CBSE Class 10 · Mathematics

Number Systems and Algebra

54 practice questions, each with the answer and an explanation.

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If two positive integers $a$ and $b$ are written as $a = x^{3} y^{2}$ and $b = x y^{3}$ , where $x$ and $y$ are prime numbers, what is the $HCF (a, b)$ ?
For what value of $k$ will the pair of linear equations $k x + 3 y - (k - 3) = 0$ and $12 x + k y - k = 0$ have infinitely many solutions?
If $α$ and $β$ are the zeroes of the quadratic polynomial $p (x) = 2 x^{2} - 5 x + 7$ , what is the value of $α^{- 1} + β^{- 1}$ ?
Which of the following numbers is an irrational number?
The roots of the quadratic equation $3 x^{2} - 6 x + 2 = 0$ are:
If the lines given by $3 x + 2 k y = 2$ and $2 x + 5 y + 1 = 0$ are parallel, what is the value of $k$ ?
If one zero of the quadratic polynomial $f (x) = 4 x^{2} - 8 k x - 9$ is negative of the other, what is the value of $k$ ?
What is the largest positive integer that divides $70$ and $125$ , leaving remainders $5$ and $8$ respectively?
If $α, β$ are the zeroes of $x^{2} - (k + 6) x + 2 (2 k - 1)$ , and $α + β = \frac{1}{2} α β$ , what is the value of $k$ ?
If two positive integers $a$ and $b$ are written as $a = x^{3} y^{2}$ and $b = x y^{3}$ , where $x$ and $y$ are prime numbers, what is $HCF (a, b)$ ?
If $HCF (306, 657) = 9$ , what is the value of $LCM (306, 657)$ ?
Which of the following rational numbers has a non-terminating repeating decimal expansion?
What is the largest number that divides 70 and 125, leaving remainders 5 and 8 respectively?
If $n$ is any natural number, then $6^{n}$ always ends with which digit?
Three bells toll together at intervals of 9, 12, and 15 minutes respectively. If they start tolling together, after what time will they toll together next?
Prove that if $p$ is irrational, which of the following is also an irrational number for any non-zero rational $q$ ?
What is the exponent of 2 in the prime factorization of 144?
If $α$ and $β$ are the zeroes of the polynomial $f (x) = x^{2} - 5 x + k$ such that $α - β = 1$ , find the value of $k$ .
If one zero of the quadratic polynomial $2 x^{2} - 3 x + k$ is reciprocal of the other, find the value of $k$ .
If the zeroes of the quadratic polynomial $x^{2} + (a + 1) x + b$ are 2 and -3, find the values of $a$ and $b$ .
What is the maximum number of zeroes a cubic polynomial can have?
If $α, β$ are the zeroes of $p (x) = 4 x^{2} + 3 x + 7$ , then the value of $\frac{1}{α} + \frac{1}{β}$ is:
Find a quadratic polynomial whose sum and product of zeroes are $- 1/4$ and $1/4$ respectively.
If one zero of the quadratic polynomial $(k - 1) x^{2} + k x + 1$ is $- 3$ , find the value of $k$ .
If the polynomial $x^{4} + 2 x^{3} + 8 x^{2} + 12 x + 18$ is divided by $x^{2} + 5$ , the remainder is $p x + q$ . Find $p$ and $q$ .
For what value of $k$ will the equations $x + 2 y = 5$ and $3 x + k y - 15 = 0$ have a unique solution?
The perimeter of a rectangular garden is 36 m, and its length is 4 m more than its width. What are the dimensions?
Solve the system: $\frac{2}{x} + \frac{3}{y} = 13$ and $\frac{5}{x} - \frac{4}{y} = - 2$ , where $x, y \neq = 0$ .
Five years ago, Nuri was thrice as old as Sonu. Ten years later, Nuri will be twice as old as Sonu. How old is Nuri now?
The sum of a two-digit number and the number obtained by reversing the digits is 66. If the digits differ by 2, how many such numbers exist?
If $2 x + 3 y = 11$ and $2 x - 4 y = - 24$ , find the value of $m$ for which $y = m x + 3$ .
For what value of $p$ does the pair of linear equations $4 x + p y + 8 = 0$ and $2 x + 2 y + 2 = 0$ have a unique solution?
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.
Places A and B are 100 km apart on a highway. One car starts from A and another from B at the same time. If they travel in the same direction at different speeds, they meet in 5 hours. If they travel…
The area of a rectangle gets reduced by 9 sq units if its length is reduced by 5 units and breadth increased by 3 units. If length is increased by 3 and breadth by 2, area increases by 67 sq units. Fi…
Find the discriminant of the quadratic equation $2 x^{2} - 4 x + 3 = 0$ and state the nature of its roots.
If the roots of the equation $(b - c) x^{2} + (c - a) x + (a - b) = 0$ are equal, prove which of the following relations holds:
An express train takes 1 hour less than a passenger train to travel 132 km between Mysore and Bangalore. If the average speed of the express train is 11 km/h more than that of the passenger train, fin…
Solve for $x$ : $4 x^{2} - 4 a x + (a^{2} - b^{2}) = 0$ .
A motor boat whose speed is 18 km/h in still water takes 1 hour more to go 24 km upstream than to return downstream to the same spot. Find the speed of the stream.
Two water taps together can fill a tank in $9 \frac{3}{8}$ hours. The tap of larger diameter takes 10 hours less than the smaller one to fill the tank separately. Find the time in which the smaller tap…
If the equation $x^{2} - (2 k + 1) x + (k + 2) = 0$ has real roots, which condition must the discriminant satisfy?
Find the positive value of $k$ for which the equation $x^{2} + k x + 64 = 0$ and $x^{2} - 8 x + k = 0$ will both have real roots.
The diagonal of a rectangular field is 60 meters more than the shorter side. If the longer side is 30 meters more than the shorter side, find the shorter side of the field.
Solve for $x$ : $3 (\frac{3 x - 1}{2 x + 3}) - 2 (\frac{2 x + 3}{3 x - 1}) = 5$ , where $x \neq = - 3/2, 1/3$ .
Which term of the AP $121, 117, 113, \dots$ is its first negative term?
If the $m$ -th term of an AP is $1/ n$ and $n$ -th term is $1/ m$ , what is the $(mn)$ -th term?
Find the sum of the first 24 terms of the AP whose $n$ -th term is given by $a_{n} = 3 + 2 n$ .
If the ratio of the sum of first $n$ terms of two APs is $(7 n + 1) : (4 n + 27)$ , find the ratio of their 9th terms.
In an AP, if $S_{n} = n (4 n + 1)$ , find the common difference $d$ .
Find the middle term of the AP $213, 205, 197, \dots, 37$ .
If 7 times the 7th term of an AP is equal to 11 times the 11th term, what is its 18th term?
The sum of three numbers in an AP is 21 and their product is 231. Find the common difference.
Kanika was given her pocket money on Jan 1st, 2008. She puts Rs 1 into her piggy bank on Day 1, Rs 2 on Day 2, Rs 3 on Day 3, and so on until the end of the month. How much total money does she save i…

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