MATHEMATICS
SECTION – A (1 mark)
1. Which term of the AP: – 4, – 1, 2, ... is 101?
2. Evaluate:
OR
Express (sin67° + cos75°) in terms of trigonometric ratios of the angle between 0° and 45°.
3. Find the values of k for which quadratic equation kx(x – 2) + 6 = 0 has two equal roots.
4. Find a rational number between √2 and √7.
OR
Write the number of zeros in the end of a number whose prime factorization is 22 × 53 × 32 × 17.
5. Find the distance between the points (a, b) and ( – a, – b).
6. Let ΔABC ~ ΔDEF and their areas be respectively, 64 cm2 and 121 cm2. If EF=15.4cm, find BC.
SECTION – B (2 marks)
7. Find the solution of the pair of equations: ; , x, y ≠ 0.
OR
Find the value(s) of k for which the pair of equations kx + 2y = 3 and 3x + 6y = 10 has a unique solution.
8. Use Euclid’s algorithm to find the HCF of 255 and 867.
9. The point R divides the line segment AB, where A (–4, 0) and B (0, 6) such that AR = AB. Find the coordinates of R.
10. How many multiples of 4 lie between 10 and 205?
OR
Determine the AP whose third term is 16 and the 7th term exceeds the 5th term by 12.
11. Three different coins are tossed simultaneously. Find the probability of getting exactly one head.
12. A die is thrown once. Find the probability of getting
(a) a prime number (b) an odd number.
SECTION – C (3 marks)
13. In figure, BL and CM are medians of a ∆ABC right angled at A. Prove that 4(BL2 + CM2) = 5BC2.
OR
Prove that the sum of the squares of the sides of a rhombus is equal to the sum of the squares of its diagonals.
14. In the given figure, two concentric circles with centre O have radii 21 cm and 42 cm. If ∠ AOB = 60°, find the area of the shaded region.
15. A cone of height 24 cm and radius of base 6 cm is made up of modeling clay. A child reshapes it in the form of a sphere. Find the radius of the sphere and hence find the surface area of this sphere.
OR
A farmer connects a pipe of internal diameter 20 cm from a canal into a cylindrical tank in his field which is 10 m in diameter and 2 m deep. If water flows through the pipe at the rate of 3 km/hr, in how much time will the tank be filled?
16. Calculate the mode of the following distribution:
17. Show that is not a rational number, given that √2 is an irrational number.
18. Obtain all other zeroes of 2x4 – 5x3 – 11x2 + 20x + 12, when 2 and – 2 are two zeroes of the above polynomial.
19. A motorboat whose speed is 18 km/hr in still water takes one hour more to go 24 km upstream than to return downstream to the same spot. Find the speed of the stream.
20. Prove that: (sinθ + 1 + cosθ)(sinθ – 1 + cosθ).secθcosecθ = 2
OR
Prove that:
21. In what ratio does the point P(–4, y) divide the line segment joining the points A(–6,10) and B(3,–8)? Hence find the value of y.
OR
Find the value of p for which the points (–5, 1), (1, p) and (4, – 2) are collinear.
22. ABC is a right triangle in which ∠B = 90°. If AB = 8 cm and BC = 6 cm, find the diameter of the circle inscribed in the triangle.
SECTION – D (4 marks)
23. In an AP., the first term is – 4, the last term is 29 and the sum of all its terms is 150. Find its common difference.
24. Draw a circle of radius 4 cm. From a point 6 cm away from its centre, construct the pair of tangents to the circle and measure their lengths.
25. Prove that:
26. Solve for x:; x ≠ 0, x ≠, a, b ≠ 0.
OR
The sum of the areas of two squares is 640 m2. If the difference of their perimeters is 64 m, find the sides of the two squares.
27. In ∆ABC, AD ⊥ BC. Prove that: AC2 = AB2 + BC2 – 2BC × BD.
28. A moving boat is observed from the top of a 150 m high cliff moving away from the cliff. The angle of depression of the boat changes from 60° to 45° in 2 minutes. Find the speed of the boat in m/min.
OR
There are two poles, one each on either bank of a river just opposite to each other. One pole is 60 m high. From the top of this pole, the angle of depression of the top of and foot of the other pole are 30° and 60° respectively. Find the width of the river and height of the other pole.
29. Calculate the mean of the following frequency distribution:
OR
The following table gives production yield in kg per hectare of wheat of 100 farms of a village:
Change the distribution to a ‘more than type’ distribution, and draw its ogive.
30. A container, opened from the top and made up of a metal sheet, is in the form of a frustum of a cone of height 16 cm with radii of its lower and upper ends as 8 cm and 20 cm, respectively. Find the cost of the milk which can completely fill the container, at the rate of ₹50 per litre. Also find the cost of metal sheet used to make the container, if it costs ₹10 per 100 cm2. (Take π = 3.14)
SOLUTIONS
1. Which term of the AP: – 4, – 1, 2, ... is 101?
Solution:
Here, a = – 4, d = 3.
Let, nth term be 101.
an = 101
⇒ a + (n – 1)d = 101
⇒ – 4 + (n – 1)3 =101
⇒ (n – 1)3 =105
⇒ n – 1 = 35
⇒ n = 36
⸫36th term is 101.
2. Evaluate:
Solution:
OR
Express (sin67° + cos75°) in terms of trigonometric ratios of the angle between 0° and 45°.
