Half Yearly Model Question Paper 2019

11th Standard

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Maths

Time : 02:30:00 Hrs
Total Marks : 90

    Part A

    Answer all the questions.

    Choose the most suitable answer from the given four alternatives and write the option code with the corresponding answer.

    20 x 1 = 20
  1. The relation R defined on a set A= {0,-1, 1, 2} by xRy if |x2+y2| ≤ 2, then which one of the following is true?

    (a)

    R = {(0,0), (0,-1), (0, 1), (-1, 0), (-1, 1), (1, 2), (1, 0)}

    (b)

    R-1 = {(0,0), (0,-1), (0, 1), (-1, 0), (1, 0)}

    (c)

    Domain of R is {0,-1, 1, 2}

    (d)

    Range of R is {0,-1, 1}

  2. Domain of the function \(y={x-1\over x+1}\) is:

    (a)

    1R

    (b)

    Q

    (c)

    R-(-1)

    (d)

    R-1

  3. If \({ log }_{ \sqrt { x } }\) 0.25 =4 ,then the value of x is

    (a)

    0.5

    (b)

    2.5

    (c)

    1.5

    (d)

    1.25

  4. \(\sqrt [ 4 ]{ { \left( -2 \right) }^{ 4 } } \times { \left( -1000 \right) }^{ \frac { 1 }{ 3 } }\)is

    (a)

    20

    (b)

    -20

    (c)

    2-10

    (d)

    100

  5. Which of the following is not true?

    (a)

    sinፀ=-\(\frac { 3 }{ 4 } \)​​​​​​​

    (b)

    cosፀ=-1

    (c)

    tanፀ=25

    (d)

    secፀ=\(\frac { 1 }{ 4 } \)

  6. If A,B,C are in A.P and B=\(\frac{\pi}{4}\) then tanA tanB tan C=

    (a)

    1

    (b)

    -1

    (c)

    0

    (d)

    None

  7. If 10 lines are drawn in a plane such that no two of them are parallel and no three are concurrent, then the total number of points of intersection are

    (a)

    45

    (b)

    40

    (c)

    10!

    (d)

    210

  8. is:

    (a)

    \(\lfloor{n}(n+2)\)

    (b)

    (c)

    (d)

    none of these

  9. \(\frac{1}{1!}+\frac{1}{3!}+\frac{1}{5!}+...\)is:

    (a)

    \(\frac{e^{-1}}{2}\)

    (b)

    \(\frac{e+e^{-1}}{2}\)

    (c)

    \(\frac{e-e^{-1}}{2}\)

    (d)

    none of these

  10. The locus of a point which moves such that it maintains equal distance from the fixed point is a

    (a)

    straight line

    (b)

    line bisector

    (c)

    circle

    (d)

    angle bisector

  11. The image of the point (1,2) with respect to the line y=x is

    (a)

    (-1,-2)

    (b)

    (2,1)

    (c)

    (2,-1)

    (d)

    (2,1)

  12. What must be the matrix X, if 2x+\(\begin{bmatrix} 1& 2 \\ 3 & 4 \end{bmatrix}=\begin{bmatrix} 3 & 8 \\ 7 & 2 \end{bmatrix}?\)

    (a)

    \(\begin{bmatrix} 1& 3 \\ 2 &-1 \end{bmatrix}\)

    (b)

    \(\begin{bmatrix} 1& -3 \\ 2 &-1 \end{bmatrix}\)

    (c)

    \(\begin{bmatrix} 2& 6 \\ 4 &-2 \end{bmatrix}\)

    (d)

    \(\begin{bmatrix} 2& -6 \\ 4 &-2 \end{bmatrix}\)

  13. The value of x, for which the matrix A=\(\begin{bmatrix} e^{x-2}& e^{7+x} \\ e^{2+x} & e^{2x+3} \end{bmatrix}\) is singular is

    (a)

    9

    (b)

    8

    (c)

    7

    (d)

    6

  14. If \(\begin{bmatrix} 4 & 3 \\ -2 & x \end{bmatrix}\) is singular then the value of x is 

    (a)

    \(\frac{3}{2}\)

