Consider three circles:
\( {C_1}:{x^2} + {y^2} = {r^2} \)
\( {C_2}:{(x - 1)^2} + {(y - 1)^2} = {r^2} \)
\( {C_3}:{(x - 2)^2} + {(y - 1)^2} = {r^2} \)
If a line L : y = mx + c be a common tangent to C1, C2 and C3 such that C1 and C3 lie on one side of line L while C2 lies on other side, then the value of \( 20({r^2} + c) \) is equal to :
If a random variable X follows the Binomial distribution B(5, p) such that P(X = 0) = P(X = 1), then \( {{P(X = 2)} \over {P(X = 3)}} \) is equal to :
Let A and B be two square matrices of order 2. If \( det\,(A) = 2 \), \( det\,(B) = 3 \) and \( \det \left( {(\det \,5(det\,A)B){A^2}} \right) = {2^a}{3^b}{5^c} \) for some a, b, c, \( \in \) N, then a + b + c is equal to :
\( \mathop {\lim }\limits_{n \to \infty } \sum\limits_{r = 1}^n {{r \over {2{r^2} - 7rn + 6{n^2}}}} \) is equal to :
The probability distribution of X is :
For the minimum possible value of d, sixty times the mean of X is equal to _______________.
Let \( {{dy} \over {dx}} = {{ax - by + a} \over {bx + cy + a}},\,a,b,c \in R \), represents a circle with center (\( \alpha \), \( \beta \)). Then, \( \alpha \) + 2\( \beta \) is equal to :
Consider a triangle ABC whose vertices are A(0, \( \alpha \), \( \alpha \)), B(\( \alpha \), 0, \( \alpha \)) and C(\( \alpha \), \( \alpha \), 0), \( \alpha \) > 0. Let D be a point moving on the line x + z \( - \) 3 = 0 = y and G be the centroid of \( \Delta \)ABC. If the minimum length of GD is \( \sqrt {{{57} \over 2}} \), then \( \alpha \) is equal to ____________.
The value of \( 1 + {1 \over {1 + 2}} + {1 \over {1 + 2 + 3}} + \,\,....\,\, + \,\,{1 \over {1 + 2 + 3 + \,\,.....\,\, + \,\,11}} \) is equal to:
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 ___________.
A hostel has 100 students. On a certain day (consider it day zero) it was found that two students are infected with some virus. Assume that the rate at which the virus spreads is directly proportional to the product of the number of infected students and the number of non-infected students. If the number of infected students on 4th day is 30, then number of infected students on 8th day will be __________.
The number of 6-digit numbers made by using the digits 1, 2, 3, 4, 5, 6, 7, without repetition and which are multiple of 15 is ____________.
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 ___________.
Let PQ be a focal chord of length 6.25 units of the parabola y2 = 4x. If O is the vertex of the parabola, then 10 times the area (in sq. units) of \( \Delta \)POQ 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 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 \( \alpha = \tan \left( {{{5\pi } \over {16}}\sin \left( {2{{\cos }^{ - 1}}\left( {{1 \over {\sqrt 5 }}} \right)} \right)} \right) \) 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 :
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 :
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 :
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 :
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 \( A = \left[ {\matrix{ 1 & { - 2} & \alpha \cr \alpha & 2 & { - 1} \cr } } \right] \) 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 :
How many of the following drugs belong to the tranquilizer class _____________ ?
Veronal, Luminal, Propanal, Seconal
100 mL of 0.3 M acetic acid is shaken with 0.8 g of wood charcoal. The final concentration of acetic acid in the solution after adsorption is 0.125 M. The mass of acetic acid adsorbed per gram of carbon is _____________ \( \times \) 10\( - \)4 g.
(Given : Molar mass of acetic acid = 60 g mol\( - \)1)