CBSE Class 10 Maths Basic 2026 Question Paper 430/4/2 with Solutions

A card is drawn from a well shuffled deck of 52 cards. The probability of getting an ace or a ten is :

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

The distance between the points (– 5, 1) and (2, 2) is :

1. (A)$2\sqrt{5}$ units
2. (B)$\sqrt{10}$ units
3. (C)$5\sqrt{2}$ units
4. (D)$3\sqrt{2}$ units

Probability of getting an irrational number at random from the numbers $\sqrt{3},\sqrt{4},3\sqrt{9},3\sqrt{8},\sqrt{5},0,4^{\frac{3}{2}}$ is :

1. (A)0
2. (B)$\frac{4}{7}$
3. (C)$\frac{3}{7}$
4. (D)1

The graph of polynomial p($x$) = k is shown here. Number of zeroes of polynomial p($x$) is :

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

Chord AB subtends an angle of $40^{\circ }$ at the centre of the circle of radius 9 cm. The length of arc AB is :

1. (A)$\frac{22}{7}$ cm
2. (B)$\frac{198}{7}$ cm
3. (C)$\frac{44}{7}$ cm
4. (D)$\frac{54}{7}$ cm

7 × 29 × 23 + 1 is :

1. (A)a prime number.
2. (B)divisible by 23.
3. (C)an odd number.
4. (D)a composite number.

In the given figure, PQ and PR are two tangents drawn to a circle with centre O. If $\angle ORQ=25^{\circ }$, then the measure of $\angle PQR$ is :

1. (A)$65^{\circ }$
2. (B)$25^{\circ }$
3. (C)$50^{\circ }$
4. (D)$75^{\circ }$

The value of $\tan 30^{\circ }\tan 60^{\circ }-\sin 90^{\circ }\cos 90^{\circ }$ is :

1. (A)1
2. (B)0
3. (C)2
4. (D)not defined

HCF of two consecutive natural numbers is :

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

In the given figure, PQ $\parallel$ BC. If AP : AB = 3 : 7 then, AQ : QC equals :

1. (A)3 : 7
2. (B)3 : 10
3. (C)7 : 3
4. (D)3 : 4

PT is tangent to the circle with centre O and radius 5 cm. OP intersects the circle at Q. If PQ = $x$, then $P{T}^2$ equals :

1. (A)$x^2+5x$
2. (B)$x^2+10x+50$
3. (C)$x^2+10x$
4. (D)$x^2-25$

Two wooden solid hemispheres of same radii r, are joined at a point, as shown in the figure. The total surface area of the object is :

1. (A)$4\pi r^2$
2. (B)$6\pi r^2$
3. (C)$3\pi r^2$
4. (D)$5\pi r^2$

While calculating mean of a grouped frequency distribution using step deviation method $\left(u=\frac{x-a}{h}\right)$ it was found that $\overline{x}$ = 62, a = 47.5, h = 5. The value of $\overline{u}$ is :

1. (A)3
2. (B)14.5
3. (C)2.9
4. (D)3.1

$n^{th}$ term of an A.P. is 5n – 15. The common difference of the A.P. is :

1. (A)5n
2. (B)5
3. (C)– 5
4. (D)10

A cone with slant height 10 cm and radius 6 cm is surmounted on a hemisphere of same radius. The height of the toy is :

1. (A)12 cm
2. (B)16 cm
3. (C)14 cm
4. (D)10 cm

$n^{th}$ term of the A.P. $-\frac{3}{2},\frac{3}{2},\frac{9}{2}$, ... is :

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

If $\sec \theta =\sqrt{3}$, then the value of $\tan \theta$ is :

1. (A)$\sqrt{3}-1$
2. (B)$60^{\circ }$
3. (C)2
4. (D)$\sqrt{2}$

The value of k for which the equation $5x^2-2x+k=0$ has equal real roots, is :

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

Assertion (A) : $4^n$ can not end with the digit zero.
Reason (R) : Prime factorisation of $4^n$ is unique.

1. (A)Both, Assertion (A) and Reason (R) are true and Reason (R) is correct explanation of Assertion (A).
2. (B)Both, Assertion (A) and Reason (R) are true, but Reason (R) is not correct explanation for 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 of an event can not be $\frac{1}{0.9}$.
Reason (R) : $0\le P(E)\le 1$ for an event E.

1. (A)Both, Assertion (A) and Reason (R) are true and Reason (R) is correct explanation of Assertion (A).
2. (B)Both, Assertion (A) and Reason (R) are true, but Reason (R) is not correct explanation for Assertion (A).
3. (C)Assertion (A) is true, but Reason (R) is false.
4. (D)Assertion (A) is false, but Reason (R) is true.

If $\sin 2A=\frac{\sqrt{3}}{2}$ and $2\tan B+1=3$, then find the value of (A + B).

If $\alpha$, $\beta$ are the zeroes of polynomial p($x$) = $-9x^2-6x+1$. Find the value of ${\alpha }^2+{\beta }^2$.

Form a quadratic polynomial whose zeroes are twice the zeroes of polynomial p($x$) = $x^2-3x-5$.

In the given figure, QR $\parallel$ CB and RP $\parallel$ AC. If BR = 10 cm, QA = 12 cm, BP = 12 cm and PC = 18 cm, then find the lengths of AR and QC.