    (b)

    -\(\frac{3}{2}\)

    (c)

    3

    (d)

    -2

  15. If \(|\overrightarrow { a } |=|\overrightarrow { b } |\) then

    (a)

    \(\overrightarrow { a } =\overrightarrow { b } \)

    (b)

    \(\overrightarrow { a } =\overrightarrow { -b } \)

    (c)

    \(\overrightarrow { a } =\pm \overrightarrow { b } \)

    (d)

    both are null vectors

  16. Assertion (A) : \(\overset { \rightarrow }{ a } ,\overset { \rightarrow }{ b } ,\overset { \rightarrow }{ c } \) are the position vector three collinear points then 2 \(\overset { \rightarrow }{ a }=\overset { \rightarrow }{ b } +\overset { \rightarrow }{ c } \)
    Reason (R): Collinear points, have same direction

    (a)

    Both A and R are true and R is the correct explanation of A

    (b)

    Both A and R are true and R is not a correct explantion of A

    (c)

    A is true but R is false

    (d)

    A is false but R is true

  17. \(lim_{\theta\rightarrow0}{Sin\sqrt{\theta}\over \sqrt{sin \theta}} \)

    (a)

    1

    (b)

    -1

    (c)

    0

    (d)

    2

  18. The rate of change of area A of a circle of radius r is

    (a)

    \(2\pi r\)

    (b)

    \(2\pi r\frac { dr }{ dt } \)

    (c)

    \(\pi { r }^{ 2 }\frac { dr }{ dt } \)

    (d)

    \(\pi \frac { dr }{ dt } \)

  19. If ,then f '(2) is 

    (a)

    0

    (b)

    1

    (c)

    2

    (d)

    does not exist

  20.  Choose the correct or the most suitable answer from the given four alternatives.
     \(If\quad y=\sin ^{ -1 }{ x } +\cos ^{ -1 }{ x } \) then\(\frac { dy }{ dx } \) is

    (a)

    1

    (b)

    \(\pi \)

    (c)

    \(\frac { \pi }{ 2 } \)

    (d)

    0

  21. Part B

    Answer any 7 questions. Question no. 30 is compulsory.

    7 x 2 = 14
  22. Let A={1,2,3,4} and B = {a,b,c,d}. Give a function from A\(\rightarrow\)B for each of the following:
    neither one- to -one and nor onto.

  23. Given log216 =4.Find log162

  24. For given Angle, find a coterminal angle with a measure \(\theta\) of such that \(0\le \theta \le 360°\) 
    3950 

  25. In how many ways can the letters of the word PENCIL be arranged so that N is always next to E.

  26. Write the first 6 terms of the exponential series \({ e }^{ \frac { 1 }{ 2 } x }\)

  27. Find the 5th term in the sequence whose first three terms are 3, 3, 6 and each term after the second is the sum of the two terms preceding it.

  28. Find |A| if A=\(\begin{bmatrix} 0& sin \alpha &cos \alpha \\ sin \alpha & 0 & sin \beta \\ cos \alpha & -sin\beta & 0 \end{bmatrix}\) .

  29. Find the value of x such that [1 x 1]\(\left[ \begin{matrix} 1 & 3 & 2 \\ 2 & 5 & 1 \\ 15 & 3 & 2 \end{matrix} \right] \left[ \begin{matrix} 1 \\ 2 \\ x \end{matrix} \right] =0\)

  30. Evaluate: \(\underset { x\rightarrow 0 }{ lim } \frac { { e }^{ 5x }-1 }{ x } \)

  31. Differentiate \(\sin { (\sqrt { 3 } \sin { x } +\cos { x } ) } \) with respect to x.

  32. Part C

    Answer any 7 questions. Question no. 40 is compulsory.

    7 x 3 = 21
  33. Compare and contrast the graph y = x2 - 1, y = 4(x2 - 1) and y = (4x)2 = 1.

