List of practice Questions

Let sets A and B have 5 elements each. Let mean of the elements in sets A and B be 5 and 8 respectively and the variance of the elements in sets A and B be 12 and 20 respectively. A new set C of 10 elements is formed by subtracting 3 from each element of A and adding 2 to each element of B. Then the sum of the mean and variance of the elements of C is:

Area of the region \((x, y) : x^2 + (y - 2)^2 \leq 4, \, x^2 \geq 2y\) is:

Let \( \mathbf{a} \) be a non-zero vector parallel to the line of intersection of the two planes described by \( i + j + k \) and \( -i - j - k \). If \( \theta \) is the angle between the vector \( \mathbf{a} \) and the vector \( \mathbf{b} = -2i - 2j + 2k \), and \( \left| \mathbf{a} \right| = 6 \), then ordered pair \( (\mathbf{a} \cdot \mathbf{b}) \) is equal to:

The number of elements in the set \(S = \{ \theta \in [0, 2\pi] : 3 \cos^4 \theta - 5 \cos^2 \theta - 2 \sin^2 \theta + 2 = 0 \}\) is:

The value of the integral \[ \int_{\log_2}^{-\log_2} e^x \left( \log \left( e^x + \sqrt{1 + e^{2x}} \right) \right) dx \] is equal to:

Let \( f: [2, 4] \to \mathbb{R} \) be a differentiable function such that \( (x \log x) f'(x) + (\log x) f(x) \geq 1 \), \( x \in [2, 4] \) with \( f(2) = \frac{1}{2} \) and \( f(4) = \frac{1}{4} \). Consider the following two statements:
\( (A) \quad f(x) \geq 1 \quad \text{for all} \quad x \in [2, 4] \)
\( (B) \quad f(x) \leq \frac{1}{8} \quad \text{for all} \quad x \in [2, 4] \) Then,

Let A be a \( 2 \times 2 \) matrix with real entries such that \( A^T = \alpha A + I \), where \( \alpha \in \mathbb{R} \setminus \{-1, 1\} \). If \( \text{det}(A^2 - A) = 4 \), then the sum of all possible values of \( \alpha \) is equal to:

The number of integral solutions of \( \log_2 \left( \frac{x - 7}{2x - 3} \right) \geq 0 \) is:

Let \(x_1, x_2, \dots, x_{100}\) be in an arithmetic progression, with \(x_1 = 2\) and their mean equal to 200. If \(y_i = (i \cdot x_i)\), then the mean of \(y_1, y_2, \dots, y_{100}\) is:

Fill in the blanks with appropriate prepositions from the given options:
In a bid to check urban apathy _______ voters, the Election comission had decided _______ hold the poll _________ the middle _________ the week to prevent people planning an outing by clubbing the poll day holiday with the weekend break.

The value of x compound ‘D’ is

Three organic compounds A, B and C were allowed to run in thin layer chromatography using hexane and gave the following result (see figure). The R_f value of the most polar compound is ____ x 10^-2.

The reaction 2NO + Br2 → 2NOBr
takes places through the mechanism given below :
NO + Br2 ⇔ NOBr2 (fast)
NOBr2 + NO → 2NOBr (slow)
The overall order of the reaction is_____.

pH of 1 litre of HCL solution is 1. How much water (in litres) is added to make pH=2?

Match the terms in Column-I with their description in Column-II and choose the correct option:

	Column I		Column II
(A).	Biodegradable	(P).	Polyacrylonitrie
(B).	Synthetic	(Q).	PHBV
(C).	Natural	(R).	Dacron
(D).	Polyester	(S).	Rubber

The incorrect statement regarding the reaction given below is

Let [t] denote the greatest integer function. If \(\int\limits_0^{2.4}[x^2]dx=α+β√2+γ√3+δ√5,\) then α+β+γ +δ is equal to _____.

Let \( k \) and \( m \) be positive real numbers such that the function} \[ f(x) = \begin{cases} 3x^2 + \frac{k}{\sqrt{x} + 1}, & 0 < x < 1, \\ mx^2 + k^2, & x \geq 1 \end{cases} \] is differentiable for all \( x > 0 \). Then \( 8f'(8) \left(\frac{1}{f(8)}\right) \) is equal to __________

Which one is a reaction to prepare CCl₂F₂(freon-12) from CCl₄?

Let \( m \) and \( n \) be the numbers of real roots of the quadratic equations \( x^2 - 12x + [x] + 31 = 0 \) and \( x^2 - 5|x+2| - 4 = 0 \), respectively, where \( [x] \) denotes the greatest integer less than or equal to \( x \). Then \( m^2 + mn + n^2 \) is equal to ___________.

The major products ‘P’ formed in the following sequence of reaction is

The negation of \( (p \land (\sim q)) \lor (\sim p) \) is equivalent to:

Let the mean and variance of 12 observations be \( \frac{9}{2} \) and 4, respectively. Later on, it was observed that two observations were considered as 9 and 10 instead of 7 and 14 respectively. If the correct variance is \( \frac{m}{n} \), where \( m \) and \( n \) are coprime, then \( m + n \) is equal to:

If the probability that the random variable \( X \) takes values \( x \) is given by \( P(X = x) = k(x + 1) 3^{-x}, x = 0, 1, 2, \dots \), where \( k \) is a constant, then \( P(X \geq 2) \) is equal to:

Let the vectors \(\mathbf{u}_1 = \hat{i} + \hat{j} + a\hat{k}, \mathbf{u}_2 = \hat{i} + b\hat{j} + \hat{k}\), and \(\mathbf{u}_3 = c\hat{i} + \hat{j} + \hat{k}\) be coplanar. If the vectors \(\mathbf{v}_1 = (a + b)\hat{i} + c\hat{j} + c\hat{k}, \mathbf{v}_2 = a\hat{i} + (b + c)\hat{j} + a\hat{k}, \mathbf{v}_3 = b\hat{i} + b\hat{j} + (c + a)\hat{k}\) are also coplanar, then \(6(a + b + c)\) is equal to: