\(y=\mathrm{e}^x(\mathrm{~A} \cos x+\mathrm{B} \sin x)\) is the solution of the differential equation
If the area of a parallelogram whose diagonals are represented by vectors \(3 \hat{i}+\lambda \hat{j}+2 \hat{k}\) and \(\hat{\mathrm{i}}-2 \hat{\mathrm{j}}+3 \hat{\mathrm{k}}\) is \(\frac{\sqrt{117}}{2}\) sq. units, then \(\lambda=\)
A family has 3 children. The probability that all the three children are girls, given that at least one of them is a girl is
In a triangle ABC , with usual notations. \(\frac{2 \cos \mathrm{~A}}{\mathrm{a}}+\frac{\cos \mathrm{B}}{\mathrm{b}}+\frac{2 \cos \mathrm{C}}{\mathrm{c}}=\frac{\mathrm{a}}{\mathrm{bc}}+\frac{\mathrm{b}}{\mathrm{ca}}\). Then \(\angle \mathrm{A}=\)
The position vectors of the points \(A, B, C\) are \(\hat{i}+2 \hat{j}-\hat{k}, \hat{i}+\hat{j}+\hat{k}, 2 \hat{i}+3 \hat{j}+2 \hat{k}\) respectively. If \(A\) is chosen as the origin, then the cross product of position vectors of \(B\) and \(C\) are
A pair of fair dice is thrown 4 times. If getting the same number on both dice is considered as a success, then the probability of two successes are
The rate of change of the volume of a sphere with respect to its surface area, when the radius is 5 m is
The maximum value of the function \(a \sin x+b \cos x\) is
If the lines \(\frac{1-x}{2}=\frac{7 y+4}{2 \lambda}=\frac{2 z-5}{2}\) and \(\frac{7-7 x}{3 \lambda}=\frac{y-1}{7}=\frac{6-\mathrm{z}}{5}\) are at right angle, then the value of \(\lambda\) is
The negation of the statement "The triangle is an equilateral or isosceles triangle and the triangle is not isosceles and it is right angled" is
If \(\mathrm{f}(x)=\frac{\sin \left(\pi \cos ^2 x\right)}{3 x^2}, x \neq 0\) is continuous at \(x=0\) then \(\mathrm{f}(0)=\)
If the statements \(p, q\) and \(r\) are true, false and true statements respectively, then the truth value of the statement pattern \([\sim \mathrm{q} \wedge(\mathrm{p} \vee \sim \mathrm{q}) \wedge \sim \mathrm{r}] \vee \mathrm{p}\) and the truth value of its dual statement respectively are
\(\mathrm{f}(x)=\frac{x}{2}+\frac{2}{x}, x \neq 0\) is strictly decreasing in
Let \(z\) be the complex number with \(\operatorname{Im}(z)=10\) and satisfying \(\frac{2 \mathrm{z}-\mathrm{n}}{2 \mathrm{z}+\mathrm{n}}=2 \mathrm{i}-1\), where \(\mathrm{i}=\sqrt{-1}\), for some natural number ' \(n\) ' then
If \(A=\left[\begin{array}{lll}3 & -3 & 4 \\ 2 & -3 & 4 \\ 0 & -1 & 1\end{array}\right]_{3 \times 3}\), then \(A^{-1}=\)
The solution of the equation \(x^2 y-x^3 \frac{\mathrm{~d} y}{\mathrm{~d} x}=y^4 \cos x\), where \(y(0)=1\), is
The length of the foot of the perpendicular from the point \(\left(1, \frac{3}{2}, 2\right)\) to the plane \(2 x-2 y+4 z+17=0\) is
A tetrahedron has vertices \(\mathrm{O}(0,0,0), \mathrm{A}(1,2,1)\), \(B(2,1,3), C(-1,1,2)\). Then the angle between the faces OAB and ABC will be
The area bounded by the parabolas \(y=9 x^2, y=\frac{x^2}{16}\) and the line \(y=1\) is
\(\int \frac{\mathrm{d} x}{3 \cos 2 x+5}\) equals
The principal value of \(\cos ^{-1}\left[\frac{1}{\sqrt{2}}\left(\cos \frac{9 \pi}{10}-\sin \frac{9 \pi}{10}\right)\right]\) is
The differential equation \(x \frac{\mathrm{~d} y}{\mathrm{~d} x}=2 y\) represents ________
Let X denote the number of hours you study on a Sunday. It is known that
\[ \mathrm{P}(\mathrm{X}=x)=\left\{\begin{array}{cc} 0.1 & , \text { if } x=0 \\ \mathrm{k} x & , \text { if } x=1 \text { or } 2 \\ \mathrm{k}(5-x) & , \text { if } x=3 \text { or } 4 \\ 0 & , \text { otherwise } \end{array}\right. \]
where k is constant. Then the probability that you study at least two hours on a Sunday is