\(\begin{array}{l} \frac{2^3-1^3}{1\times7}+\frac{4^3-3^3+2^2-1^3}{2\times 11}+\frac{6^3-5^3+4^3-3^3+2^3-1^3}{3\times 15}+\cdots+\frac{30^3-29^3+28^3-27^3+\cdots+2^3-1^3}{15\times63}\end{array}\)
is equal to _______.
Let \( {S_1} = \{ x \in [0,12\pi ]:{\sin ^5}x + {\cos ^5}x = 1\} \)
and \( {S_2} = \{ x \in [0,8\pi ]:{\sin ^7}x + {\cos ^7}x = 1\} \)
Then \( n({S_1}) - n({S_2}) \) is equal to ______________.
Let for \( f(x) = {a_0}{x^2} + {a_1}x + {a_2},\,f'(0) = 1 \) and \( f'(1) = 0 \). If a0, a1, a2 are in an arithmatico-geometric progression, whose corresponding A.P. has common difference 1 and corresponding G.P. has common ratio 2, then f(4) is equal to _____________.
The conditional statement
\( ((p \wedge q) \to (( \sim p) \vee r)) \vee ((( \sim p) \vee r) \to (p \wedge q)) \) is :
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 :
Let a vector \( \overrightarrow c \) be coplanar with the vectors \( \overrightarrow a = - \widehat i + \widehat j + \widehat k \) and \( \overrightarrow b = 2\widehat i + \widehat j - \widehat k \). If the vector \( \overrightarrow c \) also satisfies the conditions \( \overrightarrow c \,.\,\left[ {\left( {\overrightarrow a + \overrightarrow b } \right) \times \left( {\overrightarrow a \times \overrightarrow b } \right)} \right] = - 42 \) and \( \left( {\overrightarrow c \times \left( {\overrightarrow a - \overrightarrow b } \right)} \right)\,.\,\widehat k = 3 \), then the value of \( |\overrightarrow c {|^2} \) is equal to :
The distance of the point (3, 2, \( - \)1) from the plane \( 3x - y + 4z + 1 = 0 \) along the line \( {{2 - x} \over 2} = {{y - 3} \over 2} = {{z + 1} \over 1} \) 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 :
The probability distribution of X is :
For the minimum possible value of d, sixty times the mean of X 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 :
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:
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 :
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 __________.
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 ___________.
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 ___________.