CBSE Class 10 Maths Standard 2024 Question Paper 30/5/2 with Solutions

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)$\frac{2}{3}$

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

The next ($4^{th}$) term of the A.P. $\sqrt{7},\sqrt{28},\sqrt{63},\dots$ is :

1. (A)$\sqrt{70}$
2. (B)$\sqrt{84}$
3. (C)$\sqrt{97}$
4. (D)$\sqrt{112}$

If the product of two co-prime numbers is 553, then their HCF is :

1. (A)1
2. (B)553
3. (C)7
4. (D)79

If $x=a\cos \theta$ and $y=b\sin \theta$, then the value of $b^2{x}^2+a^2{y}^2$ is :

1. (A)$a^2{b}^2$
2. (B)$ab$
3. (C)$a^4{b}^4$
4. (D)$a^2+b^2$

If the quadratic equation $a{x}^2+bx+c=0$ has real and equal roots, then the value of $c$ is :

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

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

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)$150^{\circ }$

If $4\sec \theta -5=0$, then the value of $\cot \theta$ is :

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

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

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)$138^{\circ }$

If the prime factorisation of 2520 is $2^3\times 3^a\times b\times 7$, then the value of $a+2b$ is :

1. (A)12
2. (B)10
3. (C)9
4. (D)7

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

One ticket is drawn at random from a bag containing tickets numbered 1 to 40. The probability that the selected ticket has a number which is a multiple of 7 is :

1. (A)$\frac{1}{7}$
2. (B)$\frac{1}{8}$
3. (C)$\frac{1}{5}$
4. (D)$\frac{7}{40}$

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

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

The number of terms in the A.P. 3, 6, 9, 12, ..., 111 is :

1. (A)36
2. (B)40
3. (C)37
4. (D)30

The ratio of the length of a pole and its shadow on the ground is $1:\sqrt{3}$. The angle of elevation of the Sun is :

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

If the mean and mode of a data are 24 and 12 respectively, then its median is :

1. (A)25
2. (B)18
3. (C)20
4. (D)22

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.

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.

If $\alpha$ and $\beta$ are zeroes of the quadratic polynomial $p(x)=x^2-5x+4$, then find the value of $\frac{1}{\alpha }+\frac{1}{\beta }-2\alpha \beta$.

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.

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

A carton consists of 60 shirts of which 48 are good, 8 have major defects and 4 have minor defects. Nigam, a trader, will accept the shirts which are good but Anmol, another trader, will only reject the shirts which have major defects. One shirt is drawn at random from the carton. Find the probability that it is acceptable to Anmol.

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.

An arc of a circle of radius 10 cm subtends a right angle at the centre of the circle. Find the area of the corresponding major sector. (Use $\pi =3.14$)

Prove that the parallelogram circumscribing a circle is a rhombus.

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.

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.

Prove that :
$\frac{\sin \theta }{1+\cos \theta }+\frac{1+\cos \theta }{\sin \theta }=2\mathrm{cosec}\theta$

A jar contains 54 marbles, each of which is blue, green or white. The probability of selecting a blue marble at random from the jar is $\frac{1}{3}$, and the probability of selecting a green marble at random is $\frac{4}{9}$. How many white marbles does this jar contain ?

From a point on a bridge across the river, the angles of depressions of the banks on opposite sides of the river are $30^{\circ }$ and $60^{\circ }$ respectively. If the bridge is at a height of 4 m from the banks, find the width of the river.

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 $△ABC\sim △PQR$.

A tent is in the shape of a cylinder, surmounted by a conical top. If the height and diameter of the cylindrical part are 3.5 m and 6 m, and slant height of the top is 4.2 m, find the area of canvas used for making the tent. Also, find the cost of canvas of the tent at the rate of ₹ 500 per $m^2$.

A 2-digit number is such that the product of the digits is 14. When 45 is added to the number, the digits are reversed. Find the number.

The side of a square exceeds the side of another square by 4 cm and the sum of the areas of the two squares is 400 $c{m}^2$. Find the sides of the squares.

Activities like running or cycling reduce stress and the risk of mental disorders 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)

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)