Mathematics Basic Sample 2022-2023 English Medium Class 10 Question Paper Solution

If two positive integers p and q can be expressed as p = ab² and q = a³b; a, b being prime numbers, then LCM (p, q) is ______.

ab

a²b²

a³b²

a³b³

What is the greatest possible speed at which a man can walk 52 km and 91 km in an exact number of hours?

17 km/hours

7 km/hours

13 km/hours

26 km/hours

If one zero of the quadratic polynomial x² + 3x + k is 2, then the value of k is ______.

10

- 10

5 

- 5

Graphically, the pair of equations 6x – 3y + 10 = 0, 2x – y + 9 = 0 represents two lines which are ______.

Intersecting at exactly one point

Intersecting at exactly two points

Coincident

Parallel

If the quadratic equation x² + 4x + k = 0 has real and equal roots, then ______.

k < 4

k > 4

k = 4

k ≥ 4

The perimeter of a triangle with vertices (0, 4), (0, 0) and (3, 0) is ______.

5 units

12 units

10 units

11 units

`7 + sqrt(5)` units

If in triangles ABC and DEF, `(AB)/(DE) = (BC)/(FD)`, then they will be similar, when ______.

∠B = ∠E

∠A = ∠D

∠B = ∠D

∠A = ∠F

In which ratio the y-axis divides the line segment joining the points (5, – 6) and (–1, – 4)?

1 : 5

5 : 1

1 : 1

1 : 2

In the figure, if PA and PB are tangents to the circle with centre O such that ∠APB = 50°, then ∠OAB is equal to ______.

25°

30°

40°

50°

If sin A = `1/2`, then the value of sec A is ______.

`2sqrt(3)`

`1/sqrt(3)`

`sqrt(3)`

1

`sqrt(3)` cos²A + `sqrt(3)` sin²A is equal to ______.

1

`1/sqrt(3)`

`sqrt(3)`

0

The value of cos1°. cos2°. cos3°. cos4°....................... cos90° is ______.

1

0

–1

2

If the perimeter of a circle is equal to that of a square, then the ratio of their areas is ______.

22 : 7    

13 : 22

14 : 11       

7 : 22 

22 : 13

11 : 14

If the radii of two circles are in the ratio of 4 : 3, then their areas are in the ratio of ______.

4 : 3

8 : 3

16 : 9

9 : 16

The total surface area of a solid hemisphere of radius 7 cm is ______.

447π cm²

239 π cm²

174 π cm²

147 π cm²

98 π cm²

196 π cm²

`228 2/3 π  "cm"^2`

For the following distribution:

the upper limit of the modal class is:

10

15

20

25

If the mean of the following distribution is 2.6, then the value of y is:

3

8

13

24

A card is selected at random from a well shuffled deck of 52 cards. The probability of its being a red face card is ______.

`3/26`

`3/13`

`2/13`

`1/2`

Assertion: If the HCF of 510 and 92 is 2, then the LCM of 510 and 92 is 32460.

Reason: As HCF (a, b) × LCM (a, b) = a × b

Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

Assertion (A) is true but Reason (R) is false.

Assertion (A) is false but Reason (R) is true.

Assertion (A): The ratio in which the line segment joining (2, -3) and (5, 6) internally divided by x-axis is 1:2.

Reason (R): as formula for the internal division is `((mx_2 + nx_1)/(m + n) , (my_2 + ny_1)/(m + n))`

Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

Assertion (A) is true but Reason (R) is false.

Assertion (A) is false but Reason (R) is true.

For what values of k will the following pair of linear equations have infinitely many solutions?

kx + 3y – (k – 3) = 0

12x + ky – k = 0

In the following figure, altitudes AD and CE of ΔABC intersect each other at the point P. Show that:

ΔABD ∼ ΔCBE

In the following figure, altitudes AD and CE of ΔABC intersect each other at the point P. Show that:

ΔPDC ∼ ΔBEC

In the following figure, DE || AC and DF || AE. Prove that `("BF")/("FE") = ("BE")/("EC")`

Two concentric circles are of radii 5 cm and 3 cm. Find the length of the chord of the larger circle which touches the smaller circle.

If cot θ =` 7/8` evaluate `((1+sin θ )(1-sin θ))/((1+cos θ)(1-cos θ))`

Find the perimeter of a quadrant of a circle of radius 14 cm.

Find the diameter of a circle whose area is equal to the sum of the areas of the two circles of radii 24 cm and 7 cm.

