Class – XII
Subject : Mathematics 1st Paper
Time : 3 hrs. Marks. 75
Algebra -30
1.
|
For any three sets A, B and C prove that, A ´ ( BÇC) = ( A´ B) Ç ( A ´ C).
|
4
|
Or,
|  If f : R R, f(x) = 2x +1 and g: R R, g(x) = x2 -2 are any two functions, find ( gof) and (fog) | |
2.
|
Prove that,
 is an irrational number. |
3
|
Or,
|
Solve and show the solution set on a real number line: | 2-8x | £6.
| |
3.
|
Find all the cube roots of unity.
|
3
|
Or,
|
Find the square root of -8-6
 | |
4.
|
If one root is
, solve the equation x4 + 4x3+ 6x2 +4x +5 =0. |
4
|
Or,
|
If a, b, c are rational and a+b+c =0, then show that the roots of the equation ( b+c-a) x2+(c+a-b)x + ( a+b-c) = 0 are rational.
| |
5.
|      If A =  and B =  . Find AB and BA and show that AB ¹ BA. |
4
|
Or,
|
Show that,
 =  | |
6.
|
How many odd numbers of 5 significant digits can be formed with the digits 6, 5, 2, 3, 0 using each digit only once in a number?
|
4
|
Or,
|
Prove that
 | |
7.
|
Find the coefficients of x18 in the expansion of ( x2+
 . |
4
|
Or,
|
If y = 2x + 3x2+4x3 . . . . . . .. . . . .
 , show that x =  .................... | |
8.
|
Find the sum of n terms of the series 2.4. 62 + 4.6.82 + 6.8.102 +.....................
|
4
|
Or,
|
Sum to n terms of the series. 4 +44 + 444+ .................
| |
Trigonometry - 18
| ||
9.
|
Prove that, 2 sin
 =  . |
4
|
Or,
|
If a cos ( x +
) = b cos ( x - ), prove that ( a + b) tanx = ( a - b) cot . | |
10.
|
Draw the graph of y = tan x where -
 < x < . |
4
|
Or,
|
Solve graphically the equation sinx - cosx =0, where 0 £ x £
 . | |
11.
|
Solve tan x + tan 2x + tan x tan 2x =1.
|
4
|
Or,
|
Solve , sin q + cos q =
 | |
12.
|
If sin-1 x + sin -1y =
, prove that, x2 + y2 =1. |
3
|
Or,
|
Prove that sec2 ( tan -12 ) + cosec2 ( cot-1 3) = 15.
| |
13.
|
In any traingle ABC, prove that, b = c cosA + a cos C
|
3
|
Or,
|
If
 , prove that in triangle ABC, Ðc = 600. | |
Geometry - 21
| ||
14.
|
Show that the points ( 4, 4) , ( 5, 2) and ( 1, 0 ) are the vertices of a right angled triangle and find its area.
|
3
|
Or,
|
The area of the triangle whose vertices are ( 3, 4 ), (2x,5) and ( 6, x) is 19
 sq. unit. Find the value of x. | |
15.
|
If ax+by =c and x cos
+ ysin = p represent the same staight lines, express p in terms of a and b. |
3
|
Or,
|
Find the equation of the straight line which passes through ( 6,7) and makes an angle 450. with 3x +4y =11.
| |
16.
|
Find the equation of the perpendicular bisector of the straight line joining the points ( 8,5 ) and ( -4, -3)
|
3
|
Or,
|
The straight lines 4x +3y =c and 12x -5y =2 ( c+3) are at equal distances from the origin. Find the positive value of c.
| |
17.
|
The centre of the circle lies on the straight line x + 2y +3 =0 and it passes through the points
( -1, -1) and ( 3, 2). Find the equation of the circle.
|
3
|
Or,
|
Show that if
 , the circles x2 + y2 + 2ax +c =0 and x2 + y2 + 2by +c =0 touch each other. | |
18.
|
Find the equation of the tangent to the circel x2 + y2 =20 at a point on it whose abscisa is 2.
|
3
|
Or,
|
Show that the straight line lx + my =1 will touch the circle x2 + y2 -2ax =0 if a2m2 + 2al =1.
| |
19.
|
Find the vertex, focus, latus rectum and equations of the axis and directrix of the parabola
( y - 2)2 = 8 (x - 4)
|
3
|
Or,
|
Find the equation of the parabola whose focus is the point ( -8, -2) and the equation of whose directrix is 2x - y -9 =0.
| |
20.
|
Find the equations of the two directrices of the ellipse 5x2 + 4y2 =1.
|
3
|
Or,
|
Find the coordinates of the focus and eccentricity of the hyperbola
 | |
Vectors -6
| ||
21.
|
Find the value of p if
 and  are perpendicular to each other. |
3
|
Or,
|
Find the projection of the vector
 on  | |
22.
|
Show by vector method that the medians of a triangle one concurrent.
|
3
|
Or,
|
If
 and  , find the unit vector parallel to the resultont of the vectors  . |
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