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1. The quadratic equation \( 2x^2 – \sqrt{5}x + 1 = 0 \) has:

• A. two distinct real roots
• B. two equal real roots
• C. no real root
• D. more than two distinct real roots

2. Which of the following are the roots of \( x^2 – 9x + 20 = 0 \)?

3. The point which divides the line segment joining the points \( (7, -6) \) and \( (3, 4) \) in ratio 1: 2 internally, lies in:

4. A quadratic polynomial, whose zeroes are 5 and -8 is:

5. The value of k for which the quadratic equation \( 4y^2 – 3ky + 1 = 0 \) has equal roots is:

6. If \( \Delta ABC \sim \Delta PQR \), perimeter of \( \Delta ABC = 32 \text{ cm} \), perimeter of \( \Delta PQR = 48 \text{ cm} \) and \( PR = 6 \text{ cm} \), then the length of AC =?

7. If \( \alpha \) and \( \beta \) are zeroes of the polynomial \( p(x) = x^2 – x – 4 \), then the value of \( \frac{1}{\alpha} + \frac{1}{\beta} – \alpha\beta \) is:

8. If \( m = p q^3 \) and \( n = p^3 q \), then HCF \( (m, n) \) =?

9. The product of HCF and LCM of numbers 50 and 95 is:

10. For what value of \( p \), are \( 2p – 1 \), \( 7 \) and \( 3p \) three consecutive terms of an A.P.?

11. If the zeroes of the quadratic polynomial \( x^2 + (a + 1)x + b \) are 2 and \( -3 \), then:

12. If one zero of the polynomial \( x^2 + 3x + k \) is 2, then the value of \( k \) is:

13. Prime factors of the denominator of the rational number with the decimal expansion 44.123 are:

14. If the distance between the points \( A(2, -2) \) and \( B(-1, x) \) is equal to 5, then the values of \( x \) are:

15. The ratio of LCM and HCF of the least composite and the least prime number is:

16. If the zeroes of the quadratic polynomial \( ax^2 + bx + c, c \neq 0 \) are equal, then:

• A. \( c \) and \( a \) have opposite signs
• B. \( c \) and \( b \) have opposite signs
• C. \( c \) and \( a \) have same signs
• D. \( c \) and \( b \) have same signs

17. The distance of point \( P(2, 3) \) from the \( x \)-axis is:

18. O is the centre of two concentric circles of radius 3 cm and 5 cm. AB is a chord of the outer circle which touches the inner circle. The length of AB is:

19. Statement A (Assertion): The point (3, 0) lies on \( x \)-axis.
Statement R (Reason): The \( x \)-coordinate of any point lying on \( y \)-axis is zero.

• A. Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).
• B. Both Assertion (A) and Reason (R) are true and Reason (R) is not the correct explanation of Assertion (A).
• C. Assertion (A) is true but Reason (R) is false.
• D. Assertion (A) is false but Reason (R) is true.

20. Statement A (Assertion): If the pair of lines are coincident, then we say that pair of lines is consistent and it has a unique solution.
Statement R (Reason): If the pair of lines are parallel, then the pair has no solution and is called inconsistent pair of equations.

• A. Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).
• B. Both Assertion (A) and Reason (R) are true and Reason (R) is not the correct explanation of Assertion (A).
• C. Assertion (A) is true but Reason (R) is false.
• D. Assertion (A) is false but Reason (R) is true.

21. Find the zeroes of quadratic polynomial \( x^2 + 11x + 30 \).

22. Which term of the AP: \( 21, 18, 15, \dots \) is \( -81 \)?

Determine the \( 10^{\text{th}} \) term from the end of the AP : \( 4, 9, 14, \dots, 254 \).

23. Find the positive root of \( \sqrt{3x^2 + 6} = 9 \).

24. CM and RN are respectively the medians of \( \Delta ABC \) and \( \Delta PQR \). If \( \Delta ABC \sim \Delta PQR \), prove that \( \Delta AMC \sim \Delta PNR \).

25. 4 bells toll together at 9 am. They toll after 7, 8, 11 and 12 seconds respectively. How many times will they toll together again in the next 3 hours?

Prove that \( 5 + 7\sqrt{3} \) is an irrational number, if it is given that \( \sqrt{3} \) is irrational.

