\(\begin{array}{l} \frac{2^3-1^3}{1\times7}+\frac{4^3-3^3+2^2-1^3}{2\times 11}+\frac{6^3-5^3+4^3-3^3+2^3-1^3}{3\times 15}+\cdots+\frac{30^3-29^3+28^3-27^3+\cdots+2^3-1^3}{15\times63}\end{array}\)
is equal to _______.
The conditional statement
\( ((p \wedge q) \to (( \sim p) \vee r)) \vee ((( \sim p) \vee r) \to (p \wedge q)) \) is :
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 _____________.
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 :
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 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 :
The probability distribution of X is :
For the minimum possible value of d, sixty times the mean of X 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:
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 ____________.
\( \mathop {\lim }\limits_{n \to \infty } \sum\limits_{r = 1}^n {{r \over {2{r^2} - 7rn + 6{n^2}}}} \) is equal to :
