List of top Mathematics Questions

In the adjoining figure, $ \triangle CAB $ is a right triangle, right angled at A and $ AD \perp BC $. Prove that $ \triangle ADB \sim \triangle CDA $. Further, if $ BC = 10 \text{ cm} $ and $ CD = 2 \text{ cm} $, find the length of } $ AD $.

If $ \sin x + \sin^2 x = 1 $, $ x \in \left(0, \frac{\pi}{2} \right) $, then the expression \[ (\cos^2 x + \tan^2 x) + 3(\cos^4 x + \tan^4 x + \cos^4 x + \tan^4 x) + (\cos^6 x + \tan^6 x) \] is equal to:

Solve the quadratic equation $\sqrt{3}x^2 + 10x + 7\sqrt{3} = 0$ using the quadratic formula.

Let $ y^2 = 12x $ be the parabola and $ S $ its focus. Let $ PQ $ be a focal chord of the parabola such that $ (SP)(SQ) = \frac{147}{4} $. Let $ C $ be the circle described by taking $ PQ $ as a diameter. If the equation of the circle $ C $ is: \[ 64x^2 + 64y^2 - \alpha x - 64\sqrt{3}y = \beta, \] then $ \beta - \alpha $ is equal to:

For a $ 3 \times 3 $ matrix $ M $, let trace(M) denote the sum of all the diagonal elements of $ M $. Let $ A $ be a $ 3 \times 3 $ matrix such that $ |A| = \frac{1}{2} $ and $ \text{trace}(A) = 3 $. If $ B = \text{adj}(\text{adj}(2A)) $, then the value of $ |B| + \text{trace}(B) $ equals:

Let $ \hat{a} $ be a unit vector perpendicular to the vectors \[ \mathbf{b} = \hat{i} - 2\hat{j} + 3\hat{k} \quad \text{and} \quad \mathbf{c} = 2\hat{i} + 3\hat{j} - \hat{k}, \] $\text{and makes an angle of}$ $ \cos\left( -\frac{1}{3} \right) $ $\text{with the vector}$ $ \hat{i} + \alpha \hat{j} + \hat{k} $. $\text{If}$ $ \hat{a} $ $\text{makes an angle of}$ $ \frac{\pi}{3} $ $\text{with the vector}$ $ \hat{i} + \alpha \hat{j} + \hat{k} $, then the value of $ \alpha $ $\text{is:}$

Let the function $f(x) = (x^2 - 1)|x^2 - ax + 2| + \cos|x| $ be not differentiable at the two points $ x = \alpha = 2 $ and $ x = \beta $. Then the distance of the point $(\alpha, \beta)$ from the line $12x + 5y + 10 = 0$ is equal to:

Find the shortest distance between the lines: \[ \mathbf{r}_1 = (2\hat{i} - \hat{j} + 3\hat{k}) + \lambda (\hat{i} - 2\hat{j} + 3\hat{k}) \] \[ \mathbf{r}_2 = (\hat{i} + 4\hat{k}) + \mu (3\hat{i} - 6\hat{j} + 9\hat{k}) \]

Let $ A $ be a matrix of order $ 3 \times 3 $ and $ |A| = 5 $. If $ |2 \, \text{adj}(3A \, \text{adj}(2A))| = 2^{\alpha} \cdot 3^{\beta} \cdot 5^{\gamma}, \quad \alpha, \beta, \gamma \in \mathbb{N} $ then $ \alpha + \beta + \gamma $ is equal to

A person invested Rupees 10000 in a stock of a company for 6 years. The value of his investment at the end of each year is given in the following table:
\begin{tabular}{|c|c|c|c|c|c|} \hline 2018 & 2019 & 2020 & 2021 & 2022 & 2023
\hline Rupees 11000 & Rupees 11500 & Rupees 13000 & Rupees 11800 & Rupees 12200 & Rupees 14000
\hline \end{tabular}
The compound annual growth rate (CAGR) of his investment is:
Given $(1.4)^{1/6} \approx 1.058$

If a 95% confidence interval for a population mean was reported to be 132 to 160 and sample standard deviation s = 50, then the size of the sample in the study is:
(Given $Z_{0.025}$ = 1.96)

Which of the following are correct?
(A) Time series analysis does not help to understand the behavior of a variable in the past.
(B) Time series predict the future behavior of variable.
(C) Time series helps to plan future operations.
(D) The main aim of the time series analysis is to derive conclusions after arranging the time series in a systematic manner.

