CBSE Class 10 Maths Standard 2023 Question Paper 30/6/3 with Solutions

The distance between the points (0, 5) and (–3, 1) is :

1. (A)8 units
2. (B)5 units
3. (C)3 units
4. (D)25 units

If $\tan \theta =\frac{x}{y}$, then $\cos \theta$ is equal to

1. (A)$\frac{x}{\sqrt{x^2+y^2}}$
2. (B)$\frac{y}{\sqrt{x^2+y^2}}$
3. (C)$\frac{x}{\sqrt{x^2-y^2}}$
4. (D)$\frac{y}{\sqrt{x^2-y^2}}$

The zeroes of the polynomial $3x^2+11x-4$ are :

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

The circumferences of two circles are in the ratio 4 : 5. What is the ratio of their radii ?

1. (A)16 : 25
2. (B)25 : 16
3. (C)$2:\sqrt{5}$
4. (D)4 : 5

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

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

If the zeroes of the quadratic polynomial $x^2+(a+1)x+b$ are 2 and –3, then

1. (A)a = –7, b = –1
2. (B)a = 5, b = –1
3. (C)a = 2, b = – 6
4. (D)a = 0, b = – 6

If $p^2=\frac{32}{50}$, then p is a/an

1. (A)whole number
2. (B)integer
3. (C)rational number
4. (D)irrational number

If $\theta$ is an acute angle of a right angled triangle, then which of the following equation is not true ?

1. (A)$\sin \theta \cot \theta =\cos \theta$
2. (B)$\cos \theta \tan \theta =\sin \theta$
3. (C)${\text{cosec}}^2\theta -{\cot }^2\theta =1$
4. (D)${\tan }^2\theta -{\sec }^2\theta =1$

The point of intersection of the line represented by $3x-y=3$ and the y-axis is given by

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

In the given figure, PA and PB are tangents from external point P to a circle with centre C and Q is any point on the circle. Then the measure of $\angle AQB$ is

1. (A)$62{\frac{1}{2}}^{\circ }$
2. (B)$125^{\circ }$
3. (C)$55^{\circ }$
4. (D)$90^{\circ }$

If $\alpha$ and $\beta$ are the zeroes of the polynomial $x^2-1$, then the value of $(\alpha +\beta )$ is

1. (A)2
2. (B)1
3. (C)–1
4. (D)0

If $△PQR\sim △ABC$; PQ = 6 cm, AB = 8 cm and the perimeter of $△ABC$ is 36 cm, then the perimeter of $△PQR$ is

1. (A)20.25 cm
2. (B)27 cm
3. (C)48 cm
4. (D)64 cm

In the given figure, DE$\parallel$BC. If AD = 3 cm, AB = 7 cm and EC = 3 cm, then the length of AE is

1. (A)2 cm
2. (B)2.25 cm
3. (C)3.5 cm
4. (D)4 cm

The volume of a right circular cone whose area of the base is 156 cm${}^2$ and the vertical height is 8 cm, is

1. (A)2496 cm${}^3$
2. (B)1248 cm${}^3$
3. (C)1664 cm${}^3$
4. (D)416 cm${}^3$

A card is drawn at random from a well shuffled deck of 52 playing cards. The probability of getting a face card is

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

If 'p' is a root of the quadratic equation $x^2-(p+q)x+k=0$, then the value of 'k' is

1. (A)p
2. (B)q
3. (C)p + q
4. (D)pq

Cards bearing numbers 3 to 20 are placed in a bag and mixed thoroughly. A card is taken out of the bag at random. What is the probability that the number on the card taken out is an even number ?

1. (A)$\frac{9}{17}$
2. (B)$\frac{1}{2}$
3. (C)$\frac{5}{9}$
4. (D)$\frac{7}{18}$

The condition for the system of linear equations $ax+by=c$; $lx+my=n$ to have a unique solution is

1. (A)$am\ne bl$
2. (B)$al\ne bm$
3. (C)$al=bm$
4. (D)$am=bl$

Statement A (Assertion) : For $0<\theta \le 90^{\circ }$, $\text{cosec}\theta -\cot \theta$ and $\text{cosec}\theta +\cot \theta$ are reciprocal of each other.
Statement R (Reason) : ${\text{cosec}}^2\theta -{\cot }^2\theta =1$

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.

Statement A (Assertion) : If $5+\sqrt{7}$ is a root of a quadratic equation with rational co-efficients, then its other root is $5-\sqrt{7}$.
Statement R (Reason) : Surd roots of a quadratic equation with rational co-efficients occur in conjugate pairs.

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.

Find the ratio in which the y-axis divides the line segment joining the points (5, – 6) and (–1, – 4).

In the given figure, ABC is a triangle in which DE$\parallel$BC. If AD = $x$, DB = $x-2$, AE = $x+2$ and EC = $x-1$, then find the value of $x$.

