Prime Maths | Vinod Singh
Madhyamik Test Examination – Practice Paper | Subject: MATHEMATICS
Time: 3 Hours 15 Minutes
Full Marks: 90
Type: Free Practice Test Paper
General Instructions:
- The figures in the margin indicate full marks.
- Use of calculators is not permitted.
- Graph papers will be supplied if required.
1. Multiple Choice Questions [1 × 6 = 6]
Select the correct answer in each of the following cases:
iIf the mean of a frequency distribution is 8.1, \(\sum f_i x_i = 132 + 5k\), and \(\sum f_i = 20\), then the value of \(k\) is:
iiIf \(\cos\theta + \sec\theta = 2\), then the value of \(\cos^5\theta + \sin^5\theta\) is:
iiiThe volumes of two right circular cones are equal. If the radii of their circular bases are in the ratio 1:2, their heights are in the ratio:
ivIn the right-angled triangle ABC, \(\angle A\) is a right angle, AB = 12 cm, AC = 5 cm, and BC = 13 cm. AD is perpendicular to BC. The length of AD is:
vX and Y are two points on sides PQ and PR of triangle PQR respectively. If PX = 2 cm, XQ = 3.5 cm, YR = 7 cm, and PY = 4.25 cm, then:
viIf \(x \propto y\), then:
2. Fill in the blanks (any five) [1 × 5 = 5]
iIf the difference between the roots of the equation \(x^2 - px + q = 0\) is 1, then \(p^2 - 4q = \) __________.
iiIf \(x = 2 - \sqrt{3}\) and \(y = 4\sqrt{2}-2\), then \((x + y)^4 = \) __________.
iiiThe tangents drawn at the two ends of a diameter of a circle are __________ to each other.
ivThe ratio of simple interest on a fixed amount of money at the same annual rate for 4 years and 8 years is __________.
vIf the compound ratio of 2:3, 4:5, and 6:\(x\) is 2:5, then the value of \(x = \) __________.
viA tangent drawn from an external point M to a circle with center O touches the circle at point N. If ON = 9 cm and MO = 15 cm, then the length of MN = __________.
3. Write True or False (any five) [1 × 5 = 5]
iA starts a business with ₹10,000, and 6 months later B invests ₹20,000. At the end of the year, their profit shares will be equal.
iiThe ratio of the volumes of a right circular cone and a right circular cylinder with the same base and same height is 1:3.
iiiThe arithmetic mean and mode of the numbers 4, 3, 2, 5, 3, 4, 5, 1, 7, 3, 2, 1 are equal.
ivIn trapezium ABCD, AB || CD; diagonals AC and BD intersect at point O, then \(AO \cdot OD = BO \cdot OC\).
vAB is a diameter of a circle with center O. P is any point on the circumference. If \(\angle POA = 120^\circ\), then \(\angle PBO + \angle POA = 140^\circ\).
viThe current price of a mobile phone is ₹20,000. If the price depreciates by 15% every year, the price of the mobile phone after 2 years will be ₹14,450.
4. Answer the following questions (any ten) [2 × 10 = 20]
iIf the simple interest on ₹8,000 for 2 years is ₹720, find the annual rate of interest.
iiIf the compound interest on a sum of money for 2 years is ₹512 and the simple interest for the same time and at the same rate is ₹500, what is the rate of interest?
iiiIn a partnership business, the ratio of the capitals of three persons is 3 : 5 : 8. If the profit of the first person is ₹60 less than the profit of the third person, what was the total profit in the business?
ivIf 2 is a root of the equations \(x^2 + bx + 12 = 0\) and \(x^2 + bx + q = 0\), find the value of \(q\).
vIf \(a+b=2\sqrt{5}\) and \(a-b=\sqrt{2}\), what is the value of \(a^2 + b^2\)?
viThe length of each of two parallel chords AB and CD is 16 cm. If the radius of the circle is 10 cm, find the distance between the two chords.
viiIn a circle, two chords PQ and PR are perpendicular to each other. If the length of the radius of the circle is \(r\) cm, find the length of the chord QR.
viiiIf \((4a+5b):(4c+5d) = (4a-5b):(4c-5d)\), prove that a, b, c, and d are in proportion.
ixIf the volume of a right circular cone is \(V\), base area is \(A\), and height is \(H\), find the value of \(\frac{AH}{V}\).
xIf \(\theta = 30^\circ\), what is the value of \(\sin 2\theta + \cos 2\theta\)?
xiFind the median and arithmetic mean of the numbers: 2, 2, 3, 4, 6, 5, 5, 8, 9, 10, 11.
xiiThe length, breadth, and height of a rectangular cuboid room are 5 m, 4 m, and 3 m respectively. Calculate the length of the longest rod that can be kept in that room.
5. Answer any one question [5 × 1 = 5]
iA person borrowed ₹500 at 6% simple interest per annum, and exactly two years after the first loan, borrowed a second loan of ₹900 at 4% simple interest per annum. How many years after taking the first loan will the interest on these two loans be equal, and what will be the total interest on the two loans at that time?
iiA and B started a joint business with ₹6,200 and ₹10,000 respectively. They decided that A would receive 20% of the profit for managing the business, and 10% of the remaining profit would be deposited as savings. The rest of the profit would be divided in proportion to their capitals. If the total profit at the end of the year is ₹45,000, how much money will A receive in total?
