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1)
The mean of the numbers a, b, 8, 5, 10 is 6 and the
variance is 6.80. Then which one of the following
gives possible values of a and b?
| | |
1 )
a = 1, b = 6
| |
2 )
a = 3, b = 4
| |
3 )
a = 0, b = 7
| |
4 )
a = 5, b = 2
| | see the answer see the solution |
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2)
The vector
lies in the plane of the vectors
and
and bisects the angle between
and
 .
Then which one of the following gives possible values of
α and β ?
| | |
1 )
α = 2 , β = 1
| |
2 )
α = 1 , β = 1
| |
3 )
α = 2 , β = 2
| |
4 )
α = 1 , β = 2
| | see the answer see the solution |
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3)
The non-zero vectors
 ,
and
are
related by
=
8 and
=
- 7 .
Then the angle between
and
| | |
1 )
π
| |
2 )
0
| |
3 )
π/4
| |
4 )
π/2
| | see the answer see the solution |
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4)
The line passing through the points (5, 1, a) and
(3, b, 1) crosses the yz-plane at the point
( 0 , 17/2 , -13/2 ) . Then
| | |
1 )
a = 8, b = 2
| |
2 )
a = 2, b = 8
| |
3 )
a = 4, b = 6
| |
4 )
a = 6, b = 4
| | see the answer see the solution |
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5)
If the straight lines
|
x - 1
|
=
|
y - 2
|
=
|
z - 3
|
and
|
x - 2
|
=
|
y - 3
|
=
|
z - 1
|
|
k
|
2
|
3
|
3
|
k
|
2
|
intersect at a point, then the
integer k is equal to
| | |
1 )
-2
| |
2 )
-5
| |
3 )
5
| |
4 )
2
| | see the answer see the solution |
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6)
The conjugate of a complex number is
Then that complex number is
| | |
1 )
| |
2 )
| |
3 )
| |
4 )
| | see the answer see the solution |
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7)
Let R be the real line. Consider the following
subsets of the plane
R × R :
S = {(x, y) : y = x + 1 and 0 < x < 2}
T = {(x, y) : x - y is an integer}.
Which one of the following is true ?
| | |
1 )
T is an equivalence relation on R out S is not
| |
2 )
Neither S nor T is an equivalence relation on R
| |
3 )
Both S and T are equivalence relations on R
| |
4 )
S is an equivalence relation on R but T is not
| | see the answer see the solution |
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8)
Let f : N → Y be a function defined as
f(x) = 4x + 3 where
Y = { y ∈ N : y = 4x + 3 for some x ∈ N } .
Show that f is invertible and its inverse is
| | |
1 )
| |
2 )
| |
3 )
| |
4 )
| | see the answer see the solution |
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9)
Let p be the statement "x is an irrational number",
q be the statement "y is a transcendental
number", and r be the statement "x is a rational
number if y is a transcendental number".
Statement-1 :
r is equivalent to either q or p.
Statement-2 :
r is equivalent to ~(p ↔ ~q).
| | |
1 )
Statement-1 is true, Statement-2 is false
| |
2 )
Statement-1 is false, Statement-2 is true
| |
3 )
Statement-1 is true, Statement-2 is true;
Statement-2 is a correct explanation for Statement-1
| |
4 )
Statement-1 is true, Statement-2 is true;
Statement-2 is not a correct explanation for Statement-1
| | see the answer see the solution |
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10)
Let a, b, c be any real numbers. Suppose that
there are nubers x, y, z not all zero such
that
x = cy + bz, y = az + cx, and z = bx + ay.
Then a2 + b2 + c2 + 2abc
is equal to
| | |
1 )
0
| |
2 )
1
| |
3 )
2
| |
4 )
-2
| | see the answer see the solution |
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11)
Let A be a square matrix all of whose entries
are integers. Then which one of the
following is true ?
| | |
1 )
If det A = ± 1, then A-1
exists and all its entries are integers.
| |
2 )
If det A = ± 1, then A-1 need not exist.
| |
3 )
If det A = ± 1, then A-1 exists but
all its entries are not necessarily integers.
| |
4 )
If det A ≠ ± 1, then A-1
exists and all its entries are non integers.
| | see the answer see the solution |
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12)
The quadratic equations x2 - 6x + a = 0 and
x2 - cx + 6 = 0 have one root in common.
