List of practice Questions

Let \( \vec{a} = -\hat{i} + \hat{j} + 2\hat{k} \), \( \vec{b} = \hat{i} - \hat{j} - 3\hat{k} \), \( \vec{c} = \vec{a} \times \vec{b} \) and \( \vec{d} = \vec{c} \times \vec{a} \). Then \( (|\vec{a}|^2 - |\vec{b}|^2) \cdot \vec{d \) is equal to:}
Let A = \{-2, -1, 0, 1, 2, 3, 4\. Let R be a relation on A defined by xRy if and only if \(2x + y \le 2\). Let \(l\) be the number of elements in R. Let \(m\) and \(n\) be the minimum number of elements required to be added in R to make it reflexive and symmetric relations respectively. Then \(l + m + n\) is equal to :}
Let the mean and variance of 8 numbers -10, -7, -1, x, y, 9, 2, 16 be \( 2 \) and \( \frac{293}{4} \), respectively. Then the mean of 4 numbers x, y, x+y+1, |x-y| is:
A rectangle is formed by the lines \( x = 0 \), \( y = 0 \), \( x = 3 \) and \( y = 4 \). Let the line \( L \) be perpendicular to \( 3x + y + 6 = 0 \) and divide the area of the rectangle into two equal parts. Then the distance of the point \( \left(\frac{1}{2}, -5\right) \) from the line \( L \) is equal to :
Let \( y = y(x) \) be the solution of the differential equation \( x^2 dy + (4x^2 y + 2\sin x)dx = 0 \), \( x>0 \), \( y\left(\frac{\pi}{2}\right) = 0 \). Then \( \pi^4 y\left(\frac{\pi}{3}\right) \) is equal to :
A building construction work can be completed by two masons A and B together in 22.5 days. Mason A alone can complete the construction work in 24 days less than mason B alone. Then mason A alone will complete the construction work in :
Let \( f(x) = \int \frac{(2 - x^2) \cdot e^x{(\sqrt{1 + x})(1 - x)^{3/2}} dx \). If \( f(0) = 0 \), then \( f\left(\frac{1}{2}\right) \) is equal to :}
Let the domain of \( f(x) = \log_3 \log_3 \log_7 (9x - x^2 - 13) \) be (m, n). Let the hyperbola \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \) have eccentricity \( \frac{n}{3} \) and latus rectum \( \frac{8m}{3} \). Then \( b^2 - a^2 \) is equal to :
The value of the integral \( \int_{\frac{\pi}{24}}^{\frac{5\pi}{24}} \frac{dx}{1 + \sqrt[3]{\tan 2x}} \) is :
Among the statements :
I: If the given determinants are equal, then \( \cos^2\alpha + \cos^2\beta + \cos^2\gamma = \frac{3}{2} \), and
II: If the polynomial determinant equals \( px + q \), then \( p^2 = 196q^2 \), identify the truth value.
The vertices B and C of a triangle ABC lie on the line \( \frac{x}{1} = \frac{1-y}{2} = \frac{z-2}{3} \). The coordinates of A and B are (1, 6, 3) and (4, 9, 6) respectively and C is at a distance of 10 units from B. The area (in sq. units) of \( \triangle ABC \) is:
If \( \alpha \) and \( \beta \) (\( \alpha<\beta \)) are the roots of the equation \( (-2 + \sqrt{3})(\sqrt{x} - 3) + (x - 6\sqrt{x}) + (9 - 2\sqrt{3}) = 0 \), \( x \ge 0 \), then \( \sqrt{\frac{\beta}{\alpha}} + \sqrt{\alpha\beta} \) is equal to:
Let \( f(x) = \begin{cases} \frac{ax^2 + 2ax + 3}{4x^2 + 4x - 3}, & x \neq -\frac{3}{2}, \frac{1}{2} \\ b, & x = -\frac{3}{2}, \frac{1}{2} \end{cases} \) be continuous at \( x = -\frac{3}{2} \). If \( f(x) = \frac{7}{5} \), then \( x \) is equal to :
