List of practice Questions

In a triangle \(ABC\), if \(r_1 : r_2 = 3 : 4\) and \(r_1 : r_3 = 2 : 3\), then find the ratio \(a : b : c\).
Evaluate: \(\tan\left(2\tan^{-1}\left(-\frac{1}{3}\right) + \tan^{-1}\left(\frac{1}{7}\right)\right) =\)
The number of solutions of the equation $4 \cos 2\theta \cos 3\theta = \sec \theta$ in the interval $[0, 2\pi]$ is
Evaluate \[ (4 \cos^2 \frac{\pi}{20} - 1)(4 \cos^2 \frac{3\pi}{20} - 1)(4 \cos^2 \frac{5\pi}{20} - 1)(4 \cos^2 \frac{7\pi}{20} - 1)(4 \cos^2 \frac{9\pi}{20} - 1). \]
If $\cos 80^\circ + \cos 40^\circ - \cos 20^\circ = k$, then $\frac{4k}{3} =$
Assertion (A): $S_3 = 55 \times 2^9$
Reason (R): $S_1 = 90 \times 2^8$ and $S_2 = 10 \times 2^8$
If A and B are the values such that $(A + B)$ and $(A - B)$ are not odd multiples of $\frac{\pi}{2}$ and $2\tan(A+B) = 3 \tan(A-B)$, then $\sin A \cos A =$
The coefficient of $x^{10}$ in the expansion of $\left(x + \frac{2}{x} - 5 \right)^{12}$ is
The number of ways of dividing 15 persons into 3 groups containing 3, 5 and 7 persons so that two particular persons are not included into the 5 persons group is
If \[ \frac{2x^4 - 3x^2 + 4}{(x^2 + 1)(x^2 + 2)} = a + \frac{px + q}{x^2 + 1} + \frac{mx + n}{x^2 + 2}, \] then $\frac{n}{q} =$
After the roots of the equation $6x^3 + 7x^2 - 4x - 2 = 0$ are diminished by $h$, if the transformed equation does not contain $x$ term, then the product of all possible values of $h$ is
The number of integers greater than 6000 that can be formed by using the digits 0, 5, 6, 7, 8 and 9 without repetition is
If the difference of the roots of the equation $x^2 - 7x + 10 = 0$ is same as the difference of the roots of the equation $x^2 - 17x + k = 0$, then a divisor of $k$ is
The number of distinct quadratic equations $ax^2 + bx + c = 0$ with unequal real roots that can be formed by choosing the coefficients $a, b, c$ (with $a \ne 0$) from the set $\{0,1,2,4\}$ is
The product of all the real roots of the equation $|x^2 - 5||x| + 6 = 0$ is
If $\alpha, \beta$ and $\gamma$ are the roots of the equation $5x^3 - 4x^2 + 3x - 2 = 0$, then $\alpha^3 + \beta^3 + \gamma^3$ equals
If the values of $x, y,$ and $z$ satisfy the equations \[ 2x - 3y + 2z + 15 = 0,
3x + y - z + 2 = 0,
x - 3y - 3z + 8 = 0 \] simultaneously are $\alpha, \beta,$ and $\gamma$ respectively, then
Evaluate \[ \frac{1}{3 . 5} + \frac{1}{5 . 7} + \frac{1}{7 . 9} + .s \text{ up to } 24 \text{ terms}. \]
The product of the four values of the complex number $(1+i)^{3/4}$ is
If B is the inverse of a third order matrix A and det B = k, then $(\text{adj}(\text{adj} A))^{-1}=$
If $A = \begin{bmatrix} 2 & 2 & 1 \\ 1 & 3 & 1 \\ 1 & 2 & 2 \end{bmatrix}$ and $\alpha, \beta, \gamma$ are the roots of the equation represented by $|A - xI| = 0$, then $\alpha^2 + \beta^2 + \gamma^2 =$
If $x = 3 - 2\sqrt{3}i$, then evaluate $x^4 - 12x^3 + 54x^2 - 108x - 54 = $
If the range of the function $f(x) = -3x - 3$ is $\{3, -6, -9, -18\}$, then which one of the following is not in the domain of $f$?
Let $[x]$ represent the greatest integer less than or equal to $x$, $\{x\} = x - [x]$. Given \[ \sqrt{2} = 1.414
\text{and}
\sqrt{3} = 1.732. \] If \[ f(x) = x + \frac{x}{1 + x^2} \] is a real valued function, then find \[ f(\sqrt{2}) + f(-\sqrt{3}). \]
Which of the following is NOT a characteristic of cloud computing?