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

12th Standard

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Maths

Time : 02:40:00 Hrs
Total Marks : 70

      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 = \(\left[ \begin{matrix} \frac { 3 }{ 5 } & \frac { 4 }{ 5 } \\ x & \frac { 3 }{ 5 } \end{matrix} \right] \) and AT = A−1 , then the value of x is

    (a)

    \(\frac { -4 }{ 5 } \)

    (b)

    \(\frac { -3 }{ 5 } \)

    (c)

    \(\frac { 3 }{ 5 } \)

    (d)

    \(\frac { 4 }{ 5 } \)

  2. Which of the following is not an elementary transformation?

    (a)

    Ri ↔️ Rj

    (b)

    Ri ⟶ 2Ri + Rj

    (c)

    Cj ⟶ Cj + Ci

    (d)

    Ri ⟶ Ri + Cj

  3. \(\frac { 1+e^{ -i\theta } }{ 1+{ e }^{ i\theta } } \) =

    (a)

    cosθ + i sinθ

    (b)

    cosθ - i sinθ

    (c)

    sinθ - i cosθ

    (d)

    sinθ + icosθ

  4. The number of positive zeros of the polynomial \(\overset { n }{ \underset { j=0 }{ \Sigma } } { n }_{ C_{ r } }\)(-1)rxr is

    (a)

    0

    (b)

    n

    (c)

    < n

    (d)

    r

  5. Ifj(x) = 0 has n roots, thenf'(x) = 0 has __________ roots

    (a)

    n

    (b)

    n -1

    (c)

    n+1

    (d)

    (n-r)

  6. The value of sin-1 (cos x),0\(\le x\le\pi\) is

    (a)

    \(\pi-x\)

    (b)

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

    (c)

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

    (d)

    \(\pi-x\)

  7. If tan-1(3)+tan-1(x)=tan-1(8)then x= 

    (a)

    5

    (b)

    \(\cfrac { 1 }{ 5 } \)

    (c)

    \(\cfrac { 5 }{ 14 } \)

    (d)

    \(\cfrac { 14 }{ 5 } \)

  8. Let C be the circle with centre at(1,1) and radius =1. If T is the circle centered at(0, y)
    passing through the origin and touching the circleC externally, then the radius of T is equal to

    (a)

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

    (b)

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

    (c)

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

    (d)

    \(\frac { 1 }{ 4 } \)

  9. If a vector \(\vec { \alpha } \) lies in the plane of \(\vec { \beta } \) and \(\vec { \gamma } \) , then

    (a)

    \([\vec { \alpha } ,\vec { \beta } ,\vec { \gamma } ]\) = 1

    (b)

    \([\vec { \alpha } ,\vec { \beta } ,\vec { \gamma } ]\)= -1

    (c)

    \([\vec { \alpha } ,\vec { \beta } ,\vec { \gamma } ]\)= 0

    (d)

    \([\vec { \alpha } ,\vec { \beta } ,\vec { \gamma } ]\)= 2

  10. The number given by the Rolle's theorem for the functlon x3-3x2, x∈[0,3] is

    (a)

    1

    (b)

    \(\\ \\ \\ \sqrt { 2 } \)

    (c)

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

    (d)

    2

  11. If the rate of increase of s =x3-5x2+5x+8 is twice the rate of increase of x, then one value of x is

    (a)

    \(\frac{3}{5}\)

    (b)

    \(\frac{10}{3}\)

    (c)

    \(\frac{3}{10}\)

    (d)

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

  12. A circular template has a radius of 10 cm. The measurement of radius has an approximate error of 0.02 cm. Then the percentage error in calculating area of this template is

    (a)

    0.2%

    (b)

    0.4%

    (c)

    0.04%

    (d)

    0.08%

  13. If u = yx then \(\frac { \partial u }{ \partial y } \) = ............

    (a)

    xyx-1

    (b)

    yxy-1

    (c)