The other roots of the first and second equations
are integers in the ratio 4 : 3. Then the
common root is
| | |
1 )
3
| |
2 )
2
| |
3 )
1
| |
4 )
4
| | see the answer see the solution |
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13)
How many different words can be formed by
jumbling the letters in the word
MISSISSIPPI in which no two S are adjacent ?
| | |
1 )
6 . 8 . 7C4
| |
2 )
7 . 6C4 . 8C4
| |
3 )
8 . 6C4 . 7C4
| |
4 )
7 . 7. 8C4
| | see the answer see the solution |
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14)
Let
and
 .
Then which one of the following is true ?
| | |
1 )
I < 2/3 and J > 2
| |
2 )
I > 2/3 and J < 2
| |
3 )
I > 2/3 and J > 2
| |
4 )
I < 2/3 and J < 2
| | see the answer see the solution |
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15)
The area of the plane region bounded
by the curves x + 2y2 = 0
and x + 3y2 = 1 is equal
| | |
1 )
2/3
| |
2 )
4/3
| |
3 )
5/3
| |
4 )
1/3
| | see the answer see the solution |
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16)
The value of
is
| | |
1 )
x + log | sin (x - (π / 4)) | + c
| |
2 )
x - log | cos (x - (π / 4)) | + c
| |
3 )
x + log | cos (x - (π / 4)) | + c
| |
4 )
x - log | sin (x - (π / 4)) | + c
| | see the answer see the solution |
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17)
The statement p → (q → p) is equivalent to
| | |
1 )
p → (p Λ q)
| |
2 )
p → (p ↔ q)
| |
3 )
p → (p → q)
| |
4 )
p → (p V q)
| | see the answer see the solution |
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18)
The value of
|
cot ( cosec-1
|
5
|
+ tan-1
|
2
|
) is
|
|
3
|
3
|
| | |
1 )
4/17
| |
2 )
5/17
| |
3 )
6/17
| |
4 )
3/17
| | see the answer see the solution |
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19)
Let A be a 2 X 2 matrix with real entries.
Let I be 2 X 2 identity matrix. Denote by
tr(A), the sum of diagonal entries of A. Assume that A2 = I.
Statement 1 : If A ≠ I and A ≠ -I, then det A = - 1.
Statement 2 : If A ≠ I and A ≠ -I, then tr(A) ≠ 0.
| | |
1 )
Statement-1
is true, Statement-2
is true; Statement-2
is not a correct explanation
for Statement-1.
| |
2 )
Statement-1
is true, Statement-2
is false.
| |
3 )
Statement-1
is false, Statement-2
is true
| |
4 )
Statement-1
is true, Statement-2
is true; Statement-2
is a correct explanation
for Statement-1.
| | see the answer see the solution |
|
20)
In a shop there are five types of icecreams
available. A child buys six ice creams.
Statement-1 : The number of different ways the child can buy
the six ice creams is 10C5
Statement-2 : The number of different ways the
child can buy the six icecreams
is equal to the number of different ways
of arranging 6 A's and 4 B's in a row.
| | |
1 )
Statement-1
is true, Statement-2
is true; Statement-2
is not a correct explanation
for Statement-1.
| |
2 )
Statement-1
is true, Statement-2
is false.
| |
3 )
Statement-1
is false, Statement-2
is true
| |
4 )
Statement-1
is true, Statement-2
is true; Statement-2
is a correct explanation
for Statement-1.
| | see the answer see the solution |
|
21)
Statement-1 :
Statement-2 :
| | |
1 )
Statement-1
is true, Statement-2
is true; Statement-2
is not a correct explanation
for Statement-1.
| |
2 )
Statement-1
is true, Statement-2
is false.
| |
3 )
Statement-1
is false, Statement-2
is true
| |
4 )
Statement-1
is true, Statement-2
is true; Statement-2
is a correct explanation
for Statement-1.
| | see the answer see the solution |
|
22)
Statement-1 :
For every natural number n ≥ 2,
Statement-2 :
For every natural number n ≥ 2,
| | |
1 )
Statement-1
is true, Statement-2
is true; Statement-2
is not a correct explanation
for Statement-1.
| |
2 )
Statement-1
is true, Statement-2
is false.
| |
3 )
Statement-1
is false, Statement-2
is true
| |
4 )
Statement-1
is true, Statement-2
is true; Statement-2
is a correct explanation
for Statement-1.
| | see the answer see the solution |
|
23)
AB is a vertical pole with B at the
ground level and A at the top.
