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

If two positive integers a and b are written as a = x³y² and b = xy³, where x, y are prime numbers, then the result obtained by dividing the product of the positive integers by the LCM (a, b) is ______.

xy

xy²

x³y³

x²y²

The given linear polynomial y = f(x) has

2 zeros

1 zero and the zero is ‘3’

1 zero and the zero is ‘4’

No zero

The lines representing the given pair of linear equations are non-intersecting. Which of the following statements is true?

`a_1/a_2 = b_1/b_2 = c_1/c_2`

`a_1/a_2 = b_1/b_2 ≠ c_1/c_2`

`a_1/a_2 ≠ b_1/b_2 = c_1/c_2`

`a_1/a_2 ≠ b_1/b_2 ≠ c_1/c_2`

The nature of roots of the quadratic equation 9x² – 6x – 2 = 0 is ______.

No real roots

2 equal real roots

2 distinct real roots

More than 2 real roots

Two APs have the same common difference. The first term of one of these is –1 and that of the other is – 8. Then the difference between their 4^th terms is ______.

–1

– 8

7

–9

What is the ratio in which the line segment joining (2, -3) and (5, 6) is divided by x-axis?

1:2

2:1

2:5

5:2

A point (x, y) is at a distance of 5 units from the origin. How many such points lie in the third quadrant?

0

1

2

infinitely many

In ΔABC, DE || AB. If AB = a, DE = x, BE = b and EC = c. Then x expressed in terms of a, b and c is ______.

`("ac")/"b"`

`("ac")/("b" + "c")`

`("ab")/"c"`

`("ab")/("b" + "c")`

If O is centre of a circle and Chord PQ makes an angle 50° with the tangent PR at the point of contact P, then the angle subtended by the chord at the centre is ______.

130°

100°

50°

30°

A quadrilateral PQRS is drawn to circumscribe a circle. If PQ = 12 cm, QR = 15 cm and RS = 14 cm, then find the length of SP is ______.

15 cm

14 cm

12 cm

11 cm

Given that sin θ = `a/b` then cos θ is equal to ______.

`b/sqrt(b^2 - a^2)`

`b/a`

`sqrt(b^2 - a^2)/b`

`a/sqrt(b^2 - a^2)`

(sec A + tan A) (1 − sin A) = ______.

sec A

sin A

cosec A

cos A

A pole 6 m high casts a shadow `2sqrt(3)` m long on the ground, then the Sun’s elevation is ______.

60°

45°

30°

90°

Tick the correct answer in the following and justify your choice: If the perimeter and the area of a circle are numerically equal, then the radius of the circle is:

2 units

π units

4 units

2π units

7 units

It is proposed to build a single circular park equal in area to the sum of areas of two circular parks of diameters 16 m and 12 m in a locality. The radius of the new park would be ______.

10 m

15 m

20 m

24 m

There is a square board of side ‘2a’ units circumscribing a red circle. Jayadev is asked to keep a dot on the above said board. The probability that he keeps the dot on the shaded region is ______.

`π/4`

`(4 - π)/4`

`(π - 4)/4`

`4/π`

2 cards of hearts and 4 cards of spades are missing from a pack of 52 cards. A card is drawn at random from the remaining pack. What is the probability of getting a black card?

`22/52`

`22/46`

`24/52`

`24/46`

The upper limit of the modal class of the given distribution is:

165

160

155

150

Statement A (Assertion): Total Surface area of the top is the sum of the curved surface area of the hemisphere and the curved surface area of the cone.

Statement R( Reason): Top is obtained by joining the plane surfaces of the hemisphere and cone together.

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 and 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.

Statement A (Assertion): `-5, (-5)/2, 0, 5/2`, .... is in Arithmetic Progression.

Statement R (Reason): The terms of an Arithmetic Progression cannot have both positive and negative rational numbers.

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 and 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.

Prove that is `sqrt2` irrational number.

ABCD is a parallelogram. Point P divides AB in the ratio 2:3 and point Q divides DC in the ratio 4:1. Prove that OC is half of OA.

From an external point P, two tangents, PA and PB are drawn to a circle with centre O. At one point E on the circle tangent is drawn which intersects PA and PB at C and D, respectively. If PA = 10 cm, find the the perimeter of the triangle PCD.

If tan (A + B) = `sqrt3` and tan (A – B) = `1/sqrt3`; 0° < A + B ≤ 90°; A > B, find A and B.

