From a solid cylinder whose height is 2.8 cm and radius 2.1 cm, a conical cavity of the same height and same radius is hollowed out. Find the volume and the total surface area of the remaining solid.

 From a solid cylinder whose height is 2.8 cm and radius 2.1 cm, a conical cavity of the same height and same radius is hollowed out. Find the volume and the total surface area of the remaining solid.

Cylinder and conical cavity have same radius and height

r = 2.1 cm, h = 2.8 cm

.Volume of remaining solid

 Volume of cylinder − Volume of cone 

Volume of cylinder = πr²h = π × (2.1)² × 2.8 = π × 4.41 × 2.8 = 12.348π cm³  

Volume of cone = (1/3)πr²h = (1/3) × 12.348π = 4.116π cm³

Remaining volume = 12.348π − 4.116π = 8.232π cm³  

Using π = 22/7

Remaining volume  = 8.232 × 22/7 = 25.872 cm³ ≈ 25.87 cm³

. Total surface area of remaining solid

Surfaces left after hollowing:  

Bottom circular base of cylinder = πr²  

Curved surface of cylinder = 2πrh  

Curved surface of cone = πrl  

Slant height of cone: l = √(r² + h²) = √(2.1² + 2.8²) = √(4.41 + 7.84) = √12.25 = 3.5 cm 

Required surface area = πr² + 2πrh + πrl

 = π(2.1)² + 2π(2.1)(2.8) + π(2.1)(3.5)

 = 4.41π + 11.76π + 7.35π = 23.52π cm²  Using π = 22/7:

 = 23.52 × 22/7 = 73.92 cm²  

for more details watch this video 

mensuration,   surface area and volume of solids , cbse th maths old board exam question paper 2025 2026

cbse 10th maths questions

 cbse 10th maths questions

Linear equations in two unknowns 

 The sum of numerator and denominator of a fraction is 4 less than twice the denominator. If each of the numerator and denominator is decreased by 1, the fraction becomes ¹⁄₃. Find the fraction.

solution

Quadratic equations

Three consecutive positive integers are such that sum of square of the first and the product of the other two is 67, find the integers.

solution

Coordinate Geometry

Find a relation between x and y such that the point P(x, y) is equidistant from the

 points A(5, 3) and B(1, 7)

solution

The coordinates of the centre of a circle are (x − 7, 2x). Find the value(s) of ‘x’, if the circle passes through the point (−9, 11) and has radius 5√2 units.

solution

Trigonometry

 If cos A + sin A = (√2) cos A, prove that cos A − sin A = (√2) sin A

solution

Mensuration surface area and volume of solids

 From a solid cylinder whose height is 2.8 cm and radius 2.1 cm, a conical cavity of the same height and same radius is hollowed out. Find the volume and the total surface area of the remaining solid.

solution

Statistics

Find the mean of the following distribution :

Class               30 – 40    40 – 50   50 – 60      60 – 70      70 – 80

Frequency             6            13             8               12              11

solution

Probability

A boy  has a collection of balls of different colours. He has a total of 35 balls in his basket out of which seven are black in colour and eight are yellow in colour. Out of remaining balls, some are white and the rest are red.

Based on the above, answer the following questions:

(a) If the probability of drawing a red ball at random from the basket is three times that of a white ball, then find the number of red balls in the basket.

(b) Find the probability of drawing a ball at random from the basket which is either a black or a white ball

solution

While playing badminton Ravi has set the barrier chain hung between two posts at the edge of the walkway of a street. It is hung in the shape of a parabola. Based on the above information answer the following questions : (a) Which type of the polynomial (linear, quadratic, cubic etc.) is graphically represented by a parabola ? (b) If the polynomial represented by a parabola, intersects the x-axis at −2 and 3 and y-axis at −3, then write the zeroes of the parabola. (c) Find the expression for the above polynomial. OR (c) If the zeroes of the polynomial are −5 and 3, find its expression.

 While playing badminton Ravi has set the barrier chain hung between two posts at the edge of the walkway of a street. It is hung in the shape of a parabola.

