The number of real values of $$\lambda $$ for which the system of linear equations

2x + 4y $$-$$ $$\lambda $$z = 0

4x + $$\lambda $$y + 2z = 0

$$\lambda $$x + 2y + 2z = 0

has infinitely many solutions, is :

If (27)⁹⁹⁹ is divided by 7, then the remainder is :

Let A be any 3 $$ \times $$ 3 invertible matrix. Then which one of the following is not always true ?

Let p(x) be a quadratic polynomial such that p(0)=1. If p(x) leaves remainder 4 when divided by x$$-$$ 1 and it leaves remainder 6 when divided by x + 1; then :

If the arithmetic mean of two numbers a and b, a > b > 0, is five times their geometric mean, then $${{a + b} \over {a - b}}$$ is equal to :

If all the words, with or without meaning, are written using the letters of the word QUEEN and are arranged as in English dictionary, then the position of the word QUEEN is :

Let f(x) = 2¹⁰.x + 1 and g(x)=3¹⁰.x $$-$$ 1. If (fog) (x) = x, then x is equal to :

Let z$$ \in $$C, the set of complex numbers. Then the equation, 2|z + 3i| $$-$$ |z $$-$$ i| = 0 represents :

Consider an ellipse, whose center is at the origin and its major axis is along the x-axis. If its eccentricity is $${3 \over 5}$$ and the distance between its foci is 6, then the area (in sq. units) of the quadrilatateral inscribed in the ellipse, with the vertices as the vertices of the ellipse, is :

The value of tan^-1 $$\left[ {{{\sqrt {1 + {x^2}} + \sqrt {1 - {x^2}} } \over {\sqrt {1 + {x^2}} - \sqrt {1 - {x^2}} }}} \right],$$ $$\left| x \right| < {1 \over 2},x \ne 0,$$ is equal to :

If

$$S = \left\{ {x \in \left[ {0,2\pi } \right]:\left| {\matrix{ 0 & {\cos x} & { - \sin x} \cr {\sin x} & 0 & {\cos x} \cr {\cos x} & {\sin x} & 0 \cr } } \right| = 0} \right\},$$

then $$\sum\limits_{x \in S} {\tan \left( {{\pi \over 3} + x} \right)} $$ is equal to :

An unbiased coin is tossed eight times. The probability of obtaining at least one head and at least one tail is :

Three persons P, Q and R independently try to hit a target. I the probabilities of their hitting the target are $${3 \over 4},{1 \over 2}$$ and $${5 \over 8}$$ respectively, then the probability that the target is hit by P or Q but not by R is :

The mean age of 25 teachers in a school is 40 years. A teacher retires at the age of 60 years and a new teacher is appointed in his place. If now the mean age of the teachers in this school is 39 years, then the age (in years) of the newly appointed teacher is :

The area (in sq. units) of the parallelogram whose diagonals are along the vectors $$8\widehat i - 6\widehat j$$ and $$3\widehat i + 4\widehat j - 12\widehat k,$$ is :

If two parallel chords of a circle, having diameter 4units, lie on the opposite sides of the center and subtend angles $${\cos ^{ - 1}}\left( {{1 \over 7}} \right)$$ and sec^$$-$$1 (7) at the center respectivey, then the distance between these chords, is :

The integral $$\int_{{\pi \over {12}}}^{{\pi \over 4}} {\,\,{{8\cos 2x} \over {{{\left( {\tan x + \cot x} \right)}^3}}}} \,dx$$ equals :

The area (in sq. units) of the smaller portion enclosed between the curves, x² + y² = 4 and y² = 3x, is :

The locus of the point of intersection of the straight lines,

tx $$-$$ 2y $$-$$ 3t = 0

x $$-$$ 2ty + 3 = 0 (t $$ \in $$ R), is :

The curve satisfying the differential equation, ydx $$-$$(x + 3y²)dy = 0 and passing through the point (1, 1), also passes through the point :

If y = $${\left[ {x + \sqrt {{x^2} - 1} } \right]^{15}} + {\left[ {x - \sqrt {{x^2} - 1} } \right]^{15}},$$

then (x² $$-$$ 1) $${{{d^2}y} \over {d{x^2}}} + x{{dy} \over {dx}}$$ is equal to :

If a point P has co-ordinates (0, $$-$$2) and Q is any point on the circle, x² + y² $$-$$ 5x $$-$$ y + 5 = 0, then the maximum value of (PQ)² is :

The integral

$$\int {\sqrt {1 + 2\cot x(\cos ecx + \cot x)\,} \,\,dx} $$

$$\left( {0 < x < {\pi \over 2}} \right)$$ is equal to :

(where C is a constant of integration)

$$\mathop {\lim }\limits_{x \to 3} $$ $${{\sqrt {3x} - 3} \over {\sqrt {2x - 4} - \sqrt 2 }}$$ is equal to :