CBSE Class 10 Maths Basic 2023 Question Paper 430/6/2 with Solutions

The HCF of the smallest 2-digit number and the smallest composite number is

1. (A)4
2. (B)20
3. (C)2
4. (D)10

The value of ‘p’ if (–2, p) lies on the line represented by the equation $2x-3y+7=0$, is

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

Distance of the point (6, 5) from the y-axis is

1. (A)6 units
2. (B)5 units
3. (C)$\sqrt{61}$ units
4. (D)0 unit

If the mean of 6, 7, $x$, 8, y, 14 is 9, then

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

If (2, 4) is the mid-point of the line-segment joining (6, 3) and (a, 5), then the value of a is

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

If n is a natural number, then $8^n$ cannot end with digit

1. (A)0
2. (B)2
3. (C)4
4. (D)6

The time, in seconds, taken by 150 athletes to run a 100 m hurdle race are tabulated below :
Time (sec.): 13-14, 14-15, 15-16, 16-17, 17-18, 18-19
Number of Athletes: 2, 4, 5, 71, 48, 20
The number of athletes who completed the race in less than 17 seconds is

1. (A)11
2. (B)71
3. (C)82
4. (D)68

An unbiased die is thrown. The probability of getting an odd prime number is

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

In $△ABC$, right angled at C, if $\tan A=\frac{8}{7}$, then the value of $\cot B$ is

1. (A)$\frac{7}{8}$
2. (B)$\frac{8}{7}$
3. (C)$\frac{7}{\sqrt{113}}$
4. (D)$\frac{8}{\sqrt{113}}$

In the given figure, PQ and PR are tangents drawn from P to the circle with centre O such that $\angle QPR=65^{\circ }$. The measure of $\angle QOR$ is.

1. (A)$65^{\circ }$
2. (B)$125^{\circ }$
3. (C)$115^{\circ }$
4. (D)$90^{\circ }$

The prime factorisation of the number 2304 is

1. (A)$2^8\times 3^2$
2. (B)$2^7\times 3^3$
3. (C)$2^8\times 3^1$
4. (D)$2^7\times 3^2$

Area of a quadrant of a circle of radius 7 cm is

1. (A)154 cm${}^2$
2. (B)77 cm${}^2$
3. (C)$\frac{77}{2}$ cm${}^2$
4. (D)$\frac{77}{4}$ cm${}^2$

The $20^{\text{th}}$ term of an A.P, whose first term is $-2$ and the common difference is 4, is

1. (A)78
2. (B)74
3. (C)$-36$
4. (D)$-34$

The zeroes of the polynomial $p(x)=25x^2-49$ are :

1. (A)$\frac{49}{25},\frac{49}{25}$
2. (B)$-\frac{49}{25},+\frac{49}{25}$
3. (C)$\frac{7}{5},-\frac{7}{5}$
4. (D)$\frac{7}{5},\frac{7}{5}$

The mean of first ten natural numbers is

1. (A)5.5
2. (B)55
3. (C)45
4. (D)4.5

One card is drawn at random from a well-shuffled deck of 52 playing cards. What is the probability of getting a black king ?

1. (A)$\frac{1}{26}$
2. (B)$\frac{1}{13}$
3. (C)$\frac{1}{52}$
4. (D)$\frac{1}{2}$

The value of ‘k’ for which the system of equations $kx+2y=5$ and $3x+4y=1$ have no solution, is

1. (A)$k=\frac{3}{2}$
2. (B)$k\ne \frac{3}{2}$
3. (C)$k\ne \frac{2}{3}$
4. (D)$k=15$

The graph of $y=f(x)$ is shown in the figure for some polynomial $f(x)$.
The number of zeroes of $f(x)$ is

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

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

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.

Assertion (A) : The probability that a leap year has 53 Sundays is $\frac{2}{7}$.
Reason (R) : The probability that a non-leap year has 53 Sundays is $\frac{1}{7}$.

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.

A box contains 20 discs which are numbered from 1 to 20. If one disc is drawn at random from the box, then find the probability that the number on the drawn disc is a
(i) 2-digit number
(ii) number less than 10

From a point P, the length of the tangent to a circle is 24 cm and the distance of P from the centre of the circle is 25 cm. Find the radius of the circle.