    0

    (d)

    1

  14. The value of \(\int _{ 0 }^{ \frac { 2 }{ 3 } }{ \frac { dx }{ \sqrt { 4-9{ x }^{ 2 } } } } \) is 

    (a)

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

    (b)

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

    (c)

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

    (d)

    \({\pi}\)

  15. \(\int _{ 0 }^{ \frac { \pi }{ 4 } }{ { cos }^{ 3 }2x \ dx= } \)

    (a)

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

    (b)

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

    (c)

    0

    (d)

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

  16. The differential equation representing the family of curves y = Acos(x + B), where A and B are parameters, is

    (a)

    \(\frac { { d }^{ 2 }y }{ { dx }^{ 2 } } -y=0\)

    (b)

    \(\frac { { d }^{ 2 }y }{ { dx }^{ 2 } }+y=0\)

    (c)

    \(\frac { { d }^{ 2 }y }{ { dx }^{ 2 } }=0\)

    (d)

    \(\frac { { d }^{ 2 }x }{ { dy }^{ 2 } }=0\)

  17. The solution of \(\frac{dy}{dx}+y\) cot x=sin 2x is

    (a)

    y sin x=\(\frac{2}{3}\)sin3x+c

    (b)

    y sec x=\(\frac{x^2}{2}+c\)

    (c)

    y sin x =c+x

    (d)

    2y sin x=sin x-\(\frac{sin\ 3x}{3}+c\)

  18. The random variable X has the probability density function \(f(x)=\left\{\begin{array}{ll} a x+b & 0<x<1 \\ 0 & \text { otherwise } \end{array}\right.\) and \(E(X)=\cfrac { 7 }{ 12 } \) then a and b are respectively.

    (a)

    1 and \(\cfrac { 1 }{ 2 } \)

    (b)

    \(\cfrac { 1 }{ 2 } \) and 1

    (c)

    2 and 1

    (d)

    1 and 2

  19. Subtraction is not a binary operation in

    (a)

    R

    (b)

    Z

    (c)

    N

    (d)

    Q

  20. Which of the following is a contradiction?

    (a)

    p v q

    (b)

    p ∧ q

    (c)

    q v ~ q

    (d)

    q ∧ ~ q

    1. Part II

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


    7x 2 = 14
  21. Find the rank of the matrix A =\(\left[ \begin{matrix} 4 \\ 7 \end{matrix}\begin{matrix} 5 \\ -3 \end{matrix}\begin{matrix} -6 \\ 0 \end{matrix}\begin{matrix} 1 \\ 8 \end{matrix} \right] \).

  22. Find th Int rval for a for which 3x2+2(a2+1) x+(a2-3n+2) possesses roots of opposite sign.

  23. If a parabolic reflector is 24 cm in diameter and 6 cm deep, find its locus.

  24. Determine whether the three vectors \(2\hat { i } +3\hat { j } +\hat { k } \)\(\hat { i } -2\hat { j } +2\hat { k } \) and \(\hat { 3i } +\hat { j } +2\hat { k } \) are coplanar.

  25. Verify Rolle ’s Theorem for \(f(x)=\left| x-1 \right| ,O\le x\le 2\) 

  26. Let us assume that the shape of a soap bubble is a sphere. Use linear approximation to approximate the increase in the surface area of a soap bubble as its radius increases from 5 cm to 5.2 cm. Also, calculate the percentage error.

  27. Find the area of the region enclosed by the curve y = \(\sqrt x\) + 1, the axis of x and the lines x=0, x=4.

  28. Determine the order and degree (if exists) of the following differential equations: 
    \(3\left( \frac { { d }^{ 2 }y }{ { dx }^{ 2 } } \right) ={ \left[ 4+{ \left( \frac { dy }{ dx } \right) }^{ 2 } \right] }^{ \frac { 3 }{ 2 } }\)

  29. Find the binomial distribution function for each of the following.
    (i) Five fair coins are tossed once and X denotes the number of heads.
    (ii) A fair die is rolled 10 times and X denotes the number of times 4 appeared.

