1. Find the transformed equation of \(x\cos\theta + y\sin\theta = p\), when the axes are rotated through an angle \(\theta\).

1. 1. \(X = p\)
2. 2. \(Y = p\)
3. 3. \(X + Y = p\)
4. 4. \(X - Y = p\)

2. If the axes are rotated through an angle \(45^{\circ}\), then the co-ordinates of the point \((4\sqrt{2}, -6\sqrt{2})\) in the new system are

1. 1. \((-10, -2)\)
2. 2. \((-2, -10)\)
3. 3. \((10, 10)\)
4. 4. \((-2, 10)\)

3. When the origin is shifted to \((2,3)\) the transformed equation is \(x^{2} + 3xy - 2y^{2} + 17x - 7y - 11 = 0\), then the original equation of the curve is

1. 1. \(x^{2} - 2y^{2} - 3xy + 4x - y + 20 = 0\)
2. 2. \(x^{2} - 2y^{2} + 3xy + 4x - y - 20 = 0\)
3. 3. \(x^{2} - 2y^{2} - 3xy - 4x - y + 20 = 0\)
4. 4. \(x^{2} - 2y^{2} - 3xy + 4x - y - 20 = 0\)

4. The point to which the origin should be shifted so that the equation \(y^{2} - 6y - 4x + 13 = 0\) is transformed in the form \(y^{2} + Ax = 0\) is

1. 1. \((3,1)\)
2. 2. \((-1, -1)\)
3. 3. \((1,3)\)
4. 4. \((-1,3)\)

5. Find the coordinates of \(M\) in the original system if the point \(M\) changes to \((4,3)\) when the axes are rotated through an angle of \(135^{\circ}\).

1. 1. \(\left(\frac{-1}{2},\frac{7}{2}\right)\)
2. 2. \(\left(\frac{1}{2},\frac{7}{2}\right)\)
3. 3. \(\left(\frac{-1}{\sqrt{2}},\frac{7}{\sqrt{2}}\right)\)
4. 4. \(\left(\frac{1}{\sqrt{2}},\frac{7}{\sqrt{2}}\right)\)

6. The point to which the origin should be shifted so that the equation \(y^{2} - 6y - 4x + 13 = 0\) will not contain term in \(y\) and the constant term, is

1. 1. \((1,1)\)
2. 2. \((1,2)\)
3. 3. \((2,1)\)
4. 4. \((1,3)\)

7. Which of the following statement is false?

1. 1. The area of a triangle is invariant under the translation of the Axes
2. 2. The slope of a straight line is invariant under the translation of the Axes
3. 3. The shifting of origin to another point, while changing the direction of the axes, is called translation of axes.
4. 4. If \(f(x,y) = 0\) is transformed equation of a curve when the axes are translated to the point \((h,k)\) then the original equation of the curve is \(f(x - h, y - k) = 0\)

8. The transformed equation of \(3x^{2} - 4xy = r^{2}\) when the coordinate axes are rotated through an angle \(\tan^{-1}(2)\) is

1. 1. \(X^{2} - 4Y^{2} = r^{2}\)
2. 2. \(2XY + r^{2} = 0\)
3. 3. \(4Y^{2} - X^{2} = r^{2}\)
4. 4. \(XY = r^{2}\)

9. By shifting the origin to the point \((2,3)\) and then rotating the coordinate axes through an angle \(\theta\) in the counter clockwise direction, if the equation \(3x^{2} + 2xy + 3y^{2} - 18x - 22y + 50 = 0\) is transformed to \(4X^{2} + 2Y^{2} - 1 = 0\), then the angle \(\theta =\)

1. 1. \(\frac{\pi}{6}\)
2. 2. \(\frac{\pi}{2}\)
3. 3. \(\frac{\pi}{4}\)
4. 4. \(\frac{\pi}{3}\)

10. When the origin is shifted to the point \(\left(\frac{3}{2},\frac{3}{2}\right)\) by the translation of coordinate axes, then the transformed equation of \(32x^{2} + 8xy + 32y^{2} - 108x - 108y + 99 = 0\) is

1. 1. \(72X^{2} + 56Y^{2} - 63 = 0\)
2. 2. \(X^{2} - 14XY - 7Y^{2} - 2 = 0\)
3. 3. \(32X^{2} - 16XY + 32Y^{2} - 225 = 0\)
4. 4. \(32X^{2} + 8XY + 32Y^{2} - 63 = 0\)

