Let the line\(\frac{x - 3}{7} = \frac{y - 2}{-1} = \frac{z - 3}{-4}\)intersect the plane containing the lines\(\frac{x - 4}{1} = \frac{y + 1}{-2} = \frac{z}{1}\) and \(4ax-y+5z-7a = 0 = 2x-5y-z-3, a∈R\)at the point P(α, β, γ). Then the value of α + β + γ equals _____.
If y = y(x) is the solution of the differential equation
\(2x^2\frac{dy}{dx}-2xy+3y^2=0\) such that \(y(e)=\frac{e}{3},\)
then y(1) is equal to
Let ƒ :R→R be a function defined by \(f(x) = \frac{2e^{2x}}{e^{2x} + e^x}\)Then \(f\left(\frac{1}{100}\right) + f\left(\frac{2}{100}\right) + f\left(\frac{3}{100}\right) + \ldots + f\left(\frac{99}{100}\right)\) is equal to ________.
The number of solutions of the equation sin x = cos2 x in the interval (0, 10) is _____.
Let\(f(x) = 2x^2 - x - 1\ and\ S = \{ n \in \mathbb{Z} : |f(n)| \leq 800 \}\)Then, the value of ∑n∈S f(n) is equal to ________.
Let \(\stackrel{→}{a} = \hat{i} + \hat{j} + \hat{2k}, \stackrel{→}{b} = \hat{2i} - \hat{3j} + \hat{k}\)and \(\stackrel{→}{c}= \hat{i} - \hat{j} + \hat{k}\)be three given vectors.Let \(\stackrel{→}{v}\) be a vector in the plane of \(\stackrel{→}{a}\) and \(\stackrel{→}{b}\) whose projection on \(\stackrel{→}{c}\) is \(\frac{2}{\sqrt3}\).If \(\stackrel{→}{v}.\hat{j}\) = 7 , then \(\stackrel{→}{v}.(\hat{i}+\hat{k})\) is equal to :
If the two lines \(l1:\frac{(x−2)}{3}=\frac{(y+1)}{−2},z=2 \)and\( l2:\frac{(x−1)}{1}=\frac{(2y+3)}{α}=\frac{(z+5)}{2} \)are perpendicular, then an angle between the lines l2 and \(l3:\frac{(1−x)}{3}=\frac{(2y−1)}{−4}=\frac{z}{4} \)is
Let the image of the point P(1, 2, 3) in the line \(L:\frac{x−6}{3}=\frac{y−1}{2}=\frac{z−2}{3} \)be Q. Let R (α, β, γ) be a point that divides internally the line segment PQ in the ratio 1 : 3. Then the value of 22(α + β + γ) is equal to ________.
A vector \(\vec{a}\)is parallel to the line of intersection of the plane determined by the vectors\(\hat{i},\hat{i}+\hat{j} \)and the plane determined by the vectors \(\hat{i}−\hat{j},\hat{i}+\hat{k}\). The obtuse angle between \(\vec{a}\) and the vector \(\vec{b}=\hat{i}−2\hat{j}+2\hat{k}\)is
Let the slope of the tangent to a curve y = f(x) at (x, y) be given by 2 tanx(cosx – y). If the curve passes through the point (π/4, 0) then the value of \(\int_{0}^{\frac{\pi}{2}} y \,dx\)is equal to :
Let the common tangents to the curves 4(x2 + y2) = 9 and y2 = 4x intersect at the point Q. Let an ellipse, centered at the origin O, has lengths of semi-minor and semi-major axes equal to OQ and 6, respectively. If e and I respectively denote the eccentricity and the length of the latus rectum of this ellipse, then \(\frac{1}{e^2}\) is equal to
Let\(A = \{z \in \mathbb{C} : |\frac{z+1}{z-1}| < 1\}\)and\(B = \{z \in \mathbb{C} : \text{arg}(\frac{z-1}{z+1}) = \frac{2\pi}{3}\}\)Then \(A∩B\) is :