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In the following question, a Statement of Assertion (A) is given followed by a corresponding Reason (R) just below it. Read the Statements carefully and mark the correct answer
Assertion(A):If C_{r} is the coefficient of x^{r} in the expansion of (1 + x)^{20}
Reason(R) : Cr = C_{n − r} for any positive integer n
The area (in square units) bounded by the curves y^{2} = 4x and x^{2} = 4y in the plane is
In the following question, a Statement of Assertion (A) is given followed by a corresponding Reason (R) just below it. Read the Statements carefully and mark the correct answer
Assertion(A) :The inverse of does not exist.
Reason(R) :The matrix is non singular.
If the line 3x4y=λ touches the circle x^{2}+y^{2}4x8y5=0, λ can have the values
The length of the tangent from (0,0) to the circle 2x^{2} + 2y^{2} + x  y + 5 = 0 is
The differential equation which represents the family of plane curves y=exp. (cx) is
y = e^{cx}
dy/dx = c. e^{cx}
y' = cy
The fundamental period of the function f(x) = 2 cos 1/3(x  π) is
In the following question, a Statement1 is given followed by a corresponding Statement2 just below it. Read the statements carefully and mark the correct answer
Tangents are drawn from the point (17,7) to the circle x^{2}+y^{2}=169.
Statement1:
The tangents are mutually perpendicular.
Statement2:
The locus of the points from which mutually perpendicular tangents can be drawn to the given circle is x^{2}+y^{2}=338.
Clearly, m_{1}m_{2} =  1.
Hence, the two tangents arc mutually perpendicular.
Statement 1 is true.
Now, the locus of the point of intersection of two mutually perpendicular tangents to the circle x^{2} + y^{2} = r^{2} is the director circle i.e. the circle x^{2} +y^{2 }= 2r^{2}.
For the given circle r = 13. ..
Its director circle is x^{2} + y^{2} = 338.
Hence, statement 2 is true and a cogect explanation of statement as the point (17, 7) lies on the director circle of the circle (i).
The value of a for which the system of equations
a^{3}x+(a+1)^{3}y+(a+2)^{3}z = 0
ax+(a+1)y+(a+2)z = 0
x+y+z = 0
has a nonzero solution, is
The system of equation has a nonzero solution
The pole of the line 2x + 3y − 4 = 0 with respect to the parabola y^{2} = 4 x is
Total outcomes = 36
Doublet are 6 (1,1),(2,2),(3,3),(4,4),(5,5),(6,6)
Probability of getting doublet = 6/36
= 1/6
Covariance is a quantitative measure of the extent to which the deviation of one variable from its mean matches the deviation of the other from its mean. It is a mathematical relationship that is defined as:
Cov(X,Y) = E[(X − E[X])(Y − E[Y])]
Correlation between two random variables, ρ(X,Y) is the covariance of the two variables normalized by the variance of each variable. This normalization cancels the units out and normalizes the measure so that it is always in the range [0, 1]:
The two opposite vertices of a square on xyplane are A(1,1) and B(5,3), the equation of other diagonal (not passing through A and B) is
Given: Here,AB is the diagonal of square.
The vertices of a square A
Let the mid−point of AB be EThen coordinates of E are
Therefore equation of other diagonal is
If the normal to the curve y=f(x) at the point (3,4) makes an angle 3π/4 with the positive xaxis, then f'(3)
Given y = f(x)
differentiating w.r.t x
y' = f'(x) which is the slope of the tangent
Hence the slope of the normal is  1/f'(x) = 3pi/4 = 1
therefore f'(x) = 1
Hence f'(3) = 1
Assume e^{4/5} = 2/5. If x, y satisfy, y = e^{x} and the minimum value of (x^{2} + y^{2}) is expressed in the form of m/n then (2m  n)/5 equals (where m & n are coprime natural numbers)
OP^{2} = x^{2} + y^{2}
y = e^{x}, y' = e^{x},
Let ƒ(x) be nonconstant thrice differentiable function defined on (–∞, ∞) such that ƒ(x) = ƒ(6 – x) and ƒ'(0) = 0 = ƒ'(2) = ƒ'(5). If 'n' is the minimum number of roots of (ƒ"(x))^{2} + ƒ'(x)ƒ"'(x) = 0 in the interval x ∈ [0, 6] then sum of digits of n equals
ƒ(x) = ƒ(6 – x)
⇒ ƒ'(x) = –ƒ'(6 – x) .... (1)
put x = 0, 2, 5
ƒ'(0) = ƒ'(6) = ƒ'(2) = ƒ'(4) = ƒ'(5) = ƒ'(1) = 0
and from equation (1) we get ƒ'(3) = –ƒ'(3)
⇒ ƒ'(3) = 0
So ƒ'(x) = 0 has minimum 7 roots in
x ∈ [0, 6] ⇒ ƒ"(x) has min 6 roots in x ∈ [0,6]
h(x) = ƒ'(x).ƒ"(x)
h'(x) = (ƒ"(x))^{2} + ƒ'(x) ƒ"'(x)
h(x) = 0 has 13 roots in x ∈ [0, 6]
h'(x) = 0 has 12 roots in x ∈ [0, 6]
If the coordinate of the vertex of the parabola whose parametric equation is x = t^{2} – t + 1 and y = t^{2} + t + 1, t ∈ R is (a, b) then (2a + 4b) equals
x = t^{2} – t + 1 .... (1)
y = t^{2} + t + 1 .... (2)
y – x = 2t & x + y = 2(t^{2} + 1)
________on elminating 't' we get
⇒ (x + y – 2) = 2(y  x)/2^{2}
(x – y)^{2} = 2(x + y – 2)
Axis : x – y = 0
Tangent at vertex : x + y – 2 = 0
Vertex : (1, 1) = (x, y)
If a, b, c, x, y, z are nonzero real numbers and then the value of (a^{3} + b^{3} + c^{3} + abc) equals
x^{2}(y + z)y^{2}(z + x)z^{2}(x + y) = a^{3}b^{3}c^{3} = x^{3}y^{3}z^{3}
⇒ (x + y) (y + z) (z + x) = xyz
⇒ x^{2}(y + z) + y^{2}(z + y) + z^{2}(x + y) + xyz = 0
⇒ a^{3} + b^{3} + c^{3} + abc = 0
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