List of practice Questions

Two dice are thrown together. The probability that the sum of the numbers is divisible by 2 or 3 is

Evaluate the sum of the series: \[ \frac{1^2}{2} + \frac{1^2 + 2^2}{3} + \frac{1^2 + 2^2 + 3^2}{4} + \frac{1^2 + 2^2 + 3^2 + 4^2}{5} + \cdots \quad \text{upto 8 terms} \]

Evaluate the integral: \[ \int_0^{\frac{\pi}{2}} \frac{dx}{1 + \cos x} \]

Which of the following have the same value:
(a) \( \sin 120^\circ \)
(b) \( \cos 930^\circ \)
(c) \( \tan 840^\circ \)
(d) \( \cot(-1110^\circ) \)

The approximate value of \( \log_{10}99 \) is (Given \( \log_{10}e = 0.4343 \))

If the function \[ f(x)= \begin{cases} \dfrac{\log 10 + \log(0.1+2x)}{2x}, & x \neq 0 \\ k, & x=0 \end{cases} \] is continuous at \(x=0\), then \(k+2=\)

\(y = mx + \dfrac{2}{m}\) is the general solution of

In a triangle \(ABC\) with usual notations, \[ \frac{\cos A - \cos C}{a - c} + \frac{\cos B}{b} = \]

The maximum value of \(Z = 10x + 25y\) subject to \[ 0 \le x \le 3,\; 0 \le y \le 3,\; x + y \le 5,\; x \ge 0,\; y \ge 0 \] is

The equation of a line passing through the point of intersection of the lines \[ x + 2y + 8 = 0 \quad \text{and} \quad 3x - y + 4 = 0 \] and having x– and y–intercepts zero is

Evaluate the integral: \[ \int \frac{\sin x}{\sin\left(x-\frac{\pi}{4}\right)}\,dx \]

If \((\sim p \wedge q)\rightarrow r\) is false, then the truth values of \(p,q,r\) respectively are

In a triangle \(ABC\) with usual notations, if \(\tan A,\tan B,\tan C\) are in H.P., then \(a^2,b^2,c^2\) are in

If the displacement of a particle at time \(t\) is given by \[ s = 3t^2 - 12t + 14, \] then the displacement of the particle when its velocity becomes zero is

The joint equation of a pair of lines passing through \((2,3)\) and parallel to the lines \[ x^2 - y^2 = 0 \] is

The area bounded by the circle \(x^2 + y^2 = 16\) and the lines \(x = 0\) and \(x = 2\) is

Evaluate the integral \[ \int_{0}^{\pi} \frac{x \cos x \sin x}{\cos^3 x + \cos x}\,dx \]

Given \(A = \{1,2,3,4,5\}\), \(B = \{1,4,5\}\). If \(R\) is a relation from \(A\) to \(B\) such that \((x,y) \in R\) with \(x>y\), then the range of \(R\) is

The equation of a plane passing through the intersection of the planes \(x + 2y - 3z + 2 = 0\) and \(6x + y + z + 1 = 0\) and parallel to the line \(x - 1 = y + 2 = 7 - z\) is

The value of \(x\) such that the matrix \[ \begin{bmatrix} x & 2 & 3 \\ 4 & 5 & 6 \\ 2 & 3 & 5 \end{bmatrix} \] is not invertible is

If \[ A = \begin{bmatrix} 0 & 0 & -1 \\ 0 & -1 & 0 \\ -1 & 0 & 0 \end{bmatrix} \] then

Identify the number of oxygen atoms present in saccharic acid.

What is the approximate ratio of roasted ore, coke, and lime stone respectively in the charge to be added into the blast furnace for extraction of iron?

Relation between \( \Delta H \) and \( \Delta U \) for the reaction \[ \text{2SO}_3 (g) \rightarrow \text{2SO}_2 (g) + \text{O}_2 (g) \] is

At what new pressure will 100 mL of a gas at pressure 720 mm occupy volume of 84 mL, keeping temperature constant?