Jigme Sherubling Higher Secondary School
Trial Examinations - 2010
Subject: Mathematics
Class: XII
Full Mark: 100
Date: 25/10/10
Duration:3.15hrs
Answer Question 1 from section A and 14 question from section B.
All working, including rough work, should be done on the same
Sheet as, and adjacent to, the rest of the answer.
Section A
Answer all the questions.
Question 1
(i)
(2 x 15 = 30)
The sigma notation for the series 6  3 
6
a) 6
1
 (2)
i 1
i 1
6
1 i 1
5
(
)
c) 
2
i 1
3 3 3
3
  
2 4 8 16
6
1 i 1
b) 6 ( )
2
i 1
6
1 i 1
5
(
)
d) 
2
i 1
(ii)
A stone is thrown upward such that its height h metres after t seconds given
by the equation h(t)=-4.9t2 +12t+45, then the velocity after 3 seconds is
A.
-17
B.
-17.4
C.
-15.6
D.
-18.4
(iii)
The point on the curve y=cos x, xЄ [-π, π], where the tangent line is parallel
to x- axis is
A.
(-π/2, 0)
B.
(-π/2,1)
C.
(0, 0)
D.
(0,1)
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Page 1 of 8
3
2
(iv)
The function y = x – 3x + 5 has a maximum value at x equal to
A. -6
B. 0
C. 2
D. 6
(v)
The solution of the inequality │2x+5│<7 is
A.
C.
(vi)
-1<x<1
1<x <-1
B.
D.
-1> x> 1
-1≤ x <1
x3  x2  x  1
The oblique asymptote of f ( x) 
x2 1
A.
C.
y= x + 1
y= -x – 1
B. y= -x + 1
D. y = x – 1
(vii) If y=ex ln x, then the value of derivative at x=1
A.
e
B.
-e
C.
1/e
D.
2e

(viii)
1
dx
x ln x
is
A.
lnx
B.
ln(lnx)
C.
(lnx)2
D.
1/lnx
 /4
(ix)
The value of

tan 3 x.sec 2 xdx
is
0
A.
tan4 x/4
B.
¼
C.
1
D.
tan4(x/4)+tan (x/6)
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6
Page 2 of 8
(x)
The polar coordinate corresponding the rectangular coordinate (-3,4) is
A.
(5, 53.10)
B.
(5, 126.90)
C.
(4.3,1220)
D.
(5, 233.10)
x y
 x  iy  1  3i then the values of x and y are
(xi) If
i
A. x = 1, y = 2
B. x = 3, y = - 2
C .x = 1, y = -2
D. x = -3, y = 2
(xii) The ratio in which yz plane divides the line joining the points (2, 3, 4) and
(3, -4, 7) is
A.
2:-3
B.
3:2
C.
3:4
D.
4:-7
(xiii) Name the conic which defines the equation x2 + y2 + 4x – 8y – 80 is
A. Parabola
(xiv) The value of
B. Circle
C. Hyperbola
D. Ellipse
3 5
3 3

3 2
5  2 is
A.
3√10+6√3
B.
3√10-6√3
C.
-3√10- 6√3
D.
6√15-3√10
(xv) The mean deviation about the mean of the data 12, 14, 18, 20 is
a) 0
JSHSS/Trial-2010/Maths
b) 3
c) 10
d) 16
Page 3 of 8
SECTION B (70 marks)
(Answer any 14 Questions. All questions in this section have equal marks.)
Question:2
a) Simplify and find the restrictions
2 x 2  5 x  3 3x 2  13x  12

4 x 2  12 x  5 6 x 2  7 x  20
[2]
x2
y
2 x  5 where the tangent is
b) Find the points on the curve
horizontal.
[3]
Question:3
50
 (2i
 3i  5)
[2]
b) Prove that n3-n is divisible by 3
[3]
a) Evaluvate
2
i 1
Question:4
a) Find the x and y intercept of y= 4│x-8│- 12
[2]
b) Determine the equation of the tangent line to the curve
y= sinx tan(x/2) when x= π/3
[3]
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Question:5
a) Water is poured in to a conical reservoir at a constant rate of
3m3/min. The reservoir is 3m deep and has a maximum diameter
of 8m. Determine the rate at which the depth of the water is
increasing when the water level is 2m.
[5]
Question:6
a) For the function
f ( x) 
x3
x  x  6 find
2
i. Domain
ii. intercepts
iii. Asymptotes
iv. Intervals of decease and increase
v. sketch the curve.
[5]
Question:7
a) Solve the following system of linear equations by using matrix
method:
y= x+y+z=4,
2x+y+3z=4,
3x+2y+z=11.
[5]
Question:8
dy
1

y  log e ( x  x  a )
x2  a2
a) If
prove that dx
2

b)
2
[2]
2

x sin 2 xdx
0
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[3]
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Question:9
cos  log x 
dx
a) 
x
[2]
b) The half life of Radium-225 is 70 days and a sample of this
element has a mass 300mg
i) find the mass that remains after 7 days.
ii) find the mass that remains after 50 days.
iii) find the rate of decrease of the mass after 50 days.
[3]
Question:10
3(2  i)
a) Find the conjugate of the complex number (1  2i)(1  i )
[2]
x  sin x
dx

1

cos
x
b)
[3]
Question:11
a) The region enclosed by the curves y=sinx and y=cosx and x axis
between x=0 and x=π/2 is revolved about x axis. Find the volume
of the solid thus formed .
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[5]
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Question:12
a) Evaluate using de-moivres theorem (-2+3i)5.
[2]
b) Find the equation of the parabola whose focus is (1,-1) and
directrix is the line x+y-2=0.
[3]
Question: 13
4

a) Evaluate
1
x
3x  1
2
dx
[2]
b) Determine the cube root of -8
[3]
Question: 14
a) Find the Karl Pearson’s correlation coefficient for the following data
X
57
58
59
59
60
61
62
Y
67
68
65
68
72
72
69
[3]
b) Find the eccentricity of the ellipse defined by 9x2+16y2=144
[2]
Question: 15
a) Show that the four points (0,-1,0), (2,1,-1) ,(1,1,1) and (3,3,0) are
coplanar.
[3]
b) Find the length of an arc of a circle of radius 4.5cm and subtends
an angle at the centre 400.
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[2]
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Question:16
a) Find the square root of
18 + 6√5
[3]
b) Find the cost of living index number for the following:
Expense on
Price in 1995
Price in 1996
Food
35%
175
190
Rent
13%
45
45
Clothing
15%
75
85
Fuel
16%
30
32
[2]
Misc.
21%
80
75
Question:17
a) Equations of two lines of regression are 4x+3y+7=0 and
3x+4y+8=0. Find
i) Mean of x and y
ii) Regression coefficients byx and bxy.
iii) Correlation coefficient between x and y.
iv) Use the equations to find the value of x when y=4.5 and
value of y when x=7.
[5]
Question:18
2
a) Find the area of the region bounded by y = x + 1, the x-axis and the
Ordinates x= 0 and x = 2 by using limit as a sum
[5]
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