List of practice Questions

If \(\lim_{x \to 0} \frac{ax^2 e^x - b \log_e (1 + x) + c x e^{-x}}{x^2 \sin x} = 1,\)then \( 16(a^2 + b^2 + c^2) \) is equal to \(\_\_\_\_\_\).

Let \( \vec{a} = 3\hat{i} + 2\hat{j} + \hat{k} \), \( \vec{b} = 2\hat{i} - \hat{j} + 3\hat{k} \), and \( \vec{c} \) be a vector such that

\((\vec{a} + \vec{b}) \times \vec{c} = 2(\vec{a} \times \vec{b}) + 24\hat{j} - 6\hat{k}\) and \((\vec{a} - \vec{b} + \hat{i}) \cdot \vec{c} = -3.\)Then \( |\vec{c}|^2 \) is equal to \(\_\_\_\_\_.\)

Let \( A \) be a \( 3 \times 3 \) matrix and \( \det(A) = 2 \). If \(n = \det(\text{adj}(\text{adj}(\ldots(\text{adj}(A))\ldots)))\) with adjoint applied 2024 times, then the remainder when \( n \) is divided by 9 is equal to \(\_\_\_\_\_.\)

Let \( A(-2, -1) \), \( B(1, 0) \), \( C(\alpha, \beta) \), and \( D(\gamma, \delta) \) be the vertices of a parallelogram \( ABCD \). If the point \( C \) lies on \( 2x - y = 5 \) and the point \( D \) lies on \( 3x - 2y = 6 \), then the value of \( | \alpha + \beta + \gamma + \delta | \) is equal to\( \_\_\_\_\_\).

Let \( a \), \( b \), \( c \) be the lengths of three sides of a triangle satisfying the condition \( (a^2 + b^2)x^2 - 2b(a + c)x + (b^2 + c^2) = 0 \). If the set of all possible values of \( x \) is the interval \( (\alpha, \beta) \), then \( 12(\alpha^2 + \beta^2) \) is equal to \(\_\_\_\_\_\).

The shortest distance between lines \( L_1 \) and \( L_2 \), where \( L_1 : \frac{x - 1}{2} = \frac{y + 1}{-3} = \frac{z + 4}{2} \) and \( L_2 \) is the line passing through the points \( A(-4, 4, 3) \), \( B(-1, 6, 3) \) and perpendicular to the line \( \frac{x - 3}{-2} = \frac{y}{3} = \frac{z - 1}{1} \), is

Let \( A \) be a \( 3 \times 3 \) real matrix such that**
\(A \begin{pmatrix} 1 \\ 0 \\ 1 \end{pmatrix} = 2 \begin{pmatrix} 1 \\ 0 \\ 1 \end{pmatrix}, \quad A \begin{pmatrix} 0 \\ 1 \\ 1 \end{pmatrix} = 4 \begin{pmatrix} -1 \\ 0 \\ 1 \end{pmatrix}, \quad A \begin{pmatrix} 1 \\ 1 \\ 0 \end{pmatrix} = 2 \begin{pmatrix} 1 \\ 1 \\ 0 \end{pmatrix}.\)
Then, the system \( (A - 3I) \begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix} \) has

A coin is based so that a head is twice as likely to occur as a tail. If the coin is tossed 3 times, then the probability of getting two tails and one head is-

If for some \( m, n \); \( ^6C_m + 2\left(^6C_{m+1}\right) + ^6C_{m+2} > ^8C_3 \) and \( ^{n-1}P_3 \cdot ^nP_4 = 1 : 8 \), then \( ^nP_{m+1} + ^{n+1}C_m \) is equal to

If \( a = \sin^{-1} (\sin(5)) \) and \( b = \cos^{-1} (\cos(5)) \), then \( a^2 + b^2 \) is equal to

The number of solutions of the equation \(e^{\sin x} - 2e^{-\sin x} = 2\) is

Consider the function \( f : (0, \infty) \rightarrow \mathbb{R} \) defined by \(f(x) = e^{-| \log_e x |}.\)If \( m \) and \( n \) be respectively the number of points at which \( f \) is not continuous and \( f \) is not differentiable, then \( m + n \) is

Let the mean and the variance of 6 observation a,b, 68, 44, 48, 60 be 55 and 194, respectively if a > b, then a + 3b is

The area of the region enclosed by the parabola \( y = 4x - x^2 \) and \( 3y = (x - 4)^2 \) is equal to

Let \( f : \mathbb{R} \rightarrow (0, \infty) \) be a strictly increasing function such that \(\lim_{x \to \infty} \frac{f(7x)}{f(x)} = 1.\)Then, the value of \(\lim_{x \to \infty} \left[ \frac{f(5x)}{f(x)} - 1 \right]\)is equal to

The temperature \( T(t) \) of a body at time \( t = 0 \) is \( 160^\circ \)F and it decreases continuously as per the differential equation \[ \frac{dT}{dt} = -K(T - 80), \] **where \( K \) is a positive constant. If \( T(15) = 120^\circ \)F, then \( T(45) \) is equal to

Let \( P \) be a parabola with vertex \( (2, 3) \) and directrix \( 2x + y = 6 \). Let an ellipse \( E : \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \), \( a > b \), of eccentricity \( \frac{1}{\sqrt{2}} \) pass through the focus of the parabola \( P \). Then the square of the length of the latus rectum of \( E \) is

Let \( (\alpha, \beta, \gamma) \) be the mirror image of the point \( (2, 3, 5) \) in the line \(\frac{x - 1}{2} = \frac{y - 2}{3} = \frac{z - 3}{4}\). Then \( 2\alpha + 3\beta + 4\gamma \) is equal to

If n is the number of ways five different employees can sit into four indistinguishable offices where any office may have any number of persons including zero, then n is equal to:

If \(5\) moles of an ideal gas expands from \(10\) L to a volume of \(100\) L at \(300k\) under isothermal and reversible condition then work, \(W\), is \(- x J\). The value of x is \((even \;n = 8.314 \;J \;K^{-1} mol^{-1})\)

In the truth table of the above circuit the value of X and Y are

A spherical ball of radius \( 1 \times 10^{-4} \, \text{m} \) and density \( 10^5 \, \text{kg/m}^3 \) falls freely under gravity through a distance \( h \) before entering a tank of water. If after entering in water the velocity of the ball does not change, then the value of \( h \) is approximately: \[ \text{(The coefficient of viscosity of water is } 9.8 \times 10^{-6} \, \text{N s/m}^2 \text{)} \]

The energy released in the fusion of 2 kg of hydrogen deep in the sun is $E_H$ and the energy released in the fission of 2 kg of $^{235}U$ is $E_U$. The ratio $\frac{E_H}{E_U}$ is approximately:

UV light of 4.13 eV is incident on a photosensitive metal surface having work function 3.13 eV. The maximum kinetic energy of ejected photoelectrons will be