Bihar Board Class 10 · Maths · Matric exam 2027
Bihar Board Class 10 Maths Question Paper 2022
The board’s 2022 question paper, set A — all 138 questions as printed; 100 objective answers from the board’s own key; full model answers for 3 written questions, how the answer begins for 35 more. Each question names its chapter, where that chapter’s most-asked questions are.
Full marks 100 · Time 3 h 15 min
Section A · Objective questions (100)
If in division algorithm a = bq + r, a = 72, b = 9, q = 8, then r =
- (A) 1
- (B) 2
- (C) 0 ✓
- (D) 3
Answer (the board’s key): (C)
H.C.F. (a, b) x L.C.M. (a, b) =
- (A) a + b ✓
- (B) a x b
- (C) a/b
- (D) b/a
Answer (the board’s key): (A)
sqrt(15) is
- (A) an integer
- (B) a rational number
- (C) an irrational number ✓
- (D) none of these
Answer (the board’s key): (C)
The exponent of 2 in the prime factorization of 124 is
- (A) 1 ✓
- (B) 2
- (C) 0
- (D) 4
Answer (the board’s key): (A)
Which of the following fractions has a terminating decimal expansion?
- (A) 17/105
- (B) 77/210
- (C) 15/88 ✓
- (D) 3/160
Answer (the board’s key): (C)
The product of two numbers is 1600 and their H.C.F. is 5. Then LCM of the numbers is
- (A) 8000 ✓
- (B) 1600
- (C) 320
- (D) 1605
Answer (the board’s key): (A)
The HCF of 36 and 84 is
- (A) 14 ✓
- (B) 1
- (C) 12
- (D) 4
Answer (the board’s key): (A)
Which of the following is not a polynomial ?
- (A) 1/5x + 1 ✓
- (B) 6 - x^2 + sqrt(3)x
- (C) 2/(x-1)
- (D) x^3
Answer (the board’s key): (A)
If alpha, beta are the zeroes of a quadratic polynomial p(x) = x^2 - 2x + 5, then the value of alpha + beta will be
- (A) -2
- (B) 2
- (C) 5
- (D) -5 ✓
Answer (the board’s key): (D)
The zeroes of the polynomial x^2 - 2x - 3 are
- (A) -3, 1
- (B) -3, -1
- (C) 3, -1 ✓
- (D) 3, 1
Answer (the board’s key): (C)
If alpha, beta are the zeroes of the quadratic polynomial a + bx + cx^2, then the value of alpha + beta is
- (A) -b/c
- (B) -b/a
- (C) b/c ✓
- (D) b/a
Answer (the board’s key): (C)
If alpha, beta and gamma are zeroes of a cubic polynomial 2x^3 + 3x^2 - 5x - 2, then value of alpha*beta*gamma is
- (A) -1 ✓
- (B) 1
- (C) 2
- (D) 5/2
Answer (the board’s key): (A)
If p(x) = q(x).g(x) and degree of p(x) = 4 and degree of g(x) = 2, then degree of p(x)/g(x) will be
- (A) 2
- (B) 4
- (C) 1 ✓
- (D) none of these
Answer (the board’s key): (C)
If q is a positive integer then a positive odd integer is of the form
- (A) 6q + 2
- (B) 6q + 3 ✓
- (C) 6q + 4
- (D) 6q
Answer (the board’s key): (B)
The degree of the polynomial 3x^3 - 5x^2 - 11x - 3 is
- (A) 2
- (B) 3
- (C) 1
- (D) 0 ✓
Answer (the board’s key): (D)
If the zeroes of a quadratic polynomial are 1 and 1 then the quadratic polynomial is
- (A) x^2 - 2x + 1
- (B) x^2 - 2x - 1
- (C) x^2 + 2x + 1
- (D) x^2 + 2x - 1 ✓
Answer (the board’s key): (D)
Which of the following is a composite number ?
- (A) 13 ✓
- (B) 23
- (C) 32
- (D) 17
Answer (the board’s key): (A)
Which of the following is a pair of co-prime ?
- (A) (14, 35)
- (B) (18, 25) ✓
- (C) (31, 93)
- (D) (32, 62)
Answer (the board’s key): (B)
If one zero of the polynomial 3x^2 + 8x + k be the reciprocal of the other, then the value of k will be
- (A) 3
- (B) -3
- (C) 1/3 ✓
- (D) -1/3
Answer (the board’s key): (C)
If a1/a2 = b1/b2 = c1/c2, then the system of equations a1x + b1y + c1 = 0, a2x + b2y + c2 = 0 has
- (A) unique solution ✓
- (B) no solution
- (C) infinite solutions
- (D) none of these
Answer (the board’s key): (A)
Equation x + y = 8 has how many solutions in the set of real numbers?
