CBSE Class 10 Maths Basic 2025 Question Paper 430/5/1 with Solutions

$(2-5\sqrt{3}{)}^2$ is

1. (A)a negative integer
2. (B)an irrational number
3. (C)a rational number
4. (D)a positive integer

The value of k for which the roots of the quadratic equation $6x^2+4kx+k=0$ are real and equal, is

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

The distance between the points (2, 3) and (–2, –3) is

1. (A)$4\sqrt{13}$
2. (B)$\sqrt{40}$
3. (C)$2\sqrt{13}$
4. (D)5

Observe the given graph of polynomial p($x$). The number of zeroes of p($x$) is

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

In the given figure, AB is diameter of the circle with centre O. CD is tangent to the circle so that $\angle ACD=40^{\circ }$. The value of $\angle CBA$ is

1. (A)$50^{\circ }$
2. (B)$40^{\circ }$
3. (C)$80^{\circ }$
4. (D)$45^{\circ }$

$10^{th}$ term of the A.P. : –12, –19, –26, .... is

1. (A)–75
2. (B)–65
3. (C)51
4. (D)–82

The roots of the equation $x^2-8=0$ are

1. (A)rational and distinct
2. (B)irrational and distinct
3. (C)real and equal
4. (D)not real

The point ($x$, 0) divides the line segment joining the points (–4, 5) and (0, –10) in the ratio

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

A black card is lost from a deck of 52 playing cards. Rest of the cards are shuffled and one card is drawn at random from the available cards. The probability that drawn card is 'king of hearts', is

1. (A)$\frac{1}{52}$
2. (B)$\frac{1}{4}$
3. (C)$\frac{1}{51}$
4. (D)$\frac{1}{26}$

The largest possible cone is just fitted inside a hollow cube of edge 25 cm. The radius of the base of the cone is

1. (A)5 cm
2. (B)12.5 cm
3. (C)25 cm
4. (D)10 cm

If E is an event such that P(E) = 0.1, then P($\overline{E}$) is equal to

1. (A)0.9
2. (B)$\frac{1}{2}$
3. (C)0.99
4. (D)–1

If tan A = 1, then 3 sin A + cos A is equal to

1. (A)$4\sqrt{2}$
2. (B)4
3. (C)$2\sqrt{2}$
4. (D)$4\times 45^{\circ }$

A quadratic polynomial having only zero (–2) is

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

PQ and PR are tangents to a circle with centre O such that OQ = QP. The value of $\angle OPQ$ is equal to

1. (A)$45^{\circ }$
2. (B)$30^{\circ }$
3. (C)$60^{\circ }$
4. (D)$90^{\circ }$

Which of the following depends on all observations of a given data ?

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

PQ is tangent to the circle centred at O. If OQ = 3 cm, PQ = 5 cm, then OP is equal to

1. (A)5 cm
2. (B)4 cm
3. (C)$\sqrt{15}$ cm
4. (D)$\sqrt{34}$ cm

Two dice are rolled together. The probability that only one die shows number 4, is

1. (A)$\frac{11}{36}$
2. (B)$\frac{1}{3}$
3. (C)$\frac{5}{18}$
4. (D)$\frac{1}{4}$

An arc of length 22 cm subtends an angle of $x^{\circ }$ at the centre of the circle. If radius of circle is 36 cm, the value of $x$ is

1. (A)35
2. (B)40
3. (C)60
4. (D)30

Assertion (A) : When a hemisphere of same radius (r) is carved out from one side of a solid wooden cylinder, the total surface area of remaining solid is increased by $2\pi r^2$.
Reason (R) : Curved surface area of hemisphere is $2\pi r^2$.

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.

Assertion (A) : In a right angle triangle ABC, $\angle B=90^{\circ }$. Therefore the value of cos (A + C) is equal to 0.
Reason (R) : $A+B+C=180^{\circ }$ and $\cos 90^{\circ }=0$.

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.

Check whether $15^n\times 2^n$, n being a natural number, ends with the digit zero.

Evaluate : $2{\tan }^{2}45^{\circ }+{\cos }^{2}30^{\circ }-{\sin }^{2}90^{\circ }$.

Verify that $\cos 2A=\frac{1-{\tan }^{2}A}{1+{\tan }^{2}A}$ for $A=30^{\circ }$.

Find the coordinates of the points of trisection of line segment joining the points (–4, 1) and (6, 5).