Based on the above information answer the following questions :

(a) Which type of the polynomial (linear, quadratic, cubic etc.) is graphically represented by a parabola ?

(b) If the polynomial represented by a parabola, intersects the x-axis at −2 and 3 and y-axis at −3, then write the zeroes of the parabola.

(c) Find the expression for the above polynomial.

(d) If the zeroes of the polynomial are −5 and 3, find its expression.

[a] A parabola is the graph of a quadratic polynomial

Given

parabola, intersects the x-axis at −2 and 3 

therefore zeroes are [-2] and 3

Polynomial with roots α and β is given by

f(x) = k(x − α)(x − β)

or

f(x) = k[x² − (α + β)x + αβ]

where k ≠ 0 is any real number.

given that zeroes are [-2] and 3

Take them as

α = [-2]

β = 3

 f(x) = k(x + 2)(x − 3)

use distribution law

f(x) =k(x² − x − 6)

to find k use the condition that parabola cuts y-axis at −3

which means it passes through (0,-3)

so Use f(0) = −3 

k[0-0-6] = [-3]

we get k =(1/2)

and  f(x) = (1/2)(x² − x − 6)

in the last part the zeroes are given to be  −5 and 3

so g(x) = k(x + 5)(x − 3) =kx² + 2x − 15

choose k=1

g(x)  = x² + 2x − 15

see this video for more details 

cbse 10th maths old board exam question papers 2025 2026 to find the equation of a polynomial whose zeroes are given

If (sin x)ʸ = yᶜᵒˢˣ, then find dy/dx

 If (sin x)ʸ = yᶜᵒˢˣ, then find dy/dx.

differential calculus topic of logarthimic differentiation

given

(sin x)ʸ = yᶜᵒˢˣ  [power contains variables]

Take ln both sides

ln[(sin x)ʸ] = ln[yᶜᵒˢˣ]  [use properties of logarithms]

 y. ln(sin x) = cos x . ln y

 Differentiate w.r.t. x

[use product rule and chain rule]

  [dy/dx] ln(sin x) + y cot x  =  −sin x · ln y + cos x · (1/y) ·[ dy/dx]

[dy/dx ] ln(sin x) + y cot x = −sin x ln y + (cos x / y) [dy/dx]

 Collect dy/dx terms

 [dy/dx] ln(sin x) − (cos x / y)[ dy/dx ]= −sin x ln y − y cot x

[ dy/dx] [ln(sin x) − cos x / y] = −[sin x ln y + y cot x]

dy/dx = −[sin x ln y + y cot x] / [ln(sin x) − cos x / y]

  Multiply numerator and denominator by y:

dy/dx = −y[sin x ln y + y cot x] / [y ln(sin x) − cos x]

see this video for more explanation 

cbse 12th maths old board exam question paper

2025 2026 differentiation

logarthmic differentiation

 

Find a particular solution of the differential equation (x + 1) dy/dx = 2 e⁻ʸ − 1, given that y = 0 when x = 0

 Find a particular solution of the differential equation

(x + 1) dy/dx = 2 e⁻ʸ − 1,

given that y = 0 when x = 0

variable separable differential equation with initial condition

given

(x + 1) dy/dx = 2 e⁻ʸ − 1,

separating

 dy / (2 e⁻ʸ − 1) = dx / (x + 1)

multiply numerator and denominator of LHS with  eʸ 

eʸ dy / (2 − eʸ) = dx / (x + 1)

integrating

∫ eʸ / (2 − eʸ) dy = ∫ 1 / (x + 1) dx

for the LHS use the substitution

u = 2 − eʸ 

⇒ du = −eʸ dy

eʸ dy =-du

we get

 ∫ −du / u = ∫ 1 / (x + 1)  dx

 −ln|u| = ln|x + 1| + C

re substitute the value of u = 2 − eʸ 

 −ln|2 − eʸ| = ln|x + 1| + C

 ln|2 − eʸ| +ln|x + 1| = -C

using properties of logarithms,

ln|(2 − eʸ)(x + 1)| = -C

using definition of logarithm

 (x + 1)(2 − eʸ) = A-----(1)

