CBSE Class 10 Maths Basic 2024 Question Paper 430/3/2 with Solutions

A die is thrown once. The probability of getting a number less than 6, is :

1. (A)0
2. (B)$\frac{5}{6}$
3. (C)$\frac{1}{6}$
4. (D)1

In the given figure, graph of a polynomial $f(x)$ is shown. The number of zeroes of polynomial $f(x)$ is

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

In the given figure, two concentric circles of radii 5 cm and 3 cm have their centre O. OAB is a sector of outer circle making an angle of $60^{\circ }$ at the centre while OCD is the sector of smaller circle. The area of the shaded region is :

1. (A)$\frac{7\pi }{2}$ cm${}^2$
2. (B)$\frac{8\pi }{3}$ cm${}^2$
3. (C)$\frac{25\pi }{6}$ cm${}^2$
4. (D)$\frac{3\pi }{2}$ cm${}^2$

If $\sin A=\frac{1}{2}$, then $\cot A$ is equal to

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

From a solid cube of side 14 cm, a sphere of maximum diameter is carved out. The radius of sphere is :

1. (A)7 cm
2. (B)14 cm
3. (C)$\frac{7}{2}$ cm
4. (D)$\sqrt{14}$ cm

The $30^{th}$ term of the A.P. $-3,-7,-11,\dots$ is :

1. (A)113
2. (B)$-117$
3. (C)$-119$
4. (D)$-120$

In the given figure, tangents PA and PB drawn from P to circle are inclined to each other at an angle of $80^{\circ }$. The measure of $\angle PAB$ is

1. (A)$80^{\circ }$
2. (B)$60^{\circ }$
3. (C)$50^{\circ }$
4. (D)$40^{\circ }$

A quadratic polynomial whose zeroes are $-8$ and 3, is

1. (A)$(x+8)(x+3)$
2. (B)$x^2+5x+24$
3. (C)$(x-8)(x-3)$
4. (D)$x^2+5x-24$

The lines represented by linear equations $x=a$ and $y=b$ $(a\ne b)$ are

1. (A)intersecting at (a, b).
2. (B)intersecting at (b, a).
3. (C)parallel.
4. (D)coincident.

If in the given figure, $DE\parallel BC$. If $AD=2.8$ cm, $DB=2.1$ cm and $EC=4.8$ cm, then the value of $x$ is :

1. (A)3.6 cm
2. (B)2.4 cm
3. (C)6.4 cm
4. (D)4.8 cm

If the mean and median of a data are 10 and 11 respectively, then mode of the data is :

1. (A)12
2. (B)8
3. (C)20
4. (D)13

Two fair coins are tossed together. The probability of getting 2 heads, is :

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

The value(s) of k for which the quadratic equation $5x^2-9kx+5=0$ has real and equal roots, is/are :

1. (A)$\frac{-10}{9}$
2. (B)$\pm \frac{9}{10}$
3. (C)$\frac{10}{9}$
4. (D)$\pm \frac{10}{9}$

The distance between the points $A(5,-4)$ and $B(4,-5)$ is

1. (A)$\sqrt{2}$ units
2. (B)2 units
3. (C)1 unit
4. (D)$9\sqrt{2}$ units

In a right-angled triangle ABC, $\angle A=90^{\circ }$ and $AB=AC$. The value of $\sin C$ is :

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

If for a distribution, ${\sum }_1^n{f}_i{x}_i=132+5p$, ${\sum }_1^n{f}_i=20$ and the mean of the distribution is 8.1, then the value of p is :

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

From an external point P, a tangent PA is drawn to a circle. The number of tangents through P parallel to PA is :

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

If the volume of a sphere is $\frac{11}{21}$ cm${}^3$, then the radius of the sphere is :

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

Assertion (A) : $\sqrt{2}(5-\sqrt{2})$ is an irrational number.
Reason (R) : Product of two irrational numbers is always irrational.

1. (A)Both Assertion (A) and Reason (R) are true. Reason (R) is the correct explanation of Assertion (A).
2. (B)Both Assertion (A) and Reason (R) are true. 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 distance of $P(a,b)$ from origin is $a^2+b^2$.
Reason (R) : The distance between two points $A(x_1,y_1)$ and $B(x_2,y_2)$ is $\sqrt{(x_2-x_1{)}^2+(y_2-y_1{)}^2}$.

1. (A)Both Assertion (A) and Reason (R) are true. Reason (R) is the correct explanation of Assertion (A).
2. (B)Both Assertion (A) and Reason (R) are true. 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.

Prove that $-7-2\sqrt{3}$ is an irrational number, given that $\sqrt{3}$ is an irrational number.

Explain why $(7\times 11\times 13+2\times 11)$ is not a prime number.

There are 80 cards numbered from 1 to 80. One card is drawn at random from them. Find the probability that the number on the selected card is not divisible by 8.

