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12th Standard English Medium Maths Reduced Syllabus Public Exam Model Question Paper With Answer Key - 2021

12th Standard

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Maths

Time : 02:40:00 Hrs
Total Marks : 90

      Part I

      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. 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 A is a non-singular matrix such that A-1 = \(\left[ \begin{matrix} 5 & 3 \\ -2 & -1 \end{matrix} \right] \), then (AT)−1 =

    (a)

    \(\left[ \begin{matrix} -5 & 3 \\ 2 & 1 \end{matrix} \right] \)

    (b)

    \(\left[ \begin{matrix} 5 & 3 \\ -2 & -1 \end{matrix} \right] \)

    (c)

    \(\left[ \begin{matrix} -1 & -3 \\ 2 & 5 \end{matrix} \right] \)

    (d)

    \(\left[ \begin{matrix} 5 & -2 \\ 3 & -1 \end{matrix} \right] \)

  3. The solution of the equation |z|-z=1+2i is

    (a)

    \(\cfrac { 3 }{ 2 } -2i\)

    (b)

    \(-\cfrac { 3 }{ 2 } +2i\)

    (c)

    \(2-\cfrac { 3 }{ 2 } i\)

    (d)

    \(2+\cfrac { 3 }{ 2 } i\)

  4. If \(\omega \neq 1\) is a cubic root of unity and \(\left| \begin{matrix} 1 & 1 & 1 \\ 1 & { -\omega }^{ 2 } & { \omega }^{ 2 } \\ 1 & { \omega }^{ 2 } & { \omega }^{ 2 } \end{matrix} \right| \) =3k, then k is equal to 

    (a)

    1

    (b)

    -1

    (c)

    \(\sqrt { 3i } \)

    (d)

    \(-\sqrt { 3i } \)

  5. If sin1x = 2sin−1 \(\alpha\) has a solution, then

    (a)

    \(|\alpha |\le \frac { 1 }{ \sqrt { 2 } } \)

    (b)

    \(|\alpha |\ge \frac { 1 }{ \sqrt { 2 } } \)

    (c)

    \(|\alpha |<\frac { 1 }{ \sqrt { 2 } } \)

    (d)

    \(|\alpha |>\frac { 1 }{ \sqrt { 2 } } \)

  6. If \(\\ \\ \\ { cot }^{ -1 }\left( \sqrt { sin\alpha } \right) +{ tan }^{ -1 }\left( \sqrt { sin\alpha } \right) =u\), then cos2u is equal to

    (a)

    tan2\(\alpha\)

    (b)

    0

    (c)

    -1

    (d)

    tan2\(\alpha\)

  7. Tangents are drawn to the hyperbola  \(\frac { { x }^{ 2 } }{ 9 } +\frac { { y }^{ 2 } }{ 4 } =1\) 1parallel to the straight line2x−y=1. One of the points of contact of tangents on the hyperbola is

    (a)

    \(\frac { 9 }{ 2\sqrt { 2 } } ,\frac { -1 }{ \sqrt { 2 } } \)

    (b)

    \(\frac { -9 }{ 2\sqrt { 2 } } ,\frac { 1 }{ \sqrt { 2 } } \)

    (c)

    \(\frac { 9 }{ 2\sqrt { 2 } } ,\frac { 1 }{ \sqrt { 2 } } \)

    (d)

    \(\left( 3\sqrt { 3 } ,-2\sqrt { 2 } \right) \)

  8. If \(\vec { a } \) and \(\vec { b } \) are unit vectors such that \([\vec { a } ,\vec { b },\vec { a } \times \vec { b } ]=\frac { \pi }{ 4 } \), then the angle between \(\vec { a } \) and \(\vec { b } \) is

    (a)

    \(\frac { \pi }{ 6 } \)

    (b)

    \(\frac { \pi }{ 4 } \)

    (c)

    \(\frac { \pi }{ 3 } \)

    (d)

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

  9. If \(\vec { a } =\hat { i } +\hat { j } +\hat { k } \)\(\vec { b } =\hat { i } +\hat { j } \)\(\vec { c } =\hat { i } \) and \((\vec { a } \times \vec { b } )\times\vec { c } \) = \(\lambda \vec { a } +\mu \vec { b } \) then the value of \(\lambda +\mu \) is

    (a)

    0

    (b)

    1

    (c)

    6

    (d)

    3

  10. If the direction cosines of a line are \(\frac { 1 }{ c } ,\frac { 1 }{ c } ,\frac { 1 }{ c } \), then

    (a)

    \(c=\pm 3\)

    (b)

    \(c=\pm \sqrt { 3 } \)

    (c)

    c > 0

    (d)

    0 < c < 1

  11. The number given by the Mean value theorem for the function \(\cfrac { 1 }{ x } \),x∈[1,9] is

    (a)

    2

    (b)