Prove that `sqrt(5)` is an irrational number. 

Find the zeroes of the quadratic polynomial 6x² – 3 – 7x and verify the relationship between the zeroes and the coefficients.

A shopkeeper gives books on rent for reading. She takes a fixed charge for the first two days, and an additional charge for each day thereafter. Latika paid Rs 22 for a book kept for six days, while Anand paid Rs 16 for the book kept for four days. Find the fixed charges and the charge for each extra day.

Form the pair of linear equations in the following problems and find their solutions (if they exist) by any algebraic met

Places A and B are 100 km apart on a highway. One car starts from A and another from B at the same time. If the cars travel in the same direction at different speeds, they meet in 5 hours. If they travel towards each other, they meet in 1 hour. What are the speeds of the two cars?

ln Figure, PQ is a chord of length 8 cm of a circle of radius 5 cm and centre O. The tangents at P and Q intersect at point T. find the length of TP.

Prove that `tan θ/(1 - cot θ) + cot θ/(1 - tanθ)` = 1 + sec θ cosec θ

If sin θ + cos θ = `sqrt(3)`, then prove that tan θ + cot θ = 1

Two dice are thrown at the same time. What is the probability that the sum of the two numbers appearing on the top of the dice is 8?

Two dice are thrown at the same time. What is the probability that the sum of the two numbers appearing on the top of the dice is 13?

Two dice are thrown at the same time. What is the probability that the sum of the two numbers appearing on the top of the dice is less than or equal to 12?

An express train takes 1 hour less than a passenger train to travel 132 km between Mysore and Bangalore (without taking into consideration the time they stop at intermediate stations). If the average speed of the express train is 11 km/h more than that of the passenger train, find the average speed of the two trains.

Prove that If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, the other two sides are divided in the same ratio. In the figure, find EC if `(AD)/(DB) = (AE)/(EC)` using the above theorem.

A pen stand made of wood is in the shape of a cuboid with four conical depressions to hold pens. The dimensions of the cuboids are 15 cm by 10 cm by 3.5 cm. The radius of each of the depressions is 0.5 cm and the depth is 1.4 cm. Find the volume of wood in the entire stand (see the following figure).  Use [π = `22/7`]

Ramesh made a bird-bath for his garden in the shape of a cylinder with a hemispherical depression at one end. The height of the cylinder is 1.45 m and its radius is 30 cm. Find the total surface area of the bird-bath.

A life insurance agent found the following data for distribution of ages of 100 policy holders. Calculate the median age, if policies are given only to persons having age 18 years onwards but less than 60 years.

In the month of April to June 2022, the exports of passenger cars from India increased by 26% in the corresponding quarter of 2021-22, as per a report. A car manufacturing company planned to produce 1800 cars in 4th year and 2600 cars in 8th year. Assuming that the production increases uniformly by a fixed number every year.

Based on the above information answer the following questions.

1. Find the production in the 1^st year
2. Find the production in the 12^th year.
3. Find the total production in first 10 years.
   [OR]
   In how many years will the total production reach 31200 cars?

Based on the above information answer the following questions using the coordinate geometry.

1. Find the distance between Lucknow (L) to Bhuj (B).
2. If Kota (K), internally divide the line segment joining Lucknow (L) to Bhuj (B) into 3 : 2 then find the coordinate of Kota (K).
3. Name the type of triangle formed by the places Lucknow (L), Nashik (N) and Puri (P)
   [OR]
   Find a place (point) on the longitude (y-axis) which is equidistant from the points Lucknow (L) and Puri (P).

Lakshaman Jhula is located 5 kilometers north-east of the city of Rishikesh in the Indian state of Uttarakhand. The bridge connects the villages of Tapovan to Jonk. Tapovan is in Tehri Garhwal district, on the west bank of the river, while Jonk is in Pauri Garhwal district, on the east bank. Lakshman Jhula is a pedestrian bridge also used by motorbikes. It is a landmark of Rishikesh. A group of Class X students visited Rishikesh in Uttarakhand on a trip. They observed from a point (P) on a river bridge that the angles of depression of opposite banks of the river are 60° and 30° respectively. The height of the bridge is about 18 meters from the river.

Based on the above information answer the following questions.

1. Find the distance PA.
2. Find the distance PB 
3. Find the width AB of the river.
   [OR]
   Find the height BQ if the angle of the elevation from P to Q be 30°.