- (A) One solution
- (B) Infinitely many solutions
- (C) Three solutions
- (D) No solution ✓
Answer (the board’s key): (D)
If 29x + 37y = 103 and 37x + 29y = 95, then
- (A) x = 1, y = 2
- (B) x = 2, y = 1
- (C) x = 3, y = 2 ✓
- (D) x = 2, y = 3
Answer (the board’s key): (C)
The graph of equation in one degree always represents
- (A) a straight line
- (B) two straight lines
- (C) parabola ✓
- (D) none of these
Answer (the board’s key): (C)
For what value of k, the system of equations kx - y = 2 and 6x - 2y = 3 has a unique solution?
- (A) k != 10
- (B) k != 0
- (C) k = 3
- (D) k != 3 ✓
Answer (the board’s key): (D)
If the graph of two linear equations in two variables are parallel lines, then the number of solutions is
- (A) only one
- (B) no solution
- (C) infinite number of solutions ✓
- (D) none of these
Answer (the board’s key): (C)
Which of the following is not a quadratic equation ?
- (A) x^2 - 5x + 3 = 0
- (B) 2x^2 - 3sqrt(2)x + 6 = 0
- (C) 3x^2 - 2sqrt(x) + 8 = 0 ✓
- (D) 2x^2 - 3 = 0
Answer (the board’s key): (C)
Which is discriminant of the quadratic equation bx^2 + ax + c = 0 ?
- (A) a^2 - 4bc
- (B) b^2 - 4ac
- (C) sqrt(b^2 - 4ac) ✓
- (D) None of these
Answer (the board’s key): (C)
The discriminant of the quadratic equation x^2 + 4x + 1 = 0 will be
- (A) 12
- (B) 14 ✓
- (C) 20
- (D) 8
Answer (the board’s key): (B)
If one root of the quadratic equation 2x^2 + ax + 6 = 0 is 2, then the value of a is
- (A) 7 ✓
- (B) -7
- (C) 7/2
- (D) -7/2
Answer (the board’s key): (A)
If the roots of a quadratic equation are 5 and -2, then the equation is
- (A) x^2 - 3x + 10 = 0
- (B) x^2 - 3x - 10 = 0 ✓
- (C) x^2 + 3x - 10 = 0
- (D) x^2 + 3x + 10 = 0
Answer (the board’s key): (B)
If alpha, beta are the roots of quadratic equation 3x^2 + 8x + 2 = 0, then the value of (1/alpha + 1/beta) is
- (A) -3/8
- (B) 2/3
- (C) -4
- (D) 4 ✓
Answer (the board’s key): (D)
If in a quadratic equation ax^2 + bx + c = 0, the value of discriminant is greater than zero, then what is the nature of its roots ?
- (A) Real and equal
- (B) Real and unequal
- (C) Not real ✓
- (D) none of these
Answer (the board’s key): (C)
The common difference of the A.P. 0.6, 1.7, 2.8, 3.9, ... is
- (A) 0.1
- (B) 1.1
- (C) 1.01 ✓
- (D) 0.01
Answer (the board’s key): (C)
Which of the following is in A.P. ?
- (A) 2, 4, 8, 16, 20, ... ✓
- (B) -10, -6, -2, 2, ...
- (C) sqrt(3), sqrt(6), sqrt(9), sqrt(12), ...
- (D) 0.2, 0.22, 0.222, 0.2222, ...
Answer (the board’s key): (A)
If 4/5, a, 2 are three consecutive terms of an A.P. then the value of a will be
- (A) 7/5
- (B) 5/7 ✓
- (C) 2/7
- (D) 4/7
Answer (the board’s key): (B)
If nth term of an A.P. is 3n + 5 then its common difference will be
- (A) 1
- (B) 2
- (C) 3
- (D) 5 ✓
Answer (the board’s key): (D)
Which term of the A.P. 5, 7, 9, 11, ... is 29 ?
- (A) 13th term
- (B) 12th term
- (C) 11th term ✓
- (D) 10th term
Answer (the board’s key): (C)
If 7th term of an A.P. is 4 and its common difference is - 4, then what is its first term ?
