Can You Solve These 20 Tricky Math Problems for Classes IX and X CBSE ICSE

📚 20 Mathematical Challenges – CBSE, ICSE

Welcome! This slider shows two problems side by side on each page. Use Previous / Next buttons (or left/right arrow keys) to navigate through the 10 slides. Each problem has its own “Show hint” button – click to reveal a subtle nudge.

✨ Covers rational proofs, surd simplifications, Diophantine equations, and more. Suitable for grades 9–12 and competitive exams. Featuring 20 hand-picked problems covering everything from rational, irrational numbers to algebra and number theory. Whether you are prepping for a competitive exam or just love a good brain-teaser, this is for you. Suitable for students of IX and X of CBSE and ICSE. If these problems feel a bit intimidating, don't worry—they are designed to be! While standard high school mathematics focuses heavily on rote memorization and applying standard formulas to straightforward questions, this problem set bridges the gap between the regular classroom and competitive mathematics (like the AMC, math Olympiads, or advanced entrance exams). Instead of just asking you to "solve for x," these questions require creative algebraic manipulation, pattern recognition, and proof-based logical reasoning. They test whether you can combine multiple mathematical concepts to find an elegant solution, rather than just grinding through a standard algorithm. Key Mathematical Areas Covered This set of 20 problems targets several advanced domains: Advanced Algebra & Identities: Moving beyond basic factoring to use conditional identities (like the sum of cubes) and symmetric functions. Number Theory: Exploring the properties of prime numbers, divisibility rules, and solving basic Diophantine equations (finding integer solutions to polynomial equations). Complex Radicals & Surds: Tackling nested square roots, cube roots of binomial surds, and rationalizing multi-term denominators. Telescoping Series: Recognizing patterns in sequences that cancel each other out to reveal a clean, simple answer. Introductory Complex Numbers & Logarithms: Understanding the cyclical nature of imaginary numbers and manipulating logarithmic bases.

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✨ 20 problems, 10 slides, 2 per slide. Click "Show hint" for each problem individually. Left/right arrow keys navigate. Good luck! © Vinod Singh

Chapter Test Pair of Linear Equation in Two Variables CBSE ICSE

Prime Maths

25 Question MCQ Prime Maths - Class X, IX CBSE, Madhyamik and ICSE

👨‍🏫 Author: Singh

📞 WA: +91-9038126497

Prime Maths

Pair of Linear Equations in Two Variables

This quiz offers a thorough evaluation of key concepts and skills related to Linear Equations in Two Variables. 25 multiple-choice questions designed to test both theoretical understanding and practical application.

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Question 1

For the system of equations \(a_1x + b_1y + c_1 = 0\) and \(a_2x + b_2y + c_2 = 0\) to have a unique solution, the condition is:

A\(\frac{a_1}{a_2} = \frac{b_1}{b_2}\) B\(\frac{a_1}{a_2} \neq \frac{b_1}{b_2}\) C\(\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}\) D\(\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}\)

Question 2

If a pair of linear equations is consistent, then the lines representing them are:

AParallel BAlways coincident CIntersecting or coincident DAlways intersecting

Question 3

For what value of \(k\) do the equations \(3x - y + 8 = 0\) and \(6x - ky = -16\) represent coincident lines?

A\(k = 2\) B\(k = -2\) C\(k = \frac{1}{2}\) D\(k = - \frac{1}{2}\)

Question 4

The pair of equations \(x = a\) and \(y = b\) graphically represents lines which are:

AParallel BIntersecting at \((b, a)\) CCoincident DIntersecting at \((a, b)\)

Question 5

If the system \(2x + 3y = 5\) and \(4x + ky = 10\) has infinitely many solutions, then \(k\) equals:

A\(1\) B\(\frac{3}{2}\) C\(6\) D\(3\)

Question 6

The value of \(k\) for which the system \(x + 2y = 3\) and \(5x + ky + 7 = 0\) is inconsistent is:

A\(k = 10\) B\(k = \frac{2}{5}\) C\(k = -\frac{2}{5}\) D\(k = -10\)

Question 7

Solve: \(\frac{x}{a}+\frac{y}{b}=2\) and \(ax-by=a^2-b^2\)

A\(x=a, \quad y=b^2\) B\(x=a, \quad y=b\) C\(x=a^2, \quad y=b^2\) D\(x=a^2, \quad y=b\)

Question 8

The value of \(k\) for which the pair of linear equations \((3k+1)x+3y=2\) and \((k^2+1)x+(k-2)y-5=0\) has no solution.