A (– 2, 3) and B (4, 1) are end points of the diameter of a semi-circle. The semi-circle intersects y-axis at point P. Find the co-ordinates of the point P.

In the given figure, AP, AQ and BC are tangents to the circle with centre O. If AB = 6 cm, AC = 7 cm and BC = 5 cm, then what is the length of AP ?

In the given figure, TP is tangent to a circle with centre O. Diameter BA when produced meets the tangent at T. If $\angle$ ABP = $35^{\circ }$, then find the measure of $\angle$ PTA.

Two tangents TP and TQ are drawn to a circle with centre O, from an external point T.
Prove that $\angle PTQ=2\angle OPQ$.

Prove that $\sqrt{5}$ is an irrational number.

The vertices of a rhombus ABCD are A(– 3, – 4), B(5, – 3), C(1, 4) and D(– 7, 3). Find the length of both the diagonals. Hence, find area of the rhombus ABCD.

The line segment joining the points A (– 5, 1) and B (7, 6) is trisected at the points P and Q such that P is nearer to A. If P lies on the line $x$ + y = k, then find the value of k.

Prove that $\frac{(\sec \theta +\tan \theta {)}^2-1}{(\sec \theta +\tan \theta {)}^2+1}=\sin \theta$.

Prove that $\frac{\tan A}{1-\cot A}+\frac{\cot A}{1-\tan A}=1+\sec A\text{ cosec }A$.

A chord of a circle of radius 14 cm subtends a right angle at the centre. Find the area of the corresponding (i) minor segment (ii) major segment.

Two dice are rolled together. Find the probability that at least one of the numbers obtained is a multiple of 3.

The difference between two numbers is 12. The greater number is 6 less than twice the smaller one.
(i) Representing the above situation, frame two linear equations in two variables.
(ii) Show that the equations have unique solution.
(iii) Solve the equations and hence find the numbers.

Solve the following equations graphically :
$x+y=7$ and $2x-5y=7$

The third and ninth term of an A.P. are 4 and – 8 respectively.
(i) Which term of the A.P. is zero ?
(ii) Find the value of n if $S_n$ = – 36.

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.

AD and PS are respectively, the medians of $△ABC$ and $△PQR$. If $△ABC\sim △PQR$, then prove that
(i) $△ADC\sim △PSR$
(ii) $\frac{AD}{PS}=\frac{BC}{QR}$

A statue, 2 m tall, stands on the top of a pedestal. From a point on the ground, the angle of elevation of the top of the statue is $60^{\circ }$ and from the same point the angle of elevation of the bottom of the statue is $45^{\circ }$. Find the height of the pedestal and its distance from the point of observation on the ground. (use $\sqrt{3}$ = 1.73)

'Gilli Danda' is a very popular traditional game of India which is played with two wooden sticks – the larger one is called 'Danda' and smaller one 'Gilli'.
'Danda' – It is cylindrical in shape with diameter 4 cm and length 42 cm.
Gilli – It is cylindrical in middle with identical conical ends of same radius 1.5 cm and length 2.8 cm. The length of cylindrical part is 7 cm.
Based on the above, answer the following questions :
(i) Find the volume of wood used in making both the conical parts of Gilli. (1)
(ii) Find the volume of wood used in making cylindrical part of Gilli. (1)
(iii) (a) A cylindrical log of wood of radius 1.5 cm and length 14 cm is used to make Gilli. Find the volume of the wood scrapped. (2)
OR
(b) Find the total surface area of 'Danda'. (2)

Observe the figure given above. It shows six identical rectangular enclosures made by using fencing wire mesh. These enclosures are used to protect baby animals in a zoo. Dimensions of each enclosure is $x$ feet × y feet. The total length of fencing required is 152 feet and area of each enclosure is 80 square feet.
Based on the above, answer the following questions :
(i) Write an expression for length of fencing required in terms of $x$ and y. (1)
(ii) Write the area of each enclosure in terms of $x$. (1)
(iii) (a) Write the above equation in quadratic equation form and thus find the dimensions of each enclosure using factorisation method. (2)
OR
(b) Using above equation in quadratic form, solve the equation and find the dimensions of each enclosure using quadratic formula. (2)

CENTRAL POLLUTION CONTROL BOARD'S AIR QUALITY STANDARDS
AIR QUALITY INDEX (AQI) | CATEGORY
0-50 | Good
51-100 | Satisfactory
101-200 | Moderate
201-300 | Poor
301-400 | Very Poor
401-500 | Severe
The Air Quality Index (AQI) is a scale from 0 to 500 that indicates air quality, with higher numbers signifying more pollution and greater health concerns.
Mansi collected the daily data of AQI of her city for a month and presented it as given below :
AQI Range : 1 – 100 | 101 – 200 | 201 – 300 | 301 – 400 | 401 – 500
Number of Days : 3 | 9 | 12 | 4 | 2
(i) Convert the data to continuous frequency distribution. (1)
(ii) What is the quality of air in most of the days of the month ? (1)
(iii) (a) Using table formed in part (i), find mode of the data. (2)
OR
(b) Using table formed in part (i), find median of the data. (2)