\(y=\mathrm{e}^x(\mathrm{~A} \cos x+\mathrm{B} \sin x)\) is the solution of the differential equation
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
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 line passes through \(\mathrm{P}(-4,1)\) and meets the co-ordinate axes at points A and B . If P divides the segment AB internally in the ratio \(1: 2\), then the equation of the line 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 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
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 negation of the statement "The triangle is an equilateral or isosceles triangle and the triangle is not isosceles and it is right angled" is
\(\mathrm{f}(x)=\frac{x}{2}+\frac{2}{x}, x \neq 0\) is strictly decreasing in
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
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
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 area bounded by the parabolas \(y=9 x^2, y=\frac{x^2}{16}\) and the line \(y=1\) is
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
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
\(\int \frac{\mathrm{d} x}{3 \cos 2 x+5}\) equals
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
If \(y=\tan ^{-1}\left(\frac{4 x}{1+5 x^2}\right)+\cot ^{-1}\left(\frac{3-2 x}{2+3 x}\right)\), then \(\frac{\mathrm{d} y}{\mathrm{~d} x}\) is equal to
The differential equation \(x \frac{\mathrm{~d} y}{\mathrm{~d} x}=2 y\) represents ________