  34. Find the zeros of the polynomial function f(x)=9x2-36

  35. In \(\triangle\)ABC, Prove the following
    \(\frac { a+b }{ a-b } =tan\left( \frac { A+B }{ 2 } \right) cot\left( \frac { A-B }{ 2 } \right) \)

  36. If p(h) is the statement "n2 + n is even" and if p(r) is true, then p(r + 1) is true.

  37. Prove that \(\sqrt [ 3 ]{ { x }^{ 3 }+6 } -\sqrt [ 3 ]{ { x }^{ 3 }+3 } \) is approximately equal to \(\frac { 1 }{ { x }^{ 2 } } \) when x is sufficiently large.

  38. Find the equation of the line passing through the point (5, 2) and perpendicular to the line joining the points (2, 3) and (3, -1).

  39. Prove that\(\left| \begin{matrix} -2a & a+b & a+c \\ b+a & -2b & b+c \\ c+a & c+b & -2c \end{matrix} \right| \) =4(a+b)(b+c)(c+a). Using factor theorem.

  40. Let \(\overrightarrow { a } ,\overrightarrow { b } \) and \(\overrightarrow { c } \)  be non-coplanar vectors. Let A, B and C be the points whose position vectors with respect to the origin O are \(\overrightarrow { a } +2\overrightarrow { b } +3\overrightarrow { c } ,-2\overrightarrow { a } +3\overrightarrow { b } +5\overrightarrow { c } \) and \(7\overrightarrow { a } -\overrightarrow { c } \)  respectively. Then prove that A, B and C are collinear.

  41. Examine the continuity of \(f\left( x \right) =\begin{cases} \frac { \sin { 2x } }{ \sin { 3x } } \quad if\quad x\neq 0 \\ 2\quad \quad \quad if\quad x=0 \end{cases}at\quad x=0\)

  42. If xy = 4, Prove that \(x\left( \frac { dy }{ dx } +{ y }^{ 2 } \right) =3y.\)

  43. Part D

    Answer all the questions.

    7 x 5 = 35
    1. Show that the statement, "if f and gof are one-to-one, then g is one-to-one" is not true.

    2. Resolve the following rational expressions into partial fractions.
      \({{1}\over{x^4-1}}\)

    1. Find the derivative of the function with respect to corresponding independent variable: y = x sin x cos x

    2. Differentiate \({ tan }^{ -1 }(secx+tanx),\) \(-\frac{\pi}{ 2 }<x<\frac{\pi}{2}\) with respect to 'x'.

    1. Find the equation of the straight lines passing through (8, 3) and having intercepts whose sum is 1.

    2. If \(\begin{bmatrix} 0 & p& 3 \\ 2 & q^2 & -1 \\ r & 1 & 0 \end{bmatrix}\) is skew-symmetric, find the values of p,q, and r.

    1. For any natural number n, 7n - 2n is divisible by 5.

    2. If a, b, c are respectively the pth qth and rth terms of a GP. show that (q - r) log a + (r - p) log b + (p - q) log c = 0.

    1. The position vectors of the points P, Q, R, S are \(\hat{i}\) + \(\hat{j}\) + \(\hat{k}\),2 \(\hat{i}\)+5\(\hat{j}\),3\(\hat{i}\)+2\(\hat{j}\)-3\(\hat{k}\), and \(\hat{i}\)-6\(\hat{j}\)-\(\hat{k}\)respectively. Prove that the line PQ and RS are parallel.

    2. Determine k, so that  \(f\left( x \right) =\begin{cases} k{ x }^{ 2 },\quad x\le 2 \\ 3,\quad x>2 \end{cases}\) is continuous.

    1. Using the mathematical induction, show that for any natural number n > 2
      \({1\over 1+2}+{1\over 1+2+3}+{1\over 1+2+3+4}+...+{1\over 1+2+3..+n}={n-1\over n+1}\)

    2. If 22pr+1:20pr+2 = 11 : 52, find r.

    1. Resolve into partial fractions \(\frac { 9 }{ (x-1)(x+2)^{ 2 } } \)

    2. Suppose two radar stations located 100 km apart, each detect a fighter aircraft between them. The angle of elevation measured by the first station is 30°, whereas the angle of elevation measured by the second station is 45°. Find the altitude of the aircraft at that instant.

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