List of top Mathematics Questions

Given that $\sin \theta + \cos \theta = x$, prove that $\sin^4 \theta + \cos^4 \theta = \dfrac{2 - (x^2 - 1)^2}{2}$.

Nidhi received simple interest of ₹1,200 when she invested ₹x at 6% per annum and ₹y at 5% per annum for 1 year. Had she invested ₹x at 3% per annum and ₹y at 8% per annum for that year, she would have received simple interest of ₹1,260.Find the values of x and y.

In the adjoining figure, \(PQ \parallel XY \parallel BC\), \(AP=2\ \text{cm}, PX=1.5\ \text{cm}, BX=4\ \text{cm}\). If \(QY=0.75\ \text{cm}\), then \(AQ+CY =\)

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 $BC = 10$ cm and $CD = 2$ cm, find the length of AD.

Solve the following pair of equations algebraically:
\[ 101x + 102y = 304 \quad \text{(i)}\\ 102x + 101y = 305 \quad \text{(ii)} \]
Find the nature of roots of the equation \(3x^2 - 4\sqrt{3}x + 4 = 0\).
Which one of the following plots represents \( f(x) = -\frac{|x|}{x} \), where \( x \) is a non-zero real number?
Note: The figures shown are representative.
A drone is flying at a height of \(h\) metres. At an instant it observes the angle of elevation of the top of an industrial turbine as \(60^\circ\) and the angle of depression of the foot of the turbine as \(30^\circ\). If the height of the turbine is \(200\, \text{metres}\), find the value of \(h\) and the distance of the drone from the turbine.
(Use \(\sqrt{3} = 1.73\))
Let $p$, $q$ and $r$ be three distinct prime numbers. Check whether $pqr + q$ is a composite number or not. Further, give an example for three distinct primes $p$, $q$, $r$ such that
(i) $pqr + 1$ is a composite number
(ii) $pqr + 1$ is a prime number
Find the smallest value of $p$ for which the quadratic equation $x^2 - 2(p + 1)x + p^2 = 0$ has real roots. Hence, find the roots of the equation so obtained.
Solve the following linear programming problem graphically: Maximise \( Z = 20x + 30y \) Subject to the constraints: \[ x + y \leq 0, \quad 2x + 3y \geq 100, \quad x \geq 14, \quad y \geq 14. \]

A regular dodecagon (12-sided regular polygon) is inscribed in a circle of radius \( r \) cm as shown in the figure. The side of the dodecagon is \( d \) cm. All the triangles (numbered 1 to 12 in the figure) are used to form squares of side \( r \) cm, and each numbered triangle is used only once to form a square. The number of squares that can be formed and the number of triangles required to form each square, respectively, are:


 

In \(\triangle ABC, DE || BC\). If AE = (2x + 1) cm, EC = 4 cm, AD = (x + 1) cm and DB = 3 cm, then value of x is
△ABC,DE||BC. If AE = (2x + 1) cm, EC = 4 cm, AD = (x + 1)

Match List-I with List-II

List-IList-II
(A) An observed set of population selected for analysis(I) Parameter
(B) A specific characteristic of the population(II) Hypothesis
(C) A specific characteristic of the sample(III) Statistic
(D) A statement made about a population parameter for testing(IV) Sample
If X = 11 and Y = 3, then X mod Y = (X + aY) mod Y holds
Which of the following are the assumptions underlying the use of t-distribution?
(A) The variance of population is known.
(B) The samples are drawn from a normally distributed population.
(C) Sample standard deviation is an unbiased estimate of the population variance.
(D) It depends on a parameter known as degree of freedom.
Choose the correct answer from the options given below:
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)
The interval, on which the function f(x) = x2e-x is increasing, is equal to
If A = $\begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}$ and B = $\begin{bmatrix} 1 & 0 \\ 1 & 0 \end{bmatrix}$ then the matrix AB is equal to
$\int_{1}^{4} |x - 2| dx$ is equal to
If P, Q and R are three singular matrices given by \(P = \begin{bmatrix} 2 & 3a \\ 4 & 3 \end{bmatrix}\), \(Q = \begin{bmatrix} b & 5 \\ 2a & 6 \end{bmatrix}\) and \(R = \begin{bmatrix} a^2 + b^2 - c & 1-c \\ c+1 & c \end{bmatrix}\), then the value of \((2a + 6b + 17c)\) is
If the maximum value of the function f(x) = $\frac{\log_e x}{x}$, x > 0 occurs at x = a, then a2f''(a) is equal to
Let A be a non-singular matrix of order 3 and \(|A| = 15\), then \(|\text{adj } A|\) is equal to
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.
For the given 5 values, 15, 18, 21, 27, 39; the three year moving averages are: