Mathematics 1a 1
Mathematics 1a 1
Mathematics 1a 1
net
BOARD OF INTERMEDIATE EDUCATION (AP)
JUNIOR INTER MATHEMATICS - IA
t
MODEL PAPER
Time: 3 hours
ne Max. Marks: 75
a. SECTION – A
I.
b h
i) Very short answer type questions.
i
a t
ii) Answer ALL questions.
10 ´ 2 =- 20
| p|r
iii) Each question carries TWO marks.
( ) 1 + x- 2x
If f: R − {±1} → R is defined by f(x) = log then show that f = 2f(x).
u
1.
1−x 1 + x2
2.
ad
Find the domain of the real valued fuction f(x) = .
1
n
2
(x − 1)(x + 3)
3.-
4.- .e e
Define a symmetric matrix. Give one example of order 3 × 3.
( ) 2 −3
w
Find the adjoint and the inverse of the matrix .
4 6
5.
w − − − − − − −
If the position vectors of the points A, B, C are −2i + j − k; − 4i + 2j + 2k and 6i − 3j − 13k
w
− −
respectively and AB = λAC . Then find λ.
− −
6.
et − − − −
Find the vector equation of the line passing through the point 2i + 3j + k and parallel to the vector 4i
n
− −
− 2j + 3k .
7.-
− − − − −
a .− − − − − − −
If a = i + 2j − 3k, b = 3i − j + 2k then show that a + b and a − b are perpendicular to each
other.
4
ib h
8.-
5
at
If sin θ = and θ is not in the first quadrant then find the value of cos θ.-
r
3 + √5
9.- Prove that cos 48° cos 12° = .
10.-
u p 8-
For x ∈ R, show that cosh 2x = 2 cosh2x − 1.
a d SECTION - B
II.
e n
i) Short answer type questions.
. e
ii) Answer any FIVE questions.
ww
iii) Each question carries FOUR marks. 5 ´ 4 = 20
w( )
1 2 2
11. If A = 2 1 2 then show that A2 − 4A − 5I = 0.-
2 2 1
−
12. If −
a , b, −
c are non coplaner find the points of intersection of the line passing through the points
− − − − − −; with the line joining the points − − −, − −
2a + 3b − c ; 3a + 4b − 2c a − 2b + 3c a − 6b + 6c−.
AP-MP
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− − − − − − − − − − − − −
13. If a− = 2i + j − k , b = −i + 2j − 4k and c− = i + j + k then find the value of (a− × b)(b × c ).
π 3π 7π 9π
14.- ( )( )( 1
)(
Prove that 1 + cos 1 + cos 1 + cos 1 + cos = .-
10 10 10 10 16 )
15.- Solve the equation 4 cos2 θ + √ 3 = 2(√ 3 + 1) cos θ and write the general solution.-
16. ( 1
) (e t 3
Prove that cos 2 tan−1 = sin 2 tan−1 . )
n
7 4
17.
1
a.
1 3
In the triangle ABC, if + = then show that ∠c = 60°.
h
a+c b+ c a+b+c
ti b SECTION - C
a
III. i) Long answer type questions.
p r
ii) Answer any FIVE questions.
u
iii) Each question carries SEVEN marks.
d
5 ´ 7 = 35
Show that the function f : Q → Q defined by f (x) = 5x + 4 for all x ∈ Q is a bijection and find f−1.
a
18.
19.
en
Using mathematical induction prove that 1.2.3 + 2.3.4 + 3.4.5 +....up to n terms (n ∈ N)
.e
n(n + 1)(n + 2)(n + 3)
= .
4
20.
ww | || | b+c
With out expanding the determinant prove that c + a
c+a
a+b
a+b a b c
b+c =2 b c a .
21.- w et
a+b b+c c+a
Solve 2x − y + 3z = 9, x + y + z = 6, x − y + z = 2 using Cramer's rule .-
c a b
22.
− − − − − − −
.n − − − − − − − − × −b) × −
If a− = i − 2j − 3k, b = 2i + j − k and c = i + 3j − 2k verify that a × (b × c ) = (a
a
c.
23.-
S−A S−B
iC h
If A + B + C = 2S then prove that cos(S − A) + cos(S − B) + cos C =
b
t
−1 + 4 cos cos cos .
2 2 2
24.-
r a 1
In the triangle ABC, show that +
1
1 1
a2 + b2 + c2
.
p
+ + =
r2 r2 r 2 r2 ∆2
u
1 2 3
e n
. e
w w
w
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