The region represented by \(\{z=x+i y \in C:|z|-\operatorname{Re}(z) \leq 1\}\) is also given by the inequality

If the equation of the parabola, whose vertex is at \((5,4)\) and the directrix is \(3 x+y-29=0\), is \(x^{2}+a y^{2}+b x y+c x+d y+k=0\) then \(a + b + c + d + k\) is equal to

Let \(S=\{1,2,3,4,5,6\} .\) Then the probability that a randomly chosen onto function \(\mathrm{g}\) from \(\mathrm{S}\) to \(\mathrm{S}\) satisfies \(g(3)=2 g(1)\) is :

For the functions \(f(\theta) = \alpha\tan^2\theta + \beta\cot^2\theta\), and \(g(\theta) = \alpha\sin^2\theta + \beta\cos^2\theta\), \(\alpha > \beta > 0\), let \(\min_{0 < \theta < \pi/2}f(\theta) = \max_{0 < \theta < \pi}g(\theta)\). If the first term of a G.P. is \(\left(\dfrac{\alpha}{2\beta}\right)\), its common ratio is \(\left(\dfrac{2\beta}{\alpha}\right)\) and the sum of its first \(10\) terms is \(\dfrac{m}{n}\), \(\gcd(m, n) = 1\), then \(m + n\) is equal to _______.

Let \(f: R \rightarrow R\) be a function defined by \(f(x)=\left\{\begin{array}{l}\max \left\{t^{3}-3 t\right\} ; x \leq 2 \\ t \leq x \\ x^{2}+2 x-6 ; 2 < x < 3 \\ {[x-3]+9 ; 3 \leq x \leq 5} \\ 2 x+1 \quad ; \quad x > 5\end{array}\right\}\) Where \([t]\) is the greatest integer less than or equal to \(t\). Let \(m\) be the number of points where \(f\) is not differentiable and \(I =\int\limits_{-2}^{2} f( x ) dx\). Then the ordered pair \(( m , I )\) is equal to

If the area of the region {(x, y) : \( 1-2x\le y\le4-x^{2}, x\ge0,y\ge0 \)} is \( \frac{\alpha}{\beta}, \alpha, \beta \in N \), gcd(\(α,β\))=1, then the value of \( (\alpha+\beta) \) is :

A box contains \(15\) green and \(10\) yellow balls. lf \(10\)  balls are randomly drawn, one-by-one, with replacement, then the variance of the number of green balls drawn is :

A tangent and a normal are drawn at the point \(\mathrm{P}(2,-4)\) on the parabola \(\mathrm{y}^{2}=8 \mathrm{x}\), which meet the directrix of the parabola at the points \(\mathrm{A}\) and \(\mathrm{B}\) respectively. If \(Q(a, b)\) is a point such that \(A Q B P\) is a square, then \(2 \mathrm{a}+\mathrm{b}\) is equal to :

A differential equation representing the family of parabolas with axis parallel to \(\mathrm{y}\)-axis and whose length of latus rectum is the distance of the point \((2,-3)\) form the line \(3 x+4 y=5\), is given by :

If the domain of the function \( f(x)=\log_{(10x^{2}-17x+7)}(18x^{2}-11x+1) \) is \( (-\infty,a)\cup(b,c)\cup(d,\infty)-\{e\} \), then \( 90(a+b+c+d+e) \) equals:

Let \(y=y(x)\) be the solution of the differential equation \(\operatorname{cosec}^{2} x d y+2 d x=(1+y \cos 2 x) \operatorname{cosec}^{2} x d x\), with \(y\left(\frac{\pi}{4}\right)=0\). Then, the value of \((y(0)+1)^{2}\) is equal to :

A triangle \(ABC\) lying in the first quadrant has two vertices as \(A (1,2)\) and \(B (3,1)\). If \(\angle BAC =90^{\circ},\) and \(\operatorname{ar}(\Delta ABC )=5 \sqrt{5}\) sq. units then the abscissa of the vertex \(C\) is

Let \(L_1: \overrightarrow{\mathrm{r}}=(\hat{\mathrm{i}}-\hat{\mathrm{j}}+2 \hat{\mathrm{k}})+\lambda(\hat{\mathrm{i}}-\hat{\mathrm{j}}+2 \hat{\mathrm{k}}), \lambda \in \mathrm{R}\) \(L_2: \overrightarrow{\mathrm{r}}=(\hat{\mathrm{j}}-\hat{\mathrm{k}})+\mu(3 \hat{\mathrm{i}}+\hat{\mathrm{j}}+\mathrm{p} \hat{\mathrm{k}}), \mu \in \mathrm{R}\) and  \(L_3: \overrightarrow{\mathrm{r}}=\delta(\ell \hat{\mathrm{i}}+\mathrm{m} \hat{\mathrm{j}}+\mathrm{n} \hat{\mathrm{k}}) \delta \in \mathrm{R}\). Be three lines such that \(\mathrm{L}_1\) is perpendicular to \(\mathrm{L}_2\) and \(L_3\) is perpendicular to both \(L_1\) and \(L_2\). Then the point which lies on \(\mathrm{L}_3\) is