Mathematics Model Question for Class-xi-1

Class - XI

Subject: Mathematics -I

Time : 3 hrs.     Marks. 75

Group -A ( Algebra)

1.         If A and B are any two sets, prove that ( AUB) = A^/ÇB^/                                                                       4

Or,       If  f(x) = 2x - 5 and g(x) = x² + 6, what are the ralues of g (f(2) and f (g(5))?

2.         Show that  is an irrational number.                                                                                       3

Or,       Prove that |a + b| £ |a | + |b|, where a, b ÎR.

3.         If  = p + iq then show that 4 ( p²-q²) = .                                                                        3

Or,       Find the value of .

4.         If  1 is a root of the equation x³ -5x² + 17x -13 = 0, find the other two roots.                             4

Or,       If the roots of the equation ax² + bx + b =0 be in the ratio m:n, then prove that.

             =0

5.         If A =   and B =  ,  find the value of AB and BA.                                                        4

Or,       Solve :  = 0

6.         Prove that  =                                                                                                           4

Or,       Find how many arrangements can be made with the letters of the word Mathematics and in how many of them the vowels occure together?                                                                      

7.         Find the term independent of x in the expansion ( )¹⁸.                                                      4

            If          y = x +x² + x³ + ......................a then show that

                        x = y - y² + y³ - y⁴ + ............... a.

8.         Find the sum to n terms of the series.  7 + 77 + 777 + .....................                                                     4

Or,       Find the sum to n terms of the series 1.4.7 + 4.7.10 + 7.10.13 + ..........

Group -B ( Trigonometry)

9.         Prove that, tan   tan  .                                                                       4

Or,       If A +B + C = p and cosA = CosB cosC then prove that tan A = tan B + tanC.

10.       Draw the graph of function y = sin 2x when - p  < x < p                                                                       4

Or,       Solve graphically the equation cotx - tanx = 2 where 0 £ x £ p.

11.       Solve : Sin 7q -  cos4 q = sinq .                                                                                                      4

Or,       Find the general solution:  cosx + sinx = cos2x + sin2x.

12.       Prove :  2 tan^-1  + tan^-1 .                                                                                                         3

Or,       Prove :  sin^-1 cos^-1 - cot^-1 2 = tan^-1 .

13.       In any triangle ABC, Prove that.    = Cot  Cot .                                                 3

Or,       ,  Show that in the triangle ABC, C = 60⁰.

Group -C : Geometry

           

14.       The coordinates of the vertices A, B, C of the triangle ABC are respectively ( 5, b ), ( -9, 1) and  (-3, -1). Find the area of the triangle ABC.                                                                                                                      3

Or,       A ( 2, 3 ) and B ( -1, 4) are two fixed points. The point P moves in such a way that PA:PB =2:3. Find the equation of the locus of P.

15.       a)  Find the equation of the straight lines joining the origin and the points of trisection of the intercept of   the straight line 5x+4y - 20 = 0 made by the axes of coordinates.                                                       3

            b)  Find the equation of the straight line which passes through ( 6, 7) and makes an angle 45⁰ with the     straight line 3x + 4y = 11.

Or        a)  A ( h, k ) is a point on the straight line 6x -y =1 and B (k, h) is a point on the straight line 2x - 5y =5.            Find the equation of AB.

            b)  Two straight lines pass through the point ( 6, -7) and they make an angle of 60⁰ with the straight line y + x =1.  Find the equations of the straight lines.

16.       a)  Find the equation of the circle whose centre is at the point (4, 5) and which passes through the centre            of the circle x² + y² + 4x - 6y - 12 =0                                                                                                          3

            b)  The circle x² +y² -4x -6y + c = 0 touches the x - axis. Find c and the coordinates of the point of       contact.

Or,       a)  Find the equation of the circle which touches the x-axis at the point ( 4, 0) and cuts off a chord of      length 6 unit from the y axis.

            b)  Find the equation of the tangent to the circle x²+y² =20 at a point on it whose abscissa is 2.

17.       a)  Find the vertex, focus and the equation of the dircetrix of the parabola y² = 4y + 4x -16.          3

            b)  Coordinates of the focus ( ± 3, 0 ) and eccentricity is , Find the equation of the ellipse.

Or,       a)  Find the equation of the parabolla whose focus is at the point (1, 1) and the equation of whose          directrix is 3x + 4y =1.

            b)  For what value of P will the ellipse Px² + 4y² =1 pass through the point ( ±1, 0) ? Find the length of the axes of the ellipse.

18.       a)  Find the component of the vector   along the vector .                        3

            b)  Show that the medians of a triangle are concurrent. Prove it by vector method.                                 3

           

Or,       a)  Prove by vector method that in any triangle ABC,  Cos C = .

            b)  If      . Find the unit vector parallel to the resultant of the vectors             .