Mathematics Questions

Browse 839 Mathematics questions with expert-verified, step-by-step solutions — NCERT, exemplar and previous-year questions, organised by chapter.

Q1
Express each number as a product of its prime factors:
1. (i) $140$
2. (ii) $156$
3. (iii) $3825$
4. (iv) $5005$
5. (v) $7429$
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Q2
Find the LCM and HCF of the following pairs of integers and verify that $LCM \times HCF =$ product of the two numbers.
1. (i) $26$ and $91$
2. (ii) $510$ and $92$
3. (iii) $336$ and $54$
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Q3
Find the LCM and HCF of the following integers by applying the prime factorisation method.
1. (i) $12$ , $15$ and $21$
2. (ii) $17$ , $23$ and $29$
3. (iii) $8$ , $9$ and $25$
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Q4

Given that HCF (306, 657) = 9, find LCM (306, 657).

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Q5

Check whether $6^{n}$ can end with the digit $0$ for any natural number $n$ .

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Q6

Explain why $7 \times 11 \times 13 + 13$ and $7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 + 5$ are composite numbers.

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Q7

There is a circular path around a sports field. Sonia takes 18 minutes to drive one round of the field, while Ravi takes 12 minutes for the same. Suppose they both start at the same point and at the same time, and go in the same direction. After how many minutes will they meet again at the starting point?

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Q8

Prove that $5$ is irrational.

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Q9

Prove that $3 + 25$ is irrational.

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Q10
Prove that the following are irrationals:
1. (i) $\frac{1}{2}$
2. (ii) $75$
3. (iii) $6 + 2$
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Q11
The graphs of $y = p (x)$ are given in Fig. 2.10 below, for some polynomials $p (x)$ . Find the number of zeroes of $p (x)$ , in each case. (i) Find the number of zeroes of $p (x)$ from graph
1. (i).
2. (ii)Find the number of zeroes of $p (x)$ from graph (ii). (iii) Find the number of zeroes of $p (x)$ from graph (iii). (iv) Find the number of zeroes of $p (x)$ from graph (iv). (v) Find the number of zeroes of $p (x)$ from graph (v). (vi) Find the number of zeroes of $p (x)$ from graph (vi).
Graph (ii)
Graph (i)
Graph (iv)
Graph (vi)
Graph (v)
Graph (iii)
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Q12
Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients.
1. (i) $x^{2} - 2 x - 8$
2. (ii) $4 s^{2} - 4 s + 1$
3. (iii) $6 x^{2} - 3 - 7 x$
4. (iv) $4 u^{2} + 8 u$
5. (v) $t^{2} - 15$
6. (vi) $3 x^{2} - x - 4$
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Q13
Find a quadratic polynomial each with the given numbers as the sum and product of its zeroes respectively.
1. (i) $\frac{1}{4}, - 1$
2. (ii) $2, \frac{1}{3}$
3. (iii) $0, 5$
4. (iv) $1, 1$
5. (v) $- \frac{1}{4}, \frac{1}{4}$
6. (vi) $4, 1$
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Q14

Form the pair of linear equations in the following problems, and find their solutions graphically: (i) $10$ students of Class X took part in a Mathematics quiz. If the number of girls is $4$ more than the number of boys, find the number of boys and girls who took part in the quiz.

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Q15
On comparing the ratios $\frac{a _{1}}{a _{2}}$ , $\frac{b _{1}}{b _{2}}$ and $\frac{c _{1}}{c _{2}}$ , find out whether the lines representing the following pairs of linear equations intersect at a point, are parallel or coincident:
1. (i) $5 x - 4 y + 8 = 0$ and $7 x + 6 y - 9 = 0$
2. (ii) $9 x + 3 y + 12 = 0$ and $18 x + 6 y + 24 = 0$
3. (iii) $6 x - 3 y + 10 = 0$ and $2 x - y + 9 = 0$
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Q16
On comparing the ratios $\frac{a _{1}}{a _{2}}$ , $\frac{b _{1}}{b _{2}}$ and $\frac{c _{1}}{c _{2}}$ , find out whether the following pairs of linear equations are consistent or inconsistent:
1. (i) $3 x + 2 y = 5$ and $2 x - 3 y = 7$
2. (ii) $2 x - 3 y = 8$ and $4 x - 6 y = 9$
3. (iii) $\frac{3}{2} x + \frac{5}{3} y = 7$ and $9 x - 10 y = 14$
4. (iv) $5 x - 3 y = 11$ and $- 10 x + 6 y = - 22$
5. (v) $\frac{4}{3} x + 2 y = 8$ and $2 x + 3 y = 12$
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Q17
Which of the following pairs of linear equations are consistent/inconsistent? If consistent, obtain the solution graphically:
1. (i) $x + y = 5$ and $2 x + 2 y = 10$
2. (ii) $x - y = 8$ and $3 x - 3 y = 16$
3. (iii) $2 x + y - 6 = 0$ and $4 x - 2 y - 4 = 0$
4. (iv) $2 x - 2 y - 2 = 0$ and $4 x - 4 y - 5 = 0$
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Q18

Half the perimeter of a rectangular garden, whose length is $4$ m more than its width, is $36$ m. Find the dimensions of the garden.

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Q19
Given the linear equation $2 x + 3 y - 8 = 0$ , write another linear equation in two variables such that the geometrical representation of the pair so formed is:
1. (i)intersecting lines
2. (ii)parallel lines
3. (iii)coincident lines
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Q20

Draw the graphs of the equations $x - y + 1 = 0$ and $3 x + 2 y - 12 = 0$ . Determine the coordinates of the vertices of the triangle formed by these lines and the $x$ -axis, and shade the triangular region.

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