6. Answer any one question [3 × 1 = 3]
iSolve: \(\frac{2x+1}{2x-1} + \frac{2x-1}{2x+1} = 1 + \frac{7}{2x-1}, x \neq \frac{1}{2}\)
iiIf \(x \propto y\) and \(y \propto z\), prove that \(x+y+z \propto \sqrt{yz}+\sqrt{zx}+\sqrt{xy}\).
7. Solve any one question [3 × 1 = 3]
iFind the simplest value: \( (\sqrt{5}+\sqrt{3})\big( \frac{3\sqrt{3}}{\sqrt{5}+\sqrt{2}}-\frac{\sqrt{5}}{\sqrt{3}+\sqrt{2}}\big)\)
iiThe volume of a cone is in joint variation with the square of its base radius and its height. If the ratio of the base radii of two cones is 3:4 and the ratio of their heights is 6:5, find the ratio of their volumes.
8. Answer any one question [3 × 1 = 3]
iIf \(a, b, c, d\) are in continued proportion, prove that \((b-c)^2 + (c-a)^2 + (b-d)^2 = (a-d)^2\).
iiIf \(\frac{x}{b+c} = \frac{y}{c+a} = \frac{z}{a+b}\), then prove that \(\frac{a}{y+z-x} = \frac{b}{z+x-y} = \frac{c}{x+y-z}\).
9. Answer any one question [5 × 1 = 5]
iProve that the opposite angles of any cyclic quadrilateral are supplementary.
iiProve that if two tangents are drawn to a circle from an external point, the line segments joining the points of contact to the external point are equal in length and subtend equal angles at the center.
10. Answer any one question [3 × 1 = 3]
iIn an isosceles triangle ABC, AB = AC. A straight line parallel to BC intersects AB and AC at points P and Q respectively. Prove that BCQP is a cyclic quadrilateral.
iiA right-angled triangle ABC is drawn where \(\angle A\) is a right angle. P and Q are two points on sides AB and AC respectively. B, Q and C, P are joined. Prove that \(BQ^2 + PC^2 = BC^2 + PQ^2\).
11. Answer any one question [5 × 1 = 5]
iConstruct an isosceles triangle whose one angle is \(120^\circ\) and the length of each of the equal sides is 4 cm. Draw the circumcircle of the triangle.
iiConstruct a square equal in area to a rectangle with sides of length 6 cm and 3 cm. (Only traces of construction are required).
12. Solve any two questions [3 × 2 = 6]
iIf \(m = \tan\theta + \sin\theta\) and \(n = \tan\theta - \sin\theta\), prove that \(m^2 - n^2 = 4\sqrt{mn}\) (\(0^\circ < \theta < 90^\circ\)).
iiIf \(\angle A + \angle B = 90^\circ\), show that \(1 + \frac{\tan A}{\tan B} = \sec^2 A\).
iiiOne angle of a triangle is 60° and another is \(\frac{\pi}{6}\) radians. What is the measurement of the third angle in degrees?
13. Answer any one question [5 × 1 = 5]
iFrom the roof and the foot of a 16-meter high house, the angles of elevation of the top of a temple are observed to be 45° and 60° respectively. Find the height of the temple and its horizontal distance from the house.
iiThe angle of elevation of the top of a tower from a point A on the ground is 30°. After moving 20 meters from point A towards the foot of the tower to point B, the angle of elevation increases to 60°. Find the height of the tower and the distance of the tower from point A.
14. Answer any two questions [4 × 2 = 8]
iA large solid sphere is made by melting three solid copper spheres with radii of lengths 3 cm, 4 cm, and 5 cm. Calculate the length of the radius of the large sphere.
ii77 square meters of tarpaulin were required to make a right circular cone-shaped tent. If the slant height of the tent is 7 meters, calculate the base area of the tent.
iiiDetermine the ratio of the volumes of a solid cone, a solid hemisphere, and a solid cylinder having equal base diameters and equal heights.
15. Answer any two questions [4 × 2 = 8]
iCalculate the arithmetic mean from the following frequency distribution:
| Class Limit | 0-10 | 10-20 | 20-30 | 30-40 | 40-50 | 50-60 | 60-70 |
|---|---|---|---|---|---|---|---|
| Frequency | 4 | 7 | 10 | 15 | 10 | 8 | 5 |
iiFind the median of the data below:
| Class Limit | 1-5 | 6-10 | 11-15 | 16-20 | 21-25 | 26-30 | 31-35 |
|---|---|---|---|---|---|---|---|
| No. of Workers | 2 | 3 | 6 | 7 | 5 | 4 | 3 |
iiiDetermine the mode of the following frequency distribution:
| Class Limit | 3-6 | 6-9 | 9-12 | 12-15 | 15-18 | 18-21 | 21-24 |
|---|---|---|---|---|---|---|---|
| Frequency | 2 | 6 | 12 | 24 | 21 | 12 | 3 |
No comments:
Post a Comment