    2.5

    (c)

    3

    (d)

    3.5

  12. Linear approximation for g(x) = cos x at x=\(\frac{-\pi}{2}\) is

    (a)

    x + \(\frac{-\pi}{2}\)

    (b)

    - x + \(\frac{\pi}{2}\)

    (c)

    x - \(\frac{\pi}{2}\)

    (d)

    - x + \(\frac{\pi}{2}\)

  13. The integrating factor of the differential equation \(\frac { dy }{ dx } +y=\frac { 1+y }{ \lambda } \) is

    (a)

    \(\frac { x }{ { e }^{ \lambda } } \)

    (b)

    \(\frac { { e }^{ \lambda } }{ x } \)

    (c)

    \({ \lambda e }^{ x }\)

    (d)

    ex

  14. The number of arbitrary constants in the particular solution of a differential equation of third order is

    (a)

    3

    (b)

    2

    (c)

    1

    (d)

    0

  15. The differential equation of the family of parabolas y2=4ax is

    (a)

    \(2y=x\left( \frac { dy }{ dx } \right) \)

    (b)

    \(y=2x\left( \frac { dy }{ dx } \right) \)

    (c)

    \(y={ 2x }^{ 2 }\left( \frac { dy }{ dx } \right) \)

    (d)

    \({ y }^{ 2 }=2x\left( \frac { dy }{ dx } \right) \)

  16. The I.F of \(\frac{dy}{dx}-y\) tan x=cos x is

    (a)

    sec x

    (b)

    cos x

    (c)

    etan x

    (d)

    cot x

  17. A random variable X has binomial distribution with n = 25 and p = 0.8 then standard deviation of X is

    (a)

    6

    (b)

    4

    (c)

    3

    (d)

    2

  18. In eight throws of a die, 1 or 3 is considered a success. Then the mean number of success is

    (a)

    \(\cfrac { 8 }{ 3 } \)

    (b)

    \(\cfrac { 4 }{ 3 } \)

    (c)

    \(\cfrac { 2 }{ 3 } \)

    (d)

    \(\cfrac { 5 }{ 3 } \)

  19. If X is a continuous random variable then \(p(a<x<b)=\) ________

    (a)

    \(P(a<x<b)\)

    (b)

    \(P(a\le X\le b)\)

    (c)

    \(P\left( X>a \right) \)

    (d)

    \(1-P\left( X\le a-1 \right) \)

  20. If * is defined by a * b = a2 + b2 + ab + 1, then (2 * 3) * 2 is

    (a)

    20

    (b)

    40

    (c)

    400

    (d)

    445

    1. Part II

      Answer any seven questions. Question no. 23 is compulsory.


    7 x 2 = 14
  21. For the matrix A, if A3 = I, then find A-1.

  22. Show that the system of equations is inconsistent. 2x + 5y= 7, 6x + 15y = 13.

  23. If y=4x+c is a tangent to the circle x2+y2=9 , find c .

  24. Verify whether the line \(\frac { x-3 }{ -4 } =\frac { y-4 }{ -7 } =\frac { z+3 }{ 12 } \) lies in the plane 5x-y+z=8.

  25. Using the Rolle’s theorem, determine the values of x at which the tangent is parallel to the x -axis for the following functions:
    \(f(x)=\sqrt{x}-\frac{x}{3}, x\in [0,9]\)

  26. Evaluate the following integrals using properties of integration:
    \(\int _{ -\frac { \pi }{ 4 } }^{ \frac { \pi }{ 4 } }{ { sin }^{ 2 }xdx } \)

  27. A differential equation, determine its order, degree (if exists)
    \(y\left( \frac { dy }{ dx } \right) =\frac { x }{ \left( \frac { dy }{ dx } \right) +{ \left( \frac { dy }{ dx } \right) }^{ 3 } } \)

  28. Show that y = e−x + mx + n is a solution of the differential equation ex \(\left( \frac { { d }^{ 2 }y }{ { dx }^{ 2 } } \right) \) -1 = 0

  29. A random variable X has the following probability mass function.

    x 1 2 3 4 5
    f(x) k2 2k2 3k2 2k 3k
  30. Write the statements in words corresponding to ¬p, p ∧ q , p ∨ q and q ∨ ¬p, where p is ‘It is cold’ and q is ‘It is raining.’

    1. Part III

      Answer any seven questions. Question no. 32 is compulsory.

    7 x 3 = 21
  31. Prove, using mean value theorem, that \(|sin \alpha-sin\beta|\le |\alpha-\beta|, \alpha, \beta \in R\)

  32. A thermometer was taken from a freezer and placed in a boiling water. It took 22 seconds for the thermometer to raise from −10°C to 100°C. Show that the rate of change of temperature at some time t is 5°C per second.