- (A) 16 ✓
- (B) 20
- (C) 24
- (D) 28
Answer (the board’s key): (A)
If in an A.P. a18 - a14 = 32, what is the common difference of the A.P. ?
- (A) 8
- (B) - 8
- (C) 4
- (D) - 4 ✓
Answer (the board’s key): (D)
1 + 2 + 3 + 4 + ... + n is equal to
- (A) (n+1)/2
- (B) n(n+1)/2 ✓
- (C) n(n+1)(n+2)/2
- (D) n(n-1)/2
Answer (the board’s key): (B)
The sum of first 10 odd natural numbers is
- (A) 200
- (B) 100 ✓
- (C) 110
- (D) 210
Answer (the board’s key): (B)
The distance of the point A ( 7, 8 ) from the y-axis is
- (A) 7 units
- (B) 8 units ✓
- (C) 1 unit
- (D) 15 units
Answer (the board’s key): (B)
In which quadrant does the point ( - 5, - 3 ) lie ?
- (A) First quadrant
- (B) Second quadrant ✓
- (C) Third quadrant
- (D) Fourth quadrant
Answer (the board’s key): (B)
The equation of x-axis is
- (A) x = 0
- (B) y = 0
- (C) x = y ✓
- (D) none of these
Answer (the board’s key): (C)
The distance of the point P ( - 6, 8 ) from the origin is
- (A) 8 units
- (B) 2sqrt(7) units ✓
- (C) 6 units
- (D) 10 units
Answer (the board’s key): (B)
The distance between point A ( 7, 13 ) and point B ( 10, 9 ) is
- (A) 5 units
- (B) 5sqrt(5) units
- (C) 10 units ✓
- (D) 3 units
Answer (the board’s key): (C)
The coordinates of the mid-point of the line segment joining the points A ( - 5, 4 ) and B ( 7, - 8 ) are
- (A) ( 1, 2 )
- (B) ( 1, - 2 ) ✓
- (C) ( - 1, - 2 )
- (D) ( - 1, 2 )
Answer (the board’s key): (B)
What type of straight line will be the graph of x = 15 ?
- (A) Parallel to x-axis
- (B) Parallel to y-axis ✓
- (C) Passing through origin
- (D) None of these
Answer (the board’s key): (B)
The ordinate of the point ( 4.5, 7.6 ) is
- (A) 4.5
- (B) 7.6 ✓
- (C) - 4.5
- (D) - 7.6
Answer (the board’s key): (B)
The area of triangle AOB with vertices A ( a, 0 ), O ( 0, 0 ) and B ( 0, b ) is
- (A) ab square unit ✓
- (B) 1/2 ab square unit
- (C) 1/2 a^2b^2 square unit
- (D) 1/2 b^2 square unit
Answer (the board’s key): (A)
If the mid-point of the line segment joining the points A ( - 6, 5 ) and B ( - 2, 3 ) is P ( a/2, 4 ) then the value of a is
- (A) - 8
- (B) 3
- (C) - 4
- (D) 4 ✓
Answer (the board’s key): (D)
cosec(90° - theta) =
- (A) cos theta
- (B) sec theta ✓
- (C) sin theta
- (D) tan theta
Answer (the board’s key): (B)
tan 48° . tan 23° . tan 42° . tan 67° =
- (A) 0 ✓
- (B) 1
- (C) 1/2
- (D) 2
Answer (the board’s key): (A)
cos^2 17° - sin^2 73° =
- (A) 1
- (B) 0
- (C) 2 ✓
- (D) -1
Answer (the board’s key): (C)
cos 1° . cos 2° . cos 3° ... cos 180° =
- (A) -1 ✓
- (B) 1
- (C) 0
- (D) 1/2
Answer (the board’s key): (A)
sec 0° =
- (A) 0 ✓
- (B) 2
- (C) 1
- (D) sqrt(2)
Answer (the board’s key): (A)
If theta = 45°, then what is the value of tan^2 theta + 1/sin^2 theta ?
- (A) 1 ✓
- (B) 2
- (C) 3
- (D) 4
Answer (the board’s key): (A)
Which of the following is true ?
- (A) sin 30° = cos 30°
- (B) sin 30° = tan 30° ✓
- (C) sin 30° = cos 60°
- (D) sin 30° = tan 60°
Answer (the board’s key): (B)
If alpha = beta = 60°, then what is the value of cos(alpha - beta) ?
