SEE Model Question Prepared by CDC
2076
Subject: Optional Mathematics
Full marks:100 Time:
3 hours
Group ‘A 5 × (1 +
1) = 10
Attempt all the questions.
1.
a) Define trigonometric function.
b) What is arithmetic mean between two numbers a
and b.
2.
a) Write
the name of the set of numbers which is continuous.
b) If
matrix A = what is the value of |A|?
3.
a) If
the angle between two straight lines is θ and their slopes are m1and m2 respectively,
write the formula to find the value of tan θ.
b) Which
geometric figure will be formed if a plane intersects a cone parallel to its
base? write it.
4.
a) Express
Sin 2A in terms of Tan A.
b)
Define angle of elevation.
5.
a) What
is the scalar product of two vectors a⃗ and b⃗ and if the angle between them
is θ?
b) In an
inversion transformation if P' is image of P and r is radius of circle with
centre O, write the relation of OP, OP' and r.
Group ‘B' 3×
(2 + 2 +2) + 2 ×
(2 + 2) = 26
6.
a) Find f -1(x)
if f(x) = 4x + 5.
b)
If g(x) = 2x -1 and f (x) = 4x, find the value
of go f(x).
c)
What will be the points of intersection of the
curve f(x) = x2 - 1 and f(x) = 3.
7.
a) If
A = 
, find |A| and write A−1 is
defined or not.
, find |A| and write A−1 is
defined or not.
b)
According to Cramer's rule, find the values of D1
and D2 for ax + by = c and px + qy = r.
8.
a) Find
the slopes of two straight lines having equations 3x + 4y + 5 = 0 and 6x + 8y +
7 = 0 and write the relationship between them.
b)
Find the single equation for the pair of lines
represented by 3x +2y = 0 and 2x - 3y = 0.
9.
a) Convert sin6A.cos4A into sum or difference of
sine or cosine.
b)
Express 
in term of sub-multiple angle of
Tangent.
in term of sub-multiple angle of
Tangent.
c)
If 2 Sin2θ= √3, find value of θ. (0°≤θ≤ 180°)
10. a) Find the angle between vectors a⃗
and
b⃗ and if , |a|⃗=2 , |b⃗|=
12
and a⃗. b⃗
b) In the
given figure, find AP⃗ and express p⃗ in
terms of a⃗ and b⃗.
c)
If the standard deviation of a set of data is 0.25, find its variance.
Group ‘C'
11× 4 = 44
11.
Solve: x3 - 3 x3 - 4x + 12 = 0
12.
Optimize P = 5x + 4y under the constraints x –
2y ≤
1, x + y ≤ 4, x
≥
0, y ≥
0.
13.
For a real valued function f(x) = 2x + 3
a)
Find the values of f (2.95), f (2.99), f (3.01),
f (3.05) and f (3).
b)
Is this function continuous at x =
3?
14.
By
using matrix method solve the following systems of equations:
3x
+ 5y = 11, 2x − 3y = 1
15.
Find the single equation to represent the
equations of pair of straight lines passing through the origin and
perpendicular to the lines represented by 2x2 − 5xy + 2y2 = 0.
16.
Find
the value of: sin 20°.
sin 30°.
sin 40°.
sin 80°
17. If A + B + C = πc,
prove that: sin2 A − sin2B + sin2C
= 2sinAcosBsinC.
18. From a
place at the ground level in front of a tower the angle of elevations of the
top and bottom of flagstaff 6m high situated at the top of a tower are observed
60° and 45° respectively. Find the height
of the tower and the distance between the base of the tower and point of
observation.
19. Find
the 2×2 matrix which transforms unit
square to a parallelogram.
20. Find the mean deviation and its
coefficient from mean of the given data.
(Marks obtained)
|
0-10
|
10-20
|
20-30
|
30-40
|
40-50
|
(No. of students)
|
2
|
3
|
6
|
5
|
4
|
21. Find the standard deviation and
coefficient of variation from given data.
(Age)
|
0-4
|
4-8
|
8-12
|
12-16
|
16-20
|
20-24
|
(No. of Persons)
|
7
|
7
|
10
|
15
|
7
|
6
|
‘Group
D' 4× 5 = 20
22. A
contract on construction job specifies a penalty for delay of completion beyond
a certain date as follows: Rs 200 for the first day, Rs 250 for the second
dayetc, the penalty for each succeeding day being Rs 50 more than for the
preceding day. How much money the contractor has to pay as penalty, if he has
delayed the work by 30 days? find it.
23. On the
circumference of a wheel there are three points (5,7), (−1, 7) and (5,−1) located such that the distance from a fixed point to these points is always
equal. Find coordinates of the fixed point. Also, find equation representing
the locus that contains all three points.
24. By
using vector method, prove that the quadrilateral formed by joining the midpoints
of adjacent sides of a quadrilateral is a parallelogram?
25. The
coordinates of vertices of a quadrilateral ABCD are A (1,1), B (2,3), C (4, 2)
and D (3,−2).
Rotate this quadrilateral about origin through 180°. Reflect this image of quadrilateral about y =
−x. Write the name of
transformation which denotes the combined
transformation of above two transformations.
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