Find the value of x if `2 "cosec"^2 30 + x sin^2 60 - 3/4 tan^2 30` = 10

With vertices A, B and C of ΔABC as centres, arcs are drawn with radii 14 cm and the three portions of the triangle so obtained are removed. Find the total area removed from the triangle.

Find the area of the unshaded region shown in the given figure.

National Art convention got registrations from students from all parts of the country, of which 60 are interested in music, 84 are interested in dance and 108 students are interested in handicrafts. For optimum cultural exchange, organisers wish to keep them in minimum number of groups such that each group consists of students interested in the same artform and the number of students in each group is the same. Find the number of students in each group. Find the number of groups in each art form. How many rooms are required if each group will be allotted a room?

If α, β are zeroes of quadratic polynomial 5x² + 5x + 1, find the value of α² + β².

If α, β are zeroes of quadratic polynomial 5x² + 5x + 1, find the value of α^–1 + β^–1.

Solve the following systems of equations:

`2/sqrtx + 3/sqrty = 2`

`4/sqrtx - 9/sqrty = -1`

PA and PB are tangents drawn to a circle of centre O from an external point P. Chord AB makes an angle of 30° with the radius at the point of contact. If length of the chord is 6 cm, find the length of the tangent PA and the length of the radius OA.

Two tangents TP and TQ are drawn to a circle with centre O from an external point T. Prove that ∠PTQ = 2∠OPQ.

If 1 + sin²θ = 3sinθ cosθ, then prove that tanθ = 1 or `1/2`.

The lengths of 40 leaves of a plant are measured correct to the nearest millimeter, and the data obtained is represented in the following table:

Find the mean length of the leaves.

Two water taps together can fill a tank in `9 3/8`hours. The tap of larger diameter takes 10 hours less than the smaller one to fill the tank separately. Find the time in which each tap can separately fill the tank.

State and prove Basic Proportionality theorem.

In the given figure ∠CEF = ∠CFE. F is the midpoint of DC. Prove that `(AB)/(BD) = (AE)/(FD)`

Water is flowing at the rate of 15 km/h through a pipe of diameter 14 cm into a cuboidal pond which is 50 m long and 44 m wide. In what time will the level of water in pond rise by 21 cm?

What should be the speed of water if the rise in water level is to be attained in 1 hour?

A tent is in the shape of a cylinder surmounted by a conical top. If the height and radius of the cylindrical part are 3 m and 14 m respectively, and the total height of the tent is 13.5 m, find the area of the canvas required for making the tent, keeping a provision of 26 m² of canvas for stitching and wastage. Also, find the cost of the canvas to be purchased at the rate of ₹ 500 per m².

The median of the following data is 50. Find the values of p and q, if the sum of all the frequencies is 90.

Manpreet Kaur is the national record holder for women in the shot-put discipline. Her throw of 18.86m at the Asian Grand Prix in 2017 is the maximum distance for an Indian female athlete. Keeping her as a role model, Sanjitha is determined to earn gold in Olympics one day. Initially her throw reached 7.56m only. Being an athlete in school, she regularly practiced both in the mornings and in the evenings and was able to improve the distance by 9cm every week. During the special camp for 15 days, she started with 40 throws and every day kept increasing the number of throws by 12 to achieve this remarkable progress.

1. How many throws Sanjitha practiced on 11^th day of the camp?
2. What would be Sanjitha’s throw distance at the end of 6 weeks?
   (or)
   When will she be able to achieve a throw of 11.16 m?
3. How many throws did she do during the entire camp of 15 days?

1. At an instance, the midfielders and forward formed a parallelogram. Find the position of the central midfielder (D) if the position of other players who formed the parallelogram are :- A(1, 2), B(4, 3) and C(6, 6)
2. Check if the Goal keeper G(–3, 5), Sweeper H(3, 1) and Wing-back K(0, 3) fall on a same straight line.
   [or]
   Check if the Full-back J(5, –3) and centre-back I(–4, 6) are equidistant from forward C(0, 1) and if C is the mid-point of IJ.
3. If Defensive midfielder A(1, 4), Attacking midfielder B(2, –3) and Striker E(a, b) lie on the same straight line and B is equidistant from A and E, find the position of E.

1. At what distance from the foot of the tree was he observing the bird sitting on the tree?
2. How far did the bird fly in the mentioned time?
   (or)
   After hitting the tree, how far did the ball travel in the sky when Kaushik saw the ball?
3. What is the speed of the bird in m/min if it had flown `20(sqrt3 + 1) m`?