Mathematics Standard Official 2023-2024 English Medium Class 10 Question Paper Solution

If ax + by = = a² - b² and bx + ay = 0, then the value of x + y is ______.

a² - b²

a + b

a - b

a² + b²

The HCF of two numbers 65 and 104 is 13. If LCM of 65 and 104 is 40x, then the value of x is ______.

5

13

40

8

If a polynomial p(x) is given by p(x) = x² - 5x + 6, then the value of p(1) + p(4) is ______.

0

4

2

-4

If the discriminant of the quadratic equation 3x² - 2x + c = 0 is 16, then the value of c is ______.

1

0

-1

`sqrt2`

If an are subtends an angle of 90° at the centre of a circle, then the ratio of its length to the circumference of the circle is ______.

2 : 3

1 : 4

4 : 1

1 : 3

The area of the sector of a circle of radius 12 cm is 60π cm². The central angle of this sector is ______.

120°

6°

75°

150°

If the difference of Mode and Median of a data is 24, then the difference of median and mean is ______.

12

24

8

36

Two dice are tossed simultaneously. The probability of getting odd numbers on both the dice is ______.

`6/36`

`3/36`

`12/36`

`9/36`

The ratio of total surface area of a solid hemisphere to the square of its radius is ______.

2π : 1

4π : 1

3π : 1

1 : 4π

If sin θ = 1, then the value of `1/2  sin(theta/2)`is ______.

`1/(2sqrt(2))`

`1/sqrt(2)`

`1/2`

0

Two lines are given to be parallel. The equation of one of these lines is 5x - 3y = 2, The equation of the second line can be ______.

-15x - 9y = 5

15x + 9y = 5

9x - 15y = 6

-15x + 9y = 5

Three numbers in A.P. have the sum of 30. What is its middle term?

4

10

16

8

In ΔABC, DE || BC (as shown in the figure), If AD = 4 cm, AB = 9 cm and AC = 13.5 cm, then the length of EC is ______.

6 cm

7.5 cm

9 cm

5.7 cm

At some time of the day, the length of the shadow of a tower is equal to its height. Then, the sun’s altitude at that time is ______.

30°

60°

90°

45°

In the given figure, AB and AC are tangents to the circle. If ∠ABC = 42°, then the measure of ∠BAC is ______.

96°

42°

106°

86°

The fourth vertex D of a parallelogram ABCD, whose three vertices are A(–2, 3), B(6, 7) and C(8, 3), is ______.

(0, 1)

(0, –1)

(–1, 0)

(1, 0)

For an event E, if `P(E) + P(barE)` = q, then the value of q² − 4 is ______.

−3

3

5

−5

In the given figure, QR is a common tangent to the two given circles touching externally at A. The tangent at A meets QR at P. If AP = 4.2 cm, then the length of QR is ______.

4.2 cm

2.1 cm

8.4 cm

6.3 cm

Assertion (A): Mid-point of a line segment divides the line segment in the ratio 1 : 1

Reason (R): The ratio in which the point (−3, k) divides the line segment joining the points (− 5, 4) and (− 2, 3) is 1 : 2.

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): If the circumference of a circle is 176 cm, then its radius is 28 cm.

Reason (R): Circumference = 2π × radius of a circle.

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

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

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

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

Three bells toll at intervals of 9, 12 and 15 minutes respectively. If they start tolling together, after what time will they next toll together?

The length of the minute hand of a clock is 14 cm. Find the area swept by the minute hand in 5 minutes.

Find the length of the arc of a circle which subtends an angle of 60° at the centre of the circle of radius 42 cm.

Evaluate: `(5cos^2 60° +  4sec^2 30° - tan^2 45°)/(sin^2 30° + sin^2 60°)`

If `sin (A – B) = 1/2` and `cos (A + B) = 1/2`, `0^@` < A + `B <= 90^@`, A > B Find A and B.

In the given figure, O is the centre of the circle. If ∠ AOB = 145°, then find the value of x.

In the given figure, Δ AHK ∼ Δ ABC. If AK = 8 cm, BC = 3.2 cm and HK = 6.4 cm, then find the length of AC.

Prove that `(sinθ - cosθ + 1)/(sinθ + cosθ - 1) = 1/(secθ - tanθ)`

Three coins are tossed simultaneously. What is the probability of getting at least one head?

Three coins are tossed simultaneously. What is the probability of getting exactly two tails?

Three coins are tossed simultaneously. What is the probability of getting at most one tail?

A box contains 90 discs which are numbered 1 to 90. If one disc is drawn at random from the box, find the probability that it bears a 2-digit number less than 40.

A box contains 90 discs which are numbered from 1 to 90. If one disc is drawn at random from the box, find the probability that it bears a perfect square number.

Rehana went to a bank to withdraw ₹ 2000. 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 received.

Find the zeroes of the quadratic polynomial f(x) = 4x² - 4x - 3 and verify the relation between its zeroes and coefficients.

If α and β are the zeroes of the polynomial x² + x − 2, then find the value of `alpha/beta+beta/alpha`

Prove that `(2-sqrt3)/5` is an irrational number, given that `sqrt 3` is an irrational number.

Prove that a parallelogram circumscribing a circle is a rhombus.

Two pillars of equal lengths stand on either side of a road which is 100 m wide, exactly opposite to each other. At a point on the road between the pillars, the angles of elevation of the tops of the pillars are 60° and 30°. Find the length of each pillar and the distance of the point on the road from the pillars. (Use `sqrt3` = 1.732)

E is a point on the side AD produced of a parallelogram ABCD and BE intersects CD at F. Show that ΔABE ∼ ΔCFB.

Sides AB and AC and median AD of a triangle ABC are respectively proportional to sides PQ and PR and median PM of another triangle PQR. Show that ΔABC ~ ΔPQR. 

A train travels a distance of 90 km at a constant speed. Had the speed been 15 km/h more, it would have taken 30 minutes less for the journey. Find the original speed of the train.

Find the value of ‘c’ for which the quadratic equation 

(c + 1) x² - 6(c + 1) x + 3(c + 9) = 0; c ≠ - 1

has real and equal roots.

The following table shows the ages of the patients admitted in a hospital during a year:

Find the mode and the mean of the data given above. Compare and interpret the two measures of central tendency.

Ryan, from a very young age, was fascinated by the twinkling of stars and the vastness of space. He always dreamt of becoming an astronaut one day. So, he started to sketch his own rocket designs on the graph sheet. One such design is given below :

Based on the above, answer the following questions:

i. Find the mid-point of the segment joining F and G.    (1) 

ii. a. What is the distance between the points A and C?   (2)

OR

b. Find the coordinates of the points which divides the line segment joining the points A and B in the ratio 1 : 3 internally.    (2)

iii. What are the coordinates of the point D?    (1)

1. Which number is on first spot?     1
2. 
   
   1. Which spot is numbered as 112?     2
      OR
   2. What is the sum of all the numbers on the first 10 spots?    2
3. Which number is on the (n – 2)^th spot?     1

Tamper-proof tetra-packed milk guarantees both freshness and security. This milk ensures uncompromised quality, preserving the nutritional values within and making it a reliable choice for health-conscious individuals.

500 ml milk is packed in a cuboidal container of dimensions 15 cm × 8 cm × 5 cm. These milk packets are then packed in cuboidal cartons of dimensions 30 cm × 32 cm × 15 cm.

Based on the above-given information, answer the following questions:

i. Find the volume of the cuboidal carton. (1)

ii. a. Find the total surface area of the milk packet. (2)

OR

b. How many milk packets can be filled in a carton? (2)

iii. How much milk can the cup (as shown in the figure) hold? (1)