A bag contains 40 marbles out of which some are white and others are black. If the probability of drawing a black marble is $\frac{3}{5}$, then find the number of white marbles.

In a pre-primary class, a teacher put cards numbered 20 to 59 in a bowl. A student picked up a card at random and read the number. Find the probability that the number read was (i) a prime number (ii) a perfect square.

Using distance formula, prove that the points (1, 5), (2, 3) and (3, 1) are collinear.

If $\alpha$, $\beta$ are zeroes of the polynomial $3x^2-8x+4$, then form a quadratic polynomial in $x$ whose zeroes are $\frac{1}{\alpha }$ and $\frac{1}{\beta }$.

Find zeroes of the polynomial $6x^2-7x-3$ and verify the relationship between zeroes and its coefficients.

The sum of the squares of two consecutive even numbers is 452. Find the numbers.

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

The traffic lights at three different road crossings change after every 45 seconds, 75 seconds and 60 seconds respectively. If they change together at 5.00 a.m., then at what time they will change together next ?

A spherical glass vessel has a cylindrical neck 7 cm long and 8 cm in diameter. The radius of spherical part is 10 cm. Find the volume of the vessel.

From each end of a solid cylinder of height 20 cm and base radius 7 cm, a cone of base radius 2.1 cm and height 5 cm is scooped out. Find the volume of the remaining solid.

Point E lies on the extended side AD of parallelogram ABCD. BE intersects CD at F. Show that (i) $△DFE\sim △CFB$ (ii) $△AEB\sim △CBF$.

Determine graphically whether the following pair of linear equations
$2x+3y=12$ and $x-y=1$
has unique solution or infinitely many solutions.

If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, then prove that the other two sides are divided in the same ratio.

It is given that sides AB and AC and median AD of $△ABC$ are respectively proportional to sides PQ and PR and median PM of another $△PQR$. Show that $△ABC\sim △PQR$.

The angle of elevation of the top of a building from the foot of the tower is $30^{\circ }$ and the angle of elevation of the top of the tower from the foot of the building is $60^{\circ }$. If the tower is 30 m high, find the height of the building and distance between the building and the tower. (Use $\sqrt{3}=1.73$)

Find 'mean' and 'mode' of the following data :
Class: 10-25, 25-40, 40-55, 55-70, 70-85, 85-100
Number of Students: 12, 10, 15, 13, 8, 12

The following table shows the ages of 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: 7, 10, 21, 22, 15, 5
Find 'mode' and 'median' of the above data.

In a Fine Arts class, students were asked to design triangular tiles in geometric pattern.
Neelima made a circular design inside an equilateral triangle ABC. The radius of the circle is 4 cm. Observe the diagram and answer the following questions :
(i) Determine the length OB. (1)
(ii) Is DE $\parallel$ CA ? Give reason for your answer. (1)
(iii) (a) Write all angles of quadrilateral OEBD and show that it is a cyclic quadrilateral. (2)
OR
(iii) (b) Find the perimeter of $△ABC$. (Use $\sqrt{3}=1.73$) (2)

A farmer has put up a decorative windmill in his farm in which there are eight blades of equal width and equally placed in a circular arrangement. A circular wire goes through them.
The diagram shows two blades OAB and OPQ in a quarter circle with centre O. $\angle AOB=\angle POQ=30^{\circ }$, OA = 28 cm, OC = 21 cm.
O is the centre of both the circles.
(i) Determine the measure of $\angle BOP$. (1)
(ii) Find length of arc CD. (1)
(iii) (a) Find the area of region CABD. (2)
OR
(iii) (b) Find perimeter of region CABD. (2)

A tent house owner provides furniture on rent. He stacks chairs in his shop to save space.
In the diagram, the height of seat of chair from ground is represented by $h_1$, $h_2$, $h_3$, .... The height of first seat is 44 cm from ground level and gap between every two seats is 10 cm.
(i) Write the values of $h_1$, $h_2$, $h_3$, $h_4$ and $h_5$ in this order only. (1)
(ii) Show that the above values form an A.P. Write its first term and common difference. (1)
(iii) (a) If chairs can be stacked up to the maximum height of 160 cm, then find the maximum number of chairs in a stack. (2)
OR
(iii) (b) Is it possible to stack 15 chairs if maximum height of the stack can not be more than 180 cm ? Justify your answer. (2)