The number of real roots of the equation

$${e^{4x}} + 2{e^{3x}} - {e^x} - 6 = 0$$ is :

Let f be a non-negative function in [0, 1] and twice differentiable in (0, 1). If $$\int_0^x {\sqrt {1 - {{(f'(t))}^2}} dt = \int_0^x {f(t)dt} } $$, $$0 \le x \le 1$$ and f(0) = 0, then $$\mathop {\lim }\limits_{x \to 0} {1 \over {{x^2}}}\int_0^x {f(t)dt} $$ :

Let $$\overrightarrow a $$ and $$\overrightarrow b $$ be two vectors
such that $$\left| {2\overrightarrow a + 3\overrightarrow b } \right| = \left| {3\overrightarrow a + \overrightarrow b } \right|$$ and the angle between $$\overrightarrow a $$ and $$\overrightarrow b $$ is 60$$^\circ$$. If $${1 \over 8}\overrightarrow a $$ is a unit vector, then $$\left| {\overrightarrow b } \right|$$ is equal to :

The function

$$f(x) = \left| {{x^2} - 2x - 3} \right|\,.\,{e^{\left| {9{x^2} - 12x + 4} \right|}}$$ is not differentiable at exactly :

Three numbers are in an increasing geometric progression with common ratio r. If the middle number is doubled, then the new numbers are in an arithmetic progression with common difference d. If the fourth term of GP is 3 r², then r² $$-$$ d is equal to :

Which of the following is not correct for relation R on the set of real numbers ?

The integral $$\int {{1 \over {\root 4 \of {{{(x - 1)}^3}{{(x + 2)}^5}} }}} \,dx$$ is equal to : (where C is a constant of integration)

If p and q are the lengths of the perpendiculars from the origin on the lines,

x cosec $$\alpha$$ $$-$$ y sec $$\alpha$$ = k cot 2$$\alpha$$ and

x sin$$\alpha$$ + y cos$$\alpha$$ = k sin2$$\alpha$$

respectively, then k² is equal to :

cosec18$$^\circ$$ is a root of the equation :

If the following system of linear equations

2x + y + z = 5

x $$-$$ y + z = 3

x + y + az = b

has no solution, then :

The length of the latus rectum of a parabola, whose vertex and focus are on the positive x-axis at a distance R and S (> R) respectively from the origin, is :

If the function
$$f(x) = \left\{ {\matrix{ {{1 \over x}{{\log }_e}\left( {{{1 + {x \over a}} \over {1 - {x \over b}}}} \right)} & , & {x < 0} \cr k & , & {x = 0} \cr {{{{{\cos }^2}x - {{\sin }^2}x - 1} \over {\sqrt {{x^2} + 1} - 1}}} & , & {x > 0} \cr } } \right.$$ is continuous

at x = 0, then $${1 \over a} + {1 \over b} + {4 \over k}$$ is equal to :

If $${{dy} \over {dx}} = {{{2^{x + y}} - {2^x}} \over {{2^y}}}$$, y(0) = 1, then y(1) is equal to :

$$\mathop {\lim }\limits_{x \to 0} {{{{\sin }^2}\left( {\pi {{\cos }^4}x} \right)} \over {{x^4}}}$$ is equal to :

If $${a_r} = \cos {{2r\pi } \over 9} + i\sin {{2r\pi } \over 9}$$, r = 1, 2, 3, ....., i = $$\sqrt { - 1} $$, then
the determinant $$\left| {\matrix{ {{a_1}} & {{a_2}} & {{a_3}} \cr {{a_4}} & {{a_5}} & {{a_6}} \cr {{a_7}} & {{a_8}} & {{a_9}} \cr } } \right|$$ is equal to :

Let [t] denote the greatest integer $$\le$$ t. Then the value of

$$8.\int\limits_{ - {1 \over 2}}^1 {([2x] + |x|)dx} $$ is ___________.

A point z moves in the complex plane such that $$\arg \left( {{{z - 2} \over {z + 2}}} \right) = {\pi \over 4}$$, then the minimum value of $${\left| {z - 9\sqrt 2 - 2i} \right|^2}$$ is equal to _______________.

If 'R' is the least value of 'a' such that the function f(x) = x² + ax + 1 is increasing on [1, 2] and 'S' is the greatest value of 'a' such that the function f(x) = x² + ax + 1 is decreasing on [1, 2], then
the value of |R $$-$$ S| is ___________.

The mean of 10 numbers 7 $$\times$$ 8, 10 $$\times$$ 10, 13 $$\times$$ 12, 16 $$\times$$ 14, ....... is ____________.

If the variable line 3x + 4y = $$\alpha$$ lies between the two
circles (x $$-$$ 1)² + (y $$-$$ 1)² = 1
and (x $$-$$ 9)² + (y $$-$$ 1)² = 4, without intercepting a chord on either circle, then the sum of all the integral values of $$\alpha$$ is ___________.

The number of six letter words (with or without meaning), formed using all the letters of the word 'VOWELS', so that all the consonants never come together, is ___________.

If $$x\phi (x) = \int\limits_5^x {(3{t^2} - 2\phi '(t))dt} $$, x > $$-$$2, and $$\phi$$(0) = 4, then $$\phi$$(2) is __________.

If $$\left( {{{{3^6}} \over {{4^4}}}} \right)k$$ is the term, independent of x, in the binomial expansion of $${\left( {{x \over 4} - {{12} \over {{x^2}}}} \right)^{12}}$$, then k is equal to ___________.

An electric instrument consists of two units. Each unit must function independently for the instrument to operate. The probability that the first unit functions is 0.9 and that of the second unit is 0.8. The instrument is switched on and it fails to operate. If the probability that only the first unit failed and second unit is functioning is p, then 98 p is equal to _____________.