If\(\sum_{k=1}^{10} \frac{k}{k^4 + k^2 + 1} = \frac{m}{n}\)where m and n are co-prime, then m + n is equal to
LetA =\(\begin{pmatrix} 4 & -2 \\ \alpha & \beta \\ \end{pmatrix}\)If A2 + γA + 18I = 0, then det (A) is equal to ______.
An ellipse\(E:\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\)passes through the vertices of the hyperbola\(H:\frac{x^2}{49} - \frac{y^2}{64} = -1\)Let the major and minor axes of the ellipse E coincide with the transverse and conjugate axes of the hyperbola H, respectively. Let the product of the eccentricities of E and H be 1/2. If the length of the latus rectum of the ellipse E, then the value of 113l is equal to _____.
Let P1 be a parabola with vertex (3, 2) and focus (4, 4) and P2 be its mirror image with respect to the line x + 2y = 6. Then the directrix of P2 is x + 2y = _______.
Let \(S ={ (\begin{matrix} -1 & 0 \\ a & b \end{matrix}), a,b, ∈(1,2,3,.....100)}\) and let \(T_n = {A ∈ S : A^{n(n + 1)} = I}. \)Then the number of elements in \(\bigcap_{n=1}^{100}\) \(T_n \) is
Let n ≥ 5 be an integer. If 9n – 8n – 1 = 64α and 6n – 5n – 1 = 25β, then α – β is equal to
Let the solution curve y = y(x) of the differential equation (4 + x2)dy – 2x(x2 + 3y + 4)dx = 0 pass through the origin. Then y(2) is equal to _______.
If \(\frac{1}{2\times 3 \times 4} + \frac{1}{3\times 4 \times 5 } + \frac{1}{4 \times 5 \times 6 }+ \dots + \frac{1}{100 \times 101 \times 102} = \frac{k}{101}\) then 34 k is equal to ____________.
Let P be the plane passing through the intersection of the planes
r→.(i+3k−k)=5 and r→ .(2i−j+k)=3,
and the point (2, 1, –2). Let the position vectors of the points X and Y be
i−2j+4k and 5i−j+2k
respectively. Then the points