Mathematics Basic (Board Sample Paper) 2023-2024 English Medium Class 10 Question Paper Solution

If two positive integers a and b are written as a = x³ y² and b = xy³; x, y are prime numbers, then HCF (a, b) is ______.

xy

xy²

x³y³

x²y²

The LCM of smallest 2-digit number and smallest composite number is ______.

12

4

20

40

If x = 3 is one of the roots of the quadratic equation x² – 2kx – 6 = 0, then the value of k is ______.

`-1/2`

`1/2`

3

2

The pair of equations y = 0 and y = –7 has ______.

One solution

Two solutions

Infinitely many solutions

No solution

Values of k for which the quadratic equation 2x² – kx + k = 0 has equal roots is ______.

0 only

4

8 only

0, 8

The distance of the point (3, 5) from x-axis (in units) is ______.

3

– 3

5

– 5

If in two triangles ABC and PQR, `(AB)/(QR) = (BC)/(PR) = (CA)/(PQ)`, then ______.

ΔPQR ~ ΔCAB

ΔPQR ~ ΔABC

ΔCBA ~ ΔPQR 

ΔBCA ~ ΔPQR

Which of the following is NOT a similarity criterion of traingles?

AA

SAS

AAA

RHS

In the given figure, if TP and TQ are the two tangents to a circle with centre O so that ∠POQ = 110°, then ∠PTQ is equal to ______.

60°

70°

80°

90°

If cos A = `4/5`, then the value of tan A is ______.

`3/5`

`3/4`

`4/3`

`5/3`

`1/8`

If the height of the tower is equal to the length of its shadow, then the angle of elevation of the sun is ______.

30°

45°

60°

90°

(1 – cos² A) is equal to ______.

sin² A

tan² A

1 – sin² A

sec² A

The radius of a circle is same as the side of a square. Their perimeters are in the ratio

1 : 1

2 : π

π : 2

`sqrt(π) : 2`

The area of the circle is 154 cm². The radius of the circle is ______.

7 cm

14 cm

3.5 cm

17.5 cm

When a die is thrown, the probability of getting an even number less than 4 is ______.

1/4

0

1/2

1/6

For the following distribution:

The lower limit of modal class is:

15

25

30

35

A rectangular sheet of paper 40 cm × 22 cm, is rolled to form a hollow cylinder of height 40 cm. The radius of the cylinder (in cm) is ______.

 3.5

7

\[\frac{80}{7}\]

5

Consider the following frequency distribution:

The median class is:

6 – 12

12 – 18

18 – 24

24 – 30

Assertion (A): The point (0, 4) lies on y-axis.

Reason (R): The x-coordinate of a point on y-axis is zero.

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).

Assertions (A) is true but reason (R) is false.

Assertions (A) is false but reason (R) is true.

Assertion (A): The HCF of two numbers is 5 and their product is 150. Then their LCM is 40.

Reason(R): For any two positive integers a and b, 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).

Assertions (A) is true but reason (R) is false.

Assertions (A) is false but reason (R) is true.

Find whether the following pair of linear equations is consistent or inconsistent:

3x + 2y = 8

6x – 4y = 9

In the given figure, if ABCD is a trapezium in which AB || CD || EF, then prove that `(AE)/(ED) = (BF)/(FC)`.

In figure, if AD = 6cm, DB = 9cm, AE = 8cm and EC = 12cm and ∠ADE = 48°. Find ∠ABC.

The length of a tangent from a point A at distance 5 cm from the centre of the circle is 4 cm. Find the radius of the circle.

Evaluate: sin² 60° + 2tan 45° – cos² 30°.

What is the diameter of a circle whose area is equal to the sum of the areas of two circles of radii 40cm and 9cm?

A chord of a circle of radius 10 cm subtends a right angle at the centre. Find the area of the minor segment. [Use π = 3.14.]

Prove that `sqrt3` is an irrational number.

Find the zeroes of the quadratic polynomial 4s² – 4s + 1 and verify the relationship between the zeroes and the coefficients.

The coach of a cricket team buys 4 bats and 1 ball for ₹ 2050. Later, she buys 3 bats and 2 balls for ₹ 1600. Find the cost of each bat and each ball.

Form the pair of linear equation in the following problem, and find its solution (if they exist) by the elimination method:

A lending library has a fixed charge for the first three days and an additional charge for each day thereafter. Saritha paid Rs 27 for a book kept for seven days, while Susy paid Rs 21 for the book she kept for five days. Find the fixed charge and the charge for each extra day.

A circle touches all the four sides of a quadrilateral ABCD. Prove that AB + CD = BC + DA.

Prove the following identities, where the angles involved are acute angles for which the expressions are defined:

`(cosec  θ  – cot θ)^2 = (1-cos theta)/(1 + cos theta)`

Prove the following trigonometric identities.

sec A (1 − sin A) (sec A + tan A) = 1

A bag contains 6 red, 4 black and some white balls:

1. Find the number of white balls in the bag if the probability of drawing a white ball is `1/3`.
2. How many red balls should be removed from the bag for the probability of drawing a white ball to be `1/2`?

A train travels 360 km at a uniform speed. If the speed had been 5 km/h more, it would have taken 1 hour less for the same journey. Find the speed of the train.

Prove that, if a line is drawn parallel to one side of a triangle to intersect the other two sides, then the two sides are divided in the same ratio.

In ΔPQR, S and T are points on PQ and PR respectively. `(PS)/(SQ) = (PT)/(TR)` and ∠PST = ∠PRQ. Prove that PQR is an isosceles triangle.

A medicine capsule is in the shape of cylinder with two hemispheres stuck to each of its ends (see the given figure). The length of the entire capsule is 14 mm and the diameter of the capsule is 5 mm. Find its surface area. [Use π = `22/7`]

A gulab jamun, contains sugar syrup up to about 30% of its volume. Find approximately how much syrup would be found in 45 gulab jamuns, each shaped like a cylinder with two hemispherical ends with length 5 cm and diameter 2.8 cm (see the given figure).  Use [π = 22/7]

The following table gives the distribution of the life time of 400 neon lamps:

Find the average life time of a lamp.

Read the following passage:

1. In which year, the production is 29,200 sets?
2. Find the production in the 8^th year.
   OR
   Find the production in first 3 years.
3. Find the difference of the production in 7^th year and 4^th year.

Read the following passage:

Based on the above information, answer the following questions.

1. How far is Alia's house from Shagun's house?
2. How far is the library from Shagun's house?
3. Show that for Shagun, school is farther compared to Alia's house and library.
   OR
   Show that Alia’s house, shagun’s house and library for an isosceles right triangle.

Read the following passage:

A boy is standing on the top of light house. He observed that boat P and boat Q are approaching the light house from opposite directions. He finds that angle of depression of boat P is 45° and angle of depression of boat Q is 30°. He also knows that height of the light house is 100 m.

Based on the above information, answer the following questions.

1. What is the measure of ∠APD?
2. If ∠YAQ = 30°, then ∠AQD is also 30°, Why?
3. Find length of PD
   OR
   Find length of DQ