CBSE Class 10 Maths Standard 2024 Question Paper 30/3/2 with Solutions

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)$125^{\circ }$

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

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

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)$90^{\circ }$

If a pair of linear equations in two variables is consistent, then the lines represented by the two equations are :

1. (A)always intersecting
2. (B)parallel
3. (C)always coincident
4. (D)intersecting or coincident

If the area of a sector of a circle is $\frac{7}{20}$ of the area of the circle, then the angle at the centre is equal to

1. (A)$110^{\circ }$
2. (B)$130^{\circ }$
3. (C)$100^{\circ }$
4. (D)$126^{\circ }$

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)$\frac{5}{9}$

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

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

Two coins are tossed simultaneously. The probability of getting at most one tail is :

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

If the mean of five observations $x$, $x+2$, $x+4$, $x+6$ and $x+8$ is 11, then the value of $x$ is :

1. (A)4
2. (B)7
3. (C)11
4. (D)6

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)$3,\frac{3}{2}$

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 $△ABC$, 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

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

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

If the sum of first n terms of an A.P. is $3n^2+4n$ and its common difference is 6, then its first term is :

1. (A)7
2. (B)4
3. (C)6
4. (D)3

What should be subtracted from the polynomial $x^2-16x+30$, so that 15 is the zero of the resulting polynomial ?

1. (A)30
2. (B)14
3. (C)15
4. (D)16

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.

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.

Evaluate : $\frac{{\sec }^{2}45^{\circ }-{\tan }^{2}45^{\circ }}{{\sin }^{2}45^{\circ }}$

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.

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.

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

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

Prove that $\frac{\sin A+\cos A}{\sin A-\cos A}+\frac{\sin A-\cos A}{\sin A+\cos A}=\frac{2}{2{\sin }^{2}A-1}$

In what ratio does the X-axis divides the line segment joining the points(2, $-3$) and (5, 6) ? Also, find the coordinates of the point of intersection.

Find the length of the median AD of $△ABC$ having vertices A(0, $-1$), B(2, 1) and C(0, 3).

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

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

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.

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.

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.

Through the mid-point M of the side CD of a parallelogram ABCD, the line BM is drawn intersecting AC in L and AD produced in E. Prove that EL = 2BL.

The angle of elevation of a jet plane from a point A on the ground is $60^{\circ }$. After a flight of 30 seconds, the angle of elevation changes to $30^{\circ }$. If the jet plane is flying at a constant height of $3600\sqrt{3}$ m, find the speed of the jet plane.

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)

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)