11. The point \((4,1)\) undergoes the following transformations successively:
(i) reflection in the line \(x - y = 0\)
(ii) shifting through a distance of 2 units along the positive X-axis
(iii) projection on X-axis

1. 1. \((3,4)\)
2. 2. \((4,3)\)
3. 3. \((3,0)\)
4. 4. \((4,0)\)

12. Let C be a curve \(ax^{2} + 2hxy + by^{2} + 2gx + 2fy + c = 0\) in a Cartesian plane, by rotating the coordinate axes through an angle \(\frac{\pi}{4}\) in the positive direction, if the transformed equation of C is \(Y^{2} + XY - X = 0\), then \((h^{2} - ab) - 2gf =\)

1. 1. 0
2. 2. 2
3. 3. 1
4. 4. -1

13. When the coordinate axes are rotated through an angle \(\theta\) in anticlockwise direction, if the transformed equation of \(x^{2} + y^{2} + 2xy + 2x + 6y + 1 = 0\) is \((2 + \sqrt{3})X^{2} + 2XY + (2 - \sqrt{3})Y^{2} + aX + bY + 2 = 0\), then \(3a - b =\)

1. 1. 10
2. 2. \(2(1 + 2\sqrt{3})\)
3. 3. 20
4. 4. \(2(3 + \sqrt{3})\)

14. If the axes are rotated through an angle \(45^{\circ}\), the coordinates of the point \((2\sqrt{2}, -3\sqrt{2})\) in the new system are

1. 1. \((3\sqrt{3}, -5)\)
2. 2. \((-1, -5)\)
3. 3. \((5\sqrt{3}, -7)\)
4. 4. \((7, -\sqrt{3})\)

15. When the coordinate axes are rotated through an angle \(135^{\circ}\), the coordinates of a point P in the new system are known to be \((4, -3)\). Then find the coordinates of P in the original system.

1. 1. \(\left(\frac{1}{\sqrt{2}},\frac{7}{\sqrt{2}}\right)\)
2. 2. \(\left(\frac{-1}{\sqrt{2}},\frac{7}{\sqrt{2}}\right)\)
3. 3. \(\left(\frac{1}{\sqrt{2}},\frac{-7}{\sqrt{2}}\right)\)
4. 4. \(\left(\frac{-1}{\sqrt{2}},\frac{-7}{\sqrt{2}}\right)\)

16. When the axes are rotated through an angle \(45^{\circ}\), the new coordinates of a point P are \((1, -1)\). The coordinates of P in the original system are

1. 1. \((\sqrt{2},\sqrt{2})\)
2. 2. \((\sqrt{2},0)\)
3. 3. \((0,\sqrt{2})\)
4. 4. \((-\sqrt{2},0)\)

17. The point to which the origin should be shifted in order to eliminate the x and y terms from the equation \(9x^{2} + 4y^{2} + 10x + 12y + 1 = 0\) is

1. 1. \(\left(\frac{5}{9},\frac{3}{2}\right)\)
2. 2. \(\left(\frac{-5}{2},\frac{-3}{9}\right)\)
3. 3. \(\left(\frac{-5}{9},\frac{-3}{2}\right)\)
4. 4. \(\left(\frac{-3}{2},\frac{-5}{9}\right)\)

18. The transformed equation \(3x^{2} + 3y^{2} + 2xy = 2\) when the coordinate axes are rotated through an angle \(45^{\circ}\) is

1. 1. \(x^{2} + 2y^{2} = 1\)
2. 2. \(2x^{2} + y^{2} = 1\)
3. 3. \(x^{2} + y^{2} = 1\)
4. 4. \(x^{2} + 3y^{2} = 1\)

19. If a square ABCD where \(A(0,0), B(2,0), C(2,2), D(0,2)\) undergoes the following transformations successively, then the final figure would be a
(i) \(f_{1}(x,y)\to (y,x)\)
(ii) \(f_{2}(x,y)\to (x + 3y,y)\)
(iii) \(f_{3}(x,y)\to \left(\frac{x - y}{2},\frac{x + y}{2}\right)\)

1. 1. Square
2. 2. Rhombus
3. 3. Rectangle
4. 4. Parallelogram

20. Find the transformed equation of the curve \(x^{2} + 2\sqrt{3}xy - y^{2} = 8\) when the axes are rotated through an angle \(\frac{\pi}{3}\).