  33. Explain why Lagrange’s mean value theorem is not applicable to the following functions in the respective intervals:
    f(x) = \(\frac { x+1 }{ x } \), x ∈ [-1, 2]

  34. A race car driver is racing at 20th km. If his speed never exceeds 150 km/hr, what is the maximum distance he can cover in the next two hours.

  35. Write the Maclaurin series expansion of the following function
    cos x

  36. Evaluate the following limit, if necessary use l’Hôpital Rule
    \(\underset { x\rightarrow 0 }{ lim } \left( \frac { 1 }{ sinx } -\frac { 1 }{ x } \right) \)

  37. Evaluate the following limit, if necessary use l’Hôpital Rule
    \(\underset { x\rightarrow { 1 }^{ + } }{ lim } \left( \frac { 2 }{ { x }^{ 2 }-1 } -\frac { x }{ x-1 } \right) \)

  38. Find the intervals of monotonicity and hence find the local extrema for the function \(f(x)=x^{\frac{2}{3}}\).

  39. Determine the intervals of concavity of the curve y = 3+ sin x .

  40. Find the slant (oblique) asymptote for the function \(f(x)=\frac { { x }^{ 2 }-6x+7 }{ x+5 } \)

    1. Part IV

      Answer all the questions.


    7 x 5 = 35
    1. If \({ tan }^{ -1 }\left( \cfrac { \sqrt { 1+{ x }^{ 2 } } -\sqrt { 1-{ x }^{ 2 } } }{ \sqrt { 1+{ x }^{ 2 } } +\sqrt { 1-{ x }^{ 2 } } } \right) =a\) than prove that x2=sin2a

    2. A kho-kho player In a practice Ion while running realises that the sum of tne distances from the two kho-kho poles from him is always 8m. Find the equation of the path traced by him of the distance between the poles is 6m.

    1. 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

    2. The length x of a rectangle is decreasing at the rate of 3cm/min and the width is increasing at the rate of 2cm/min. When x = 10cm, y = 6cm, find the rate of change of (i) the perimeter (ii) the area of the rectan

    1. Find the points on the curve y=2x2-2x2 at which the tangent lines are parallel to the line y=3x-2.

    2. Solve the Linear differential equation:
      ​​​​​​​(2x- 10y3) dy +ydx =0

    1. Solve the Linear differential equation:
      \(\left( y-{ e }^{ sin^{ -1 }x } \right) \frac { dx }{ dy } +\sqrt { 1-{ x }^{ 2 } } =0\)

    2. In a murder investigation, a corpse was found by a detective at exactly 8 p.m. Being alert, the detective also measured the body temperature and found it to be 70oF. Two hours later, the detective measured the body temperature again and found it to be 60oF. If the room temperature is 50oF, and assuming that the body temperature of the person before death was 98.6oF, at what time did the murder occur?
      [log(2.43)=0.88789; log(0.5)=-0.69315]

    1. Assume that the rate at which radioactive nuclei decay is proportional to the number of such nuclei that are present in a given sample. In a certain sample 10% of the original number of radioactive nuclei have undergone disintegration in a period of 100 years. What percentage of the original radioactive nuclei will remain after 1000 years?

    2. If X is the random variable with distribution function F(x) given by,
      \(F(x)=\begin{cases} 0,\qquad -\infty <x<0 \\ \frac { 1 }{ 2 } \left( { x }^{ 2 }+x \right) \qquad 0\le x<1\quad \\ 1,\qquad 1\le x<\infty \qquad \end{cases}\)
      then find (i) the probability density function f{x) 
      (ii) P(0.3 ≤ X ≤ 0.6)

    1. A six sided die is marked ‘1’ on one face, ‘2’ on two of its faces, and ‘3’ on remaining three faces. The die is rolled twice. If X denotes the total score in two throws.
      (i) Find the probability mass function.
      (ii) Find the cumulative distribution function.
      (iii) Find P(3 ≤ X< 6) (iv) Find P(X ≥ 4) .

    2. Let X be a random variable denoting the life time of an electrical equipment having probability density function
      \(f(x)=\begin{cases} \begin{matrix} { ke }^{ -2x } & forx>0 \end{matrix} \\ \begin{matrix} 0 & forx\le 0 \end{matrix} \end{cases}\) 
      Find
      (i) the value of k
      (ii) Distribution function 
      (iii) P(X < 2)
      (iv) calculate the probability that X is at least for four unit of time 
      (v) P(X = 3)

    1. Two balls are chosen randomly from an urn containing 8 white and 4 black balls. Suppose that we win Rs 20 for each black ball selected and we lose Rs10 for each white ball selected. Find the expected winning amount and variance 

    2. Verify
      (i) closure property,
      (ii) commutative property,
      (iii) associative property,
      (iv) existence of identity, and
      (v) existence of inverse for the operation +5 on Z5 using table corresponding to addition modulo 5.

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