A man finds that the
angle of elevation of the point A from
a certain point C on the ground is 60°.
He moves
away from the pole along the line BC to a
point D such that CD = 7 m. From D the angle
of elevation of the point A is 45°.
Then the height of the pole is
| |
1 )
7 
|
( - 1 ) m
|
|
|
2
|
|
|
|
2 )
|
7
|
|
1
|
|
2
|
+ 1
|
|
3 )
|
7
|
|
1
|
|
2
|
- 1
|
|
4 )
|
7
|
( + 1 ) m
|
|
|
2
|
|
|
| | see the answer see the solution |
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24)
A die is thrown. Let A be the event that
the number obtained is greater than 3. Let B be
the event that the number obtained is less than 5.
Then P(A ∪ B) is
| | |
1 )
1
| |
2 )
2/5
| |
3 )
3/5
| |
4 )
0
| | see the answer see the solution |
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25)
It is given that the events A and B
are such that P(A) = 1 / 4, P(A | B) = 1 / 2 and
P(B | A) = 2 / 3. Then P(B) is
| | |
1 )
2/3
| |
2 )
1/2
| |
3 )
1/6
| |
4 )
1/3
| | see the answer see the solution |
|
26)
A focus of an ellipse is at the origin.
The directrix is the line x = 4 and the eccentricity is
1/22. Then the length of the semimajor
axis is
| | |
1 )
4/3
| |
2 )
5/3
| |
3 )
8/3
| |
4 )
2/3
| | see the answer see the solution |
|
27)
A parabola has the origin as its focus and
the line x = 2 as directrix. Then the vertex of
the parabola is at
| | |
1 )
( 0 , 1 )
| |
2 )
( 2 , 0 )
| |
3 )
( 0 , 2 )
| |
4 )
( 1 , 0 )
| | see the answer see the solution |
|
28)
The point diametrically opposite to the
point P(1, 0) on the circle
x2 + y2 + 2x + 4y - 3 = 0 is
| | |
1 )
( -3 , -4 )
| |
2 )
( 3 , 4 )
| |
3 )
( 3 , -4 )
| |
4 )
( -3 , 4 )
| | see the answer see the solution |
|
29)
The perpendicular bisector of the line
segment joining P(1, 4) and Q(k, 3) has
y-intercept -4. Then a possible value of k is
| | |
1 )
-2
| |
2 )
-4
| |
3 )
1
| |
4 )
2
| | see the answer see the solution |
|
30)
The first two terms of a geometric
progression add up to 12. The sum of the third and
the fourth terms is 48. If the terms of the
geometric progression are alternately positive
and negative, then the first term is
| | |
1 )
12
| |
2 )
4
| |
3 )
-4
| |
4 )
-12
| | see the answer see the solution |
|
31)
Suppose the cubic x3 - px + q has three
distinct real roots where p > 0 and q > 0.
Then which one
of the following holds ?
| |
1 )
The cubic has maxima at both
 and
|
2 )
The cubic has minima at
and maxima at
|
3 )
The cubic has minima at
and maxima
|
4 )
The cubic has minima at both
and
| | see the answer see the solution |
|
32)
How many real solutions does the equation
x7 + 17x5 + 16x3 +
30x - 560 = 0 have ?
| | |
1 )
3
| |
2 )
5
| |
3 )
7
| |
4 )
1
| | see the answer see the solution |
|
33)
Let
Then which one of the following is true ?
