Class 10 CBSE Maths: Real Numbers MCQs

In a frequency distribution, if the mode is significantly affected by extreme values while the median remains relatively stable, what can be inferred about the distribution's characteristics?

In a grouped data set, the modal class and median class are different. Which of the following statements must be true?

Given a grouped data frequency distribution, under what condition would the calculated mode using the formula l + [(f₁ - f₀) / (2f₁ - f₀ - f₂)] * h be considered unreliable?

Consider two events, A and B, within the same sample space. If P(A) = 0.6, and P(A and B) = 0.4, what is the minimum possible value for P(B)?

A bag contains an unknown number of marbles, some red and some blue. You are told that the probability of drawing a red marble is $x$ and the probability of drawing a blue marble is $y$. Given that $x = y^2 - y +1 $, determine the range of possible values for $y$.

Given that $p$ is a prime number and $a$ is a positive integer such that $p$ divides $a^3$, which of the following statements must be true?

Consider two distinct irrational numbers, $x$ and $y$. Which of the following statements is always true?

A polynomial $p(x)$ has a factor of $(x - a)^2$. What can be concluded about the behavior of the graph of $y = p(x)$ at $x = a$?

If the zeroes of the quadratic polynomial $ax^2 + bx + c$ are reciprocals of each other, what can be said about the relationship between the coefficients?

Given the pair of linear equations $a_1x + b_1y + c_1 = 0$ and $a_2x + b_2y + c_2 = 0$, if $a_1/a_2 = b_1/b_2 \neq c_1/c_2$, what is the geometric interpretation of these equations and the nature of their solution?

Consider the system of equations: $kx + 3y = k - 3$ $12x + ky = k$ For what value(s) of k will this system have infinitely many solutions?

Let $p(x) = x^4 + ax^3 + bx^2 + cx + d$ be a polynomial such that when $p(x)$ is divided by $x - 1$, the remainder is 5; when divided by $x - 2$, the remainder is 11; and when divided by $x - 3$, the remainder is 21. What is the remainder when $p(x)$ is divided by $x - 4$?

Determine the number of real solutions to the equation $\sqrt{x+3-4\sqrt{x-1}} + \sqrt{x+8-6\sqrt{x-1}} = 1$.

Consider two arithmetic progressions. The first has a first term $a_1$ and common difference $d_1$, and the second has a first term $a_2$ and common difference $d_2$. If the ratio of the sums of their first $n$ terms is constant for all $n$, what can be concluded about the relationship between $a_1$, $d_1$, $a_2$, and $d_2$?

Given a triangle ABC, point D lies on side AB and point E lies on side AC such that DE is parallel to BC. AD = x cm, DB = (x - 3) cm, AE = (x + 3) cm, and EC = (x - 2) cm. What is the value of x?

A line segment joining points A(2, 3) and B(6, -5) is divided by point P internally in the ratio 3:1 and by point Q externally in the ratio 1:2. What is the midpoint of the line segment PQ?

If $\sin \theta + \cos \theta = \sqrt{3}$, then what is the value of $\tan \theta + \cot \theta$?

From the top of a cliff 20 m high, the angle of depression of a boat is $60^\circ$. Find the distance of the boat from the foot of the cliff.

Two tangents TP and TQ are drawn to a circle with center O from an external point T. If $\angle TPQ = 65^\circ$, what is the measure of $\angle POQ$?

A wire when bent in the form of a square encloses an area of 121 cm². If the same wire is bent into the form of a circle, then find the area of the circle.

A solid metallic sphere of radius 10.5 cm is melted and recast into several smaller cones, each of radius 3.5 cm and height 3 cm. Find the number of cones so formed.

Consider the equation $(x-a)(x-b) = c$, with $c \neq 0$. What can be said about the solutions to this equation?

Let $S_n$ denote the sum of the first $n$ terms of an arithmetic progression. If $S_{2n} = 3S_n$, then what is the value of $S_{3n}/S_n$?

In triangle $ABC$, $D$ and $E$ are points on sides $AB$ and $AC$ respectively such that $DE \parallel BC$. If the area of triangle $ADE$ is $k$ times the area of the trapezoid $BCED$, what is the ratio $AE:EC$?

Two vertices of a triangle are at $(1,1)$ and $(3,5)$. If the centroid of the triangle is at $(-1,1)$, what are the coordinates of the third vertex?

What is the value of $\frac{\sin^4 \theta - \cos^4 \theta}{\sin^2 \theta - \cos^2 \theta}$?

A ladder rests against a wall at an angle $\alpha$ to the horizontal. Its foot is pulled away from the wall through a distance $x$, so that its upper end slides a distance $y$ down the wall and the ladder makes an angle $\beta$ to the horizontal. What is the length of the ladder?

A cone is inscribed in a cylinder such that the base of the cone is the base of the cylinder, and the apex of the cone is on the top circular edge of the cylinder. If the cylinder has radius $r$ and height $h$, what fraction of the cylinder's volume is occupied by the cone?