, where A is another arbitrary constant

use the initial condition   y = 0 when x = 0 to solve for A

we get A =1

(1)⇒ (x + 1)(2 − eʸ) =1

or

2 − eʸ = 1 / (x + 1)

eʸ = 2 − [ 1 / (x + 1)]

eʸ =  (2x + 1) / (x + 1)

using definition of log

y = ln[(2x + 1) / (x + 1)]

for more details watch this video 

cbse 12th past year question papers 2025 2026 differential equations

variable separable differential equation dy/dx = (2 − y) / (x + 1)

 Solve the following variable separable differential equation

dy/dx = (2 − y) / (x + 1)

dy/dx = (2 − y) / (x + 1)

clearly it is variable separable type

dy / (2 − y) = dx / (x + 1)

integrating both sides

∫ dy / (2 − y) = ∫ dx / (x + 1)

− ln|2 − y| = ln|x + 1| + lnC [ note the ( -1), coefficient of y when integrating]

0 = ln|2 − y| + ln|x + 1| + lnC

use the property of logarithms

0=ln [ |2 − y| |x + 1| C ]

or

[ |2 − y| |x + 1| C ] = 1

for more details refer to this video

cbse  12th applied mathematics 2025 2026 old board exam question papers applied mathematics

variable sepaprable differential equation

Three consecutive positive integers are such that sum of square of the first and the product of the other two is 67, find the integers.

 Three consecutive positive integers are such that sum of square of the first and the product of the other two is 67, find the integers.

Let the three consecutive positive integers be n, n+1, n+2

Given that

that sum of square of the first and the product of the other two is 67

n² + (n+1)(n+2) = 67

n² + (n² + 2n + n + 2) = 67  

n² + n² + 3n + 2 = 67  

2n² + 3n + 2 = 67  

2n² + 3n + 2 - 67  = 0

2n² + 3n − 65 = 0

Solve by quadratic formula or factorisation

Factorisation method

Find two numbers that gives a product of 2 × (−65) = −130 and add to 3.

by trial and error the numbers13 and −10 satisfy the condition

use this to split the middle term of

2n² + 3n − 65 = 0

.2n² + 13n − 10n − 65 = 0

n(2n + 13) − 5(2n + 13) = 0

(n − 5)(2n + 13) = 0

solving

n − 5 = 0 ⇒ n = 5

2n + 13 = 0 ⇒ n = −13/2 (rejecct)

required positive integers are

n=5

n+1 =6

n+2=7

for a video of the solution

cbse 10th maths pervious years old question papers 2025 2026 standard mathematics

   quadratic equation word problem  formation and solution by splitting the middle term

∫ cos x / [(2 + sin x)(4 + sin x)] dx

 find ∫ cos x / [(2 + sin x)(4 + sin x)] dx 

integration by substitution and partial fractions

Substitution

Let t = sin x

Then dt = cos x dx

 ∫ cos x / [(2 + sin x)(4 + sin x)] dx =  ∫ 1 / [(2 + t)(4 + t)] dt

Method of partial fractions

 1 / [(t + 2(t + 4)] = A/(t + 2) + B/(t + 4)

multiply [t+2][t+4]

1 = A(t + 4) + B(t + 2)

Put t = −2

1 = A(2) 

⇒ A = ½

Put t = −4:

1 = B(−2) 

⇒ B = −½

1 / [(t + 2)(t + 4)] = ½[1/(t + 2) − 1/(t + 4)]

integrate

I = ½  ∫ [1/(t + 2)  − 1/(t + 4)] dt 

= ½[ln|t + 2| − ln|t + 4|] + C

= ½ ln|(t + 2)/(t + 4)| + C   use  t = sin x

=½ ln[(2 + sin x)/(4 + sin x)] + C

for  explanation watch this video 

The sum of numerator and denominator of a fraction is 4 less than twice the denominator. If each of the numerator and denominator is decreased by 1, the fraction becomes ¹⁄₃. Find the fraction.

 The sum of numerator and denominator of a fraction is 4 less than twice the denominator. If each of the numerator and denominator is decreased by 1, the fraction becomes ¹⁄₃. Find the fraction.

