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 :
Let\(f(x) = \begin{vmatrix} a & -1 & 0\\ ax & a & -1\\ ax^2 & ax & a \end{vmatrix}\)a ∈ R. Then the sum of the square of all the values of a, for which 2f′(10) –f′(5) + 100 = 0, is
Let y = y(x), x > 1, be the solution of the differential equation\((x-1)\frac{dy}{dx} + 2xy = \frac{1}{x-1}\)with \(y(2) = \frac{1+e^4}{2e^4}\). If \(y(3) = \frac{e^α + 1}{βe^α}\) , then the value of α + β is equal to ____.
Let f(x) and g(x) be two real polynomials of degree 2 and 1 respectively. If f(g(x)) = 8x2 – 2x and g(f(x)) = 4x2 + 6x + 1, then the value of f(2) + g(2) is ____________ .
Let the abscissae of the two points P and Q on a circle be the roots of x2 – 4x – 6 = 0 and the ordinates of P and Q be the roots of y2 + 2y – 7 = 0. If PQ is a diameter of the circle x2 + y2 + 2ax + 2by + c = 0, then the value of (a + b – c) is
Let\(M = \begin{bmatrix} 0 & -\alpha \\ \alpha & 0 \\ \end{bmatrix}\)where α is a non-zero real number an\(N = \sum\limits_{k=1}^{49} M^{2k}. \) If \((I - M^2)N = -2I\)then the positive integral value of α is ____ .
The sum of the infinite series\(1 + \frac{5}{6} + \frac{12}{6^2} + \frac{22}{6^3} + \frac{35}{6^4} +\frac{51}{6^5} + \frac{70}{6^6}+…..\)is equal to