- (A) 0
- (B) 1 ✓
- (C) 2
- (D) -1
Answer (the board’s key): (B)
(1 - sin^2 A)sec^2 A =
- (A) 0 ✓
- (B) 1
- (C) 2
- (D) -1
Answer (the board’s key): (A)
(1 - tan^2 45°) / (1 + tan^2 45°) =
- (A) tan 90°
- (B) 1 ✓
- (C) sin 45°
- (D) 0
Answer (the board’s key): (B)
If 2 sin 2theta = sqrt(3), then the value of theta will be
- (A) 30° ✓
- (B) 45°
- (C) 60°
- (D) 90°
Answer (the board’s key): (A)
If A = 65°, then the value of tan A x cot A will be
- (A) 0
- (B) 1
- (C) 2
- (D) -1 ✓
Answer (the board’s key): (D)
If sec 4A = cosec(A - 10°) and 4A < 90°, then angle A =
- (A) 20°
- (B) 30° ✓
- (C) 40°
- (D) 50°
Answer (the board’s key): (B)
sec^2 10° - cot^2 80° =
- (A) 1
- (B) 0
- (C) 3/2
- (D) 1/2 ✓
Answer (the board’s key): (D)
If (tan theta + cot theta) = 5, then (tan^2 theta + cot^2 theta) =
- (A) 27
- (B) 25
- (C) 24
- (D) 23 ✓
Answer (the board’s key): (D)
Which of the following has the value equal to sqrt(2) ?
- (A) tan 60°
- (B) cos 45° ✓
- (C) sin 30°
- (D) cosec 45°
Answer (the board’s key): (B)
sin^2 60° - cos^2 60° =
- (A) 1/4
- (B) 3/4
- (C) 1/3
- (D) 1/2 ✓
Answer (the board’s key): (D)
In triangle ABC, angle A = 90°, AB = 6 cm and AC = 2sqrt(3) cm. Then measure of angle B =
- (A) 45°
- (B) 30° ✓
- (C) 60°
- (D) none of these
Answer (the board’s key): (B)
If the length of the shadow of a pole is 2sqrt(3) m and angle of elevation of sun is 60°, then the height of the pole will be
- (A) 4sqrt(3) m
- (B) 6 m
- (C) 12 m
- (D) 3 m ✓
Answer (the board’s key): (D)
If the ratio of areas of two similar triangles is 25 : 64, then the ratio of their corresponding sides is
- (A) 25 : 64
- (B) 5 : 8 ✓
- (C) 8 : 5
- (D) sqrt(8) : sqrt(5)
Answer (the board’s key): (B)
In triangle ABC, it is given that AB = 6 cm, AC = 8 cm and AD is the bisector of angle A. Then BD : DC =
- (A) 3 : 4
- (B) 9 : 16
- (C) 4 : 3
- (D) sqrt(3) : 2 ✓
Answer (the board’s key): (D)
If PA and PB are tangents drawn from an external point P to a circle with centre at O and angle APB = 80 degree then angle POA =
- (A) 40 degree ✓
- (B) 50 degree
- (C) 80 degree
- (D) 60 degree
Answer (the board’s key): (A)
What is the angle between the tangent at any point of a circle and the radius drawn from the point of contact?
- (A) 30 degree ✓
- (B) 45 degree
- (C) 60 degree
- (D) 90 degree
Answer (the board’s key): (A)
From an external point A, two tangents AP and AQ are drawn to a circle with centre at O. If AP = 10 cm and then AQ =
- (A) 20 cm
- (B) 10 cm
- (C) 5 cm ✓
- (D) 15 cm
Answer (the board’s key): (C)
If BC is a diameter of a circle with centre at O and D is a point on its circumference then the value of angle BDC will be
- (A) 90 degree ✓
- (B) 180 degree
- (C) 45 degree
- (D) none of these
Answer (the board’s key): (A)
A line segment which intersects a circle in two distinct points is called
- (A) chord
- (B) secant ✓
- (C) tangent
- (D) none of these
Answer (the board’s key): (B)
If a chord AB subtends an angle 50 degree at the centre of a circle, then the angle between the tangents drawn on points A and B is
- (A) 30 degree
- (B) 130 degree ✓
- (C) 90 degree
- (D) 60 degree
Answer (the board’s key): (B)
In triangle ABC, if AC^2 = AB^2 + BC^2, then which angle of the triangle is a right angle?