A\(k=-2\) B\(k=2\) C\(k=1\) D\(k=-1\)

Question 9

Solve for \(x\) and \(y\): \((a-b)x + (a+b)y = a^2 -2ab-b^2\) and \((a+b)(x+y) = a^2 + b^2\)

A\(x = a+b, y = -\frac{2ab}{a+b}\) B\(x = a+b, y = \frac{2ab}{a+b}\) C\(x = a-b, y = -\frac{2ab}{a+b}\) D\(x = a-b, y = \frac{2ab}{a+b}\)

Question 10

The value of \(x\) satisfying \(\frac{5}{x-1}+\frac{1}{y-2}=2\) and \(\frac{6}{x-1}-\frac{3}{y-2}-1=0, \quad x \neq 1, y \neq 2\) is:

A2 B3 C4 D5

Question 11

If \(37x + 43y = 123\) and \(43x + 37y = 117\), then:

A\(x = 2, y = 3\) B\(x = 1, y = 2\) C\(x = 3, y = 2\) D\(x = -1, y = -2\)

Question 12

The pair of equations \(225x + 881y = 2^{10}\) and \(450x + 1762y = 2048\) has:

ANo solution BUnique solution CTwo solutions DInfinitely many solutions

Question 13

If \(\frac{b}{a}x + \frac{a}{b}y = a^2+b^2\) and \(x + y = 2ab\) then

A\(x=2y\) B\(x=-y\) C\(x=y\) D\(2x=y\)

Question 14

If \(2^x + 3^y = 17\) and \(2^{x+2} - 3^{y+1} = 5\), find \(x\) and \(y\).

A\(x = 9, y = 5\) B\(x = 3, y = 2\) C\(x = 5, y = 9\) D\(x = 8, y = 6\)

Question 15

Solve for \(x\): \(\frac{2}{x} + \frac{3}{y} = 13\) and \(\frac{5}{x} - \frac{4}{y} = -2\).

A\(x = 1/2\) B\(x = 1/3\) C\(x = 2\) D\(x = 3\)

Question 16

For what value of \(p\) does the pair of equations \(4x + py + 8 = 0\) and \(2x + 2y + 2 = 0\) have a unique solution?

A\(p = 4\) B\(p \neq 4\) C\(p = 2\) D\(p \neq 2\)

Question 17

If \(49x + 51y = 499\) and \(51x + 49y = 501\), then \(x\) is:

A3 B5.5 C4 D6.5

Question 18

The difference between two numbers is 26 and one number is three times the other. Find them.

A30, 4 B36, 10 C39, 13 D40, 14

Question 19

Solve: \( \sqrt{2}x + \sqrt{3}y = 0 \) and \( \sqrt{3}x - \sqrt{8}y = 0 \).

A\(x=1, y=1\) B\(x=\sqrt{2}, y=\sqrt{3}\) C\(x=0, y=0\) D\(x=2, y=3\)

Question 20

Find \(k\) if the lines \(x + 2y - 1 = 0\) and \(2x + ky + 2 = 0\) are parallel.

A\(k=2\) B\(k=3\) C\(k=4\) D\(k=-4\)

Question 21

Five years ago, Ramesh was thrice as old as Sonu. Ten years later, Ramesh will be twice as old as Sonu. What is Ramesh's present age?

A40 years B50 years C60 years D45 years

Question 22

A fraction becomes 9/11 if 2 is added to both numerator and denominator. If 3 is added to both, it becomes 5/6. Find the fraction.