Find a point which is equidistant from the points $A(-1,5)$ and $B(2,1)$. How many such points are there ?

If $Q(0,2)$ is equidistant from $P(5,-3)$ and $R(x,7)$, find the value(s) of $x$.

If $A(1,1)$ and $B(7,9)$ are the end points of a diameter of a circle, then find the co-ordinates of the centre of the circle.

Evaluate : $3{\sec }^{2}30^{\circ }\text{cosec}30^{\circ }+{\tan }^{2}60^{\circ }\cdot {\tan }^{2}45^{\circ }$

Find the zeroes of the polynomial $2x^2+3x-9$ and verify the relationship between the zeroes and the co-efficients of polynomial.

In the given figure, $\angle ABC=\angle ACB$ and $\frac{BC}{BE}=\frac{BD}{AC}$.
Show that $△ABE\sim △DBC$ and $AE\parallel DC$.

The altitude of a right-angled triangle is 7 cm less than its base. If its hypotenuse is 17 cm long, then
(a) represent the above information in the form of a quadratic equation;
(b) find the length of the sides of the triangle.

Find the HCF and LCM of 96 and 404, using prime-factorisation method.

Prove that $(\sin \theta +\text{cosec}\theta {)}^2+(\cos \theta +\sec \theta {)}^2=7+{\tan }^2\theta +{\cot }^2\theta$.

If $\cos A=\frac{5}{13}$, then verify that $\frac{\cos A}{1-\tan A}+\frac{\sin A}{1-\cot A}=\cos A+\sin A$.

A vessel is in the form of a hollow hemisphere surmounted by a hollow cylinder. The diameter of the hemisphere is 14 cm and the total height of the vessel is 13 cm. Find the inner surface area of the vessel.

A solid toy is in the form of a hemisphere surmounted by a right circular cone. The height of the cone is 2 cm and the diameter of the base is 4 cm. Determine the volume of the toy.

Using graphical method, solve the following pair of equations :
$x+2y=8$ and $3x-2y=12$

The sum of the digits of a 2-digit number is 9. Also, nine times this number is twice the number obtained by reversing the order of the digits. Find the number.

A man standing on the deck of a ship, which is 10 m above water level, observes the angle of elevation of the top of a hill as $60^{\circ }$ and the angle of depression of the base of the hill as $30^{\circ }$. Find the distance of the hill from the ship and the height of the hill. (Take $\sqrt{3}=1.73$)

In the given figure, AB is chord of length 6 cm of a circle of radius 5 cm. The tangents at A and B intersect at a point P. Find the length of PB.

Prove that the parallelogram circumscribing a circle is a rhombus. Also, find area of the rhombus, if radius of circle is 3 cm and length of one side of the rhombus is 10 cm.

Find the mean and median of the following distribution :
Class: 0 – 20, 20 – 40, 40 – 60, 60 – 80, 80 – 100, 100 – 120
Frequency: 5, 8, 10, 12, 7, 8

In the figure given below, a folding table is shown :
The legs of the table are represented by line segments AB and CD intersecting at O. Join AC and BD.
Considering table top is parallel to the ground, and $OB=x$, $OD=x+3$, $OC=3x+19$ and $OA=3x+4$, answer the following questions :
(i) Prove that $△OAC$ is similar to $△OBD$. (1)
(ii) Prove that $\frac{OA}{AC}=\frac{OB}{BD}$. (1)
(iii) Observe the figure and find the value of $x$. Hence, find the length of OC. (2)
OR Observe the figure and find $\frac{BD}{AC}$. (2)

While preparing for a competitive examination, Akbar came across a match-stick pattern based question. The pattern is given below :
Based on the above information, answer the following questions :
(i) Write first term and common difference of the A.P. formed by number of squares in each figure. (1)
(ii) Write first term and common difference of the A.P. formed by number of sticks used in each figure. (1)
(iii) How many squares are there in Fig. (10) ? Also, write the number of sticks used in Fig. (10). (2)
OR If 88 sticks are used to make $m^{th}$ figure (Fig. (m)), find the value of m. How many squares are formed in this figure ? (2)

NSS (National Service Scheme) aims to connect the students to the community and to involve them in problem solving process.
NSS symbol is based on the ‘Rath’ wheel of the Konark Sun Temple situated in Odisha. The wheel signifies the progress cycle of life.
The diagramatic representation of the symbol is given below :
Observe the figure given above. The diameters of inner circle are equally placed. Given that OP = 21 cm, OS = 10 cm.
Based on the above information, answer the following questions :
(i) Find $m\angle ROS$. (1)
(ii) Find the perimeter of sector OPQ. (1)
(iii) Find the area of shaded region PQRS. (2)
OR Find the area of shaded region ACB i.e. the segment ACB. (2)