List of practice Questions

Bacteria increases at the rate proportional to the number of bacteria present. If the original number \( N \) doubles in \( 4 \) hours, then the number of bacteria will be \( 4N \) in

The length of the perpendicular from the point \( P(a,b) \) to the line \( \dfrac{x}{a} + \dfrac{y}{b} = 1 \) is

If \( f(x) = \begin{cases \dfrac{81^x - 9^x}{k^x - 1}, & x \neq 0
2, & x = 0 \end{cases} \) is continuous at \( x = 0 \), then the value of \( k \) is}

If \( \frac{1-\tan\theta}{1+\tan\theta} = \frac{1}{\sqrt{3}}, \) where \( \theta \in \left(0,\frac{\pi}{2}\right) \), then \( \theta = \)

The direction cosines of a line which lies in the \( ZOX \) plane and makes an angle of \( 30^\circ \) with the \( Z \)-axis are

If \( \dfrac{d^2y}{dx^2} = \sin x + e^x \), \( y(0) = 3 \) and \( \left.\dfrac{dy}{dx}\right|_{x=0} = 4 \), then the equation of the curve is

The general solution of \( \tan 3x = 1 \) is

If \( A = \begin{pmatrix} 1 & 0 & 2 \\ 2 & 1 & 3 \\ 0 & 3 & -5 \end{pmatrix} \) where \( A_{ij} \) is the cofactor of the element \( a_{ij} \) of matrix \( A \), then \( a_{21}A_{21} + a_{22}A_{22} + a_{23}A_{23} = \)

Evaluate \( \displaystyle \int_{0}^{1} \tan^{-1}\!\left(\frac{2x}{1-x^2}\right)\,dx \)

If the lines given by \( \vec{r} = 2\hat{i} + \lambda(\hat{i} + 2\hat{j} + m\hat{k}) \) and \( \vec{r} = \hat{i} + \mu(2\hat{i} + \hat{j} + 6\hat{k}) \) are perpendicular, then the value of \( m \) is

If \( A(-1,2,3) \), \( B(3,-2,1) \), \( C(2,1,3) \) and \( D(-1,-2,4) \) are the vertices of a tetrahedron, then its volume is

If \( \cosec\theta + \cot\theta = 5 \), then \( \sin\theta = \)

A metal wire \(108\) meters long is bent to form a rectangle. If the area of the rectangle is maximum, then its dimensions are

The eccentricity of the hyperbola \( 16x^2 - 3y^2 - 32x - 12y - 44 = 0 \) is

\( y = c^2 + \dfrac{c}{x} \) is the solution of the differential equation

If \( y\sqrt{1-x^2} + x\sqrt{1-y^2} = 1 \), then \( \dfrac{dy}{dx} = \)

If the vectors \( (2\mathbf{i}-q\mathbf{j}+3\mathbf{k}) \) and \( (4\mathbf{i}-5\mathbf{j}+6\mathbf{k}) \) are collinear, then the value of \( q \) is

If the vectors \( \vec{a}, \vec{b}, \vec{c} \) are non-coplanar, then \( \dfrac{\left| \vec{a} + 2\vec{b} \;\; \vec{b} + 2\vec{c} \;\; \vec{c} + 2\vec{a} \right|} {\left| \vec{a} \;\; \vec{b} \;\; \vec{c} \right|} \) is

The logical expression \( [p \wedge (q \vee r)] \vee [\neg r \wedge \neg q \wedge p] \) is equivalent to

Evaluate the integral \( \displaystyle \int \frac{e^x}{\sqrt{x}}(1+2x)\,dx \)

Simplify \( \cos\left(\dfrac{3\pi}{4}+x\right) - \sin\left(\dfrac{\pi}{4}-x\right) \)

Evaluate \( \displaystyle \int_{-\pi}^{\pi} \frac{2x}{1+\cos^2 x}\,dx \)

If \( f(x) = [x]^2 - 5[x] + 6 = 0 \), where \( [x] \) denotes the greatest integer function, then \( x \in \)

If \( \int_{0}^{1} (5x^2 - 3x + k) \, dx = 0 \), then \( k = \)

For a sequence \( (t_n) \), if \( s_n = 7(3^n - 1) \), then \( t_n = \)