The angle between the straight lines, whose direction cosines are given by the equations 2l + 2m $$-$$ n = 0 and mn + nl + lm = 0, is :

Let $$A = \left( {\matrix{ {[x + 1]} & {[x + 2]} & {[x + 3]} \cr {[x]} & {[x + 3]} & {[x + 3]} \cr {[x]} & {[x + 2]} & {[x + 4]} \cr } } \right)$$, where [t] denotes the greatest integer less than or equal to t. If det(A) = 192, then the set of values of x is the interval :

Let M and m respectively be the maximum and minimum values of the function
f(x) = tan^$$-$$1 (sin x + cos x) in $$\left[ {0,{\pi \over 2}} \right]$$, then the value of tan(M $$-$$ m) is equal to :

If two tangents drawn from a point P to the
parabola y² = 16(x $$-$$ 3) are at right angles, then the locus of point P is :

If the solution curve of the differential equation (2x $$-$$ 10y³)dy + ydx = 0, passes through the points (0, 1) and (2, $$\beta$$), then $$\beta$$ is a root of the equation :

Let [$$\lambda$$] be the greatest integer less than or equal to $$\lambda$$. The set of all values of $$\lambda$$ for which the system of linear equations
x + y + z = 4,
3x + 2y + 5z = 3,
9x + 4y + (28 + [$$\lambda$$])z = [$$\lambda$$] has a solution is :

The set of all values of K > $$-$$1, for which the equation $${(3{x^2} + 4x + 3)^2} - (k + 1)(3{x^2} + 4x + 3)(3{x^2} + 4x + 2) + k{(3{x^2} + 4x + 2)^2} = 0$$ has real roots, is :

A box open from top is made from a rectangular sheet of dimension a $$\times$$ b by cutting squares each of side x from each of the four corners and folding up the flaps. If the volume of the box is maximum, then x is equal to :

Let Z be the set of all integers,

$$A = \{ (x,y) \in Z \times Z:{(x - 2)^2} + {y^2} \le 4\} $$

$$B = \{ (x,y) \in Z \times Z:{x^2} + {y^2} \le 4\} $$

$$C = \{ (x,y) \in Z \times Z:{(x - 2)^2} + {(y - 2)^2} \le 4\} $$

If the total number of relation from A $$\cap$$ B to A $$\cap$$ C is 2^p, then the value of p is :

The area of the region bounded by the parabola (y $$-$$ 2)² = (x $$-$$ 1), the tangent to it at the point whose ordinate is 3 and the x-axis is :

If $$y(x) = {\cot ^{ - 1}}\left( {{{\sqrt {1 + \sin x} + \sqrt {1 - \sin x} } \over {\sqrt {1 + \sin x} - \sqrt {1 - \sin x} }}} \right),x \in \left( {{\pi \over 2},\pi } \right)$$, then $${{dy} \over {dx}}$$ at $$x = {{5\pi } \over 6}$$ is :

The value of the integral $$\int\limits_0^1 {{{\sqrt x dx} \over {(1 + x)(1 + 3x)(3 + x)}}} $$ is :

If $$\mathop {\lim }\limits_{x \to \infty } \left( {\sqrt {{x^2} - x + 1} - ax} \right) = b$$, then the ordered pair (a, b) is :

The probability distribution of random variable X is given by :

X	1	2	3	4	5
P(X)	K	2K	2K	3K	K

Let p = P(1 < X < 4 | X < 3). If 5p = $$\lambda$$K, then $$\lambda$$ equal to ___________.

Let z₁ and z₂ be two complex numbers such that $$\arg ({z_1} - {z_2}) = {\pi \over 4}$$ and z₁, z₂ satisfy the equation | z $$-$$ 3 | = Re(z). Then the imaginary part of z₁ + z₂ is equal to ___________.

Let S = {1, 2, 3, 4, 5, 6, 9}. Then the number of elements in the set T = {A $$ \subseteq $$ S : A $$\ne$$ $$\phi$$ and the sum of all the elements of A is not a multiple of 3} is _______________.

3 $$\times$$ 7²² + 2 $$\times$$ 10²² $$-$$ 44 when divided by 18 leaves the remainder __________.

An online exam is attempted by 50 candidates out of which 20 are boys. The average marks obtained by boys is 12 with a variance 2. The variance of marks obtained by 30 girls is also 2. The average marks of all 50 candidates is 15. If $$\mu$$ is the average marks of girls and $$\sigma$$² is the variance of marks of 50 candidates, then $$\mu$$ + $$\sigma$$² is equal to ________________.

If $$\int {{{2{e^x} + 3{e^{ - x}}} \over {4{e^x} + 7{e^{ - x}}}}dx = {1 \over {14}}(ux + v{{\log }_e}(4{e^x} + 7{e^{ - x}})) + C} $$, where C is a constant of integration, then u + v is equal to _____________.