CBSE Class 10 Maths Standard 2024 Question Paper 30/3/1 with Solutions

The pair of linear equations $x+2y+5=0$ and $-3x=6y-1$ has

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

The common difference of the A.P.
$\frac{1}{2x},\frac{1-4x}{2x},\frac{1-8x}{2x}$, ................is :

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

Two dice are thrown together. The probability that they show different numbers is :

1. (A)$1/6$
2. (B)$5/6$
3. (C)$1/3$
4. (D)$2/3$

The probability of guessing the correct answer to a certain test question is $\frac{x}{6}$. If the probability of not guessing the correct answer to this question is $\frac{2}{3}$, then the value of $x$ is :

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

If $a=2^2\times 3^x$, $b=2^2\times 3\times 5$, $c=2^2\times 3\times 7$ and LCM (a, b, c) = 3780, then $x$ is equal to

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

The zeroes of the quadratic polynomial $2x^2-3x-9$ are :

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

From a point on the ground, which is 30 m away from the foot of a vertical tower, the angle of elevation of the top of the tower is found to be $60^{\circ }$. The height (in metres) of the tower is :

1. (A)$10\sqrt{3}$
2. (B)$30\sqrt{3}$
3. (C)60
4. (D)30

If $\cos \theta =\frac{\sqrt{3}}{2}$ and $\sin \varphi =\frac{1}{2}$, then $\tan (\theta +\varphi )$ is :

1. (A)$\sqrt{3}$
2. (B)$\frac{1}{\sqrt{3}}$
3. (C)1
4. (D)not defined

Maximum number of common tangents that can be drawn to two circles intersecting at two distinct points is :

1. (A)4
2. (B)3
3. (C)2
4. (D)1

In the given figure, if PT is a tangent to a circle with centre O and $\angle TPO=35^{\circ }$, then the measure of $\angle x$ is :

1. (A)$110^{\circ }$
2. (B)$115^{\circ }$
3. (C)$120^{\circ }$
4. (D)$125^{\circ }$

If the diagonals of a quadrilateral divide each other proportionally, then it is a :

1. (A)parallelogram
2. (B)rectangle
3. (C)square
4. (D)trapezium

In $△ABC$, DE $\parallel$ BC (as shown in the figure). If AD = 2 cm, BD = 3 cm, BC = 7.5 cm, then the length of DE (in cm) is :

1. (A)2.5
2. (B)3
3. (C)5
4. (D)6

Given HCF (2520, 6600) = 40, LCM (2520, 6600) = $252\times k$, then the value of k is :

1. (A)1650
2. (B)1600
3. (C)165
4. (D)1625

A pair of irrational numbers whose product is a rational number is :

1. (A)$(\sqrt{16},\sqrt{4})$
2. (B)$(\sqrt{5},\sqrt{2})$
3. (C)$(\sqrt{3},\sqrt{27})$
4. (D)$(\sqrt{36},\sqrt{2})$

If a digit is chosen at random from the digits 1, 2, 3, 4, 5, 6, 7, 8, 9; then the probability that this digit is an odd prime number is :

1. (A)$\frac{1}{3}$
2. (B)$\frac{2}{3}$
3. (C)$\frac{4}{9}$
4. (D)$\frac{5}{9}$

The mean of five observations is 15. If the mean of first three observations is 14 and that of the last three observations is 17, then the third observation is

1. (A)20
2. (B)19
3. (C)18
4. (D)17

Perimeter of a sector of a circle whose central angle is $90^{\circ }$ and radius 7 cm is :

1. (A)35 cm
2. (B)11 cm
3. (C)22 cm
4. (D)25 cm

In the given figure, O is the centre of the circle. MN is the chord and the tangent ML at point M makes an angle of $70^{\circ }$ with MN. The measure of $\angle MON$ is :

1. (A)$120^{\circ }$
2. (B)$140^{\circ }$
3. (C)$70^{\circ }$
4. (D)$90^{\circ }$

Assertion (A) : The point which divides the line segment joining the points A (1, 2) and B($-1$, 1) internally in the ratio 1 : 2 is $\left(\frac{-1}{3},\frac{5}{3}\right)$
Reason (R) : The coordinates of the point which divides the line segment joining the points A ($x_1$, $y_1$) and B($x_2$, $y_2$) in the ratio $m_1:m_2$ are $\left(\frac{m_1{x}_2+m_2{x}_1}{m_1+m_2},\frac{m_1{y}_2+m_2{y}_1}{m_1+m_2}\right)$

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) : In a cricket match, a batsman hits a boundary 9 times out of 45 balls he plays. The probability that in a given ball, he does not hit the boundary is $\frac{4}{5}$.
Reason (R) : P(E) + P(not E) = 1

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.

One card is drawn at random from a well shuffled deck of 52 cards. Find the probability that the card drawn
(i) is queen of hearts;
(ii) is not a jack.

