CBSE Class 10 Maths Standard 2024 Question Paper 30/4/3 with Solutions

The area of the sector of a circle of radius 12 cm is $60\pi$ cm${}^2$. The central angle of this sector is :

1. (A)$120^{\circ }$
2. (B)$6^{\circ }$
3. (C)$75^{\circ }$
4. (D)$150^{\circ }$

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 :

1. (A)$-15x-9y=5$
2. (B)$15x+9y=5$
3. (C)$9x-15y=6$
4. (D)$-15x+9y=5$

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 :

1. (A)$30^{\circ }$
2. (B)$45^{\circ }$
3. (C)$60^{\circ }$
4. (D)$90^{\circ }$

The pair of linear equations y = 0 and y = $-7$ have

1. (A)exactly one solution
2. (B)two solutions
3. (C)infinitely many solutions
4. (D)no solution

The area of the square inscribed in a circle of radius $5\sqrt{2}$ cm is :

1. (A)50 cm${}^2$
2. (B)100 cm${}^2$
3. (C)25 cm${}^2$
4. (D)200 cm${}^2$

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 :

1. (A)4.2 cm
2. (B)2.1 cm
3. (C)8.4 cm
4. (D)6.3 cm

In the given figure, AB and AC are tangents to the circle. If $\angle ABC=42^{\circ }$, then the measure of $\angle BAC$ is :

1. (A)$96^{\circ }$
2. (B)$42^{\circ }$
3. (C)$106^{\circ }$
4. (D)$86^{\circ }$

Two identical solid cubes of side ‘a’ are joined end-to-end. The total surface area of the resulting cuboid is :

1. (A)$6a^2$
2. (B)$10a^2$
3. (C)$5a^2$
4. (D)$4a^2$

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

1. (A)5
2. (B)13
3. (C)40
4. (D)8

For an event E, if $P(E)+P(\overline{E})=q$, then the value of $q^2-4$ is :

1. (A)$-3$
2. (B)3
3. (C)5
4. (D)$-5$

In $△ABC$, DE $\parallel$ BC (as shown in the figure). If AD = 4 cm, AB = 9 cm and AC = 13.5 cm, then the length of EC is :

1. (A)6 cm
2. (B)7.5 cm
3. (C)9 cm
4. (D)5.7 cm

If a polynomial $p(x)$ is given by $p(x)=x^2-5x+6$, then the value of $p(1)+p(4)$ is :

1. (A)0
2. (B)4
3. (C)2
4. (D)$-4$

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

1. (A)(0, 1)
2. (B)(0, $-1$)
3. (C)($-1$, 0)
4. (D)(1, 0)

If the difference of mode and median of a data is 24, then the difference of its median and mean is :

1. (A)12
2. (B)24
3. (C)8
4. (D)36

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

1. (A)4
2. (B)10
3. (C)16
4. (D)8

The probability of throwing a number greater than 2 with a fair die is :

1. (A)$\frac{2}{3}$
2. (B)$\frac{1}{3}$
3. (C)$\frac{1}{2}$
4. (D)$\frac{5}{6}$

If y = 1 is one of the solutions of the quadratic equation $p{y}^2+py+3=0$, then the value of p is :

1. (A)$-3$
2. (B)2
3. (C)$-\frac{3}{2}$
4. (D)$-2$

The value of $\theta$ for which $2{\sin }^2\theta =\frac{1}{2}$; $0^{\circ }\le \theta \le 90^{\circ }$ is :

1. (A)$30^{\circ }$
2. (B)$60^{\circ }$
3. (C)$45^{\circ }$
4. (D)$90^{\circ }$

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.

1. (A)Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A).
2. (B)Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).
3. (C)Assertion (A) is true, but Reason (R) is false.
4. (D)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 = $2\pi \times$ radius of a circle.

1. (A)Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A).
2. (B)Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).
3. (C)Assertion (A) is true, but Reason (R) is false.
4. (D)Assertion (A) is false, but Reason (R) is true.

In the given figure, $△ABC$ is circumscribing a circle. Find the length of BC, if AR = 4 cm, BR = 3 cm and AC = 11 cm.

