_{x \rightarrow 3}\left\lfloor x \right\rfloor =\)

    (a)

    2

    (b)

    3

    (c)

    does not exist

    (d)

    0

  15. The points of discontinuity of the function \(\frac { { x }^{ 2 }+6x+8\quad }{ { x }^{ 2 }-5x+6\quad } is\)

    (a)

    3,2

    (b)

    3,-2

    (c)

    -3,2

    (d)

    -3,-2

  16. If  ,then which one of the following is true?

    (a)

    f(x)is not differentiable at x = a

    (b)

    f(x) is discontinuous at x = a

    (c)

    f(x) is continuous for all x in R

    (d)

    f(x) is differentiable for all x\(\ge\)a

  17. \(\int sin \sqrt{xdx}\) is

    (a)

    \(2(-\sqrt{x}cos\sqrt{x}+sin\sqrt{x})+c\)

    (b)

    \(2(-\sqrt{x}cos\sqrt{x}-sin\sqrt{x})+c\)

    (c)

    \(2(-\sqrt{x}sin\sqrt{x}-cos\sqrt{x})+c\)

    (d)

    \(2(-\sqrt{x}sin\sqrt{x}+cos\sqrt{x})+c\)

  18. \(\int { \frac { sin\sqrt { x } }{ x } } \) dx = ________ +c.

    (a)

    2 cos \(\sqrt { x } \)

    (b)

    2 sin \(\sqrt { x } \)

    (c)

    -2 sin \(\sqrt { x } \)

    (d)

    -2 cos \(\sqrt { x } \)

  19. There are three events A, B, and C of which one and only one can happen. If the odds are 7 to 4 against A and 5 to 3 against B, then odds against C is

    (a)

    23: 65

    (b)

    65: 23

    (c)

    23: 88

    (d)

    88: 23

  20. A box contains 10 good articles and 6 with defects. One item is drawn at random. The probability that it is either good or has a defect is

    (a)

    \(\frac { 64 }{ 64 } \)

    (b)

    \(\frac { 49 }{ 64 } \)

    (c)

    \(\frac { 40 }{ 64 } \)

    (d)

    \(\frac { 24 }{ 64 } \)

  21. II. Answer any 7 questions. Q.no 30 is compulsory: 
    7 x 2 = 14
  22. Let C be the set of all circles in a plane and define a circle C is related to a circle C', if the radius of C is equal to the radius of C'

  23. Prove that \(\sin { \left( \pi +\theta \right) } =-\sin { \theta } \)

  24. How many 3-digit even numbers can be made using the digits 1,2,3,4,6,7 if no digit is repeated?

  25. In the binomial expansion of (1+a)m+n,Prove  that the coefficients of am and an are equal.

  26. For what value of k does the equation 12x2 + 2kxy + 2y2 + 11x - 5y + 2 = 0 represent two straight lines.

  27. Find \(\overrightarrow{a}\).\(\overrightarrow{b}\)when \(\overrightarrow{a}=2\hat{i}+2\hat{j}-\hat{k}\) and \(\overrightarrow{b}=6\hat{i}-3\hat{j}+2\hat{k}\)

  28. Evaluate\(\lim _{ x\rightarrow 0 }{ \frac { { x }^{ \frac { 2 }{ 3 } }-9 }{ x-27 } } \)

  29. Show that the greatest integer function \(f(x)=\left\lfloor x \right\rfloor \) is not differentiable at any integer?

  30. Integrate the following with respect to x:\((x+4)^5+{5\over (2-5x)^4}-cosec^2(3x-1)\)

  31. The ratio of the number of boys to the number of girls in a class is 1:2. It is known that the probability of a girl and a boy getting a first class are 0.25 and 0.28 respectively. Find the probability that a student chosen  at random will get first class?

  32. III. Answer any 7 questions. Q.no 40 is compulsory: 

    7 x 3 = 21
  33. Check whether the following functions are one-to-one and onto.
    \(f:N\cup{-1,0}\rightarrow N\) defined by f(n) = n + 2.

