If \((\mathrm{a}+\mathrm{b} x) \mathrm{e}^{\frac{y}{x}}=x\), then \(x^3 \frac{\mathrm{~d}^2 y}{\mathrm{~d} x^2}\) is equal to
If the function \(\mathrm{f}(x)=x(x+3) \mathrm{e}^{-\frac{x}{2}}\) satisfies all the conditions of Rolle's theorem in \([-3,0]\), then c is
In a triangle ABC , with usual notations if \(\mathrm{a}=4, \mathrm{~b}=8, \angle \mathrm{C}=60^{\circ}\), then the value of \(\angle \mathrm{B}\) and the ratio \(\cos \mathrm{A}: \cos \mathrm{C}\) respectively are,
The third element in the second row of adjoint of a matrix \(A=\left[a_{i j}\right]_{3 \times 3}\) (where \(a_{i j}=2 i+j\) ) is
If \(y=\frac{\mathrm{K}^{\cos ^{-1} x}}{1+\mathrm{K}^{\cos ^{-1} x}}\) and \(\mathrm{t}=\mathrm{K}^{\cos ^{-1} x}\), then \(\frac{\mathrm{d} y}{\mathrm{dt}}=\)
The function f defined by \(\mathrm{f}(x)=(x+2) \mathrm{e}^{-x}\) is
\(\int \sqrt{x^2-6 x-16} \mathrm{~d} x\) equals
The value of \(2 \tan ^{-1} \frac{1}{2}+\tan ^{-1} \frac{3}{8}\) is
With usual notation, in triangle ABC , \(\mathrm{m} \angle \mathrm{A}=30^{\circ}\) then the value of \(\left(1+\frac{\mathrm{a}}{\mathrm{c}}+\frac{\mathrm{b}}{\mathrm{c}}\right)\left(1+\frac{\mathrm{c}}{\mathrm{b}}-\frac{\mathrm{a}}{\mathrm{b}}\right)\) is equal to
The greatest possible number of points of intersection of 8 distinct straight lines and 4 distinct circles is
In \(\triangle A B C\), with usual notations, \(a \cos B=b \cos A, a \cos C \neq c \cos A\) then \(\mathrm{A}(\triangle \mathrm{ABC})\) \(\qquad\) sq. units.
The radius of the base of a cone is increasing at the rate \(3 \mathrm{~cm} /\) minute and the altitude is decreasing at the rate \(4 \mathrm{~cm} /\) minute . The rate at which the lateral surface area is changing, when the radius is 7 cm and altitude is 24 cm is
Three vectors \(\hat{\mathrm{i}}-\hat{\mathrm{k}}, \lambda \hat{\mathrm{i}}+\hat{\mathrm{j}}+(1-\lambda) \hat{\mathrm{k}}\) and \(\mu \hat{\mathrm{i}}+\lambda \hat{\mathrm{j}}+(1+\lambda-\mu) \hat{\mathrm{k}}\) represents coterminous edges of a parallelopiped, then the volume of the parallelopiped depends on.
\[ \int \frac{\mathrm{d} x}{x\left(x^2+1\right)}= \]
If \(\sin ^{-1} \frac{1}{3}+\sin ^{-1} \frac{3}{5}+\sin ^{-1} x=\frac{\pi}{2}\) then \(x=\)
The joint equation of the bisectors of the angles between the lines \(x=5\) and \(y=3\) is
The value of \(\int_0^2\left[x^2\right] \mathrm{d} x\) is (where \([x]\) denotes the greatest integer function not greater than \(x\) )
If \(\overline{\mathrm{a}}\) and \(\overline{\mathrm{b}}\) are unit vectors such that \(|\overline{\mathrm{a}}+\overline{\mathrm{b}}|=\sqrt{3}\), then the angle between \(\bar{a}\) and \(\bar{b}\) is
The area inside the parabola \(y^2=4 \mathrm{a} x\), between the lines \(x=\mathrm{a}\) and \(x=4 \mathrm{a}\) is equal to
\(\int_2^4 \frac{\log x^2}{\log x^2+\log \left(36-12 x+x^2\right)} \mathrm{d} x\) is equal to
Let \(\bar{a}, \bar{b}\), and \(\bar{c}\) be unit vectors. Suppose that \(\overline{\mathrm{a}} \cdot \overline{\mathrm{b}}=\overline{\mathrm{a}} \cdot \overline{\mathrm{c}}=0\) and if the angle between \(\overline{\mathrm{b}}\) and \(\overline{\mathrm{c}}\) is \(\frac{\pi}{6}\), then \(\overline{\mathrm{a}}\) is
If \(\quad \int \frac{3 \sin x \cos x}{4 \sin x+7} \mathrm{~d} x=\mathrm{A} \sin x-\mathrm{Blog}(4 \sin x+7)+\mathrm{c}\) where c is the constant of integration, then the value of \(\mathrm{A}+\mathrm{B}\) is equal to
Which of the following is not a homogeneous function?
The general solution of \(\frac{\mathrm{d} y}{\mathrm{~d} x}=2 x y \mathrm{e}^{x^2}\) is