Let W denote the words in the English dictionary. Define the relation R by :
R = {(x, y) ∈ W × W | the words x and y have at least one letter in common}. Then R is
not reflexive, symmetric and transitive
reflexive, symmetric and not transitive
reflexive, symmetric and transitive
reflexive, not symmetric and transitive
B.
reflexive, symmetric and not transitive
Clearly (x, x) ∈ R ∀ x ∈ W. So, R is reflexive. Let (x, y) ∈ R,
then (y, x) ∈ R as x and y have at least one letter in common. So, R is symmetric. But R is not transitive for example
Let x = DELHI, y = DWARKA and z = PARK then
(x, y) ∈ R and (y, z) ∈ R but (x, z) ∉ R
A body falling from rest under gravity passes a certain point P. It was at a distance of 400 m from P, 4s prior to passing through P. If g = 10 m/s^{2} , then the height above the point P from where the body began to fall is
720 m
900 m
320 m
680 m
A.
720 m
In an ellipse, the distance between its foci is 6 and minor axis is 8. Then its eccentricity is
3/5
1/5
2/5
4/5
A.
3/5
2ae = 6 ⇒ ae = 3
2b = 8 ⇒ b = 4
b^{2}= a^{2}(1 − e^{2})
16 = a2 − a^{2}e^{2}
a^{2}= 16 + 9 = 25
a = 5
∴e = 3/a = 3/5
Suppose a population A has 100 observations 101, 102, … , 200, and another population B has 100 observations 151, 152, … , 250. If VA and VB represent the variances of the two populations, respectively, then V_{A}/V_{B} is
1
9/4
4/9
2/3
A.
1
If the roots of the quadratic equation x^{2} + px + q = 0 are tan30° and tan15°, respectively then the value of 2 + q − p is
2
3
0
1
B.
3
x^{2} + px + q = 0
tan 30° + tan 15° = − p
tan 30° ⋅ tan 15° = q
⇒ − p = 1 − q
⇒ q − p = 1
∴ 2 + q − p = 3
A straight line through the point A(3, 4) is such that its intercept between the axes is bisected at A. Its equation is
x + y = 7
3x − 4y + 7 = 0
4x + 3y = 24
3x + 4y = 25
C.
4x + 3y = 24
The equation of axes is xy = 0
⇒ the equation of the line is
All the values of m for which both roots of the equations x^{2} − 2mx + m^{2} − 1 = 0 are greater than −2 but less than 4, lie in the interval
−2 < m < 0
m > 3
−1 < m < 3
1 < m < 4
C.
−1 < m < 3
Equation x^{2} − 2mx + m^{2} − 1 = 0
(x − m)^{2} − 1 = 0
(x − m + 1) (x − m − 1) = 0
x = m − 1, m + 1 − 2 < m − 1 and m + 1 < 4
m > − 1 and m < 3 − 1 < m < 3.
The two lines x = ay + b, z = cy + d; and x = a′y + b′, z = c′y + d′ are perpendicular to each other if
aa′ + cc′ = −1
aa′ + cc′ = 1
A.
aa′ + cc′ = −1
Equation of lines
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