This section contains multiple choice questions in straight objective type format. There are 9 questions testing concepts related to mathematics, including functions, geometry, probability, limits, and differential equations. The document also includes an assertion-reason type question and a comprehension section containing word problems and geometry concepts related to engineering drawing.

1. SECTION – I
Straight Objective Type
This section contains 9 multiple choice questions numbered 45 to 53. Each question has 4 choices
(a), (b), (c), (d), out of which ONLY ONE is correct.
1. [x] + [2x] + [3x] = n, for every real x and n ≤ 2007. How many positive integral values of n are
possible? ( [] is Greatest Integer Function)
A. 1340
B. 1336
C. 1339
D. 1338
2. Find the number of points with integral coordinates (2a, a - 1) that fall in the interior of the
larger segment of the circle x2 + y2 =25 cut off by the parabola x2 + 4y=0.
A. one
B. two
C. three
D. zero
e 37 П sin (П lnx )
3. The value of 1
dx is
x
A. 1
B. -1
C. 2
D. None of these
4. At what x does the following expression minimise?
Y(x) = (x2 -2x +2)(2x2 – 4x + 3) (3x2 – 6x + 4) (4x2 – 8x + 5)...............
A. 1
B. 1/2
C. 2
D. Cannot be determined
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2. 5. A box has 5 balls. It is equally likely that there are 1, 2, 3, 4, 5 black balls in the box. If a ball
is drawn randomly what is the probability that the drawn ball is black.
A. ½
B. 5/9
C. 3/5
D. 2/5
1 x2 1 π
6. The point P on the curve y= x tan α - 2 u 2 (cos α)2 , α ε (0,2 ) has a tangent parallel to y = x+5.
u2
If the ordinate of P is then α=
4
П
A. 12
П
B. 6
П
C. 4
П
D. 3
a+bx sin x+c cos x
7. If limx→0 =2, then the values of a, b, c are
x4
A. a=48, b= -24, c=48
B. a=48, b= -24, c= -48
C. a=48, b=48, c=24
D. None of these
8. The stiffness of a rectangular beam is proportional to the product of its breadth and
the cube of its depth. Find the stiffest beam that can be cut from a cylindrical log of
diameter = a.
3a 3������
A. depth = 2 , breadth = 4
√3a a
B. depth = , breadth =2
2
√3a √3a
C. depth = , breadth =
2 4
a
D. depth = a, breadth =3
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3. Assertion Reason
A. Statement I and Statement II both are correct and Statement II is the reason for
Statement I.
B. Statement I and Statement II both are correct but Statement II is not the reason for
Statement I.
C. Statement I is false and Statement II is true.
D. Statement II is false and Statement I is true.
E. Both statements are false.
1. Statement I: When x is positive, the minimum value of x x is e−1 e
Statement II: y is max. or min. according as z=log y is max. or min.
������ 4 −10������ 3 +39������ 2 −70������+50
2. Statement I: The least value of attained at x=2 and x=3
������ 2 −5������+7
Statement II: Harmonic mean of two positive numbers is less than their arithmetic mean.
3. Statement I: The curve y2 = 4(x+h) is orthogonal to the curve y = ke-x/2
Statement II: Their tangents at the common point of intersection are perpendicular to each
other
������
4. Statement I: F(x) = ������������ is discontinuous at x = 1
������−2 (������ 2 +1)
Statement II: If f(x) is discontinuous at x = a then ������ ������ ������������ is also discontinuous at x = a
More than one option correct
1. The solution of the following differential equation is
[y + x f(x2+y2)]dx + [y f(x2+y2) – x]dy = 0
������ ������(������)
A. 2tan−1 ������ = ������������
������
������ ������(������ 2 )
B. tan−1 ������ = ������������
������
������ ������(������ 2 )
C. 2tan−1 ������ = ������������
������
������ ������(������)
D. tan−1 ������ = ������������
������
2. The relation ‘Brotherhood’ can be called…
A. Equivalence
B. Transitive
C. Reflexive
D. Symmetric
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4. Comprehension
1. 8 players play against each other exactly once. The winner of each game gets one point and
the loser gets zero or each get half if match is drawn.