A\(7/9\) B\(9/7\) C\(5/9\) D\(3/7\)

Question 23

The sum of the digits of a two-digit number is 9. Also, nine times this number is twice the number obtained by reversing the order of the digits. Find the number.

A18 B27 C81 D72

Question 24

A taxi charge consists of a fixed charge together with the charge for the distance covered. For 10 km, the charge paid is ₹105 and for 15 km, the charge paid is ₹155. What are the fixed charges?

A₹ 10 B₹ 5 C₹ 15 D₹ 8

Question 25

Ritu can row downstream 20 km in 2 hours, and upstream 4 km in 2 hours. Find her speed of rowing in still water.

A6 km/h B4 km/h C8 km/h D10 km/h

Prime Maths - Excellence in Mathematical Education

© 2026 Prime Maths | Created by Vinod Singh

Real Numbers : Assessment for classes IX and X

Prime Maths

25 Question MCQ Chapter on Real Numbers for CBSE, ICSE, Madhyamik and other State Boards Class IX & X

👨‍🏫 Author: Singh

📞 WA: +91-9038126497

Prime Maths

Real Numbers Quiz

Test your understanding of rational, irrational numbers and their properties. The study of Real Numbers in Classes IX and X establishes the foundation for all higher-level mathematics.Real Numbe: The set including all Rational and Irrational numbers. Rational Numbers: Numbers expressible as p/q, q is non-zero . Their decimal expansions are either terminating or non-terminating recurring.Irrational Numbers: Numbers that cannot be written as fractions. Their decimal expansions are non-terminating non-recurring.Number Line: Every point on a number line represents a unique real number.

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Question 1

Which of the following is an irrational number?

A\(\sqrt{25}\) B\(\sqrt{2} + \sqrt{3}\) C\(\frac{3}{7}\) D\(0.\overline{3}\)

Question 2

The decimal expansion of \(\frac{13}{625}\) will terminate after how many decimal places?

A2 B3 C4 D5

Question 3

Which of the following has a terminating decimal expansion?

A\(\frac{77}{210}\) B\(\frac{23}{30}\) C\(\frac{125}{441}\) D\(\frac{129}{2^2 \times 5^7}\)

Question 4

The simplified value of \(\sqrt{12} + \sqrt{27} - \sqrt{75}\) is:

A\(\sqrt{3}\) B\(2\sqrt{3}\) C\(3\sqrt{3}\) D\(0\)

Question 5

If \(p\) and \(q\) are co-prime numbers, then \(p^2\) and \(q^2\) are:

ACo-prime BNot co-prime CEven numbers DOdd numbers

Question 6

A rational number between \(\sqrt{2}\) and \(\sqrt{3}\) is:

A\(\frac{\sqrt{2} + \sqrt{3}}{2}\) B\(\frac{\sqrt{2} \times \sqrt{3}}{2}\) C\(1.5\) D\(1.8\)

Question 7

The value of \(1.\overline{36} + 0.\overline{9}\) is:

A\(2.\overline{27}\) B\(2.\overline{26}\) C\(2.\overline{3}\) D\(2.3\)

Question 8

Which of the following is not an irrational number?

A\(\sqrt{8} \times \sqrt{2}\) B\(\sqrt{18} \div \sqrt{2}\) C\(\sqrt{5} + \sqrt{10}\) D\(\sqrt{7} \times \sqrt{28}\)

Question 9

If \(n\) is a natural number, then \(\sqrt{n}\) is:

AAlways rational BAlways irrational CRational if \(n\) is a perfect square DIrrational if \(n\) is a perfect square

Question 10

The product of a non-zero rational and an irrational number is:

AAlways rational BAlways irrational CCan be rational or irrational DAlways an integer

Question 11

Which of the following rational numbers have terminating decimal representation?