Let \( \alpha \) and \( \beta \) respectively be the maximum and the minimum values of the function \( f(\theta) = 4\left(\sin^4\left(\frac{7\pi}{2} - \theta\right) + \sin^4(11\pi + \theta)\right) - 2\left(\sin^6\left(\frac{3\pi}{2} - \theta\right) + \sin^6(9\pi - \theta)\right) \). Then \( \alpha + 2\beta \) is equal to :
Let the direction cosines of two lines satisfy the equations : \( 4l + m - n = 0 \) and \( 2mn + 5nl + 3lm = 0 \). Then the cosine of the acute angle between these lines is :
Number of solutions of \( \sqrt{3} \cos 2\theta + 8 \cos \theta + 3\sqrt{3} = 0, \theta \in [-3\pi, 2\pi] \) is:
Let the line \( y - x = 1 \) intersect the ellipse \( \frac{x^2}{2} + \frac{y^2}{1} = 1 \) at the points A and B. Then the angle made by the line segment AB at the center of the ellipse is:
What are the constituents of Natalite?
Consider the following redox reaction taking place in acidic medium: \[ \mathrm{BH_4^- (aq) + ClO_3^- (aq) \rightarrow H_2BO_3^- (aq) + Cl^- (aq)} \] If the Nernst equation for the above balanced reaction is \[ E_{\text{cell}} = E^\circ_{\text{cell}} - \frac{RT}{nF}\ln Q, \] then the value of \(n\) is ________ (Nearest integer).
\(0.53\ \text{g}\) of an organic compound \(X\) when heated with excess concentrated nitric acid and then with silver nitrate gave \(0.75\ \text{g}\) of silver bromide precipitate. \(1.0\ \text{g}\) of \(X\) gave \(1.32\ \text{g}\) of \(\mathrm{CO_2}\) on combustion. Find the percentage of hydrogen in compound \(X\). (Nearest integer) [Given: Atomic masses (g mol\(^{-1}\)): H = 1, C = 12, Br = 80, Ag = 108, O = 16]
Consider the dissociation equilibrium of the following weak acid: \[ \mathrm{HA \rightleftharpoons H^+(aq) + A^-(aq)} \] If the \(pK_a\) of the acid is \(4\), then the pH of a \(10\ \text{mM}\) HA solution is ________ (Nearest integer). (Given: The degree of dissociation can be neglected with respect to unity)
500 mL of \(1.2\) M KI solution is mixed with 500 mL of \(0.2\) M \( \mathrm{KMnO_4} \) solution in basic medium. The liberated iodine is titrated with standard \(0.1\) M \( \mathrm{Na_2S_2O_3} \) solution in the presence of starch indicator till the blue colour disappears. The volume (in L) of \( \mathrm{Na_2S_2O_3} \) consumed is ________ (nearest integer).

In the given pentapeptide, find out an essential amino acid (\(Y\)) and the sequence present in the pentapeptide.
 Choose the correct answer from the options given below:

The correct statement among the following is:

Given below are the four isomeric compounds \(P, Q, R, S\):
\(P\): Aromatic compound containing an \(-\mathrm{OH}\) group 
\(Q\): Aromatic compound containing an \(-\mathrm{CHO}\) group (aldehyde) 
\(R\): Aromatic compound containing a ketone group 
\(S\): Aromatic compound containing a ketone group Identify the correct statements from below: 
[A.] \(Q, R\) and \(S\) will give precipitate with \(2,4\)-DNP. 
[B.] \(P\) and \(Q\) will give positive Baeyer’s test. 
[C.] \(Q\) and \(R\) will give sooty flame. 
[D.] \(R\) and \(S\) will give yellow precipitate with \(I_2/\mathrm{NaOH}\). 
[E.] \(Q\) alone will deposit silver with Tollens’ reagent. Choose the correct option.