List of practice Questions

Two views of a solid are shown. Which of the options is/are the view/s of the same solid? 

Find the digit that is not written in the same font as others.
Let \( P = \begin{bmatrix} 3 & -1 & -2 \\ 2 & 0 & \alpha \\ 3 & -5 & 0 \end{bmatrix} \), where \( \alpha \in \mathbb{R} \). Suppose \( Q = [q_{ij}] \) is a matrix satisfying \( PQ = k I_3 \) for some non-zero \( k \in \mathbb{R} \). If \( q_{23} = -\dfrac{k}{8} \) and \( |Q| = \dfrac{k^3}{2} \), then \( \alpha^2 + k^2 \) is equal to __________.
The equation of the plane passing through the point (1, 2, -3) and perpendicular to the planes $3x + y - 2z = 5$ and $2x - 5y - z = 7$, is :
The system of linear equations 3x - 2y - kz = 10, 2x - 4y - 2z = 6, x + 2y - z = 5m is inconsistent if :
If \( f(x) = [x - 1]\cos\!\left( \frac{2x - 1}{2}\pi \right) \), then \( f \) is :
The function \( f(x) = \dfrac{4x^3 - 3x^2}{6} - 2 \sin x + (2x - 1)\cos x \) :
\( \displaystyle \lim_{x \to 0} \frac{\int_{0}^{x^2} \sin(\sqrt{t}) \, dt}{x^3} \) is equal to :
Let \( p \) and \( q \) be two positive numbers such that \( p + q = 2 \) and \( p^4 + q^4 = 272 \). Then \( p \) and \( q \) are roots of the equation :

Identify products A and B. $CH_3 - CH(CH_3) - CH_3 \xrightarrow{\text{dil. } KMnO_4, 273 K} A \xrightarrow{CrO_3} B$ 

The product formed in the first step of the reaction of $CH_3 - CH_2 - CH(Br) - CH_2 - CH(Br) - CH_3$ with excess $Mg/Et_2O$ is :

What is the major product formed by HI reaction with

The electrode potential of \( \mathrm{M^{2+}/M} \) of 3d-series elements shows positive value for :
\( \mathrm{Al_2O_3} \) was leached with alkali to get \( X \). The solution of \( X \), on passing the gas \( Y \), forms \( Z \). \( X, Y \) and \( Z \) respectively are :

In Freundlich adsorption isotherm, slope of AB line is : 

In the given figure, energy levels of H-atom are shown. Transitions A ($n=\infty \to 1$), B ($n=4 \to 2$), C ($n=4 \to 3$) represent : 

If the velocity-time graph has the shape AMB (where M is a peak), what would be the shape of the corresponding acceleration-time graph ? 

If \(\int_{-a}^{a} (|x| + |x-2|) dx = 22, (a>2)\) and \([x]\) denotes the greatest integer \(\le x\), then \(\int_{a}^{-a} (x + [x]) dx\) is equal to ______
\(\lim_{n \to \infty} \tan \left\{ \sum_{r=1}^{n} \tan^{-1} \left( \frac{1}{1 + r + r^2} \right) \right\}\) is equal to __________.
The statement among the following that is a tautology is :
If the least and the largest real values of \(\alpha\), for which the equation \(|z| + \alpha(z - 1) + 2i = 0\) (\(z \in C\) and \(i = \sqrt{-1}\)) has a solution, are \(p\) and \(q\) respectively; then \(4(p^2 + q^2)\) is equal to _________
Let \(M\) be any \(3\times3\) matrix with entries from the set {0, 1, 2}. The maximum number of such matrices, for which the sum of diagonal elements of \(M^T M\) is seven, is __________.
Let \(A = \{n \in N : n \text{ is a 3-digit number}\}\), \(B = \{9k + 2 : k \in N\}\) and \(C = \{9k + l : k \in N\}\) for some \(l\) (0<l<9) If the sum of all the elements of the set \(A \cap (B \cup C)\) is \(274 \times 400\), then \(l\) is equal to __________
The minimum value of \(\alpha\) for which the equation \(\frac{4}{\sin x} + \frac{1}{1 - \sin x} = \alpha\) has at least one solution in \((0, \frac{\pi}{2})\) is ____________
The tangent to the curve $y = x^3$ at the point $P(t, t^3)$ meets the curve again at Q, then the ordinate of the point which divides PQ internally in the ratio 1:2 is :