A\(\frac{16}{125}\) B\(\frac{27}{250}\) C\(\frac{343}{2^3 \times 5^3}\) DAll of these

Question 12

The ascending order of \(\sqrt[3]{2}, \sqrt{3}, \sqrt[6]{5}\) is:

A\(\sqrt[3]{2} < \sqrt[6]{5} < \sqrt{3}\) B\(\sqrt[6]{5} < \sqrt[3]{2} < \sqrt{3}\) C\(\sqrt[6]{5} < \sqrt{3} < \sqrt[3]{2}\) D\(\sqrt{3} < \sqrt[3]{2} < \sqrt[6]{5}\)

Question 13

The decimal expansion of the rational number \(\frac{14587}{1250}\) will terminate after:

AOne decimal place BTwo decimal places CThree decimal places DFour decimal places

Question 14

After rationalizing the denominator of \(\frac{7}{3\sqrt{3} - 2\sqrt{2}}\), we get:

A\(3\sqrt{3} + 2\sqrt{2}\) B\(3\sqrt{3} - 2\sqrt{2}\) C\(7(3\sqrt{3} + 2\sqrt{2})\) D\(7(3\sqrt{3} - 2\sqrt{2})\)

Question 15

An irrational number between \(\frac{1}{7}\) and \(\frac{2}{7}\) is:

A\(\frac{1}{2}\left(\frac{1}{7} + \frac{2}{7}\right)\) B\(\sqrt{\frac{1}{7} \times \frac{2}{7}}\) C\(\frac{1}{\sqrt{14}}\) D\(\sqrt{0.14}\)

Question 16

If \(x = 2 + \sqrt{3}\), then \(x + \frac{1}{x}\) equals:

A\(4\) B\(2\sqrt{3}\) C\(-4\) D\(-2\sqrt{3}\)

Question 17

Which of the following is a rational number?

A\(\pi\) B\(\sqrt{2}\) C\(0.\overline{101}\) D\(\sqrt{5} + 1\)

Question 18

The simplest rationalizing factor of \(\sqrt[3]{54}\) is:

A\(\sqrt[3]{2}\) B\(\sqrt[3]{3}\) C\(\sqrt[3]{4}\) D\(\sqrt[3]{9}\)

Question 19

If \(\frac{p}{q}\) is a rational number with terminating decimal expansion, where \(p\) and \(q\) are co-prime, then \(q\) must be of the form:

A\(2^m \times 3^n\) B\(2^m \times 5^n\) C\(3^m \times 5^n\) D\(2^m \times 3^n \times 5^k\)

Question 20

The sum of two irrational numbers is:

AAlways irrational BAlways rational CCan be rational or irrational DAlways an integer

Question 21

The value of \(\frac{1}{1 + \sqrt{2}} + \frac{1}{\sqrt{2} + \sqrt{3}} + \frac{1}{\sqrt{3} + \sqrt{4}} + \cdots + \frac{1}{\sqrt{99} + \sqrt{100}}\) is:

A\(1\) B\(9\) C\(10\) D\(99\)

Question 22

If \(a\) and \(b\) are rational numbers and \(\frac{3 + 2\sqrt{3}}{3 - 2\sqrt{3}} = a + b\sqrt{3}\), then \(a + b =\)

A\(11\) B\(-11\) C\(13\) D\(-13\)

Question 23

Which of the following is not true?

AEvery integer is a rational number BEvery whole number is a natural number CEvery irrational number is a real number DEvery rational number is a real number

Question 24

The product \(\sqrt[3]{2} \times \sqrt[4]{2} \times \sqrt[12]{32}\) equals:

A\(\sqrt{2}\) B\(2\) C\(\sqrt[12]{2}\) D\(\sqrt[12]{32}\)

Question 25

The value of \(0.\overline{6} + 0.\overline{7} + 0.\overline{4}\) is:

A\(2\) B\(\frac{17}{9}\) C\(\frac{53}{22}\) D\(\frac{59}{22}\)

Prime Maths - Excellence in Mathematical Education

© 2026 Prime Maths | Created by Vinod Singh