CNS/
CERT- ECE FINAL COACHING
Module Mod ule 1
MATHEMATICS
Algebra/Discrete Algebra/Discrete Mathematics/Number Theory/Probability Theory/Probability & Statistics
--------------------------------------------------------------------------------------------------------------------------------------1.
The co-ordinates of the middle points of the sides sides of of a triangle are (4, 2) Then the co-ordinates of the centroid are Answer: (3, 7/3) Solution: Coordinate of the centroid: x1
+
x2
+
x3
3
2.
3.
4.
,
y1
+
y2
+
3
y3
2
2
sin x dx is sin sin x cos x
equal to
log log (sin sin 2x)
+
4
625
3
125
x
x
x ®5
is equal to
2
1 + tan 165°
sin sin (log x) x
3 2
9.
The equation of the straight line line which makes equal intercepts on the axes and passes through the point (1, 2) is Answer: x+y=3 10. Area of the triangle whose vertices are (a, b) (a, a + b), (-a, -a + b) is Answer: a2 11. 1 cosx is Lim 2 x
x ®0
Answer: ½
3
sin x
3
cos x
2
2
sin x cos x
dx is
equal to
Answer: secx + cosec x /2
25. The value of the
3
sin xdx is
equal to
Answer: 2/3 15. Solution of differential differential equation equation
dy dx
=
x y
e
is
Answer: ex – ey = const
CERTC: MANILA-CEBU-BAGUIO-DAVAO-ILOILO
lim
sin xe
x
is
x
x ®0
Answer: 1 1
26. The integral
0
x
xe dx is
equal to
Answer: 1 27. The area under the curve y = 1 is Answer: 1/3 28. The solution of
2 12. The point on the curve y = 4x at which the tangent to the curve is parallel to y = x is Answer: (1,2)
0
b2 −ac
22. The circles x2 y2 px py−7 0 and x2 y2 −10x 2py 2p y 1 0 cut cut orth orthog ogon onal ally ly ifif the the valu value e of p is Answer: 3 23. The eccentricity eccentricity of the ellipse 16x2 25y2 400 is Answer: 3/5 24. The derivative of –cos(log x) is Answer:
Answer:
14.
3
c−a
2
13.
...−1002 1012
Answer:
is
tan 165°
1
32 −42
B 60, then
Answer: 20/3 8.
19. Sum of the series S 12 −22 is equal to Answer: 5151 20. The value value of tan 15 is Answer:
∠
Solution: cos2x = cos2x – sin2x sin2x = 2sinxcosx If 20Cr 20Cr-10 then 18Cr is equal to
Lim
18. If pth, (2p)th and (3p)th terms of a G.P. are x, y, z respectively, then x, y, z are in Answer: G.P.
21. In a triangle ABC, let a = BC, b = CA and c=AB. c=AB. If
C
Answer: 816 7.
1
2
Answer:
6.
17. The value value of Sin 105 105 + Cos 105 is Answer: 2
nCr then the value of r is
cos x
5.
Answer:
÷
If x, 2x+2, 3x+3 3x+3 are first three terms of a G.P.then G.P.then its th 4 term is Answer: -13.5 The angle made by any any diameter of a circle circle at at any point on the circumference is Answer: 90 degrees If nPr 720 Answer: 6
16. Period of of Sin (2x + 3) is
dy =
dx
2
y , y(1)
x2 between x = 0 and x
=
1 is
Answer: y = -1/x 2 29. If one root of the equation equation 2x −10x K 0 is 2/3 of the other root, then K is Answer: 12 30. The centroid of the triangle triangle formed by the straight lines y x 3, y− x 3, y 0 is Answer: (0,1) 31. The distance between the parallel lines 3x + 4y + 5 = 0 and 3x + 4y + 15 = 0 is Answer: 2 32. lim
sin mx
x ®0
sin nx x
, where m ≠ n is equal to
Answer: m – n (0932-175-1218/0917-302-8824
Page 1
CNS/
CERT- ECE FINAL COACHING
Module 1
MATHEMATICS
Algebra/Discrete Mathematics/Number Theory/Probability & Statistics
--------------------------------------------------------------------------------------------------------------------------------------Solution: 33. If y
=
(1
2 sin 2x , then dy/dx is equal to
Answer: 2sin4x 1 + sin x
(2
is equal to
Answer: tan x – sec x 0
2x
dx
1 + e
is equal to
Answer: /4 36. The solution of the differential equation ydx
=
1
n (n
nx +
xdy
+
=
3
+
1)
2
2
x
+ ....
)
52. If cos A = Answer: 7/25
x
e
35.
n
51. Value of tan105 is Answer:
dx
34.
x)
53.
then the value of cos 2A is
The value of ‘x’ such that PQ = QR, where P, Q and R are (6, -1), (1, 3) and (x, 8) respectively is given by Answer: 5, -3
0
Answer: x
y
+
=
54. Slope of the line passing through the points (5/2, 3) & (0, ¾) is Answer: 9/10
const
37. The value of sin 75 − cos 75 is equal to Answer: 1
2
55.