  30. Form the truth table of (~q)^p.

  31. Obtain the Cartesian form of the locus of z in
    |2z-3-i|=3

    1. Part III

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


    7 x 3 = 21
  32. Solve: x + y + 3z = 4, 2x + 2y + 6z = 7, 2x + y +  z = 10.

  33. Find the condition that the roots of x3+ax2+bx+c = 0 are in the ratio p:q:r.

  34. Find the number of positive integral solutions of (pairs of positive integers satisfying) x2 - y2 =353702.

  35. If \(sin\left( { sin }^{ -1 }\cfrac { 1 }{ 5 } +{ cos }^{ -1 }x \right) =1\) then find the value ofx.

  36. Find the equation of the ellipse with foci (±2,0) , vertices (±3,0) .

  37. Dot product of a vector with vector \(\overset { \wedge }{ 3i } -5\overset { \wedge }{ k } \)\(2\overset { \wedge }{ i } +7\overset { \wedge }{ j } \) and \(\overset { \wedge }{ i } +\overset { \wedge }{ j } +\overset { \wedge }{ k } \) are respectively -1, 6 and 5. Find the vector.

    ()

    \(\overset { \rightarrow }{ b } \)\(\overset { \rightarrow }{ d } \)

  38. Write the Taylor series expansion of \(\frac{1}{x}\) about x = 2 by finding the first three non-zero terms.

  39. Find the area bounded by the curves y=x3,y=x2 and the ordinates x=1 and x=2.

  40. If in a pair (S,*) where S is a nonemty set and * is a binary operation defined on S as a2=e for all a દ s, then that * is commutative given that e is the identify element.

    1. Part IV

      Answer all the questions.


    7 x 5 = 35
    1. The sum of three numbers is 20. If we multiply the third number by 2 and add the first number to the result we get 23. By adding second and third numbers to 3 times the first number we get 46. Find the numbers using Cramer's rule.

    2. Show that \(\left( \frac { i+\sqrt { 3 } }{ -i+\sqrt { 3 } } \right) ^{ 2\omega }+\left( \frac { i-\sqrt { 3 } }{ i+\sqrt { 3 } } \right) ^{ 2\omega }\)=-1

    1. Solve:
      (x-5)(x-7)(x+6)(x+4)=504

    2. Find the principal value of cos−1\(\left( \frac { \sqrt { 3 } }{ 2 } \right) \)

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

    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. Find the shortest distance between the following pairs of lines \(\frac { x-3 }{ 3 } =\frac { y-8 }{ -1 } =\frac { z-3 }{ 1 } \)and \(\frac { x+3 }{ -3 } =\frac { y+7 }{ 2 } =\frac { z-6 }{ 4 } \) 

    2. Find the asymptotes of the following curve \(f(x)=\cfrac { { x }^{ 2 } }{ { x }^{ 2 }-1 } \)

    1. For each of the following functions find the fx, fy, and show that fxy =fyx
      f(x,y) = \(\frac { 3x }{ y+sinx \ } \) 

    2. 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.

    1. Solve the Linear differential equation:
      \(\frac { dy }{ dx } +\frac { 3y }{ x } =\frac { 1 }{ { x }^{ 2 } } \), given that y=2 when x=1 

    2. The slope of the tangent at p(x,y) on the curve is -\(\left( \frac { y+3 }{ x+2 } \right) \). If the curve passes through the origin, find the equation of the curve.

    1. On the average, 20% of the products manufactured by ABC Company are found to be defective. If we select 6 of these products at random and X denote the number of defective products find the probability that (i) two products are defective (ii) at most one product is defective (iii) at least two products are defective.

    2. Verify (p ∧ -p) ∧ (~q ∧ p) is a tautlogy, contradiction or contingency.

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