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12th Standard English Medium Maths Subject Creative 5 Mark Questions with Solution Part - I

12th Standard

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Maths

Time : 01:00:00 Hrs
Total Marks : 124

    5 Marks

    25 x 5 = 125
  1. Solve: \(\frac { 2 }{ x } +\frac { 3 }{ y } +\frac { 10 }{ z } =4,\frac { 4 }{ x } -\frac { 6 }{ y } +\frac { 5 }{ z } =1,\frac { 6 }{ x } +\frac { 9 }{ y } -\frac { 20 }{ z } \) = 2

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

  3. Using Gaussian Jordan method, find the values of λ and μ so that the system of equations 2x - 3y + 5z = 12, 3x + y + λz =μ, x - 7y + 8z = 17 has
    (i) unique solution
    (ii) infinite solutions and
    (iii) no solution.

  4. 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

  5. Verify that arg(1+i) + arg(1-i) = arg[(1+i) (1-i)]

  6. If a, b, c, d and p are distinct non-zero real numbers such that (a2+b2+c2) p2-2 (ab+bc+cd) p+(b2+c2+d2)≤ 0 then prove that a, b, c, d are in G.P and ad = bc

  7. If the equation x2 + bx + ca = 0 and x2 + cx + ab = 0 have a comnion root and b≠c, then prove that their roots will satisfy the equation x2 + ax + bc = 0.

  8. Simplify \({ sin }^{ -1 }\left( \frac { sinx+cosx }{ \sqrt { 2 } } \right) ,\frac { \pi }{ 4 }\) 

  9. Prove that \({ tan }^{ -1 }\left( \frac { 1-x }{ 1+x } \right) -{ tan }^{ -1 }\left( \frac { 1-y }{ 1+y } \right) ={ sin }^{ -1 }\left( \frac { y-x }{ \sqrt { 1+{ x }^{ 2 } } .\sqrt { 1+{ y }^{ 2 } } } \right)\)

  10. An equilateral triangle is inscribed in the parabola y2 = 4ax whose vertex is at the vertex of the parabola. Find the length of its side.

  11. The foci of a hyperbola coincides with the foci of the ellipse \(\frac { { x }^{ 2 } }{ 25 } +\frac { y^{ 2 } }{ 9 } =1\). Find the equation of the hyperbola if its eccentricity is 2.

  12. Show that the points A, B, C with position vector \(2\overset { \wedge }{ i } -\overset { \wedge }{ j } +\overset { \wedge }{ k } ,\overset { \wedge }{ i } -3\overset { \wedge }{ j } -5\overset { \wedge }{ k } \) and \(3\overset { \wedge }{ i } -4\overset { \wedge }{ j } +4\overset { \wedge }{ k } \) respectively are the vector of a right angled, triangle. Also, find the remaining angles of the triangle.

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    points on the plane

  13. 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

  14. Find the vector and Cartesian equation of the plane passing through the point (1,1, -1) and perpendicular to the planes x + 2y + 3z - 7 = 0 and 2x - 3y + 4z = 0

  15. Find the angle of intersection of the curves 2y2 = x3 and y2 = 32x.

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

  17. A water tank has a shape of an inverted cone with its axis vertical and vertex lower most. Its semi vertical angle is tan−1(0.5). Water is poured into it at a constant rate of 5 cm3/hr. Find the rate at which the level of the water is rising at that instant when the depth of the water is 4 m.

  18. If f(x)=alogx+bx2+x has exterme at x=1, x=2 then find a and b.

  19. Find the intervals for which the function f(x)=2x2-9x2-12x+1 is increasing or decfreasing and find the local extermems.

  20. Find the local maximum and local minimum values for f(x)=12x2-2x2-x4.

  21. Find the intervals of concavity and the points of inflection of f(x)=12x2-2x3-x4.

  22. Find the equation of the tangent and the normal to the curve \(y=2sin^{ 2 }3x,x=\frac { \pi }{ 6 } \)

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