Diagonals AC and BD of trapezium ABCD with AB$\parallel$DC intersect each other at point O. Show that $\frac{OA}{OC}=\frac{OB}{OD}$.

Show that $6^n$ can not end with digit 0 for any natural number 'n'.

Find the LCM and HCF of 72 and 120

Find the length of the shadow on the ground of a pole of height 18 m when angle of elevation $\theta$ of the sun is such that $\tan \theta =\frac{6}{7}$.

In the given figure, PA is a tangent to the circle drawn from the external point P and PBC is the secant to the circle with BC as diameter. If $\angle AOC=130^{\circ }$, then find the measure of $\angle APB$, where O is the centre of the circle.

Prove that $(\sin \theta +\cos \theta )(\tan \theta +\cot \theta )=\sec \theta +\text{cosec}\theta$.

A natural number, when increased by 12, equals 160 times its reciprocal. Find the number.

If one root of the quadratic equation $x^2+12x-k=0$ is thrice the other root, then find the value of k.

Find the ratio in which the line segment joining the points A(6, 3) and B(–2, –5) is divided by $x$-axis.

In a circle of radius 21 cm, an arc subtends an angle of $60^{\circ }$ at the centre. Find the area of the sector formed by the arc. Also, find the length of the arc.

In the given figure, E is a point on the side CB produced of an isosceles triangle ABC with AB = AC. If AD $\perp$ BC and EF $\perp$ AC, then prove that $△ABD\sim △ECF$.

Find the HCF and LCM of 26, 65 and 117, using prime factorisation.

Prove that $\sqrt{2}$ is an irrational number.

The sum of first seven terms of an A.P. is 182. If its 4${}^{\text{th}}$ term and the 17${}^{\text{th}}$ term are in the ratio 1 : 5, find the A.P.

The sum of first q terms of an A.P. is $63q-3q^2$. If its p${}^{\text{th}}$ term is –60, find the value of p. Also, find the 11${}^{\text{th}}$ term of this A.P.

Prove that a parallelogram circumscribing a circle is a rhombus.

In the given figure, tangents PQ and PR are drawn to a circle such that $\angle RPQ=30^{\circ }$. A chord RS is drawn parallel to the tangent PQ. Find the measure of $\angle RQS$.

250 apples of a box were weighed and the distribution of masses of the apples is given in the following table :
Mass (in grams): 80 – 100, 100 – 120, 120 – 140, 140 – 160, 160 – 180
Number of apples: 20, 60, 70, $x$, 60
(i) Find the value of $x$ and the mean mass of the apples. (3)
(ii) Find the modal mass of the apples (2)

A solid is in the shape of a right-circular cone surmounted on a hemisphere, the radius of each of them being 7 cm and the height of the cone is equal to its diameter. Find the volume of the solid.

Radio towers are used for transmitting a range of communication services including radio and television. The tower will either act as an antenna itself or support one or more antennas on its structure. On a similar concept, a radio station tower was built in two Sections A and B. Tower is supported by wires from a point O.
Distance between the base of the tower and point O is 36 cm. From point O, the angle of elevation of the top of the Section B is $30^{\circ }$ and the angle of elevation of the top of Section A is $45^{\circ }$.
Based on the above information, answer the following questions :
(i) Find the length of the wire from the point O to the top of Section B. (1)
(ii) Find the distance AB. (2)
OR Find the area of $△OPB$.
(iii) Find the height of the Section A from the base of the tower. (1)

“Eight Ball” is a game played on a pool table with 15 balls numbered 1 to 15 and a “cue ball” that is solid and white. Of the 15 numbered balls, eight are solid (non-white) coloured and numbered 1 to 8 and seven are striped balls numbered 9 to 15.
The 15 numbered pool balls (no cue ball) are placed in a large bowl and mixed, then one ball is drawn out at random.
Based on the above information, answer the following questions :
(i) What is the probability that the drawn ball bears number 8 ?
(ii) What is the probability that the drawn ball bears an even number ?
OR What is the probability that the drawn ball bears a number, which is a multiple of 3 ?
(iii) What is the probability that the drawn ball is a solid coloured and bears an even number ?

A coaching institute of Mathematics conducts classes in two batches I and II and fees for rich and poor children are different. In batch I, there are 20 poor and 5 rich children, whereas in batch II, there are 5 poor and 25 rich children. The total monthly collection of fees from batch I is ₹ 9000 and from batch II is ₹ 26,000. Assume that each poor child pays ₹ $x$ per month and each rich child pays ₹ y per month.
Based on the above information, answer the following questions :
(i) Represent the information given above in terms of $x$ and y. (1)
(ii) Find the monthly fee paid by a poor child. (2)
OR Find the difference in the monthly fee paid by a poor child and a rich child.
(iii) If there are 10 poor and 20 rich children in batch II, what is the total monthly collection of fees from batch II ? (1)