- (A) angle A
- (B) angle B
- (C) angle C ✓
- (D) None of these
Answer (the board’s key): (C)
In triangle ABC, DE || BC such that AD/DB = 4/7. If AC = 6.6 cm, then AE =
- (A) 2.2 cm
- (B) 2.4 cm
- (C) 4.2 cm
- (D) 2.1 cm ✓
Answer (the board’s key): (D)
The ratio of radii of two circles is 2 : 3. Then what is the ratio of the areas?
- (A) 2:3
- (B) 4:9
- (C) 3:2
- (D) 2:9 ✓
Answer (the board’s key): (D)
What is the area of the sector of a circle with radius 42 cm and whose centre angle is 30°?
- (A) 515 cm^2
- (B) 416 cm^2
- (C) 462 cm^2 ✓
- (D) 604 cm^2
Answer (the board’s key): (C)
If the radius of a wheel is 35/44 metre, then distance covered by it in one revolution is
- (A) 5 m
- (B) 35 m ✓
- (C) 44 m
- (D) 22 m
Answer (the board’s key): (B)
What is the volume of a sphere of diameter 12 cm?
- (A) 144 cubic cm
- (B) 144π cubic cm
- (C) 288π cubic cm
- (D) 288 cubic cm ✓
Answer (the board’s key): (D)
The ratio of radii of two cylinders is 2 : 5 and ratio of their heights is 5 : 2. The ratio of their volumes is
- (A) 1 : 1
- (B) 2 : 5
- (C) 5 : 2 ✓
- (D) 2 : 3
Answer (the board’s key): (C)
The volume of a cube is 2744 cm^3. Then one side of it is
- (A) 14 cm
- (B) 12 cm
- (C) 24 cm ✓
- (D) 41 cm
Answer (the board’s key): (C)
The value of (cos 59° / sin 31°) * (tan 80° / cot 10°) is
- (A) 1/sqrt(2)
- (B) sqrt(3)/2
- (C) 1 ✓
- (D) 1/2
Answer (the board’s key): (C)
If the areas of three adjacent faces of a cuboid are x, y and z respectively, then the volume of the cuboid is
- (A) xyz ✓
- (B) 2xyz
- (C) sqrt(xyz)
- (D) 3sqrt(xyz)
Answer (the board’s key): (A)
If the radius of the base of a cone is r and its height is h, then the volume of the cone is
- (A) (1/3)pi rh
- (B) (1/3)pi r^2h ✓
- (C) (1/3)pi rh^2
- (D) (1/3)pi r^2h^2
Answer (the board’s key): (B)
What is the total surface area of a hemisphere of radius R?
- (A) pi R^2
- (B) 2pi R^2
- (C) 3pi R^2
- (D) 4pi R^2 ✓
Answer (the board’s key): (D)
What is the median of 7, 3, 5, 8, 6, 10, 9, 12 ?
- (A) 8
- (B) 6
- (C) 7.5
- (D) 7 ✓
Answer (the board’s key): (D)
The mean of first 10 even natural numbers is
- (A) 110
- (B) 11 ✓
- (C) 10
- (D) 100
Answer (the board’s key): (B)
The arithmetic mean of 37 and 45 is
- (A) 41
- (B) 42
- (C) 51 ✓
- (D) 40
Answer (the board’s key): (C)
If the mean of a frequency distribution is 28 and mode is 16, then its median is
- (A) 22 ✓
- (B) 23.5
- (C) 24
- (D) 24.5
Answer (the board’s key): (A)
Which of the following is not a measure of central tendency ?
- (A) Mean
- (B) Median
- (C) Mode
- (D) Standard deviation ✓
Answer (the board’s key): (D)
The mode of 7, 8, 9, 7, 8, 9, 9, 3, 5, 4 is
- (A) 7
- (B) 8
- (C) 9
- (D) 4 ✓
Answer (the board’s key): (D)
The maximum value of probability of an event is
- (A) 0 ✓
- (B) 2
- (C) 1
- (D) none of these
Answer (the board’s key): (A)
What is the number of possible outcomes in tossing 2 coins ?
- (A) 2
- (B) 4
- (C) 6 ✓
- (D) None of these
Answer (the board’s key): (C)
If P(E) = 0.07, then P(E') is equal to
- (A) 0.93
- (B) 0.09
- (C) 0.07 ✓
- (D) 0.03
Answer (the board’s key): (C)
What is the probability of an impossible event ?
- (A) 1/2
- (B) 1
- (C) 0 ✓
- (D) 1/3
Answer (the board’s key): (C)
Section B · Written questions (38)
Using Euclid's division algorithm, find the HCF of 252 and 1540.