Let numerator = x, denominator = y. Fraction = x⁄y

Given

sum of numerator and denominator of a fraction is 4 less than twice the denominator

 x + y = 2y − 4

⇒ x = y − 4  --- (1)

Given

If each of the numerator and denominator is decreased by 1, the fraction becomes ¹⁄₃

 (x − 1)⁄(y − 1) = ¹⁄₃

⇒ 3(x − 1) = y − 1

⇒ 3x − 3 = y − 1

⇒ 3x = y + 2  --- (2)

Solve using substitution method

Substitute (1) into (2):

3(y − 4) = y + 2

3y − 12 = y + 2

2y = 14

y = 7 

 Then 

x = y − 4 

x = 7 -4

x= 3  

x =3 

y=7

 The fraction is ³⁄₇

linear equations in two unknowns

formation of linear equations and solution by substitution method

cbse 10th maths previous years question papers 

2025 2026

for a detailed video of the solution using elimination method

\refer  

If cos A + sin A = (√2) cos A, prove that cos A − sin A = (√2) sin A

 If cos A + sin A = (√2) cos A, prove that cos A − sin A = (√2) sin A

given

cos A + sin A = (√2) cos A

sin A =  (√2) cos A - cos A

sin A = (√2 - 1 ) cos A

sin A/ (√2 - 1 )  = cos A

rationalising

(√2 + 1 ) sin A/ [ (√2 - 1 ) (√2 + 1 ) ]  = cos A

(√2 + 1 ) sin A / [2-1] = cosA

(√2 + 1 ) sin A / [1] = cosA

(√2 + 1 ) sin A = cosA

apply distribution law

(√2) sin A + 1 sinA = cosA

(√2) sin A + sin A = cosA  

(√2) sin A  = cosA - sin A

OR

cos A − sin A = (√2) sin A

cbse 10th maths old board exam question paper 2025 2026 trigonometry

video of an alternate method 

AB is a chord of length 24 cm of a circle of radius 15 cm. The tangents at A and B intersect at a point P. Find the length PA.

 AB is a chord of length 24 cm of a circle of radius 15 cm. The tangents at A and B intersect at a point P. Find the length PA.

Let O be the centre of the circle and Q the midpoint of chord /AB

Given: OA = OB = 15 cm, AB = 24 cm

Q is midpoint of AB

AQ = QB = 24/2 = 12 cm,

also OQ ⊥ AB  

In right triangle  ΔOQA:

OA² = OQ² + AQ²

15² = OQ² + 12²

225 = OQ² + 144

OQ² = 81 

OQ = 9 cm

  ∠OAP = 90° since radius ⊥ tangent

so ΔOAQ ~ ΔOPA

using corresponding sides

OA/OP = OQ/OA

OA² = OQ × OP

15² = 9 × OP

225 = 9 × OP 

 OP = 25 cm 

In right triangle  ΔOPA:

 PA² = OP² − OA²

PA² = 25² − 15² = 625 − 225 = 400

PA = 20 cm  

cbse 10th maths old board exam question paper previous years 2025 2026 stansdard mathematics

circles tangents to circles

a detailed video on an alternative method to solve the problem

Find a relation between x and y such that the point P(x, y) is equidistant from the points A(5, 3) and B(1, 7).

Find a relation between x and y such that the point P(x, y) is equidistant from the

 points A(5, 3) and B(1, 7)

Given

P(x, y) is equidistant from A and B, 

 PA = PB

squaring both sides

PA² = PB²

Using distance formula to find PA and PB

(x - 5)² + (y - 3)² = (x - 1)² + (y - 7

Expand both sides using identity

x² - 10x + 25 + y² - 6y + 9 = x² - 2x + 1 + y² - 14y + 49

-10x - 6y + 34 = -2x - 14y + 50

-10x + 2x - 6y + 14y = 50 - 34

-8x + 8y = 16

or

-x + y = 2

or 

y = x + 2

watch this video for more details

cbse 10th maths 2025 2026 old board exam question paper coordinate geometry distance formula