Let \( m \) and \( n \) be the numbers of real roots of the quadratic equations \( x^2 - 12x + [x] + 31 = 0 \) and \( x^2 - 5|x+2| - 4 = 0 \), respectively, where \( [x] \) denotes the greatest integer less than or equal to \( x \). Then \( m^2 + mn + n^2 \) is equal to ___________.
The negation of \( (p \land (\sim q)) \lor (\sim p) \) is equivalent to:
If the probability that the random variable \( X \) takes values \( x \) is given by \( P(X = x) = k(x + 1) 3^{-x}, x = 0, 1, 2, \dots \), where \( k \) is a constant, then \( P(X \geq 2) \) is equal to:
The area of the quadrilateral having vertices as (1,2), (5,6), (7,6), (-1,-6) is?
Let \( A = \{1, 2, 3, 4, 5, 6, 7\} \). Then the relation \( R = \{(x, y) \in A \times A : x + y = 7\} \) is:
Let \((\alpha, \beta, \gamma)\) \(\text{ be the image of the point }\) \(P(3, 3, 5) \text{ in the plane } 2x + y - 3z = 6\). \(\text{ Then } \alpha + \beta + \gamma \text{ is equal to:}\)
Consider the word INDEPENDENCE. The number of words such that all the vowels are together is?
For two groups of 15 sizes each, mean and variance of first group is 12, 14 respectively, and second group has mean 14 and variance of σ2. If combined variance is 13 then find variance of second group?
Let $a_1, a_2, \ldots, a_n$ be in AP If $a_5=2 a_7$ and $a_{11}=18$, then $12\left(\frac{1}{\sqrt{a_{10}}+\sqrt{a_{11}}}+\frac{1}{\sqrt{a_{11}}+\sqrt{a_{12}}}+\ldots+\frac{1}{\sqrt{a_{17}}+\sqrt{a_{18}}}\right)$ is equal to
The remainder on dividing $5^{99}$ by 11 is
Let $\theta$ be the angle between the planes $P_1: \vec{r} \cdot(\hat{i}+\hat{j}+2 \hat{k})=9$ and $P_2: \hat{r} \cdot(2 \hat{i}-\hat{j}+\hat{k})=15$ Let $L$ be the line that meets $P_2$ at the point $(4,-2,5)$ and makes an angle $\theta$ with the normal of $P_4$ If $\alpha$ is the angle between $L$ and $P_2$, then $\left(\tan ^2 \theta\right)\left(\cot ^2 \alpha\right)$ is equal to
Let for $x \in R$ $f(x)=\frac{x+|x|}{2} \text { and } g(x)=\begin{cases}x, & x<0 \\x^2, & x \geq 0\end{cases} $
Then area bounded by the curve $y=(f \circ g)(x)$ and the lines $y=0,2 y-x=15$ is equal to
If the variance of the frequency distribution
is 3 , then $\alpha$ is equal to
The number of real roots of the equation $\sqrt{x^2-4 x+3}+\sqrt{x^2-9}=\sqrt{4 x^2-14 x+6}$, is:
$( S 1)(p \Rightarrow q) \vee(p \wedge(\sim q))$ is a tautology(S2) $((\sim p) \Rightarrow(\sim q)) \wedge((\sim p) \vee q)$ is a contradictionThen
Let $A=\begin{pmatrix}1 & 0 & 0 \\ 0 & 4 & -1 \\ 0 & 12 & -3\end{pmatrix}$ Then the sum of the diagonal elements of the matrix $(A+I)^{11}$ is equal to :
If $\sin ^{-1} \frac{\alpha}{17}+\cos ^{-1} \frac{4}{5}-\tan ^{-1} \frac{77}{36}=0,0<\alpha<13$, then $\sin ^{-1}(\sin \alpha)+\cos ^{-1}(\cos \alpha)$ is equal to
A pair of dice is rolled 5 times. let getting a total of 5 in a single throw is considered as success. If probability of getting atleast four success is \(\frac{x}{3}\) then x is equal to
If the maximum distance of normal to the ellipse \(\frac{x^2}{4}+\frac{y^2}{b^2}=1, b<2\), from the origin is 1 , then the eccentricity of the ellipse is
Among the statements : \((S1)\) \((( p \vee q ) \Rightarrow r ) \Leftrightarrow( p \Rightarrow r )\)\((S2)\)\((( p \vee q ) \Rightarrow r ) \Leftrightarrow(( p \Rightarrow r ) \vee( q \Rightarrow r ))\)