2 -1
Lim
2 sin x
sin 2x 3
x
x ®0
y
x
=
2 and Y≠
cos2x
dx is 2 2 cos x sin x
x
sin 3x sin 2x dx
is equal to
Answer: ½ (sinx – (sin5x)/5) 58. Order and degree of the differential equation 3
dy 3
dx
2
dy
+
2
dx
÷ +
dy dx
+
4y
= sin
x is given by
Answer: 3,1
equal to
60. How many terms are there in the expansion of [( x −5y ) ] Answer:
46. The solution of differential equation
dy dx
=
x y
e
+
16 61. If sin
y
, find the value of
62. Expansion of 5 sin sin 8 is equal to Answer: 5 2
63.
(cos 7
cos 9
)
For what value of k do the points (−1 ,4),(− 3,8) & (− k +1, 3k) lie on a straight line. Answer: 0
The number of terms in the sequence 5/2, 5, 10,…., 640 64.
are Answer: 9 *
and sec =
cot Answer: -4/3
2xe is
Answer: ey = ex + x2 + C −1 −1 47. Value of sin x + cos x is Answer: pi/2 48. value of (sin3A−sinA) cosA − (cos3A+cosA)sinA is Answer: 0 *
Mid point of the line joining (3, 5) and (− 7,−3) is given by Answer: (-2,1)
50. First three terms in the expansion of (1 Answer: 1 – 11x3 + 99/2 x6
is equal to
59. Which term of the series 37+32+27+22+.............. is –103? Answer: 29th
Answer: - sec x cosec x 45. The area bounded by the parabola y2 = 4ax and its latus rectum is Answer: 8/3 a2
49.
3
x, then dy/dx is equal to
Answer: 1 44.
2x
2
1
57.
y
+
x
3
is equal to
Answer: bx = ay 42. The area of the triangle formed by the lines y = a + x, y = a – x, y = 0, where a > 0, is Answer: a2 x
4x + 3
2
56. If y=sin-1 (3x−4x3)then dy/dx is equal to Answer:
If the point P(x, y) is equidistant from the points A(a + b,b − a) and B(a − b,a + b), then
43. If
x
Answer: ½
Answer: 1 41.
x ®3
-1
38. The value of tan 1/2 + tan 1/3 is Answer: pi/4 39. If nC12 = nC8 , then n is equal to Answer: 20 40.
Lim
3
2x
11/2
)
are
+ …..
CERTC: MANILA-CEBU-BAGUIO-DAVAO-ILOILO
65.
Lim x ®2
x x
2 2
is equal to
Answer: (0932-175-1218/0917-302-8824
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CNS/
CERT- ECE FINAL COACHING
Module 1
MATHEMATICS
Algebra/Discrete Mathematics/Number Theory/Probability & Statistics
--------------------------------------------------------------------------------------------------------------------------------------2 2
2
67.
tan x dx is
equal to
z
70.
(5
84. If ax b, by Answer: 1
)
If α, β be the roots of ax 2 + bx + c = 0,a
71. The 12th term in the binomial expansion of
x
1 x
15
÷
1 + i 1
i
n
= 1
÷
is
Answer: 4 is
72. The area of the triangle formed by the coordinate axes and the line 2x + 3y=6 is Answer: 3 sq. units
87. If cos2 3 sin Answer: 90 degrees a
88.
73. The eccentricity of the ellipse if its latus rectum is equal to one half of its minor axis is Answer: (√3)/2
( )
f x dx 0
3, then is equal to
is equal to
Answer: a
(
)
f a 0
x dx
89. Integrating factor of the differential equation
dy
2
(1 x) dx +
y = (1+ x) is
Answer:
74. In a triangle ABC, sin A – cos B = cos C, then angle B is Answer: pi/2 e
a, then the value of xyz is
86. The smallest positive integer n for which
Answer: -15C11x-7
1
c, cz
0,then the
2
quadratic equation whose roots are , is Answer: a2x2 – (b2 – 2ac)x + c2 = 0
75.
on
85. The equation whose roots are the reciprocals of the roots of the equation ax2 bx c 0 is Answer: a + bx + cx2 = 0
i
2
1 lie
=
Answer: The x-axis
69. The square root of 12 – 5i is Answer: 2
5i
z + 5i
68. The solution of the differential equation (1+y2)dx+(1+x2)dy=0 is Answer: (x+y) = k (1 – xy)
±
where w is a cube root of
83. The complex number z = x +iy which satisfies the equation
Answer: tan x – x + C *
1
3 w 3w2
82. The expression unity, equals Answer: 16 w
66. If y = x sinx, then dy/dx is equal to Answer: x cos x + sin x
1 1 + x
90. The distance between two parallel lines 3x + 4y = 5 and 6x + 8y = 35 is Answer: 2.5
log x dx is equal to
Answer: 1 76. If xy = ex – y , then dy/dx is equal to Answer: log x (1 + log x)-2
91. The angle between the vectors
and
is 77. The point α, β 4) if Answer: = 3 , = 0
is equidistant from points (7,6) and (-3,
Answer: 60 degrees 92. The value of
lim x ®0
78. The value of sin 45 Answer: 0
θ
− cos 45 −θ
is
79. The equation of a line through point (2, -3) and parallel to y-axis is Answer: x=2 2 2 80. The length of tangent from point (5,1) to the circle x y 6x−4y−3 0 is Answer: 7 81. The differential coefficient of log tan x is Answer: 2 cosec 2x
(0932-175-1218/0917-302-8824
cos 2x x
2
is
Answer: 2 93. 12 22 32 Answer: n ( n + 1) (2n
.. .. ... . n2 is equal to + 1
)
