Gyanpoints
10th Mathematics Worksheet
Q. If (—5) is a root of the quadratic equation 2x2 + px + 15 = 0 and the quadratic equation p(x2+x) +k=0 has equal roots, then find the values of p and k.
Q. If the last term of an AP. of 30 terms is 119 and the 8th term from the end (towards the first term) is 91, then find the common difference of the A.P. Hence, find the sum of all the terms of the A.P.
Q. If two pipes function simultaneously, a reservoir will be filled in 12 hours. One pipe fills the reservoir 10 hours faster than the other. How many hours will the second pipe take to fill the reservoir?
Q. In Figure, XY and X’Y’ are two parallel tangents to a circle with centre O and another tangent AB with point of contact C intersects XY at A and X’Y’ at B .
Prove that ∠AOB 90° .
Q. A triangle ABC is drawn to circumscribe a circle of radius 4 cm such that the segments BD and DC into which BC is divided by the point of contact D are of lengths 6 cm and 8cm respectively. Find the lengths of the sides AB and AC.
Q. Some students planned a picnic. The total budget for food was Rs. 2000. But, 5 students failed to attend the picnic and thus the cost for food for each member increased by Rs. 20. How many students attended the picnic and how much did each student pay for the food?
Q. The sum of n term of an A.P is 3n2 + 5n. Find the A.P and its 15th term.
Q. Which term of A.P. 3,15,27,39,……. will be 132 more than its 54th term.
Q. Draw the graphs of the equations x – y + 1 = 0 and 3x + 2y – 12 = 0. Write the co-ordinates of the vertices of the triangle formed by these lines and the x -axis and shade the corresponding triangular region.
Q. If the point P(k−1,2) is equidistant from the points A(3,k) and B(k,5), find the values of k.
Q. Two ships are there in the sea on either side of a light house in such a way that the ships and the light house are in the same straight line. The angles of depression of the two ships as observed from the top of the light house are 60° and 45°. If the height of the light house is 200 m , fi nd the distance between the two ships. [Use √3=1.73]
Q. In the given figure, OACB is a quadrant of a circle with centre O and radius 3.5cm . If OD = 2cm, find the area of the shaded region.
Q. In Figure, ABCD is a square with side 7 cm . A circle is drawn circumscribing the square. Find the area of the shaded region.
Q. The largest possible hemisphere is drilled out from a wooden cubical block of side 21 cm such that the base of the hemisphere is on one of the faces of the cube. Find:
(i) the volume of wood left in the block,
(ii) the total surface area of the remaining solid.
Q. In the given figure, two concentric circles have radii 3 cm and 5 cm . Two tangents TR and TP are drawn to the circles from an external point T such that TR touches the inner circle at and TP touches the outer circle at P. If, TR = 4√10 cm, then find the length of TP.
Q. The sum of first and eighth terms of an A.P. is 32 and their product is 60 . Find the first term and common difference of the A.P. Hence, also find the sum of its first 20 terms.
Q. A solid is in the shape of a hemisphere surmounted by a cone. If the radius of hemisphere and base radius of cone is 7 cm and height of cone is 3.5 cm, find the volume of the solid. (Take π = 22/7)
Q. The sum of squares of two consecutive multiples of 7 is 637. Find the multiples. 2.
Q. Prove that: 
Q. Prove that the lengths of tangents drawn from an external point to a circle are equal.
Using above result, find the length BC of △ABC. Given that, a circle is inscribed in △ABC touching the sides AB, BC, and CA at R,P and Q respectively AB = 10cm and AQ = 7cm,CQ = 5cm
Q. The sum of two numbers is 18 and the sum of their reciprocals is 9/40. Find the numbers.
Q. If α, β are zeroes of the quadratic polynomial x2 + 9x + 20, form a quadratic polynomial whose zeroes are (α +1) and (β+1).
Q. Two different dice are rolled together. Find the probability of getting (i) the sum of numbers on two dice to be 5, (ii) even number on both dice, (iii) a doublet.
Q. Find the acute angle θ, when 
Q. A construction company will be penalized each day of delay in the construction of the bridge. The penalty will be ` 4000 for the first day and will increase by 1000 for each following day. Based on its budget, the company can afford to pay a maximum of ` 165000 toward penalty.
(i) The penalty amount paid by the construction company from the first day as a sequence
(ii) First-term and difference respectively of the above series is
(iii) Find the maximum number of days by which the completion of work can be delayed (take Sn = 165000)
OR
The penalty will be charged on the tenth day
Q. A group of students conducted a survey to find out about the preferred mode of transportation to school among their classmates. They surveyed 200 students from their school. The results of the survey are as follows:
120 students preferred to walk to school.
25% of the students preferred to use bicycles.
10% of the students preferred to take the bus.
Remaining students preferred to be dropped off by car.
Based on the above information, answer the following questions:
(i) What is the probability that a randomly selected student does not prefer to walk to school?
(ii) Find the probability of a randomly selected student who prefers to walk or use a bicycle.
(iii)(A) One day 50% of walking students decided to come by bicycle. What is the probability that a randomly selected student comes to school using a bicycle on that day?
OR
(B) What is the probability that a randomly selected student prefers to be dropped off by car?
