The area bounded by the curves y = |x2 – 1| and y = 1 is
Let \(S = z ∈ C: |z-3| <= 1\) and \(z (4+3i)+z(4-3)≤24.\)If α + iβ is the point in S which is closest to 4i, then 25(α + β) is equal to ______.
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
The sum of all the elements of the set {α ∈ {1, 2, …, 100} : HCF(α, 24) = 1} is
The remainder on dividing 1 + 3 + 32 + 33 + … + 32021 by 50 ____ is
Let a circle C : (x – h)2 + (y – k)2 = r2, k > 0, touch the x-axis at (1, 0). If the line x + y = 0 intersects the circle C at P and Q such that the length of the chord PQ is 2, then the value of h + k + r is equal to ____.
In an examination, there are 10 true-false type questions. Out of 10, a student can guess the answer of 4 questions correctly with probability 3/4 and the remaining 6 questions correctly with probability ¼. If the probability that the student guesses the answers of exactly 8 questions correctly out of 10 is \(\frac{27k}{4^{10}}\),then k is equal to
Let the hyperbola \(H:\frac{x^2}{a^2}−y^2=1\)and the ellipse \(E:3x^2+4y^2=12\) be such that the length of latus rectum of H is equal to the length of latus rectum of E. If eH and eE are the eccentricities of H and E respectively, then the value of \(12 (e^{2}_H+e^{2}_E)\) 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 f: ℝ → ℝ be defined as\(f(x) = \left\{ \begin{array}{ll} [e^x] & x < 0 \\ [a e^x + [x-1]] & 0 \leq x < 1 \\ [b + [\sin(\pi x)]] & 1 \leq x < 2 \\ [[e^{-x}] - c] & x \geq 2 \\ \end{array} \right.\)Where a, b, c ∈ ℝ and [t] denotes greatest integer less than or equal to t. Then, which of the following statements is true?
Let the solution curve y = y(x) of the differential equation\([ \frac{x}{\sqrt{x² -y²}} + e^\frac{y}{x} ] x \frac{dy}{dx} = x + [ \frac{x}{\sqrt{x² -y²}} + e^\frac{y}{x} ]y\)pass through the points (1, 0) and (2α, α), α> 0. Then α is equal to
Let the eccentricity of the hyperbola\(H : \frac{x²}{a²} - \frac{y²}{b²} = 1\)be √(5/2) and length of its latus rectum be 6√2, If y = 2x + c is a tangent to the hyperbola H. then the value of c2 is equal to
If two distinct points Q, R lie on the line of intersection of the planes –x + 2y – z = 0 and 3x – 5y + 2z = 0 and\(PQ = PR = \sqrt{18}\)where the point P is (1, –2, 3), then the area of the triangle PQR is equal to