CBSE Class 10 Maths Basic 2023 Question Paper 430/6/3 with Solutions

The prime factorisation of the number 5488 is

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

The Empirical relation between the three measures of central tendency is

1. (A)Mode = 3 Mean – 2 Median
2. (B)Mode = 2 Median – 3 Mean
3. (C)Mode = 2 Mean – 3 Median
4. (D)Mode = 3 Median – 2 Mean

In the given figure, $△PQR$ is a right triangle right angled at Q. If PQ = 4 cm and PR = 8 cm, then $\angle P$ is

1. (A)$60^{\circ }$
2. (B)$45^{\circ }$
3. (C)$30^{\circ }$
4. (D)$15^{\circ }$

The median of first 10 natural numbers is

1. (A)5
2. (B)6
3. (C)5.5
4. (D)6.5

The zeroes of the polynomial $p(x)=2x^2-x-3$ are

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

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

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

The distance of the point (5, 0) from the origin is

1. (A)0
2. (B)5
3. (C)$\sqrt{5}$
4. (D)$5^2$

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

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

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${}^2$

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)$90^{\circ }$

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, if (6, k) lies on the line represented by $x-3y+6=0$, is

1. (A)$-4$
2. (B)12
3. (C)$-12$
4. (D)4

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)$-2$

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)$\frac{1}{3}$

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)$k=15$

If $-5,x,3$ are three consecutive terms of an A.P., then the value of $x$ is

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

If HCF (72, 120) = 24, then LCM (72, 120) is

1. (A)72
2. (B)120
3. (C)360
4. (D)9640

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 bag contains 30 discs numbered from 1 to 30. One disc is drawn at random from the bag. Find the probability that it bears a number
(a) divisible by 6.
(b) greater than 25.

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.

Evaluate : $5{\mathrm{cosec}}^{2}45^{\circ }-3{\sin }^{2}90^{\circ }+5\cos 0^{\circ }$.

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

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

Prove that $7+4\sqrt{5}$ is an irrational number, given that $\sqrt{5}$ is an irrational number.

Solve for $x$ :
$\frac{1}{x}-\frac{1}{x-2}=3;x\ne 0,2$

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 $△AMC\sim △PNR$.

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.

Find the co-ordinates of the points of trisection of the line-segment joining the points (5, 3) and (4, 5).

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 $\sqrt{3}=1.73$)

The sum of the $4^{\text{th}}$ and $8^{\text{th}}$ term of an A.P. is 24 and the sum of the $6^{\text{th}}$ and $10^{\text{th}}$ term of the A.P. is 44. Find the A.P. Also, find the sum of first 25 terms of the A.P.

A wooden article was made by scooping out a hemisphere from each end of a solid cylinder (as shown in the figure).
If the height of the cylinder is 10 cm and its base is of radius 3.5 cm, find the total surface area of the article.

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)

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)