CBSE Class 10 Maths Standard 2024 Question Paper 30/4/2 with Solutions

If $\sin \theta =1$, then the value of $\frac{1}{2}\sin \left(\frac{\theta }{2}\right)$ is :

1. (A)$\frac{1}{2\sqrt{2}}$
2. (B)$\frac{1}{\sqrt{2}}$
3. (C)$\frac{1}{2}$
4. (D)0

The ratio of total surface area of a solid hemisphere to the square of its radius is :

1. (A)$2\pi :1$
2. (B)$4\pi :1$
3. (C)$3\pi :1$
4. (D)$1:4\pi$

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

If in the given figure, $\angle C=\angle D=90^{\circ }$, $\angle B=60^{\circ }$ and AP = 3 cm, then the area of the shaded region is :

1. (A)$3\pi$ cm${}^2$
2. (B)$6\pi$ cm${}^2$
3. (C)$7\pi$ cm${}^2$
4. (D)$9\pi$ cm${}^2$

If the mean of 6, 7, p, 8, q, 14 is 9, then

1. (A)$p-q=19$
2. (B)$p+q=19$
3. (C)$p-q=21$
4. (D)$p+q=21$

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)$90^{\circ }$

If an arc subtends an angle of $90^{\circ }$ at the centre of a circle, then the ratio of its length to the circumference of the circle is :

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

If $ax+by=a^2-b^2$ and $bx+ay=0$, then the value of $x+y$ is :

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

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)$86^{\circ }$

If the discriminant of the quadratic equation $3x^2-2x+c=0$ is 16, then the value of $c$ is :

1. (A)1
2. (B)0
3. (C)$-1$
4. (D)$\sqrt{2}$

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)

Two dice are tossed simultaneously. The probability of getting odd numbers on both the dice is :

1. (A)$\frac{6}{36}$
2. (B)$\frac{3}{36}$
3. (C)$\frac{12}{36}$
4. (D)$\frac{9}{36}$

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)$-15x+9y=5$

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

From the letters of the word “MOBILE”, a letter is selected at random. The probability that the selected letter is a vowel, is :

1. (A)$\frac{3}{7}$
2. (B)$\frac{1}{6}$
3. (C)$\frac{1}{2}$
4. (D)$\frac{1}{3}$

If $3825=3^x\times 5^y\times 17^z$, then the value of $x+y-2z$ is :

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

A quadratic polynomial, one of whose zeroes is $2+\sqrt{5}$ and the sum of whose zeroes is 4, is :

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

If the first three terms of an A.P. are $3p-1$, $3p+5$, $5p+1$ respectively; then the value of p is :

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

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.

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.

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.

In a school, there are two sections of class X. There are 40 students in the first section and 48 students in the second section. Determine the minimum number of books required for their class library so that they can be distributed equally among students of both sections.

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.

In the given figure, O is the centre of the circle. If $\angle AOB=145^{\circ }$, then find the value of x.

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.

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

If $\sin \theta +\cos \theta =p$ and $\sec \theta +\text{cosec}\theta =q$, then prove that $q(p^2-1)=2p$.

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 $\frac{\alpha }{\beta }+\frac{\beta }{\alpha }$.

A man starts his job with a certain monthly salary and earns a fixed increment every year. If his salary was ₹ 15,000 after 4 years of service and ₹ 18,000 after 10 years of service, what was his starting salary and what was the annual increment ?

Find the value of ‘k’ for which the quadratic equation $(k+1)x^2-2(3k+1)x+(8k+1)=0$ has real and equal roots.

A 2-digit number is such that the product of its digits is 18. When 63 is subtracted from the number, the digits interchange their places. Find the number.

The following table shows the ages of the patients admitted in a hospital during a year :
Age (in years): 5–15, 15–25, 25–35, 35–45, 45–55, 55–65
Number of patients: 6, 11, 21, 23, 14, 5
Find the mode and mean of the data given above.

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 50 m high building from the top of a tower are $45^{\circ }$ and $60^{\circ }$, respectively. Find the height of the tower. (Use $\sqrt{3}$ = 1.73)

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)

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)