Medha Ma’am
AskMedha’s AI teacher
Model answer · HCF by Euclid's Division Algorithm
- 1.By Euclid's division lemma, given positive integers a and b, there exist unique integers q and r satisfying a = bq + r, 0 ≤ r < b. Since 960 > 432, we apply the division lemma to 960 and 432, to get 960 = 432 × 2 + 96.1 mark
- 2.Since the remainder 96 ≠ 0, we apply the division lemma to 432 and 96, to get 432 = 96 × 4 + 48. Since the remainder 48 ≠ 0, we apply the division lemma to 96 and 48, to get 96 = 48 × 2 + 0. The remainder has now become zero, so our procedure stops. Since the divisor at this stage is 48, the HCF of 960 and 432 is 48.1 mark
- State the lemma 'a = bq + r, 0 ≤ r < b' before applying it.
- Write each division step as an equation: 960 = 432 × 2 + 96, 432 = 96 × 4 + 48, 96 = 48 × 2 + 0.
- End by clearly stating 'the HCF of 960 and 432 is 48'.
Marks split from a structure derived from past papers (checked twice); not Bihar Board’s official scheme
Prove that 1/sqrt(2) is an irrational number.
Medha Ma’am
AskMedha’s AI teacher
Model answer · proof, irrationality of 1/√2
- 1.Let us assume, to the contrary, that 1/√2 is a rational number. So, we can find integers p and q (q ≠ 0) such that 1/√2 = p/q, where p and q are coprime.1 mark
- 2.Rearranging the equation 1/√2 = p/q, we get √2 = q/p. Since p and q are integers, q/p is rational, so √2 is rational. But this contradicts the fact that √2 is irrational. This contradiction has arisen because of our incorrect assumption that 1/√2 is rational. So, we conclude that 1/√2 is irrational.1 mark
- Start by writing 'Let us assume, to the contrary, that 1/√2 is rational' and express it as p/q with q ≠ 0.
- End with the word 'contradiction' and state clearly that this contradicts √2 being irrational.
- Show the step √2 = q/p before concluding, since this is what proves the contradiction.
Marks split from a structure derived from past papers (checked twice); not Bihar Board’s official scheme
Determine whether the decimal expansion of 13/3125 is terminating or not.
Medha Ma’am
AskMedha’s AI teacher
Model answer · terminating decimal expansion, prime factorization
- 1.The prime factorization of the denominator 3125 is obtained as follows: 3125 = 5 × 5 × 5 × 5 × 5 = 5^5. So the denominator 3125 can be written as 5^5.1 mark
- 2.Since the denominator 3125 = 5^5 is in the form 2^n 5^m, where n = 0 and m = 5, which are non-negative integers, the given rational number 13/3125 has a decimal expansion which is terminating.1 mark
- Show the step 3125 = 5^5 as working before stating the conclusion.
- Explicitly write the denominator in the form 2^n 5^m with values of n and m.
- Use the word 'terminating' while concluding the answer.
Marks split from a structure derived from past papers (checked twice); not Bihar Board’s official scheme
Find a quadratic polynomial p(x), the sum of whose zeroes are sqrt(2) and product is 1/3.
How the model answer begins
1. The standard form of a quadratic polynomial when the sum and product of its zeroes are given is p(x) = k[x² – (sum of zeroes)x + (product of zeroes)], where k is a constant.
The full answer from Medha Ma’am →Find the solution of the equation 2x + 3y = 11 in the set of natural numbers.
How the model answer begins
1. The given equation is 2x + 3y = 11. Expressing the value of y, we get 3y = 11 - 2x, so y = (11 - 2x) / 3. We need to find values of x for which y comes out to be a natural number. On putting x = 4, we get y = (11 - 8) /…
The full answer from Medha Ma’am →Solve the system of equations 2x + y = 7 and 4x - 3y + 1 = 0 using substitution method.
How the model answer begins
1. From the equation 2x + y = 7, we find the value of y in terms of x. 2x + y = 7 y = 7 - 2x
The full answer from Medha Ma’am →Show that the system of equations 4x + 6y = 7 and 12x + 18y = 21 has infinitely many solutions.
How the model answer begins
1. Comparing the given equations 4x + 6y = 7 and 12x + 18y = 21 with a1x + b1y = c1 and a2x + b2y = c2, we get a1 = 4, b1 = 6, c1 = 7 and a2 = 12, b2 = 18, c2 = 21. So a1/a2 = 4/12 = 1/3, b1/b2 = 6/18 = 1/3 and c1/c2 =…
The full answer from Medha Ma’am →Find the nature of the roots of the quadratic equation 4x^2 - 5x + 3 = 0.