Evaluate : $\frac{5{\cos }^{2}60^{\circ }+4{\sec }^{2}30^{\circ }-{\tan }^{2}45^{\circ }}{{\sin }^{2}30^{\circ }+{\sin }^{2}60^{\circ }}$

If $\sin (A-B)=\frac{1}{2}$, $\cos (A+B)=\frac{1}{2}$; $0<A+B\le 90^{\circ }$, $A>B$; find $\angle A$ and $\angle B$.

In the given figure, $△AHK\sim △ABC$. If AK = 8 cm, BC = 3.2 cm and HK = 6.4 cm, then find the length of AC.

The minute hand of a clock is 14 cm long. Find the area on the face of the clock described by the minute hand in 5 minutes.

Find the length of the arc of a circle which subtends an angle of $60^{\circ }$ at the centre of the circle of radius 42 cm.

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 ?

Three coins are tossed simultaneously. What is the probability of getting
(i) at least one head ?
(ii) exactly two tails ?
(iii) 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 :
(i) 2-digit number less than 40.
(ii) number divisible by 5 and greater than 50.
(iii) a perfect square number.

Prove that the parallelogram circumscribing a circle is a rhombus.

Find the zeroes of the polynomial $4x^2+4x-3$ and verify the relationship between zeroes and coefficients of the polynomial.

If $\alpha$ and $\beta$ are the zeroes of the polynomial $x^2+x-2$, then find the value of $\frac{\alpha }{\beta }+\frac{\beta }{\alpha }$.

Prove that $\frac{2-\sqrt{3}}{5}$ is an irrational number, given that $\sqrt{3}$ is an irrational number.

A part of monthly hostel charges is fixed and the remaining depends on the number of days one has taken food in the mess. When a student A takes food for 20 days, she has to pay ₹ 1,500 as hostel charges while another student B, who takes food for 26 days, pays ₹ 1,800. Find the fixed charges and the cost of food.

Prove that $\sqrt{{\sec }^2\theta +{\text{cosec}}^2\theta }=\tan \theta +\cot \theta$.

The following distribution shows the daily pocket allowance of children of a locality. The mean daily pocket allowance is ₹ 36.10. Find the missing frequency, f.
Daily pocket allowance (in ₹): 20–25, 25–30, 30–35, 35–40, 40–45, 45–50, 50–55
Number of children: 7, 6, 9, 13, f, 5, 4

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
$(c+1)x^2-6(c+1)x+3(c+9)=0$; $c\ne -1$
has real and equal roots.

E is a point on the side AD produced of a parallelogram ABCD and BE intersects CD at F. Show that $△ABE\sim △CFB$.

Sides AB, BC and the median AD of $△ABC$ are respectively proportional to sides PQ, QR and the median PM of another $△PQR$. Prove that $△ABC\sim △PQR$.

The angles of depression of the top and the bottom of a 8 m tall building from the top of a multi-storeyed building are $30^{\circ }$ and $45^{\circ }$ respectively. Find the height of the multi-storeyed building and the distance between the two buildings.

Treasure Hunt is an exciting and adventurous game where participants follow a series of clues/numbers/maps to discover hidden treasures. Players engage in a thrilling quest, solving puzzles and riddles to unveil the location of the coveted prize.
While playing a treasure hunt game, some clues (numbers) are hidden in various spots collectively forming an A.P. If the number on the $n^{th}$ spot is $20+4n$, then answer the following questions to help the players in spotting the clues :
(i) Which number is on first spot ? (1)
(ii) Which spot is numbered as 112 ? (2)
OR What is the sum of all the numbers on the first 10 spots ? (2)
(iii) 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 $\times$ 8 cm $\times$ 5 cm. These milk packets are then packed in cuboidal cartons of dimensions 30 cm $\times$ 32 cm $\times$ 15 cm.
Based on the above given information, answer the following questions :
(i) Find the volume of the cuboidal carton. (1)
(ii) Find the total surface area of a milk packet. (2)
OR 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)

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) What is the distance between the points A and C ? (2)
OR Find the coordinates of the point 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)