Suppose \( \mathop {\lim }\limits_{x \to 0} {{F(x)} \over {{x^3}}} \) exists and is equal to L, where
\( F(x) = \left| {\matrix{ {a + \sin {x \over 2}} & { - b\cos x} & 0 \cr { - b\cos x} & 0 & {a + \sin {x \over 2}} \cr 0 & {a + \sin {x \over 2}} & { - b\cos x} \cr } } \right| \).
Then, \( - \)112 L is equal to ___________.
Let \( f(t) = \int\limits_0^t {{e^{{x^3}}}\left( {{{{x^8}} \over {{{({x^6} + 2{x^3} + 2)}^2}}}} \right)dx} \). If \( f(1) + f'(1) = \alpha e - {1 \over 6} \), then the value of 150\( \alpha \) is equal to ___________.
If for some \( \alpha \) > 0, the area of the region \( \{ (x,y):|x + \alpha | \le y \le 2 - |x|\} \) is equal to \( {3 \over 2} \), then the area of the region \( \{ (x,y):0 \le y \le x + 2\alpha ,\,|x| \le 1\} \) is equal to ____________.
Let \( \alpha = \tan \left( {{{5\pi } \over {16}}\sin \left( {2{{\cos }^{ - 1}}\left( {{1 \over {\sqrt 5 }}} \right)} \right)} \right) \) and \( \beta = \cos \left( {{{\sin }^{ - 1}}\left( {{4 \over 5}} \right) + {{\sec }^{ - 1}}\left( {{5 \over 3}} \right)} \right) \) where the inverse trigonometric functions take principal values. Then, the equation whose roots are \( \alpha \) and \( \beta \) is :
Let the eccentricity of the ellipse \( {x^2} + {a^2}{y^2} = 25{a^2} \) be b times the eccentricity of the hyperbola \( {x^2} - {a^2}{y^2} = 5 \), where a is the minimum distance between the curves y = ex and y = logex. Then \( {a^2} + {1 \over {{b^2}}} \) is equal to :
Let \( \alpha \)1, \( \alpha \)2 (\( \alpha \)1 < \( \alpha \)2) be the values of \( \alpha \) fo the points (\( \alpha \), \( - \)3), (2, 0) and (1, \( \alpha \)) to be collinear. Then the equation of the line, passing through (\( \alpha \)1, \( \alpha \)2) and making an angle of \( {\pi \over 3} \) with the positive direction of the x-axis, is :
For two positive real numbers a and b such that \( {1 \over {{a^2}}} + {1 \over {{b^3}}} = 4 \), then minimum value of the constant term in the expansion of \( {\left( {a{x^{{1 \over 8}}} + b{x^{ - {1 \over {12}}}}} \right)^{10}} \) is :
Let \( f(x) = 4{x^3} - 11{x^2} + 8x - 5,\,x \in R \). Then f :
If xy4 attains maximum value at the point (x, y) on the line passing through the points (50 + \( \alpha \), 0) and (0, 50 + \( \alpha \)), \( \alpha \) > 0, then (x, y) also lies on the line :
Let m and M respectively be the minimum and the maximum values of \( f(x) = {\sin ^{ - 1}}2x + \sin 2x + {\cos ^{ - 1}}2x + \cos 2x,\,x \in \left[ {0,{\pi \over 8}} \right] \). Then m + M is equal to :
Let S be the set of all integral values of \( \alpha \) for which the sum of squares of two real roots of the quadratic equation \( 3{x^2} + (\alpha - 6)x + (\alpha + 3) = 0 \) is minimum. Then S :
Let \( {S_1} = \left\{ {x \in R - \{ 1,2\} :{{(x + 2)({x^2} + 3x + 5)} \over { - 2 + 3x - {x^2}}} \ge 0} \right\} \) and \( {S_2} = \left\{ {x \in R:{3^{2x}} - {3^{x + 1}} - {3^{x + 2}} + 27 \le 0} \right\} \). Then, \( {S_1} \cup {S_2} \) is equal to :
The real part of the complex number \( {{{{(1 + 2i)}^8}\,.\,{{(1 - 2i)}^2}} \over {(3 + 2i)\,.\,\overline {(4 - 6i)} }} \) is equal to :
Let \( A = \left[ {\matrix{ 1 & { - 2} & \alpha \cr \alpha & 2 & { - 1} \cr } } \right] \) and \( B = \left[ {\matrix{ 2 & \alpha \cr { - 1} & 2 \cr 4 & { - 5} \cr } } \right],\,\alpha \in C \). Then the absolute value of the sum of all values of \( \alpha \) for which det(AB) = 0 is :
The sum of all the elements of the set {α ∈ {1, 2, …, 100} : HCF(α, 24) = 1} is
The number of 5-digit natural numbers, such that the product of their digits is 36, is _____ .
The equations of the sides AB, BC and CA of a triangle ABC are 2x + y = 0, x + py = 15a and x – y = 3, respectively. If its orthocentre is\((2, a),−\frac{1}{2}<a<2 \)then p is equal to _______.
Let S be the set of all the natural numbers, for which the line \(\frac{x}{a}+\frac{y}{b}=2 \)is a tangent to the curve\((\frac{x}{a})^n+(\frac{y}{b})^n=2 \)at the point (a, b), ab ≠ 0. Then :
Let P and Q be any points on the curves (x – 1)2 + (y + 1)2 = 1 and y = x2, respectively. The distance between P and Q is minimum for some value of the abscissa of P in the interval
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 _______.