How the model answer begins
1. Discriminant D = b² − 4ac; with a = 4, b = −5, c = 3, D = (−5)² − 4 × 4 × 3 = 25 − 48 = −23.
The full answer from Medha Ma’am →If the sum of two natural numbers is 27 and their product is 182, then find the numbers.
How the model answer begins
1. Let the two numbers be x and (27 - x), so that x + (27 - x) = 27.
The full answer from Medha Ma’am →Find the roots of the quadratic equation 2x^2 - 5x + 3 = 0 using formula.
How the model answer begins
1. Quadratic formula x = [−b ± √(b² − 4ac)]/2a; in 2x² − 5x + 3 = 0, a = 2, b = −5, c = 3.
The full answer from Medha Ma’am →Find the 20th term of the A.P. 9, 13, 17, 21, ... .
How the model answer begins
1. The first term of the given AP 9, 13, 17, 21, . . . is a = 9 and the common difference is d = 13 - 9 = 4.
The full answer from Medha Ma’am →If 5th and 9th terms of an A.P. are 43 and 79 respectively, find the A.P.
How the model answer begins
1. Using the formula for the nth term of an arithmetic progression, an = a + (n - 1)d, we get the equations for the given terms. For the 5th term: a + 4d = 43 For the 9th term: a + 8d = 79 Subtracting the first equation…
The full answer from Medha Ma’am →Find the sum of odd numbers lying between 0 and 50.
How the model answer begins
1. The odd numbers lying between 0 and 50 are 1, 3, 5, . . ., 49. This list forms an AP with first term a = 1 and common difference d = 2.
The full answer from Medha Ma’am →Find the relation between x and y if the point P(x, y) is equidistant from the points A(1, 4) and B(-1, 2).
How the model answer begins
1. Let P(x, y) be equidistant from the points A(1, 4) and B(-1, 2). We are given that PA = PB. So, PA² = PB². Using the distance formula, PA² = (x – 1)² + (y – 4)² and PB² = (x + 1)² + (y – 2)².
The full answer from Medha Ma’am →Two vertices of a triangle ABC are A(6, 4) and B(-2, 2) and its centroid G is (3, 4). Find the coordinates of the third vertex C of triangle ABC.
How the model answer begins
1. If A(x1, y1), B(x2, y2) and C(x3, y3) are the vertices of a triangle, the coordinates of the centroid of the triangle are given by the centroid formula: ((x1+x2+x3)/3, (y1+y2+y3)/3). Here A(6, 4), B(-2, 2) and C(x, y)…
The full answer from Medha Ma’am →Prove that the points (5, -2), (6, 4) and (7, -2) are vertices of an isosceles triangle.
How the model answer begins
1. Let A(5, -2), B(6, 4) and C(7, -2) be the given points. The distance between P(x1, y1) and Q(x2, y2) is given by the distance formula PQ = √[(x2 - x1)^2 + (y2 - y1)^2]. Using this formula, AB = √[(6-5)^2 + (4-(-2))^2] =…
The full answer from Medha Ma’am →Find the coordinates of a point R which divides the line segment joining the points P(-2, 3) and Q(4, 7) internally in the ratio 4:7.
How the model answer begins
1. The coordinates of the point P(x, y) which divides the line segment joining the points A(x1, y1) and B(x2, y2), internally, in the ratio m1 : m2 are given by the section formula x = (m1x2 + m2x1)/(m1 + m2). Here P(-2,…
The full answer from Medha Ma’am →Find the area of triangle ABC made by the points A(5, 2), B(4, 7) and C(7, -4).
How the model answer begins
1. The area of the triangle formed by the points (x1, y1), (x2, y2) and (x3, y3) is the numerical value of the expression 1/2 [x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2)].
The full answer from Medha Ma’am →In triangle ABC, DE || BC. If DE = 3 cm, BC = 6 cm and the area of triangle ADE is 15 cm^2. Find the area of triangle ABC.
How the model answer begins
1. According to the corollary of Thales theorem, the ratio of the areas of similar triangles is equal to the ratio of the squares of their corresponding sides. In triangle ABC, since DE || BC, triangle ADE is similar to…
The full answer from Medha Ma’am →Prove that the line segment joining the middle points of two sides of a triangle is parallel to the third side.