Let F(Z) be the cumulative density function of the standard normal variate Z, then which of the following are correct?
(A) $F(Z) = \int_{-\infty}^{Z} \frac{1}{\sqrt{2\pi}} e^{-z^2/2} dz, -\infty < Z <\infty$
(B) $F(-Z) = 1 - F(Z)$
(C) $F(0) = 0$
(D) $F(\infty) = 1$
Choose the correct answer from the options given below:

The probability distribution of the random variable X is given by
\begin{tabular}{|l|l|l|l|l|} \hline X & 0 & 1 & 2 & 3 \\ \hline P(X) & 0.2 & k & 2k & 2k \\ \hline \end{tabular}
The variance of the random variable X is

If $\int \frac{(1 + x \log x)}{xe^{-x}} dx = e^x f(x) + C$, where C is constant of integration, then f(x) is

Which of the following inequalities holds true?
(A) $\sqrt{5} + \sqrt{3}>\sqrt{6} + \sqrt{2}$
(B) If $a>b$ and $c<0$, then $\frac{a}{c}<\frac{b}{c}$
(C) $\frac{1}{x^2}>\frac{1}{x}>1$, if $0<x<1$
(D) If a and b are positive integers and $\frac{a-b}{6.25} = \frac{4}{2.5}$ then $b>a$
Choose the correct answer from the options given below:

The least non-negative remainder when $3^{128}$ is divided by 7 is:

Which one of the following set of constraints does the given shaded region represent?

Consider the line \[ \vec{r} = (\hat{i} - 2\hat{j} + 4\hat{k}) + \lambda(-\hat{i} + 2\hat{j} - 4\hat{k}) \]

Match List-I with List-II:

List-I	List-II
(A) A point on the given line	(I) $\left(-\tfrac{1}{\sqrt{21}}, \tfrac{2}{\sqrt{21}}, -\tfrac{4}{\sqrt{21}}\right)$
(B) Direction ratios of the line	(II) (4, -2, -2)
(C) Direction cosines of the line	(III) (1, -2, 4)
(D) Direction ratios of a line perpendicular to given line	(IV) (-1, 2, -4)

Let $\vec{a} = \hat{i} + 4\hat{j}$, $\vec{b} = 4\hat{j} + \hat{k}$ and $\vec{c} = \hat{i} - 2\hat{k}$. If $\vec{d}$ is a vector perpendicular to both $\vec{a}$ and $\vec{b}$ such that $\vec{c} \cdot \vec{d} = 16$, then $|\vec{d}|$ is equal to

If $\hat{i}$, $\hat{j}$ and $\hat{k}$ are unit vectors along co-ordinates axes OX, OY and OZ respectively, then which of the following is/are true?
(A) $\hat{i} \times \hat{i} = \vec{0}$
(B) $\hat{i} \times \hat{k} = \hat{j}$
(C) $\hat{i} . \hat{i} = 1$
(D) $\hat{i} . \hat{j} = 0$
Choose the correct answer from the options given below:

The area (in sq. units) of the region bounded by y = $2\sqrt{1-x^2}$, x $\in$ [0,1] and x-axis is equal to

$\int_{\pi/6}^{\pi/3} \frac{\tan x}{\tan x + \cot x} dx$ is equal to

Let A = [aij]n x n be a matrix. Then Match List-I with List-II
List-I
(A) AT = A
(B) AT = -A
(C) |A| = 0
(D) |A| $\neq$ 0
List-II
(I) A is a singular matrix
(II) A is a non-singular matrix
(III) A is a skew symmetric matrix
(IV) A is a symmetric matrix
Choose the correct answer from the options given below:

The general solution of the differential equation $ x\,dy + \left(y - e^x\right) dx = 0 $ is:

List-I	List-II
(A) A point on the given line	(I) \(\left(-\tfrac{1}{\sqrt{21}}, \tfrac{2}{\sqrt{21}}, -\tfrac{4}{\sqrt{21}}\right)\)
(B) Direction ratios of the line	(II) (4, -2, -2)
(C) Direction cosines of the line	(III) (1, -2, 4)
(D) Direction ratios of a line perpendicular to given line	(IV) (-1, 2, -4)