1. 1. \(x^{2} + y^{2} + 2\sqrt{3}xy = 8\)
2. 2. \(x^{2} + y^{2} - 2\sqrt{3}xy = 8\)
3. 3. \(x^{2} - y^{2} + 2\sqrt{3}xy = 8\)
4. 4. \(x^{2} - y^{2} - 2\sqrt{3}xy = 8\)

21. The equation obtained by transforming \(x^{2} + y^{2} - 6x + 10y - 2 = 0\) to the parallel axis through \((3, -5)\) is

1. 1. \(x^{2} + y^{2} = 16\)
2. 2. \(x^{2} + y^{2} = 9\)
3. 3. \(x^{2} + y^{2} = 25\)
4. 4. \(x^{2} + y^{2} = 36\)

22. If the axes are transformed to the point \((-1,1)\) then the equation \(3x^{2} + y^{2} + 2x + 4y + 15 = 0\) would transform to

1. 1. \(3x^{2} + 2y^{2} - 4x + 6y + 23 = 0\)
2. 2. \(3x^{2} + y^{2} - 4x + 6y + 21 = 0\)
3. 3. \(3x^{2} + y^{2} + 4x - 6y - 21 = 0\)
4. 4. \(3x^{2} + y^{2} + 4x + 6y + 21 = 0\)

23. If \(P(a,b)\) is the point to which the origin is to be shifted by translation of axes so as to remove the first degree terms from the equation \(4x^{2} + 2xy + y^{2} - 8x - 4y - 12 = 0\) and \(\theta\) is the angle through which the axes are to be rotated about the origin so as to remove the xy-term from the above equation, then \(a + b + 3\tan 2\theta =\)

1. 1. 2
2. 2. 4
3. 3. 8
4. 4. 6

24. The transformed equation of the curve \(2x^{2} + y^{2} - 3x + 5y - 8 = 0\) translated to the point \((-1,2)\) is

1. 1. \(2x^{2} + y^{2} - 7x + 9y + 11 = 0\)
2. 2. \(2x^{2} + y^{2} + 7x + 9y + 11 = 0\)
3. 3. \(2x^{2} + y^{2} - x + y + 11 = 0\)
4. 4. \(2x^{2} + y^{2} + 7x - 9y + 11 = 0\)

25. When the origin is shifted to \((-1,2)\) by the translation of axes, the transformed equation of \(x^{2} + y^{2} + 2x - 4y + 1 = 0\) is

1. 1. \(X^{2} + Y^{2} = 4\)
2. 2. \(X^{2} + Y^{2} = 16\)
3. 3. \(X^{2} + 2X + Y^{2} = 4\)
4. 4. \(X^{2} - 2X + Y^{2} = 16\)

26. The angle by which axes are to be rotated without changing the origin so that the transformed equation of \(x^{2} + 4xy - y^{2} = 0\) in new coordinates \((X,Y)\) does not contain XY term is

1. 1. \(\frac{1}{2}\tan^{-1}(2)\)
2. 2. \(\tan^{-1}(2)\)
3. 3. \(\frac{\pi}{8}\)
4. 4. \(\frac{\pi}{4}\)

27. The equation of a curve \(C\) is transformed to \(X^{2} + Y^{2} - 6X + 8Y + 21 = 0\) by the rotation of coordinate axes about the origin through an angle of \(\frac{\pi}{4}\) in the positive direction of X-axis. If \(ax^{2} + by^{2} + cx + dy + e = 0\) is the equation of the curve \(C\) before the transformation, then \((a + b + c^{2} + d^{2} - 5e)^{2} =\)

1. 1. 4
2. 2. 9
3. 3. 16
4. 4. 25

28. If the coordinate axes are rotated in positive direction by \(45^{\circ}\) without changing the origin, then the transformed equation of \(3x^{2} + 3y^{2} + 2xy - 2 = 0\) is

1. 1. \(2X^{2} + Y^{2} = 1\)
2. 2. \(X^{2} + 2Y^{2} = 1\)
3. 3. \(X^{2} - 2Y^{2} = 1\)
4. 4. \(2X^{2} - Y^{2} = 1\)

29. When the coordinate axes are rotated about the origin in the positive direction through an angle \(\frac{\pi}{4}\), if the equation \(49x^{2} + 25y^{2} = 1225\) is transformed to \(px^{2} + qxy + ry^{2} = t\) and the G.C.D of \(p,q,r,t\) is 1, then

