10th Mathematics Online Test

Chapter 1 Real Numbers

Class 10 Mathematics Multiple Choice Questins Quiz / MCQ Quiz online test base on NCERT useful for NTSE Maths Olympiad Board Exam Important Questions.

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10th Maths Online Test: Real Numbers

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Q. 1. The sum of zeros of quadratic equation $x^{2}-2 x-8$ is:

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Q. 2. A polynomial with two degree is called:

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Q. 3. Which one is polynomial?

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Q. 4. The product of zeros of quadratic equation $x^{2}-2 x-8$ is:

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Q. 5. The zeros of $\sqrt{3} x^{2}+10 x+7 \sqrt{3}$ are:

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Q. 6. If $a, \beta, \gamma$ are the zeros of cubic polynomial $a x^{3}+b x^{2}+c x+d$, then the value of $\alpha \beta \gamma$ is:

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Q. 7. Which one is polynomial?

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Q. 8. The zeros of $x^{2}+7 x+10$ are:

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Q. 9. The zeros of $3 x^{2}-4-x$ are:

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Q. 10. The number of polynomials having zeros as -2 and 4 is/are:


Real Numebrs Chapter 1 class 10 Basic Concepts

1. Euclid's division lemma :
Given positive integers $a$ and $b$, there exist whole numbers $q$ and $r$ satisfying $a=b q+r$, $0 \leq r < b$


2. Euclid's division algorithm : This is based on Euclid's division lemma. According to this, the HCF of any two positive integers $a$ and $b$, with $a > b$, is obtained as follows:
Step 1: Apply the division lemma to find $q$ and $r$ where $a=b q+r, 0 \leq r< b$.
Step 2 : If $r=0$, the HCF is $b$. If $r \neq 0$, apply Euclid's lemma to $b$ and $r$.
Step 3 : Continue the process till the remainder is zero. The divisor at this stage will be $\operatorname{HCF}(a, b)$. Also, $\operatorname{HCF}(a, b)=\operatorname{HCF}(b, r)$.


3. The Fundamental Theorem of Arithmetic :
Every composite number can be expressed (factorised) as a product of primes, and this factorisation is unique, apart from the order in which the prime factors occur.


4. If $p$ is a prime and $p$ divides $a^{2}$, then $p$ divides $a$, where $a$ is a positive integer.


5.  $\sqrt{2}, \sqrt{3}$ are irrationals.


6. Let $x$ be a rational number whose decimal expansion terminates. Then we can express $x$ in the form $\frac{p}{q}$, where $p$ and $q$ are coprime, and the prime factorisation of $q$ is of the form $2^{n} 5^{m}$, where $n, m$ are non-negative integers.


7. Let $x=\frac{p}{q}$ be a rational number, such that the prime factorisation of $q$ is of the form $2^{n} 5^{m}$, where $n, m$ are non-negative integers. Then $x$ has a decimal expansion which terminates.


8. Let $x=\frac{p}{q}$ be a rational number, such that the prime factorisation of $q$ is not of the form $2^{n} 5^{m}$, where $n, m$ are non-negative integers. Then $x$ has a decimal expansion which is non-terminating repeating (recurring).



$\mathrm{HCF}(p, q, r) \times \operatorname{LCM}(p, q, r) \neq p \times q \times r$, where $p, q, r$ are positive integers. However, the following results hold good for three numbers $p, q$ and $r:$
$$ \begin{aligned} & \operatorname{LCM}(p, q, r)=\frac{p \cdot q \cdot r \cdot \mathrm{HCF}(p, q, r)}{\operatorname{HCF}(p, q) \cdot \mathrm{HCF}(q, r) \cdot \operatorname{HCF}(p, r)} \\ & \operatorname{HCF}(p, q, r)=\frac{p \cdot q \cdot r \cdot \operatorname{LCM}(p, q, r)}{\operatorname{LCM}(p, q) \cdot \operatorname{LCM}(q, r) \cdot \operatorname{LCM}(p, r)} \end{aligned} $$