  34. Given log 2 = 0.310, find the position of the first significant digit in the value of (0.5)10.

  35. Solve the following equations sin θ + cos θ = \(\sqrt2\)

  36. If the mth term of a H.P. is n and nth term is m, then show that its pth  term is \(\frac{mn}{p}\).

  37. Find the equation of the straight line upon which the length of perpendicular from origin is \(3\sqrt{2}\) units and this perpendicular makes an angle of 75° with the positive direction of x-axis.

  38. Prove that \(\begin{vmatrix} 1 &x &x \\ x & 1 &x \\ x &x &1 \end{vmatrix}^2=\begin{vmatrix}1-2x^2 & -x^2 &-x^2 \\ -x^2 &-1 &x^2-2x \\ -x^2 &x^2-2x &-1 \end{vmatrix}\)

  39. Evaluate the following limits :
    \(lim_{x-1}{3\sqrt{7+x^3}-\sqrt{3+x^2}\over x-1}\)

  40. If \({ x }^{ 2 }+2xy+{ y }^{ 3 }=42,\quad find\quad \frac { dy }{ dx } \)

  41. Find the integrals of the following :\(1\over \sqrt{x^2+4x+2}\)

  42. The chances of A, B, and C becoming manager of a certain company are 5 : 3: 2. The probabilities that the office canteen will be improved if A, B, and C become managers are 0.4, 0.5 and 0.3 respectively. If the office canteen has been improved, what is the probability that B was appointed as the manager?

  43. IV. Answer all in detail:
    5 x 5 = 25
    1. If f:R \(\rightarrow\) R is defined by f(x) = 3x - 5, prove that f is a bijection and find its inverse.

    2. \(\left| x-\frac { 1 }{ 4 } \right| <\left| \frac { 1 }{ 2 } x-\frac { 3 }{ 4 } \right| \)

    1. Using the Mathematical induction, show that for any integer
      n\(\ge\) 2, 3n2 > (n + 1)2

    2. Compute the sum of first n terms of 1 + (1 + 4) + (1 + 4 + 42) + (1 + 4 + 42 + 43) + ...

    1. Find the length of the perpendicular and the coordinates of the foot of the perpendicular form (-10, -2) to the line x + y - 2 = 0

    2. Using cofactors of elements of second row, evaluate | A |, where A=\(\begin{bmatrix} 5 & 3 &8 \\ 2 & 0 & 1 \\1 &2 &3 \end{bmatrix}\)

    1. Find the value or values of m for which m (\(\hat{i}+\hat{j}+\hat{k}\)) is a unit vector

    2. \(If\quad x=4{ z }^{ 2 }+5,y=6{ z }^{ 2 }+7z+3,\quad find\quad \frac { d^{ 2 }y }{ dx^{ 2 } } \)

    1. Integrate the following functions with respect to x:\({1\over 1+36x^2}\)

    2. An insurance company insured 2000 scooter drivers, 4000 car drivers and 6000 truck drivers. The probability of an accident are 0.01, 0.03 and 0.15 respectively. One of the insured person meets with an accident. What is the probability that he is a scooter driver?

    1. Prove that \(sin\frac { \theta }{ 2 } sin\frac { 7\theta }{ 2 } +sin\frac { 3\theta }{ 2 } sin\frac { 11\theta }{ 2 } =sin2\theta sin5\theta \)

    2. Evaluate \(\int { \frac { 1 }{ { x }^{ \frac { 1 }{ 2 } }+{ x }^{ \frac { 1 }{ 3 } } } } \)dx

    1. Which of the following functions f has a removable discontinuity at x = x0? If the discontinuity is removable, find a function g that agrees with f for x ≠ x0 and is continuous on R
      \(f(x)={x^2-2x-8\over x+2},x_o=-2\)

    2. show that \(\left| \begin{matrix} x & y & z \\ { x }^{ 2 } & { y }^{ 2 } & { z }^{ 2 } \\ { x }^{ 3 } & { y }^{ 3 } & { z }^{ 3 } \end{matrix} \right| \)=xyz(x-y)(y-z)(z-x)

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