A 5-digit number divisible by 3 is to be formed using the digits 0, 1, 2, 3, 4 and 5 without repetition. The total number of ways in which this can be done is:

A 5-digit number divisible by 3 is to be formed using the digits …

Let $\quad S=\left\{z=x+i y: \frac{2 z-3 i}{4 z+2 i}\right.$ is a real number $\}$. Then which of the following is NOT correct?

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Circles ${x^2} + {y^2} - 2x - 4y = 0$ and ${x^2} + {y^2} - 8y - 4 = 0$

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$\sin 12^\circ \sin 24^\circ \sin 48^\circ \sin 84^\circ = $

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The sum of all values of $x$ in $[0,2 \pi]$, for which $\sin x+\sin 2 x+\sin 3 x+\sin 4 x=0$, is equal to:

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A circle with center $(3, 8)$ contains the point $(2, -1)$. Another point on the circle is:

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The eccentricity of the hyperbola can never be equal to

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If $3{p^2} = 5p + 2$ and $3{q^2} = 5q + 2$ where $p \ne q$, then the equation whose roots are $3p - 2q$ and $3q - 2p$ is

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Let $S=\{1,2,3,5,7,10,11\}$. The number of nonempty subsets of $S$ that have the sum of all elements a multiple of $3$ , is $........$

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The value of $\cos^248^\circ-\sin^212^\circ$ is:
$[$Hint: Use $\cos^2\text{A}-\sin^2\text{B}=\cos(\text{A + B})\cos(\text{A}-\text{B})]$

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Let $S$ be the set of all $\alpha \in R$ such that the equation, $cos\,2 x + \alpha \,sin\, x = 2\alpha -7$ has a solution. Then $S$ is equal to

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