If it is known that the sum of the scores of four bottom players is same as the score of the
player placing second. (Also each player has different score.)
i) Find the score of the player placing second:
A. 8
B. 7
C. 6
D. 10
ii) Find the outcome of the match between the player placed 3rd and player placed 5th:
A. Cannot be commented
B. 5 wins
C. Draw
D. 3 wins
iii) Find the total score of top four players:
A. 20
B. 22
C. 18
D. 21
2. An important concept applied in Engineering Drawing is projecting a given line on two given
planes (an easier choice can be the vertical and the horizontal planes) and then dealing with the
2-D representations instead of the more complicated 3-D figures. The front view is the projection
of the line in the vertical plane where as the top view is the projection of the line in the horizontal
plane (See adjoining figure). While representing the top view together with the front view, we
rotate the top view clockwise 90 degrees and the front view is represented in the first quadrant
and the top view in the fourth quadrant in the resultant plane.
rotate 900
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5. Consider a line with one end X in the vertical plane and one end Y in the horizontal plane
represented this way.
The coordinates of the end X’ in the front view is (a, b) and the coordinates of the end Y’ in the
top view is (c, d). Now answer the following questions:-
(i) What is the true length of the line segment?
A. (������ − ������)2 + (������ − ������)2
B. (������ − ������)2 + ������ 2 + ������ 2
C. (������ − ������)2 + ������2 + ������ 2
D. Information insufficient
(ii) The angle the line makes with the horizontal plane is
������
A. tan−1 ( ������ )
������
B. sin−1 ( )
������−������ 2 + ������−������ 2
������
C. tan−1
������−������ 2 +������ 2
(������−������)
D. tan−1 (������−������)
(iii) If the angle the line makes with the horizontal plane is α & that with the vertical plane is β
then which of them is always true?
A. α+β ≥π/2
B. α+β= π/2
C. α+β< π/2
D. none of these
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6. Match the columns
1. Please match the entries of column I and II
������ ������
A. Let z = ii. Then, z and ż are the roots of P. ������ 2 ������ 2 − 1 + ������ ������ ������ + ������ 2
the quadratic expression
3������ ������
B. 1 - ������ 2 is the minimum value of Q. ������ ������ ������ 2 − 2������ 2 ������ + 1
quadratic expression
������
C. If a quadratic expression Q<0 in (ii, i-i ) R. x2 - 2������ 2 ������ + ������ ������
and Q >0 in (-∞,ii)U (i-i,∞), then Q is
������
D. Let ri = i. Then, (x-r)2 is S. ������ 2 ������ 2 − 2������ ������ ������ + 1
П
20
2. A. If In = 4
0
tann x dx, (n>0, n ε N), then In + In-2 = 1. 99
B. The value of I12 + 2I10 + I8 = 2. H.P
1
C. I2 + I4, I3 + I5, I4 + I6 are in 3. n−1
π П
sin 2 nx
D. If Jn = 0
2 dx, then I2 – I1, I3 – I2, I4 – I3,..... 4. 4
0
tann x + tann−2 x d( x )
sin x
are in 5. A.P.
3. Match the following column entries based on the concept of co-ordinate geometry.
A. The length of common chord of the circles P. 15
x2 + y2 + 4x – 6y – 4=0 and x2 + y2 – x + 6y + 2=0
B. Length of the tangent drawn from (16,7) to circle Q. 6
x2 + y2 – 2x – 4y – 20=0
C. Length of the intercept on X-axis by the circle R. 2
x2 + y2 + 10x – 6y +9 =0
D. Length of intercept on the line 2x – 5y = -18 by the circle S. 8
x2 + y2 – 6x + 2y = 28
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