HSC Mathematics 1st
Paper (Theoretical) Model Question for Class XII
1. State and prove the De Morgan's
Law. 4
Or,
F:
R→R defined by F(x) = 
. Find F(5), F(0), and F(-2).
. Find F(5), F(0), and F(-2).
2. If a, b
R then prove that │ a + b│ 
│ a│ +│b│. 3
R then prove that │ a + b│ │ a│ +│b│. 3
Or,
(i) Solve│2x+1│‹ 3 and show the solution set on
real line.
(ii) Express the inequalities -1 ‹ 2x-3‹ 5 using
absolute value sign.
3. Find the value 0f 
. 3
. 3
Or,
If x+y+z = 0 and 
is an imaginary cube root of unity then show that
is an imaginary cube root of unity then show that
(x+y+z2)3 +(x+y2+z)3 = 27xyz.
+z2)3 +(x+y2+z)3 = 27xyz.
4. Find the condition that the two expressions px2+qx+1
and qx2+px+1 may have a
common factor.
4
Or,
If
α,β,γ are the roots of the equation x3+px2+qx+r=0 then
find the value of ∑α3 .
5. If
A=
, B=
and C=
then prove that, (AB)C
= A(BC). 4
, B= and C= then prove that, (AB)C
= A(BC). 4
Or,
Prove
that 
6. Find number of permutations of n
different things taken r at a time where n and r 4
positive
integers and n ≥
r.
r.
Or,
Find
the number of combinations of the letters of the word THESIS taken 4 letters at
a time.
7. If the 21st and 22nd terms are equal in
the expansion of (1+x)44 find the value of
x. 4
Or,
If y = 2x+3x2+4x3+……………..∞, show that
x = 
y-
y2+
y3- ………………∞.
y-y2+y3- ………………∞.
8. Find the sum of the squares of the first
n natural number. 4
Or,
Sum to n terms of series; 
9. If A+B+C = π, then prove that
4
4
Or,
Prove that: secx = 
.
.
10.
Draw the graph of y = sinx.cosx, x= 
to x = 
. 4
to x = . 4
Or,
Solve graphically:
x – tanx =0, where 0 ≤ x ≤ 
.
.
11.
Solve: sec4
-sec2 = 2, when 0≤ 
≤3600. 4
-sec2 = 2, when 0≤ ≤3600. 4
Or,
Find general solution of sin7x
- √3cos4x = sinx.
12. If
sin(πcos θ) = cos(πsin θ), then show that θ = ±sin-1
. 3
sin-1. 3
Or,
Prove that,
cos-1
-- cos-1
= 
.
-- cos-1 = .
13. For any triangle ABC prove that 
= 
= 
. 3
= = . 3
Or,
If (a+b+c)(b+c-a) = 3bc, then find the value of the
angle A.
14. If A(2, 5), B(5, 9) and D(6, 8) are three vertices of the
rhombus ABCD, then find the coordinates of the fourth vertex C and the area of
the rhombus. 3
Or,
Co-ordinates of the points A and B are (-2,4) and
(4,-5) respectively. AB is producedup to
C such that AB=3BC .Find the co-ordinates of C.
15. a) A straight line passes through the point
(-2,-5) and intersect the x and y-axes at A and B respectively such that
OA+2.OB=0 where o is the origin . Find its equation. 3
b) Find the equation of the straight line passing
through the origin which makes equal
angles
with the straight lines 2x + 3y - 5 = 0 and 3x + 2y - 7 = 0. 3
Or,
a) A(h,
k) is a point on the straight line 6x-y = 1 and B(h, k) is a point on the
straight line
2x- 5y =5. Find the equation of AB. 3
b) If
the straight lines ax+by+c=0; bx+cy+a=0; cx+ay+b=0 are concurrent, then
establish
the
relation among a, b and c. 3
16.
a) Find the equation of the circle which
cuts off intercept of length 5 and 2 units from
x and y - axes respectively and whose
centre lies on the straight line 2x-y = 6. 3
b) Find the equation of the tangent to the circle x2 + y2
=a2 which forms a triangle of area a2
with the axes of coordinates. 3
Or,
a) Find the equation o OR f the circle which
passes through origin and the points of intersection
of
the circle x2+y2 - 2x - 4y- 4 = 0 and the straight line
2x+3y +1 = 0.
3
b) Find the equation of the chord of the
circle x2+y2=144,
whose middle point is (4,-6). 3
17. a) Equation of directrix of the parabola is x – c
= 0 and its vertex is at the point (c
, 0). Show that the equation of the parabola is y2
= 4(c –c)(x –cz). 3
, 0). Show that the equation of the parabola is y2
= 4(c –c)(x –cz). 3
b) The distance between the focus and the
corresponding directrix of an ellipse 16 inches
and
its eccentricity is 
; find the equation of
the ellipse.
3
; find the equation of
the ellipse.
3
Or,
a) Find the equation of the parabola whose vertex is at the point
(4,-3), directrix is parallel to the
x-axis and which passes through the point (-4, -7). 3
b) Find the equation of the hyperabola whose centre is at origin,
transeverse axis is along x-axis and passes through the points (6, 4) and (-3,
1).
3
18. a) 
;find a unit vector 
coplaner with the vectors 
and 
and perpendicular to the vector .
3
;find a unit vector coplaner with the vectors and and perpendicular to the vector .
3
b) With the help of the vectors, prove that the
diagonals of a parallelogram bisect each-other.
3
Or,
a) If
then find │AB│.
3
then find │AB│.
3
b) In any triangle ABC, prove that cosA = 
.
3
.
3
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