Find the value of k for which the roots of the quadratic equation $5x^2-10x+k=0$ are real and equal.

If one root of the quadratic equation $3x^2-8x-(2k+1)=0$ is seven times the other, then find the value of k.

Find a quadratic polynomial whose zeroes are 6 and $-3$.

Find the zeroes of the polynomial $x^2+4x-12$.

Evaluate : $\frac{5{\mathrm{cosec}}^{2}30^{\circ }-\cos 90^{\circ }}{4{\tan }^{2}60^{\circ }}$

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

If A and B are (–2, –2) and (2, –4) respectively; then find the co-ordinates of the point P such that $\frac{AP}{AB}=\frac{3}{7}$.

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

Prove that $(\sec \theta +\tan \theta )(1-\sin \theta )=\cos \theta$

E is a point on the side AD produced of a parallelogram ABCD and BE intersects CD at F. Show that $△ABE\sim △CFB$.

In the given figure, CM and RN are respectively the medians of $△ABC$ and $△PQR$. If $△ABC\sim △PQR$, then prove that $△AMC\sim △PNR$.

The sum of the reciprocals of Varun’s age (in years) 3 years ago and 5 years from now is $\frac{1}{3}$. Find his present age.

A survey conducted on 20 households in a locality by a group of students resulted in the following frequency table for the number of family members in a household :
Family size: 1-3, 3-5, 5-7, 7-9, 9-11
Number of Families: 7, 8, 2, 2, 1
Find the median of this data.

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.

A solid is in the shape of a cone standing on a hemisphere with both their diameters being equal to 1 cm and the height of the cone is equal to its radius. Find the volume of the solid. [Use $\pi =3.14$]

Prove that a line drawn parallel to one side of a triangle to intersect the other two sides in distinct points, divides the two sides in the same ratio.

In the given figure, $\frac{QR}{QS}=\frac{QT}{PR}$ and $\angle 1=\angle 2$. Prove that $△PQS\sim △TQR$.

From a point on a bridge across a river, the angles of depression of the banks on opposite sides of the river are $30^{\circ }$ and $45^{\circ }$ respectively. If the bridge is at a height of 3 m from the banks, find the width of the river. (Use $\sqrt{3}=1.73$)

From a point on the ground, the angle of elevation of the bottom and 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. (Use $\sqrt{3}=1.73$)

Lokesh, a production manager in Mumbai, hires a taxi everyday to go to his office. The taxi charges in Mumbai consists of a fixed charges together with the charges for the distance covered. His office is at a distance of 10 km from his home. For a distance of 10 km to his office, Lokesh paid ₹ 105. While coming back home, he took another route. He covered a distance of 15 km and the charges paid by him were ₹ 155.
Based on the above information, answer the following questions :
(i) What are the fixed charges ? (1)
(ii) What are the charges per km ? (1)
(iii) If fixed charges are ₹ 20 and charges per km are ₹ 10, then how much Lokesh have to pay for travelling a distance of 10 km ? (2)
OR (iii) Find the total amount paid by Lokesh for travelling 10 km from home to office and 25 km from office to home. [Fixed charges and charges per km are as in (i) & (ii). (2)

People of a circular village Dharamkot want to construct a road nearest to it. The road cannot pass through the village. But the people want the road at a shortest distance from the centre of the village. Suppose the road starts from A which is outside the circular village (as shown in the figure) and touch the boundary of the circular village at B such that AB = 20 m. Also the distance of the point A from the centre O of the village is 25 m.
Based on the above information, answer the following questions :
(i) If B is the mid-point of AC, then find the distance AC. (1)
(ii) Find the shortest distance of the road from the centre of the village. (1)
(iii) Find the circumference of the village. (2)
OR (iii) Find the area of the village. (2)

For the inauguration of ‘Earth day’ week in a school, badges were given to volunteers. Organisers purchased these badges from an NGO, who made these badges in the form of a circle inscribed in a square of side 8 cm.
O is the centre of the circle and $\angle AOB=90^{\circ }$ :
Based on the above information, answer the following questions :
(i) What is the area of square ABCD ? (1)
(ii) What is the length of diagonal AC of square ABCD ? (1)
(iii) Find the area of sector OPRQO. (2)
OR (iii) Find the area of remaining part of square ABCD when area of circle is excluded. (2)