List of top Questions asked in TS EAMCET

With respect to the roots of the equation \( 3x^3 + bx^2 + bx + 3 = 0 \), match the items of List-I with those of List-II.

The number of ways of arranging all the letters of the word "COMBINATIONS" around a circle so that no two vowels come together is 

A man has 7 relatives, 4 of them are ladies and 3 gents; his wife has 7 other relatives, 3 of them are ladies and 4 gents. The number of ways they can invite them to a party of 3 ladies and 3 gents so that there are 3 of man's relatives and 3 of wife's relatives, is 

If the coefficient of \( x^r \) in the expansion of \( (1 + x + x^2)^{100} \) is \( a_r \), and \( S = \sum\limits_{r=0}^{300} a_r \), then

\[ \sum\limits_{r=0}^{300} r a_r = \]

Given below are two statements, one is labelled as Assertion (A) and the other one labelled as Reason (R).
Assertion (A): \[ 1 + \frac{2.1}{3.2} + \frac{2.5.1}{3.6.4} + \frac{2.5.8.1}{3.6.9.8} + \dots \infty = \sqrt{4} \] Reason (R): \[ |x| <1, \quad (1 - x)^{-1} = 1 + nx + \frac{n(n+1)}{1.2} x^2 + \frac{n(n+1)(n+2)}{1.2.3} x^3 + \dots \] 

If \[ \frac{1}{x^4 + x^2 + 1} = \frac{Ax + B}{x^2 + ax + 1} + \frac{Cx + D}{x^2 - ax + 1} \] then \( A + B - C + D = ? \)

Identify the correct statements:

  • (i) LiF has more covalent character compared to KF.
  • (ii) Dipole moment of NF3 is greater than that of NH3.
  • (iii) The bond order is same for F2 and O2.
  • (iv) Ionic compounds possess low melting and boiling points.

If \( 0 <\theta <\frac{\pi}{4} \) and \( 8\cos\theta + 15\sin\theta = 15 \), then \( 15\cos\theta - 8\sin\theta = \)

Evaluate: \[ \sin 20^\circ (4 + \sec 20^\circ) = ? \]
If \( A \) is the solution set of the equation \( \cos^2 x = \cos^2 \frac{\pi}{6} \) and \( B \) is the solution set of the equation \( \cos^2 x = \log_{10} P \) where \( P + \frac{16}{P} = 10 \), then \( B - A = ? \)
The trigonometric equation \( \sin^{-1}x = 2\sin^{-1}a \) has a solution. Find the valid range for \( a \).
If \( \sinh x = \frac{12}{5} \), then \( \sinh 3x + \cosh 3x \) = ?
If \( \triangle ABC \) is an isosceles triangle with base \( BC \), then \( r_1 = ? \)
In \( \triangle ABC \), if \( r_1 + r_2 = 3R \), \( r_2 + r_3 = 2R \), then what type of triangle is \( \triangle ABC \)?

Let \( \bar{n} \) be a unit vector normal to the plane \( \pi \) containing the vectors \( \bar{T} + 3\bar{k} \) and \( 2\bar{i} + \bar{j} - \bar{k} \). If this plane \( \pi \) passes through the point \( (-3,7,1) \) and \( p \) is the perpendicular distance from the origin to this plane, then \( \sqrt{p^2 + 5} \) is:

If \( \bar{a} = \bar{i} - \bar{j} + 3\bar{k} \), \( \bar{c} = -\bar{k} \) are position vectors of two points and \( \bar{b} = 2\bar{i} - \bar{j} + \lambda \bar{k} \),

\( \bar{d} = \bar{i} + \bar{j} - \bar{k} \) are two vectors, then the lines \( r = \bar{a} + t \bar{b} \), \( r = \bar{c} + s \bar{d} \) are:

Let \( \bar{a} \) be a vector perpendicular to the plane containing non-zero vectors \( \bar{b} \) and \( \bar{c} \). If \( \bar{a}, \bar{b}, \bar{c} \) are such that

\[ |\bar{a} + \bar{b} + \bar{c}| = \sqrt{|\bar{a}|^2 + |\bar{b}|^2 + |\bar{c}|^2}, \]

then

\[ |(\bar{a} \times \bar{b}) \cdot (\bar{a} \times \bar{c})| = \]

The real-valued function \[ f(x) = \frac{|x - a|}{x - a} \] is analyzed as follows:

If the function

\[ f(x) = \begin{cases} \frac{(e^x - 1) \sin kx}{4 \tan x}, & x \neq 0 \\ P, & x = 0 \end{cases} \]

is differentiable at \( x = 0 \), then:

If

\[ A = \{ P(\alpha, \beta) \mid \text{the tangent drawn at P to the curve } y^3 - 3xy + 2 = 0 \text{ is a horizontal line} \} \]

and

\[ B = \{ Q(a, b) \mid \text{the tangent drawn at Q to the curve } y^3 - 3xy + 2 = 0 \text{ is a vertical line} \} \]

then \( n(A) + n(B) = \)

In a \( \triangle ABC \), the sides \( b, c \) are fixed. In measuring angle \( A \), if there is an error of \( \delta A \), then the percentage error in measuring the length of the side \( a \) is:
Consider the curves \( y = f(x) \) and \( x = g(y) \), and let \( P(x,y) \) be a common point of these curves. If at \( P \), on the curve \( y = f(x) \), \[ \frac{dy}{dx} = Q(x), \] and at the same point \( P \) on the curve \( x = g(y) \), \[ \frac{dx}{dy} = -Q(x), \] then:

If Rolle's Theorem is applicable for the function:

\[ f(x) = \begin{cases} x^p \log x, & x \neq 0 \\ 0, & x = 0 \end{cases} \]

on the interval \([0,1]\), then a possible value of \( p \) is:

If \[ \int (1 + x - x^x) e^{x + x^x} dx = f(x) + c, \] then \( f(1) - f(-1) = \)
If \[ \int \sqrt{\csc x + 1} \, dx = k \tan^{-1} (f(x)) + c, \] then \[ \frac{1}{k} f\left(\frac{\pi}{6}\right) = ? \]