CBSE Class 10 Maths Standard 2024 Question Paper 30/5/3 with Solutions

The LCM of the smallest prime number and the smallest odd composite number is :

1. (A)10
2. (B)6
3. (C)9
4. (D)18

If the mean of the first $n$ natural numbers is $\frac{5n}{9}$, then the value of $n$ is :

1. (A)5
2. (B)4
3. (C)9
4. (D)10

If $5\tan \theta -12=0$, then the value of $\sin \theta$ is :

1. (A)$\frac{5}{12}$
2. (B)$\frac{12}{13}$
3. (C)$\frac{5}{13}$
4. (D)$\frac{12}{5}$

The next ($4^{th}$) term of the A.P. $\sqrt{18},\sqrt{50},\sqrt{98},\dots$ is :

1. (A)$\sqrt{128}$
2. (B)$\sqrt{140}$
3. (C)$\sqrt{162}$
4. (D)$\sqrt{200}$

In the given figure, in $△ABC$, DE $\parallel$ BC. If AD = 2.4 cm, DB = 4 cm and AE = 2 cm, then the length of AC is :

1. (A)$\frac{10}{3}$ cm
2. (B)$\frac{3}{10}$ cm
3. (C)$\frac{16}{3}$ cm
4. (D)1.2 cm

In the given figure, RJ and RL are two tangents to the circle. If $\angle RJL=42^{\circ }$, then the measure of $\angle JOL$ is :

1. (A)$42^{\circ }$
2. (B)$84^{\circ }$
3. (C)$96^{\circ }$
4. (D)$138^{\circ }$

The perimeter of the sector of a circle of radius 21 cm which subtends an angle of $60^{\circ }$ at the centre of circle, is :

1. (A)22 cm
2. (B)43 cm
3. (C)64 cm
4. (D)462 cm

The ratio of the sum and product of the roots of the quadratic equation $5x^2-6x+21=0$ is :

1. (A)5 : 21
2. (B)2 : 7
3. (C)21 : 5
4. (D)7 : 2

The $14^{th}$ term from the end of the A.P. –11, –8, –5, ..., 49 is :

1. (A)7
2. (B)10
3. (C)13
4. (D)28

The length of the shadow of a tower on the plane ground is $\sqrt{3}$ times the height of the tower. The angle of elevation of the Sun is :

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

What is the probability that a number selected randomly from the numbers 1, 2, 3, ..., 15 is a multiple of 4 ?

1. (A)$\frac{4}{15}$
2. (B)$\frac{6}{15}$
3. (C)$\frac{3}{15}$
4. (D)$\frac{5}{15}$

If $\frac{x}{3}=2\sin A$, $\frac{y}{3}=2\cos A$, then the value of $x^2+y^2$ is :

1. (A)36
2. (B)9
3. (C)6
4. (D)18

If $\alpha$ and $\beta$ are the zeroes of the polynomial $p(x)=k{x}^2-30x+45k$ and $\alpha +\beta =\alpha \beta$, then the value of $k$ is :

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

The length of an arc of a circle with radius 12 cm is $10\pi$ cm. The angle subtended by the arc at the centre of the circle, is :

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

The LCM of three numbers 28, 44, 132 is :

1. (A)258
2. (B)231
3. (C)462
4. (D)924

A chord of a circle of radius 10 cm subtends a right angle at its centre. The length of the chord (in cm) is :

1. (A)$5\sqrt{2}$
2. (B)$10\sqrt{2}$
3. (C)$\frac{5}{\sqrt{2}}$
4. (D)5

Which out of the following type of straight lines will be represented by the system of equations $3x+4y=5$ and $6x+8y=7$ ?

1. (A)Parallel
2. (B)Intersecting
3. (C)Coincident
4. (D)Perpendicular to each other

The greatest number which divides 281 and 1249, leaving remainder 5 and 7 respectively, is :

1. (A)23
2. (B)276
3. (C)138
4. (D)69

Assertion (A) : Degree of a zero polynomial is not defined.
Reason (R) : Degree of a non-zero constant polynomial is 0.

1. (A)Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).
2. (B)Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of 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) : ABCD is a trapezium with DC $\parallel$ AB. E and F are points on AD and BC respectively, such that EF $\parallel$ AB. Then $\frac{AE}{ED}=\frac{BF}{FC}$.
Reason (R) : Any line parallel to parallel sides of a trapezium divides the non-parallel sides proportionally.

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

The king, queen and ace of clubs and diamonds are removed from a deck of 52 playing cards and the remaining cards are shuffled. A card is randomly drawn from the remaining cards. Find the probability of getting
(i) a card of clubs.
(ii) a red coloured card.

If two tangents inclined at an angle of $60^{\circ }$ are drawn to a circle of radius 3 cm, then find the length of each tangent.

Prove that the tangents drawn at the ends of a diameter of a circle are parallel.

