Mathematics Basic - 430/2/2 2025-2026 English Medium Class 10 Question Paper Solution

The sum of the age (in years) of a father and three times the age of his daughter is 59. If the age of the father is x years and that of his daughter is y years, the equation representing the given information is ______.

3x + y = 59

x + y = 59

x + 3y = 59

x + y = 56

The 10^th term of the A.P. `sqrt(2), sqrt(8), sqrt(18), ...` is ______.

`sqrt(162)`

`sqrt(200)`

`sqrt(54)`

`sqrt(94)`

The value of (sec 45° – cos 45°) is equal to the value of:

sin 45°

sin 90°

cos 90°

tan 45°

One of the possible values of A, for which cos 2A = cos A, is ______.

0°

30°

45°

90°

At an instant, the length of shadow of a stick is found to be `sqrt(3)` times the length of the stick as shown in the figure below. The Sun’s altitude at that instant is:

30°

45°

60°

90°

The largest possible right circular cone is carved out of a solid hemisphere of radius ‘r’ as shown in the figure below. The slant height of the cone is:

r

2r

`sqrt(2)r`

`sqrt(3)r`

If for a frequency distribution, the mean is `3/4` times the median, then mode is ______.

equal to median

`3/2` times the median

equal to mean

3 times the mean

Consider the given data:

The difference of lower limit of the median class and upper limit of the modal class is:

0

20

40

10

An unbiased die is thrown once. The probability of getting an odd prime number less than 3 is ______.

`1/2`

`1/3`

`1/6`

0

If the HCF of the smallest odd prime number and the greatest 2-digit number is expressed as 3^m . 11ⁿ, then the values of m and n respectively are:

0, 0

1, 0

1, 1

2, 1

Which of the following statements is true for HCF and LCM of two distinct natural numbers a and b?

1. HCF is always greater than LCM.
2. HCF is a factor of LCM.
3. LCM is a factor of HCF.

(i) only

(i) and (iii)

(i) and (ii)

(ii) only

Which of the following system of equations has a unique solution?

x = 0, x = 1

x + y = 0, 2x + 2y = 0

x + y = 2, x – y = 3

x + y = 5, x + y = 10

If the quadratic equation lx² – mx + n = 0 has roots which are reciprocal of each other, then which of the following is true?

l = n

l = m

m = n

`l = 1/n`

If x = –1 is a root of the equation ax² – bx + 3 = 0, then:

– a + b – 3 = 0

a – b – 3 = 0

– a – b + 3 = 0

a + b + 3 = 0

In the given figure, the length of a diagonal is:

3 units

4 units

5 units

25 units

In the given figure, if ME || SU, then which of the following statements is correct?

ΔMOE ~ ΔSOU

ΔMOE ~ ΔSUO

ΔOEM ~ ΔUSO

ΔOEM ~ ΔOSU

In the given figure, if ΔACP ~ ΔDCP, then:

∠DCP = 60°

∠DCP = 30°

∠DCP = 90°

∠DCP = 30°

The graph of a quadratic polynomial is shown in the given figure. The sum of the zeroes of the polynomial is:

–2

–1

–3

3

Assertion (A): `(sqrt(2) + sqrt(3))` is an irrational number.

Reason (R): The sum of two irrational numbers is always an irrational number.

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

Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of Assertion (A).

Assertion (A) is true, but Reason (R) is false.

Assertion (A) is false, but Reason (R) is true.

Assertion (A): The probability of a certain event E is 1.

Reason (R): The sum of probabilities of all elementary events of an experiment is 1.

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

Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of Assertion (A).

Assertion (A) is true, but Reason (R) is false.

Assertion (A) is false, but Reason (R) is true.

Triangle PQR is an isosceles right triangle, right angled at Q. Find the value of sec² P + cosec² R.

Evaluate: sin 45°. cos 45° + cosec² 30°

A square dart board is divided into four equal squares as shown in the figure given below. If a dart hits square APOS, a player wins ₹ 100. If a dart hits square BPOQ, a player loses ₹ 50. If a dart hits square OQCR, a player loses ₹ 100 and if it hits square ORDS, the player will win ₹ 25. A player takes a turn and hits the dart board. What is the probability that

1. the player loses money?
2. the player wins 100?

The point P(x, y) lies on a semi-circular arc having diameter AB as shown in the given figure. The coordinates of points A and B are (3, 0) and (0, 4) respectively. Find the relation between x and y, if PA² + PB² = AB².

Find the coordinates of the points of trisection P and Q of the line-segment AB as shown in the given figure.

Verify the relationship between the zeroes and the coefficients of the quadratic polynomial 25x² – 16.

In the figure given below, ∠1 = ∠2 and `(BE)/(BC) = (CD)/(AB)`. Prove that ΔBDE ∼ ΔBAC.