Q. An age-wise list of number of literate people in a block is prepared in the following table. There are total 100 people and their median age is 41.5 years. Information about two groups are missing, which are denoted by x and y. Find the value of x and y.
| Age (in years) | Number of literate people |
| 10-20 | 15 |
| 20-30 | X |
| 30-40 | 12 |
| 40-50 | 20 |
| 50-60 | Y |
| 60-70 | 8 |
| 70-80 | 10 |
Q. The largest possible hemisphere is drilled out from a wooden cubical block of side 21 cm such that the base of the hemisphere is on one of the faces of the cube. Find:
(i) the volume of wood left in the block,
(ii) the total surface area of the remaining solid.
Q. The sum of the digits of a 2 -digit number is 12 . Seven times the number is equal to four times the number obtained by reversing the order of the digits. Find the number.
Q. In the given figure PA, QB and RC are each perpendicular to AC . If
AP = x, BQ = y and CR = z, then prove that 
Q. In an A.P. of 40 terms, the sum of first 9 terms is 153 and the sum of last 6 terms is 687. Determine the fi rst term and common difference of A.P. Also, find the sum of all the terms of the A.P.
Q. Prove that : 
Q. ABCD is a rectangle formed by the points A(−1,−1), B(−1,6) C(3,6) and
D(3,−1). P, Q, R and S are mid-points of sides AB, BC, CD and DA respectively. Show that diagonals of the quadrilateral PQRS bisect each other.
Q. In an A.P., the sum of three consecutive terms is 24 and the sum of their squares is 194. Find the numbers.
Q. Prove that √5 is an irrational number.
Q. If (−5) is a root of the quadratic equation 2x2 +px+15 = 0 and the quadratic equation p(x2 +x)+k = 0 has equal roots, then find the values of p and k .
Q.A tent is in the shape of a right circular cylinder up to a height of 3 m and then aright circular cone, with a maximum height of 13.5 m above the ground. Calculate the cost of painting the inner side of the tent at the rate of ₹ 2 per square metre, if the radius of the base is 14 m.
Q. A straight highway leads to the foot of a tower. A man standing at the top of the tower observes a car at an angle of depression of 30°, which is approaching the foot of the tower with a uniform speed. Six seconds later, the angle of depression of the car is found to be 60°. Find the further time taken by the car to reach the foot of the tower from this point.
Q. If the median of the following frequency distribution is 32.5 . Find the values of f1 and f2.
| Class | 0- 10 | 10 -20 | 20-30 | 30-40 | 40-50 | 50-60 | 60-70 | Total |
| Frequency | f1 | 5 | 9 | 12 | f2 | 63 | 2 | 40 |
Q. The sum of 5th and 9th terms of an A.P. is 72 and the sum of 7th and 12th terms is 97 .Find the A.P.
Q. If the last term of an A.P. of 30 terms is 119 and the 8th term from the end (towards the first term) is 91 , then fi nd the common difference of the A.P. Hence, find the sum of all the terms of the A.P.
Q. Find the greatest number that will divide 445, 572 and 699 leaving remainders 4,5 and 6 respectively.
Q. The ratio of incomes of two persons is 11 : 7 and the ratio of their expenditures is 9 : 5. If each of them manages to save Rs 400 per month, find their monthly incomes.
Q. In a cylindrical vessel of radius 10 cm, containing some water, 9000 small spherical balls are dropped which are completely immersed in water which raises the water level. If each spherical ball is of radius 0.5 cm , then find the rise in the level of water in the vessel.
Q. Find the lengths of the medians of a △ABC having vertices at A(0,−1),B(2,1) and C(0,3).
Q. If 2x + y = 13 and 4x – y = 17 , find the value of (x−y).
Q. The age of a man is twice the square of the age of his son. Eight years hence, the age of the man will be 4 years more than three times the age of his son. Find their present ages.
Q . Prove that the parallelogram circumscribing a circle is a rhombus. 3.
Q. Three unbiased coins are tossed simultaneously. Find the probability of getting :
(i) at least one head.
(ii) exactly one tail.
(iii) two heads and one tail.
Q. E is a point on the side AD produced of a parallelogram ABCD and BE intersects CD at F. Show that △ABE ∼ ΔCFB
Q. Rehana went to a bank to withdraw ₹ 2,000 . She asked the cashier to give her ₹50 and ₹ 100 notes only. Rehana got 25 notes in all. Find how many notes of ₹ 50 and ₹ 100 did she receive.
Q. A box contains 90 discs which are numbered 1 to 90 . If one disc is drawn at random from the box, fi nd the probability that it bears a :
(i) 2-digit number less than 40 .
(ii) number divisible by 5 and greater than 50 .
(iii) a perfect square number.
Q. In a flight of 2800 km , an aircraft was slowed down due to bad weather. Its average speed is reduced by 100 km/h and by doing so, the time of flight is increased by 30 minutes. Find the original duration of the flight.
Q. State and prove Basic Proportionality theorem.
Q. The following table shows the ages of the patients admitted in a hospital during a year:
| Age (in years) | 5-15 | 15-25 | 25-35 | 35-45 | 45-55 | 55-65 |
| Number of patients | 6 | 11 | 21 | 23 | 14 | 5 |
Find the mode and mean of the data given above.