If $2x+y=13$ and $4x-y=17$, find the value of $(x-y)$.

Sum of two numbers is 105 and their difference is 45. Find the numbers.

Find a relation between $x$ and y such that the point P($x$, y) is equidistant from the points A(7, 1) and B(3, 5).

Points A($-1$, y) and B(5, 7) lie on a circle with centre O(2, $-3y$) such that AB is a diameter of the circle. Find the value of y. Also, find the radius of the circle.

In the given figure, $\frac{EA}{EC}=\frac{EB}{ED}$, prove that $△EAB\sim △ECD$

Evaluate : $\frac{\cos 45^{\circ }+\sin 60^{\circ }}{\sec 30^{\circ }+\mathrm{cosec}30^{\circ }}$.

If the sum of first 7 terms of an A.P. is 49 and that of first 17 terms is 289, find the sum of its first 20 terms.

The ratio of the $10^{th}$ term to its $30^{th}$ term of an A.P. is 1 : 3 and the sum of its first six terms is 42. Find the first term and the common difference of A.P.

Find the zeroes of the quadratic polynomial $x^2-15$ and verify the relationship between the zeroes and the coefficients of the polynomial.

Solve the following system of linear equations graphically :
$x-y+1=0$
$x+y=5$

Find the ratio in which the line segment joining the points (5, 3) and ($-1$, 6) is divided by Y-axis.

P($-2$, 5) and Q(3, 2) are two points. Find the coordinates of the point R on line segment PQ such that PR = 2QR.

Prove that $\frac{\sin \theta -2{\sin }^3\theta }{2{\cos }^3\theta -\cos \theta }=\tan \theta$.

Prove that the tangents drawn at the end points of a chord of a circle makes equal angles with the chord.

In a flight of 2800 km, an aircraft was slowed down due to bad weather. Its average speed is reduced by 100 km/h and by doing so, the time of flight is increased by 30 minutes. Find the original duration of the flight.

The denominator of a fraction is one more than twice the numerator. If the sum of the fraction and its reciprocal is $2\frac{16}{21}$, find the fraction.

State and prove Basic Proportionality theorem.

From a point on the ground, the angles of elevation of the bottom and the top of a transmission tower fixed at the top of a 20 m high building are $45^{\circ }$ and $60^{\circ }$ respectively. Find the height of the tower.

A solid iron pole consists of a solid cylinder of height 200 cm and base diameter 28 cm, which is surmounted by another cylinder of height 50 cm and radius 7 cm. Find the mass of the pole, given that 1 cm${}^3$ of iron has approximately 8 g mass.

A medicine capsule is in the shape of a cylinder with two hemispheres stuck to each of its ends. The length of the entire capsule is 14 mm and the diameter of the capsule is 4 mm, find its surface area. Also, find its volume.

A stable owner has four horses. He usually tie these horses with 7 m long rope to pegs at each corner of a square shaped grass field of 20 m length, to graze in his farm. But tying with rope sometimes results in injuries to his horses, so he decided to build fence around the area so that each horse can graze.
Based on the above, answer the following questions :
(i) Find the area of the square shaped grass field. (1)
(ii) Find the area of the total field in which these horses can graze. (2)
OR If the length of the rope of each horse is increased from 7 m to 10 m, find the area grazed by one horse. (Use $\pi$ = 3.14) (2)
(iii) What is area of the field that is left ungrazed, if the length of the rope of each horse is 7 cm ? (1)

Vocational training complements traditional education by providing practical skills and hands-on experience. While education equips individuals with a broad knowledge base, vocational training focuses on job-specific skills, enhancing employability thus making the student self-reliant. Keeping this in view, a teacher made the following table giving the frequency distribution of students/adults undergoing vocational training from the training institute.
Age (in years): 15-19, 20-24, 25-29, 30-34, 35-39, 40-44, 45-49, 50-54
Number of participants: 62, 132, 96, 37, 13, 11, 10, 4
From the above answer the following questions :
(i) What is the lower limit of the modal class of the above data ? (1)
(ii) Find the median class of the above data. (2)
OR Find the number of participants of age less than 50 years who undergo vocational training. (2)
(iii) Give the empirical relationship between mean, median and mode. (1)

Teaching Mathematics through activities is a powerful approach that enhances students’ understanding and engagement. Keeping this in mind, Ms. Mukta planned a prime number game for class 5 students. She announces the number 2 in her class and asked the first student to multiply it by a prime number and then pass it to second student. Second student also multiplied it by a prime number and passed it to third student. In this way by multiplying to a prime number, the last student got 173250.
Now, Mukta asked some questions as given below to the students :
(i) What is the least prime number used by students ? (1)
(ii) How many students are in the class ? (2)
OR What is the highest prime number used by students ? (2)
(iii) Which prime number has been used maximum times ? (1)