PQRS is a trapezium with PQ || SR. U and V are points on the non-parallel sides PS and QR respectively as shown in the given figure. If UV || SR. prove that `(PU)/(US) = (QV)/(VR)`.

Prove that the tangent at any point of a circle is perpendicular to the radius through the point of contact.

Neha claimed that there does not exist any irrational number between 1 and 2. Raunak claimed that `sqrt(2)` lies between 1 and 2 and `sqrt(2)` is an irrational number. Who do you think is correct? Justify by proving either `sqrt(2)` as an irrational number or otherwise.

Prove that the system of equations given as 2x – 3y = 7 and 4x + ky = 9, is inconsistent for k = – 6. Also, obtain the solution of the system of equations, if k = – 1.

Represent the following system of equations graphically and conclude whether the system is consistent or inconsistent.

2x + 3y = 6

4x + 6y = 24

Show that the quadrilateral ‘HOPE’ with vertices H(–2, 1), О(–1, 2), P(0, 1) and E(–1, 0) is a rhombus. Is it a square? Justify.

If sin A + sin² A = 1, find the value of cos² A + cos⁴ A. Also, using the above, prove that tan² A . sec² A = 1.

Prove that: `(1 + "cosec"  θ)/("cosec"  θ) = (cos^2 θ)/(1 - sin θ)`

ABCD is a square. With centres A, B, C and D, four quadrants (each touching two of the remaining three) are drawn inside the square ABCD as shown in the figure. If the area of the shaded region is 42 cm², find the side of the square ABCD. Also, find the perimeter of the shaded region.

A rectangle ABCD with diagonal 14 cm is inscribed in a circle with centre O as shown in the given figure. If the area of the shaded portion is expressed as `a + bsqrt(3)`, find the values of a and b. Also, find the perimeter of the sector OАВО.

The median of the given data is 28.5.

Find the values of a and b.

Express `x - 1/x = 3` as a quadratic equation in standard form and hence find its roots. Also, find the value of ‘a’ for which the equation `x + 1/x = a`, when expressed as a quadratic equation, has real and equal roots.

Observe the following pattern in which each small square represents a unit square (square of side 1 unit).

Fig (i)	Fig (ii)	Fig (iii)

If the sum of number of unit squares in the n^th figure and (n + 2)^th figure is 290, find the value of n.

In the given figure, ΔABC circumscribes a circle of radius of 4 cm. If AD = 7 cm, BD = 8 cm and area (ΔАBC) = 84 cm², find the lengths of BC and AC.

Climate change and global warming are influencing storm behaviour, particularly in terms of intensity and rainfall. Strong winds and storms often cause uprooting and/or breaking of trees, which damage the vehicles standing underneath the trees. On a particular day, during a high intensity storm, a tree broke such that its broken part formed an angle of 30° with the ground. The distance between the base of the tree to the point where the top touches the ground is found to be 10 m.

Based on the above information, answer the following questions:

(i) Represent the given information with the help of a neat and well-labelled diagram.   [1]

(ii) Find the height above the ground at which the tree is broken.   [1]

(iii) (a) Find the height of the tree before it broke. (Use `sqrt(3) = 1.732`)   [2]

OR

(iii) (b) If another tree broke from the same height as in part (ii), but the broken part made a 60° angle with the ground, find the total height of the tree.   [2]

Based on the above information, answer the following questions:

(i) Find the total height of the trophy excluding the wooden part.   [1]

(ii) Find the difference between the radius of sphere and that of cylinder.   [1]

(iii) (a) If the cylindrical part and spherical part are separated and gold plated overall, find the total surface area to be gold plated.   [2]

OR

(iii) (b) Find the volume of the metal used in making the trophy, assuming that the metal is completely filled in it.   [2]

A multistorey building is constructed with stilt parking. There is a provision of lift, as well as staircase from the ground floor to the top floor. The number of stairs from the ground floor to the first floor is 10, from the first floor to the second floor is 24, from the second floor to the third floor is 38 and so on.

Based on the above information, answer the following questions:

1. Does 10, 24, 38, ... form an A.P.? Justify your answer.   [1]
2. What will be the total number of stairs from the ground floor to the eleventh floor?   [1]
3. A person supplies water cans to people living in the building. As water cans are heavy, he supplies water cans on each floor, carrying one at a time. He supplied the water can from the ground floor to the first floor, came back and supplied water can to the second floor, again came back then supplied water can to the third floor and so on.

1. Find the total number of stairs he climbed up and down to supply water till the sixth floor, using A.P.   [2]
   OR
2. The next day, following the same process, if a person climbed up and down a total of 380 stairs, till which floor did he supply water cans?   [2]