| | |
1 )
f is differentiable at x = 0 but not at x = 1
| |
2 )
f is differentiable at x = 1 but not at x = 0
| |
3 )
f is neither differentiable at x = 0 nor at x = 1
| |
4 )
f is differentiable at x = 0 and at x = 1
| | see the answer see the solution |
|
34)
The solution of the differential equation
satisfying the condition y(1) = 1 is
| | |
1 )
y = xe(x - 1)
| |
2 )
y = x ln x + x
| |
3 )
y = ln x + x
| |
4 )
y = x ln x + x2
| | see the answer see the solution |
|
35)
The differential equation of the family of circles
with fixed radius 5 units and centre of
the line y = 2 is
| | |
1 )
(y - 2)2 y'2 = 25 - (y - 2)2
| |
2 )
(x - 2)2 y'2 = 25 - (y - 2)2
| |
3 )
(x - 2) y'2 = 25 - (y - 2)2
| |
4 )
(y - 2) y'2 = 25 - (y - 2)2
| | see the answer see the solution | ANSWERS | 1)
2 | 2)
2 | 3)
1 | 4)
4 | | 5)
2 | 6)
4 | 7)
1 | 8)
1 | | 9)
2 | 10)
2 | 11)
1 | 12)
2 | | 13)
2 | 14)
4 | 15)
2 | 16)
1 | | 17)
4 | 18)
3 | 19)
2 | 20)
3 | | 21)
4 | 22)
1 | 23)
4 | 24)
1 | | 25)
4 | 26)
3 | 27)
4 | 28)
1 | | 29)
2 | 30)
4 | 31)
2 | 32)
4 | | 33)
1 | 34)
2 | 35)
1 |
Solution
1 back to 1
Mean of the numbers: (a+b+8+5+10)/5=6
a+b=7 ......... (i)
variance = 6.8 = {(a-6)2+(b-6)2+(8-6)2+(5-6)2+(10-6)2}
a2+b2+93-12(a+b)=34 ........... (ii)
By solving (i) and (ii)
a=3 , b=4
| 2 back to 2
a, b and c
are coplanar, so
a.(b x c) = 0
So, (αi + 2j + βk).(i - j + k) = 0
=> α + β = 2
a bisect the angle between b and c
So a = (λ/ )
(b + c)
=> a = (λ/)
(i+2j+k)
=> (αi + 2j + βk) =
(λ/)
(i+2j+k)
=> So, λ =
=> So, α=1 , β=1
| 3 back to 3
The vector and
are parallel and are
in opposite direction.
So angle between them is 180° = π
| 4 back to 4
Let A = (5, 1, a)
Let B = (3, b, 1)
Let C = ( 0 , 17/2 , -13/2 )
Point A , B and C are in straight line
So,
=> 51a - 65b = 46
So, a = 6, b = 4
| 5 back to 5
As the two given lines are intersecting, so
So
=> 1(4-3k)-1(2k-9)-2(k2-6)=0
=> 2k2 + 5k - 25 = 0
=> k=-5 OR k=5/2
| 6 back to 6
The conjugate of a complex number is given by changing the sign of the imaginary part.
For example: conjugate of a complex number a+ib is a-ib
z = 1/(i-1) = (i+1)/((i-1)(i+1)) = (i+1)/(-2) = -1/2-i/2
So conjugate complex number = -1/2+i/2 = (i-1)/2
So conjugate complex number = ((i-1)(i+1))/(2(i+1)) = -2/2((i+1)) = -1/(i+1)
| 7 back to 7
In equivalence relation a set satisfies following three conditions:
a. Every member of the set is related to itself.
b. Whenever a is related to b then b is also related to a.
c. If a is related to b and b is related to c,
then a is related to c.
| 8 back to 8
y = 4x + 3
=> x = (y-3)/4
=> f(y) = (y-3)/4
| 9 back to 9
p : x is an irrational number
q : y is a transcendental number
r : x is a rational number iff y is a transcendental number
=> r : ~ p ↔ q
s1 : q or p
s2 : ~ (p↔~q)
| p |
q |
~p |
~q |
r:~p↔q |
s1:q or p |
p↔~q |
s2:~(p↔~q)
|
| T | T | F | F | F | T | F | T |
| T |
F |
F |
T |
T |
T |
T |
F |
| F | T | T | F | T | T | T | F |
| F | F | T | T | F | F | F | T |
Based upon above chart
s1 and r are not equivalent
s2 and r are not equivalent
| 10 back to 10
x - cy - bz = 0
cx - y - az = 0
bx + ay - z = 0
So,
(1 + a2) + c(-c - ab) - b(ca + b) = 0
a2 + b2 + c2 + 2abc = 0
| 11 back to 11
As det A = ±1
So inverse of matrix A exists
As all entries of matrix A are integers and
Adjoint A is matrix of transpose of co-factors of matrix A.
