\(\begin{array}{l} \frac{2^3-1^3}{1\times7}+\frac{4^3-3^3+2^2-1^3}{2\times 11}+\frac{6^3-5^3+4^3-3^3+2^3-1^3}{3\times 15}+\cdots+\frac{30^3-29^3+28^3-27^3+\cdots+2^3-1^3}{15\times63}\end{array}\)
is equal to _______.
Let 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 :
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 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 :