Find the ratio in which the point P(–4, 6) divides the line segment joining the points A(–6, 10) and B(3, –8).

Prove that the points (3, 0), (6, 4) and (–1, 3) are the vertices of an isosceles triangle.

If $\alpha$, $\beta$ are zeroes of the polynomial $p(x)=5x^2-6x+1$, then find the value of $\alpha +\beta +\alpha \beta$.

Evaluate :
$\frac{2\tan 30^{\circ }\cdot \sec 60^{\circ }\cdot \tan 45^{\circ }}{1-{\sin }^{2}60^{\circ }}$

Prove that :
$\frac{\tan \theta -\cot \theta }{\sin \theta \cos \theta }={\sec }^2\theta -{\mathrm{cosec}}^2\theta$

A sector is cut from a circle of radius 21 cm. The central angle of the sector is $150^{\circ }$. Find the length of the arc of this sector and the area of the sector.

Prove that $\sqrt{3}$ is an irrational number.

Prove that ${\left(\sqrt{2}+\sqrt{3}\right)}^2$ is an irrational number, given that $\sqrt{6}$ is an irrational number.

Three unbiased coins are tossed simultaneously. Find the probability of getting :
(i) at least one head.
(ii) exactly one tail.
(iii) two heads and one tail.

Prove that the parallelogram circumscribing a circle is a rhombus.

If the sum of the first 14 terms of an A.P. is 1050 and the first term is 10, then find the $20^{th}$ term and the $n^{th}$ term.

The first term of an A.P. is 5, the last term is 45 and the sum of all the terms is 400. Find the number of terms and the common difference of the A.P.

In the given figure, $△FEC\cong △GDB$ and $\angle 1=\angle 2$.
Prove that $△ADE\sim △ABC$.

Sides AB and AC and median AD of a $△ABC$ are respectively proportional to sides PQ and PR and median PM of another $△PQR$. Show that $△ABC\sim △PQR$.

Find the value of 'k' for which the quadratic equation $(k+1)x^2-6(k+1)x+3(k+9)=0$, $k\ne -1$ has real and equal roots.

The age of a man is twice the square of the age of his son. Eight years hence, the age of the man will be 4 years more than three times the age of his son. Find their present ages.

From a window 15 metres high above the ground in a street, the angles of elevation and depression of the top and the foot of another house on the opposite side of the street are $30^{\circ }$ and $45^{\circ }$ respectively. Find the height of the opposite house. (Use $\sqrt{3}$ = 1.732)

A juice seller was serving his customers using glasses as shown in the figure. The inner diameter of the cylindrical glass was 5.6 cm, but the bottom of the glass had a hemispherical raised portion which reduced the capacity of the glass. If the height of the glass was 10 cm, find the apparent capacity and the actual capacity of the glass.

A garden is in the shape of a square. The gardener grew saplings of Ashoka tree on the boundary of the garden at the distance of 1 m from each other. He wants to decorate the garden with rose plants. He chose a triangular region inside the garden to grow rose plants. In the above situation, the gardener took help from the students of class 10. They made a chart for it which looks like the given figure.
Based on the above, answer the following questions :
(i) If A is taken as origin, what are the coordinates of the vertices of $△PQR$ ? (1)
(ii) (a) Find distances PQ and QR. (2)
OR
(b) Find the coordinates of the point which divides the line segment joining points P and R in the ratio 2 : 1 internally. (2)
(iii) Find out if $△PQR$ is an isosceles triangle. (1)

Activities like running or cycling reduce stress and the risk of mental disorder like depression. Running helps build endurance. Children develop stronger bones and muscles and are less prone to gain weight. The physical education teacher of a school has decided to conduct an inter school running tournament in his school premises. The time taken by a group of students to run 100 m, was noted as follows :
Time (in seconds): 0–20, 20–40, 40–60, 60–80, 80–100
Number of students: 8, 10, 13, 6, 3
Based on the above, answer the following questions :
(i) What is the median class of the above given data ? (1)
(ii) (a) Find the mean time taken by the students to finish the race. (2)
OR
(b) Find the mode of the above given data. (2)
(iii) How many students took time less than 60 seconds ? (1)

Essel World is one of India’s largest amusement parks that offers a diverse range of thrilling rides, water attractions and entertainment options for visitors of all ages. The park is known for its iconic “Water Kingdom” section, making it a popular destination for family outings and fun-filled adventure. The ticket charges for the park are ₹ 150 per child and ₹ 250 per adult.
On a day, the cashier of the park found that 300 tickets were sold and an amount of ₹ 55,000 was collected.
Based on the above, answer the following questions :
(i) If the number of children visited be $x$ and the number of adults visited be $y$, then write the given situation algebraically. (1)
(ii) (a) How many children visited the amusement park that day ? (2)
OR
(b) How many adults visited the amusement park that day ? (2)
(iii) How much amount will be collected if 250 children and 100 adults visit the amusement park ? (1)