1. 1. \((p - q + r - 32)^{2} = 4t\)
2. 2. \((p - q - r + 12)^{2} = t\)
3. 3. \((p + q + r - 15)^{2} = t\)
4. 4. \((p - q - r + 13)^{2} = t\)

30. The transformed equation of \(3x^{2} + 4xy + y^{2} - 8x - 4y - 4 = 0\) is \(f(X,Y) = aX^{2} + 2hXY + bY^{2} + c = 0\) by translation of axes. Then \(f(1,1) =\)

1. 1. 0
2. 2. 1
3. 3. -1
4. 4. -8

31. The point to which the origin is to be shifted by translation of axes so that the transformed equation of \(y^{2} + 4y + 8x - 2 = 0\) will not contain \(y\) term and constant term is

1. 1. \(\left(\frac{3}{4}, -2\right)\)
2. 2. \(\left(\frac{-3}{4}, -2\right)\)
3. 3. \(\left(2,\frac{3}{4}\right)\)
4. 4. \(\left(-2, - \frac{3}{4}\right)\)

32. If \(x^{2} = 8ay\) is the transformed equation of \(x^{2} - 4y + 6x + 15 = 0\) when the origin is shifted to the point \((\alpha ,\beta)\) by translation of axes, then \(2\alpha +8\beta^{2} =\)

1. 1. 8
2. 2. 18
3. 3. 12
4. 4. 16

33. By rotating the axes through an angle of \(30^{0}\) in the anti-clockwise direction about the origin, the equation \(4x^{2} + 12xy + 9y^{2} + 6x + 9y + 2 = 0\) becomes \(ax^{2} + 2hxy + by^{2} + 2gx + 2fy + c = 0\), then

1. 1. \(a = 21 - 6\sqrt{3}\)
2. 2. \(g / f = \frac{3 + 2\sqrt{3}}{3\sqrt{3} - 2}\)
3. 3. \(b = 31 + 6\sqrt{3}\)
4. 4. \(c = 6\)

34. The transformed equation of \(2x^{2} + 3y^{2} - z^{2} - 8x + 18y + 2z + 9 = 0\) when the axes are translated to the point \((2, - 3, 1)\) is

1. 1. \(2x^{2} + 3y^{2} - z^{2} = 25\)
2. 2. \(2x^{2} + 3y^{2} + z^{2} = 25\)
3. 3. \(2x^{2} - 3y^{2} - z^{2} = 25\)
4. 4. \(2x^{2} + 3y^{2} - z^{2} = 50\)

35. The angle by which the coordinate axes are to be rotated about the origin so that the transformed equation of \(\sqrt{3} x^{2} + \left(\sqrt{3} - 1\right)xy - y^{2} = 0\) would be free from \(xy\) term is

1. 1. \(45^{\circ}\)
2. 2. \(22.5^{\circ}\)
3. 3. \(15^{\circ}\)
4. 4. \(7.5^{\circ}\)

36. Let P be the point to which origin has to be shifted by the translation of axes so as to remove the first degree terms from the equation \(3x^{2} + y^{2} - 6x + 4y + 4 = 0\). If the origin is shifted to P by the translation of axes, then the transformed equation of \(2x^{2} + 3xy - 5y^{2} + 2x - 23y - 24 = 0\) is

1. 1. \(x^{2} + 4xy - 3y^{2} - 4x + 20y + 23 = 0\)
2. 2. \(2x^{2} - 3xy + 5y^{2} = 0\)
3. 3. \(2x^{2} + 3xy - 5y^{2} = 0\)
4. 4. \(2x^{2} + 3xy - 5y^{2} - 13 = 0\)

37. When the origin is shifted to the point \((h,k)\) by translating the coordinate axes, the equation \(S\equiv 2x^{2} - xy + y^{2} + 2x + 3y + 1 = 0\) is changed to \(S^{1}\equiv ax^{2} + 2hxy + by^{2} - 3 = 0\). Again by rotating the coordinate axes about the new origin through the angle \(\theta\) in the positive direction, \(S^{1} = 0\) is changed to \(Ax^{2} + By^{2} + C = 0\). Then \(h + k + \tan 2\theta =\)

1. 1. -4
2. 2. 0
3. 3. 1
4. 4. -1