So, Entries of Adjoint A is also integers.
| 12 back to 12
Let α and β are roots of: x2 - 6x + a = 0
Let α and γ are roots of: x2 - cx + 6 = 0
So, α + β = 6 ........ (i)
α.β = a ........ (ii)
α + γ = c ........ (iii)
α.γ = 6 ........ (iv)
β/γ = 4/3 ........ (v)
Now using (ii) and (iii)
α.β/α.γ = a/6
=> β/γ = a/6
=> using (v), 4/3 = a/6
=> So, a = 8
Now , x2 - 6x + a = 0
x2 - 6x + 8 = 0
x = 2 OR 6
As α.β = 8
So α = 2
| 13 back to 13
Count of M = 1
Count of I = 4
Count of S = 4
Count of P = 2
Number of ways MISSISSIPPI can be jumbled without using S = (7!)/(4! x 2!)
Now number od ways four S can be placed in MIIIPPI = 8C4
So total number of ways = (7!)/(4! x 2!) x 8C4
So total number of ways = 7 . 6C4 x 8C4
| 14 back to 14
|
sinx
|
< 1 when x ∈ (0, 1) => So,
|
sin
|
<
|
|
x
|
|
So,
|
cosx
|
<
|
1
|
when x ∈ (0, 1)
|
|
|
|
So,
| 15 back to 15
x + 2y2 = 0 .............(i)
x + 3y2 = 1 .............(ii)
By solving (i) and (ii)
x = -2 and y = ±1
| 16 back to 16
| 17 back to 17
| p |
q |
p V q |
q→q |
p→(q→q) |
p→(p V q) |
| F |
F |
F |
T |
T |
T |
| F |
T |
T |
F |
T |
T |
| T |
F |
T |
T |
T |
T |
| T |
T |
T |
T |
T |
T |
| 18 back to 18
cot(cosec-15/3 + tan-12/3) = cot(tan-13/4 + tan-12/3)
= cot(tan-1(3/4 + 2/3)/(1-(3/4)x(2/3))) = cot(tan-117/6) = 6/17
| 19 back to 19
| 20 back to 20
| 21 back to 21
| 22 back to 22
| 23 back to 23
| 24 back to 24
A = {4, 5, 6}
B = {1, 2, 3, 4}
A ∪ B = {1, 2, 3, 4, 5, 6}
P(A ∪ B) = 1
| 25 back to 25
| 26 back to 26
| 27 back to 27
| 28 back to 28
| 29 back to 29
Let the equation of perpendicular bisector is y = mx + c
c = -4
So, y = mx - 4 .............(i)
Center point between P(1, 4) and Q(k, 3) will be ((1+k)/2, 7/2)
In equation (i) passes thru ((1+k)/2, 7/2)
So, 7/2 = m((1+k)/2) - 4
7 = m + mk - 8
m + mk - 15 = 0 .................(ii)
m = gradient of the perpendiculat line
m = - (gradient of PQ) = -(1-k)/(4-3) = k - 1
Replace value of m in equation (ii)
k - 1 + k(k - 1) - 15 = 0
k2 = 16
k = ±4
| 30 back to 30
a + ar = 12
a(1 + r) = 12
ar2 + ar3 = 48
ar2 (1 + r) = 48
r2 = 4
r = ±2
=> r = -2
=> a = -12
| 31 back to 31
f(x) = x3 - px + q
So, f'(x) = 3x2 - p = 0
=> x = ±
=> f"(x) = 6x
=> f"(x) at x = is positive
=> So minima is at x =
=> f"(x) at x = is negative
=> So maxima is at x =
| 32 back to 32
| 33 back to 33
| 34 back to 34
| 35 back to 35
|
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