Test - Real Numbers, Coordinate Geometry

1 marks

Q1. What is the HCF of the smallest prime number and the smallest composite number?

Q2. If the HCF of two numbers $a$ and $b$ is 12 and their product is 1800, find the LCM of $a$ and $b$.

Q3. Express 156 as a product of its prime factors.

Q4. Is it possible for two numbers to have 18 as their HCF and 380 as their LCM? Give a one-line reason for your answer.

2 marks

Q5. Check whether $6^n$ can end with the digit 0 for any natural number $n$. Justify your answer.

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.

Q7. Find the HCF and LCM of 90 and 144 using the prime factorization method.

Q8. Given that $\sqrt{3}$ is an irrational number, prove that $5 - 2\sqrt{3}$ is an irrational number.

Q9. Find the largest number which divides 70 and 125, leaving remainders 5 and 8, respectively. (Hint: Subtract the remainders first!)

5 marks

Q10. Prove that $\sqrt{5}$ is an irrational number.

Q11. There is a circular path around a sports field. Sonia taes 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?

Q12. Three sets of English, Hindi, and Mathematics books have to be stacked in such a way that all the books are stored subject-wise and the height of each stack is the same. The number of English books is 96, the number of Hindi books is 240, and the number of Mathematics books is 336. Assuming that all books are of the same thickness, determine the total number of stacks formed.

Q13. Two tankers contain 850 liters and 680 liters of petrol respectively. Find the maximum capacity of a container which can measure the petrol of either tanker in an exact number of times.

1 marks

1. Find the distance of the point $P(3, 4)$ from the origin.

2. Find the coordinates of the midpoint of the line segment joining the points $(-2, 4)$ and $(6, 8)$.

3. If the distance between the points $(4, p)$ and $(1, 0)$ is $5$ units, find the possible values of $p$.

3 marks

1. Determine if the points $A(1, 5)$, $B(2, 3)$, and $C(-2, -11)$ are collinear using the distance formula.

2. Find a point on the x-axis which is equidistant from the points $(2, -5)$ and $(-2, 9)$.

3. If $(1, 2), (4, y), (x, 6)$, and $(3, 5)$ are the vertices of a parallelogram taken in order, find the values of $x$ and $y$.

4. Find the ratio in which the point $(-4, 6)$ divides the line segment joining the points $A(-6, 10)$ and $B(3, -8)$.

5. Find the coordinates of the points of trisection (points dividing into three equal parts) of the line segment joining the points $A(2, -2)$ and $B(-7, 4)$.

5 marks

1. Case Study Problem: Four friends are seated at points $A(3, 4), B(6, 7), C(9, 4)$, and $D(6, 1)$ in a classroom coordinate grid. (a) Using the distance formula, prove what specific type of quadrilateral $ABCD$ forms. (Is it a square, rhombus, or rectangle?) (3 Marks) (b) Find the coordinates of the point where the diagonals of this quadrilateral intersect. (2 Marks)

2. If $A(-2, 1), B(a, 0), C(4, b)$, and $D(1, 2)$ are the vertices of a parallelogram, find the values of $a$ and $b$. Hence, find the lengths of its distinct sides.