6
94. If log x 9 A. 27
1.5 then x is equal to
95. The value of Sin 75 Cos 15 Answer: 1 96.
CERTC: MANILA-CEBU-BAGUIO-DAVAO-ILOILO
1
If α, β are the roots of ax 2
Cos 75 Sin 15 is equal to
bx
c
0 then α2
β2 is Page 3
CNS/
CERT- ECE FINAL COACHING
Module 1
MATHEMATICS
Algebra/Discrete Mathematics/Number Theory/Probability & Statistics
--------------------------------------------------------------------------------------------------------------------------------------2
A.
b
2ac
9
1
97.
2x
2
a
( )
lim x x ®1
x 1
is equal to
106.
Answer: e 98.
y
=
1
Find the equation of a line passing
perpendicular to the line 3x – y +5 =0
Answer:
xlogx−x c 99. The maximum value of y = 2 cos 2x – cos 4x, 0 ≤ x Answer: 3/2
≤ pi/2 is
100. The equation of the l ine which is perpendicular to the line 3x – 4y +7=0 and passes through the point (-3, 2) is Answer: 4x + 3y + 6 = 0 2 re the roots of the equation ax 0 . Find the equation whose roots are
bx
Answer:
x + 3y + 14 = 0 107. Find the equation of the circle whose centre lies on the line x – 4y = 1 and which passes through the points (3, 7) and (5,5).
c
Answer: x2 + y2 + 6x + 2y – 90 = 0
and
Answer: a4x2 – a2 (2b2 – 4ac)x + (b2 – 4ac)b2 = 0
108. 102.
12
through 2,4 and
log x dx is equal to
101. If
5
+
Find the term independent of x in
Find the equation of tangent to
the expansion of
16x2 9y2 144 at x1,y1,where
Answer: 8064
2x
1 x
10
÷
x1 2 and y1 0. Answer: 1/a 103.
109.
Differentiate f x Cos x by the 2
first principle.
Evaluate
/2
sin x sin x
0
+
dx
cos x
Answer: pi/4
Answer: - sin
2x 2
x
110. Solve
tan
1
6
3
x
dx
1 + x
Answer: 104.
Find the area bounded by the
(tan
1
3
x
)
6
curve x2 4y and the straight line
+
C
x 4y 2
105.
Answer: 5/6 units
111. Evaluate
Find the equation of tangent to
112.
16x2 9y2 144 at x1,y1,where
given y = 1 when x = 1. Answer: x2 + y2 – 2x = 0 113. Find the differential equation of
sin x
1
cos x
dx
Answer: -xcot(x/2) – log2 +C
x1 2 and y1 0. Answer: CERTC: MANILA-CEBU-BAGUIO-DAVAO-ILOILO
x
(0932-175-1218/0917-302-8824
Solve x2 y2dx 2xydy 0
,
Page 4
CNS/
CERT- ECE FINAL COACHING
MATHEMATICS
Module 1
Algebra/Discrete Mathematics/Number Theory/Probability & Statistics
--------------------------------------------------------------------------------------------------------------------------------------x 3x 5x
which y Ae Be solution.
Ce
is a
Answer: 3
dy 3
dx
114.
2
9
d y 2
dx
+
23
dy dx
15y
=
0
Find the term independent of x in the expansion of (x – 1/x)12. Answer: 924
115.
Solve the equation
x
dy dx
3y
=
2
x
Answer: y = -x2 + cx3
Find the equation of a straight line when p is the length of perpendicular on it from the origin and the inclination of this perpendicular to the x – axis is
116.
Answer: x cos y sin p
Find the equation of the straight line which passes through the intersection of the straight lines 2x – 3y + 4 = 0 and 3x + 4y + 5 = 0 and is perpendicular to the straight line 6x – 7y + 8 = 0.
117.
Answer: 119x + 102y + 205 = 0
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(0932-175-1218/0917-302-8824
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CNS/
CERT- ECE FINAL COACHING
MATHEMATICS
Module 1
Algebra/Discrete Mathematics/Number Theory/Probability & Statistics
---------------------------------------------------------------------------------------------------------------------------------------
CERTC: MANILA-CEBU-BAGUIO-DAVAO-ILOILO
(0932-175-1218/0917-302-8824
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CNS/
CERT- ECE FINAL COACHING
MATHEMATICS
Module 1
Algebra/Discrete Mathematics/Number Theory/Probability & Statistics
---------------------------------------------------------------------------------------------------------------------------------------
CERTC: MANILA-CEBU-BAGUIO-DAVAO-ILOILO
(0932-175-1218/0917-302-8824
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