3. Find the values of k for which quadratic equation kx(x – 2) + 6 = 0 has two equal roots.
Solution:
We have,
kx(x – 2) + 6 = 0
⇒ kx2 – 2kx + 6 = 0
For equal roots,
b2 – 4ac = 0
⇒ (– 2k)2 – 4k.6 = 0
⇒ 4k2 – 24k = 0
⇒ 4k(k – 6) = 0
⇒ 4k = 0 or k – 6 = 0
⇒ k = 0 or k = 6
4. Find a rational number between √2 and √7.
OR
Write the number of zeros in the end of a number whose prime factorization is 22 × 53 × 32 × 17.
5. Find the distance between the points (a, b) and ( – a, – b).
6. Let ΔABC ~ ΔDEF and their areas be respectively, 64 cm2 and 121 cm2. If EF=15.4cm, find BC.
SECTION – B (2 marks)
7. Find the solution of the pair of equations: ; , x, y ≠ 0.
OR
Find the value(s) of k for which the pair of equations kx + 2y = 3 and 3x + 6y = 10 has a unique solution.
8. Use Euclid’s algorithm to find the HCF of 255 and 867.
9. The point R divides the line segment AB, where A (–4, 0) and B (0, 6) such that AR = AB. Find the coordinates of R.
10. How many multiples of 4 lie between 10 and 205?
OR
Determine the AP whose third term is 16 and the 7th term exceeds the 5th term by 12.
11. Three different coins are tossed simultaneously. Find the probability of getting exactly one head.
12. A die is thrown once. Find the probability of getting
(a) a prime number (b) an odd number.
SECTION – C (3 marks)
13. In figure, BL and CM are medians of a ∆ABC right angled at A. Prove that 4(BL2 + CM2) = 5BC2.
OR
Prove that the sum of the squares of the sides of a rhombus is equal to the sum of the squares of its diagonals.
14. In the given figure, two concentric circles with centre O have radii 21 cm and 42 cm. If ∠ AOB = 60°, find the area of the shaded region.
15. A cone of height 24 cm and radius of base 6 cm is made up of modeling clay. A child reshapes it in the form of a sphere. Find the radius of the sphere and hence find the surface area of this sphere.
OR
A farmer connects a pipe of internal diameter 20 cm from a canal into a cylindrical tank in his field which is 10 m in diameter and 2 m deep. If water flows through the pipe at the rate of 3 km/hr, in how much time will the tank be filled?
16. Calculate the mode of the following distribution:
17. Show that is not a rational number, given that √2 is an irrational number.
18. Obtain all other zeroes of 2x4 – 5x3 – 11x2 + 20x + 12, when 2 and – 2 are two zeroes of the above polynomial.
19. A motorboat whose speed is 18 km/hr in still water takes one hour more to go 24 km upstream than to return downstream to the same spot. Find the speed of the stream.
20. Prove that: (sinθ + 1 + cosθ)(sinθ – 1 + cosθ).secθcosecθ = 2
Solution:
LHS = (sinθ + 1 + cosθ)(sinθ – 1 + cosθ).secθcosecθ
= (sinθ + cosθ + 1)(sinθ + cosθ – 1).secθcosecθ
= {(sinθ + cosθ)2 – 12}.secθcosecθ
= (sin2θ + cos2θ + 2sinθcosθ – 1).secθcosecθ
= (1 + 2sinθcosθ – 1) .secθcosecθ
= 2sinθcosθ.secθcosecθ
= 2sinθcosecθcosθ.secθ
= 2
= RHS
OR
Prove that:
21. In what ratio does the point P(–4, y) divide the line segment joining the points A(–6,10) and B(3,–8)? Hence find the value of y.
OR
Find the value of p for which the points (–5, 1), (1, p) and (4, – 2) are collinear.
22. ABC is a right triangle in which ∠B = 90°. If AB = 8 cm and BC = 6 cm, find the diameter of the circle inscribed in the triangle.
SECTION – D (4 marks)
23. In an AP., the first term is – 4, the last term is 29 and the sum of all its terms is 150. Find its common difference.
24. Draw a circle of radius 4 cm. From a point 6 cm away from its centre, construct the pair of tangents to the circle and measure their lengths.
25. Prove that:
Solution:
26. Solve for x:; x ≠ 0, x ≠, a, b ≠ 0.
Solution:
OR
The sum of the areas of two squares is 640 m2. If the difference of their perimeters is 64 m, find the sides of the two squares.
27. In ∆ABC, AD ⊥ BC. Prove that: AC2 = AB2 + BC2 – 2BC × BD.
28. A moving boat is observed from the top of a 150 m high cliff moving away from the cliff. The angle of depression of the boat changes from 60° to 45° in 2 minutes. Find the speed of the boat in m/min.
OR
There are two poles, one each on either bank of a river just opposite to each other. One pole is 60 m high. From the top of this pole, the angle of depression of the top of and foot of the other pole are 30° and 60° respectively. Find the width of the river and height of the other pole.
29. Calculate the mean of the following frequency distribution:
OR
The following table gives production yield in kg per hectare of wheat of 100 farms of a village:
Change the distribution to a ‘more than type’ distribution, and draw its ogive.
30. A container, opened from the top and made up of a metal sheet, is in the form of a frustum of a cone of height 16 cm with radii of its lower and upper ends as 8 cm and 20 cm, respectively. Find the cost of the milk which can completely fill the container, at the rate of ₹50 per litre. Also find the cost of metal sheet used to make the container, if it costs ₹10 per 100 cm2. (Take π = 3.14)
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