26. Solve graphically: \( x – y = 1 \), \( 2x + y = 8 \).

27. Find the coordinates of the point which lies on the perpendicular bisector of the line segment joining the points \( A(-2, -5) \) and \( B(2, 5) \).

28. Solve for x and y: \( (a – b)x + (a + b)y = a^2 – 2ab – b^2 \) ; \( (a + b)(x + y) = a^2 + b^2 \).

Solve \( 2x + 3y = 11 \) and \( 2x – 4y = -24 \) and hence find the value of \( m \) such that \( y = mx + 3 \).

29. ABCD is a trapezium in which \( AB \parallel DC \) and its diagonals intersect each other at O. Show that \( \frac{AO}{BO} = \frac{CO}{DO} \).

30. PA and PB are tangents to a circle with centre O from an external point P. If \( \angle APB = 120^\circ \), prove that \( OP = 2AP \).

AP is a tangent at point A of the circle with centre O from an external point P. OP intersects the circle at point B. If \( PA = 4 \text{ cm} \), \( PB = 2 \text{ cm} \), Find the radius of the circle.

31. Two numbers are in the ratio 5: 6. If 8 is subtracted from each of the numbers, the ratio becomes 4: 5. Find the numbers.

32. Prove that \( \sqrt{2} \) is an irrational number.

33. ABC is a right triangle with \( BC = 5 \text{ cm} \), \( AC = 13 \text{ cm} \). Find the radius of the incircle.

Prove that the lengths of tangents drawn to a circle from an external point are equal. Apply it to find the perimeter of \( \Delta ABC \) shown in the figure below, if \( PC = 7 \text{ cm} \).

34. In the figure, if \( \angle ABD = \angle XYD = \angle CDB = 90^\circ \), \( AB = a \), \( XY = c \) and \( CD = b \), then prove that: \( c(a + b) = ab \).

35. If m times the \( m^{\text{th}} \) term of an AP is equal to n times its \( n^{\text{th}} \) term, then find the \( (m + n)^{\text{th}} \) term of the AP.

If the \( p^{\text{th}} \), \( q^{\text{th}} \) and \( r^{\text{th}} \) terms of an AP are x, y and z respectively, then show that:

36. Aditya, Ritesh and Damodar are fast friend since childhood. They always want to sit in a row in the classroom. But teacher doesn’t allow them and rotate the seats row-wise everyday. Ritesh is very good in maths and he does distance calculation everyday. He considers the centre of class as origin and marks their position on a paper in a co-ordinate system. One day Ritesh make the following diagram of their seating position marked Aditya as A, Ritesh as B and Damodar as C.

Based on the above information, answer the following questions:

1. What is the distance between A and B?
2. A point D lies on the line segment between points A and B such that \( AD:DB = 4:3 \). What are the coordinates of point D?
   OR

   If the point \( P(k, 0) \) divides the line segment joining the points \( A(2, -2) \) and \( B(-7, 4) \) in the ratio 1: 2, then find the value of k.

3. What is the distance between B and C?

37. Rohita visited a temple in Gwalior. On the way she sees the Agra Fort. The entrance gate of the fort has a shape of parabola representing a quadratic polynomial as shown in the figure.

Based on the above information, answer the following questions:

1. If one zero of the polynomial \( x^2 – 12x + (3k – 1) \) representing the shape above is five times than other, then write the value of k.
2. If both the zeroes of the quadratic polynomial \( ax^2 + bx + c \) are equal and opposite in sign, then write the value of b.
   OR

   If the polynomial \( x^2 + kx – 15 \) represents such a curve, with one of its zeroes as 3, then write the value of k.

3. State the zeroes of the polynomial represented in the graph above.

38. Your elder brother wants to buy a car and plans to take loan from a bank for it. He pays his total loan of ₹18,000 by paying every month starting from the first instalment of ₹ 1000. He increases the instalment by ₹ 100 every month.

Based on the above information, answer the following questions:

1. What is the amount paid by him in 30^th instalment?
2. What is the amount paid by him upto 30 instalments?
   OR

   If total instalments are 40, find the amount paid in the last instalment.

3. What amount does he still have to offer after 30^th instalment?