The negation of the statement “If I become a teacher, then I will open a school” is
I will become a teacher and I will not open a school
Either I will not become a teacher or I will not open a school
Neither I will become a teacher nor I will open a school
I will not become a teacher or I will open a school
A.
I will become a teacher and I will not open a school
Let us assume that
p: I become a teacher' and
q: I will open a school
Then, we can easily as certain that
Negation of (p →q)
~(~p ∨ q) = p ∧ ~q
Which means that ' l' will become a teacher and I will not open a school.
Let x1, x2, ......, x_{n} be n observations, and let be their arithmetic mean and σ2 be their variance.
Statement 1: Variance of 2x1, 2x2, ......, 2xn is 4 σ^{2}.
Statement 2: Arithmetic mean of 2x1, 2x2, ......, 2xn is 4.
Statement 1 is false, statement 2 is true
Statement 1 is true, statement 2 is true; statement 2 is a correct explanation for statement 1
Statement 1 is true, statement 2 is true; statement 2 is not a correct explanation for statement 1
Statement 1 is true, statement 2 is false
D.
Statement 1 is true, statement 2 is false
If 100 times the 100^{th} term of an AP with non zero common difference equals the 50 times its 50^{th} term, then the 150^{th} term of this AP is
–150
150
times its 50th term
0
D.
0
The 150 th term of this AP
Let a be the first term and d be the common difference of the given AP, then
T_{100} = a+ (100-1)d = a + 99d
T_{50} = a +(50-1)d = a +49 d
T_{150} = a + (150-1) d = a +149 d
Now, according to the question,
100 x T_{100} = 50 x T_{50}
⇒ 100 (a +99d) = 50(a +49d)
2(a +99d) = (a+ 49d)
2a +198 d =a +49d
a +149d = 0
Statement 1: The sum of the series 1 + (1 + 2 + 4) + (4 + 6 + 9) + (9 + 12 + 16) + ...... + (361 + 380 +400) is 8000.
Statement 2: , for any natural number n.
Statement 1 is false, statement 2 is true
Statement 1 is true, statement 2 is true; statement 2 is a correct explanation for statement 1
Statement 1 is true, statement 2 is true; statement 2 is not a correct explanation for statement 1
Statement 1 is true, statement 2 is false
B.
Statement 1 is true, statement 2 is true; statement 2 is a correct explanation for statement 1
Statement 1 has 20 terms whose sum is 8000 And statement 2 is true and supporting statement 1.
k^{th} bracket is (k – 1)^{2} + k(k – 1) + k^{2} = 3k^{2} – 3k + 1.
An equation of a plane parallel to the plane x – 2y + 2z – 5 = 0 and at a unit distance from the origin is
x – 2y + 2z – 3 = 0
x – 2y + 2z + 1 = 0
x – 2y + 2z – 1 = 0
x – 2y + 2z + 5 = 0
A.
x – 2y + 2z – 3 = 0
Perpendicular distance of the plane ax +by + cz +d =0 from the point
(x,y,z) is d =
Equation of plane parallel to x – 2y + 2z – 5 = 0 is x – 2y + 2z + k = 0 ...... (1)
perpendicular distance from O(0, 0, 0) to (1) is 1
Statement I An equation of a common tangent to the parabola and the ellipse 2x^{2} +y^{2} =4 is .
Statement II If the line is a common tangent to the parabola and the ellipse 2x^{2} +y^{2} =4, then m satisfies m^{4} +2m^{2} =24
Statement 1 is false, statement 2 is true
Statement 1 is true, statement 2 is true; statement 2 is a correct explanation for statement 1
Statement 1 is true, statement 2 is true; statement 2 is not a correct explanation for statement 1
Statement 1 is true, statement 2 is false
C.
Statement 1 is true, statement 2 is true; statement 2 is not a correct explanation for statement 1
If n is a positive integer, then is
an irrational number
an odd positive integer
an even positive integer
a rational number other than positive integers
A.
an irrational number
The equation e^{sinx}-e^{-sinx }-4 = 0 has
infinite number of real roots
No real root
exactly one real root
exactly four real roots
B.
No real root
In a ∆PQR, if 3 sin P + 4 cos Q = 6 and 4 sin Q + 3 cos P = 1, then the angle R is equal to
5π/6
π/6
π/4
3π/4
B.
π/6
3 sin P + 4 cos Q = 6 ...... (1)
4 sin Q + 3 cos P = 1 ...... (2)
From (1) and (2) ∠P is obtuse.
(3 sin P + 4 cos Q)2+ (4 sin Q + 3 cos P)2= 37
⇒9 + 16 + 24 (sin P cos Q + cos P sin Q) = 37
⇒ 24 sin (P + Q) = 12
If the line 2x + y = k passes through the point which divides the line segment joining the points (1, 1) and (2, 4) in the ratio 3 : 2, then k equals
29/5
5
6
11/5
C.
6
Line L : 2x +y = k passes through the point (say P) which divides a lie segment (say AB) in ratio 3:2 where A (1,1) and B (2,4).
Using section formula, the coordinates of the point P which divides AB internally in the ratio 3:2 are
Also, since the line L passes through P, hence substituting the coordinates of in the equation of line L: 2x +y = k,
we get
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