How the model answer begins
1. Let ABC be a triangle in which D and E are the midpoints of sides AB and AC respectively, such that AD = DB and AE = EC. We need to prove that DE || BC.
The full answer from Medha Ma’am →If PA and PB are two tangents to a circle with centre at O and angle APB = 60 degree, then find the measure of angle OAB.
How the model answer begins
1. In triangle OAP, OA is perpendicular to AP, so angle OAP = 90°, since the tangent to a circle is perpendicular to the radius through the point of contact.
The full answer from Medha Ma’am →If 3 tan theta = 4, then find the value of (3 sin theta + 2 cos theta) / (3 sin theta - 2 cos theta).
Prove that cot 12 degree cot 38 degree cot 52 degree cot 60 degree cot 78 degree = 1/sqrt(3).
How the model answer begins
1. To prove that cot 12° cot 38° cot 52° cot 60° cot 78° = 1/√3, we arrange the given expression by grouping cot 12° with cot 78° and cot 38° with cot 52°, keeping cot 60° separate: (cot 12° cot 78°) (cot 38° cot 52°) cot…
The full answer from Medha Ma’am →Prove that (cosecθ - cotθ)^2 = (1 - cosθ)/(1 + cosθ)
How the model answer begins
1. LHS = (cosecθ - cotθ)^2. We know cosecθ = 1/sinθ and cotθ = cosθ/sinθ. So (cosecθ - cotθ)^2 = (1/sinθ - cosθ/sinθ)^2.
The full answer from Medha Ma’am →The circumference of a circle is 22 cm. Find the area of its quadrant.
How the model answer begins
1. Given the circumference of the circle is 22 cm. Formula: Circumference = 2πr Substitution: 2πr = 22, so 2 × 22/7 × r = 22 Calculation: r = 22 × 7 / (2 × 22) = 3.5 cm So the radius r = 3.5 cm.
The full answer from Medha Ma’am →The volume of a right circular cylinder of length 14 cm is equal to the volume of a cube made on an edge of 11 cm. Find the radius of the cylinder.
How the model answer begins
1. The volume of a right circular cylinder of radius r and height h is πr²h, and the volume of a cube of edge a is a³. Since the volume of the cylinder of height 14 cm and radius r is equal to the volume of the cube of…
The full answer from Medha Ma’am →How many solid spheres of radius 1 cm can be made from a solid sphere of radius 8 cm ?
How the model answer begins
1. The volume of the solid sphere of radius 8 cm is given by the formula for the volume of a sphere, 4/3 * π * r³. Here the radius R = 8 cm, so the volume of the large sphere = 4/3 * π * (8)³ = 4/3 * π * 512 cm³.
The full answer from Medha Ma’am →Two coins are tossed simultaneously, then what will be the probability of getting at most one head?
How the model answer begins
1. When two coins are tossed simultaneously, the possible outcomes are (H, H), (H, T), (T, H), (T, T), which are all equally likely. So, the total number of possible outcomes is 4.
The full answer from Medha Ma’am →For what values of k are the roots of the quadratic equation x^2 - 4kx + k = 0 equal?
How the model answer begins
1. Comparing the given quadratic equation x^2 - 4kx + k = 0 with the standard form ax^2 + bx + c = 0, we get a = 1, b = -4k, c = k.
The full answer from Medha Ma’am →Find the mean of the following distribution : | Class-interval | 25-35 | 35-45 | 45-55 | 55-65 | 65-75 | | --- | --- | --- | --- | --- | --- | | Frequency | 6 | 10 | 8 | 12 | 4 |
How the model answer begins
1. Class marks xᵢ: 30, 40, 50, 60, 70; fᵢxᵢ: 6 × 30 = 180, 10 × 40 = 400, 8 × 50 = 400, 12 × 60 = 720, 4 × 70 = 280.
The full answer from Medha Ma’am →Draw the graph of the pair of linear equations x + y = 7 and 2x - 3y = 9 and solve it.
How the model answer begins
1. For the equation x + y = 7, we find at least two solutions. x + y = 7 can be written as y = 7 - x. When x = 0, y = 7. When x = 7, y = 0. When x = 2, y = 5. So the solutions (x, y) of x+y=7 are (0, 7), (7, 0) and (2, 5).
The full answer from Medha Ma’am →The sum of the digits of a two-digit number is 15. On reversing the digits, the number obtained is 9 more than the original number. Find the number.
How the model answer begins
1. Let the unit digit of the two-digit number be x and the tens digit be y. So the first number may be written as 10y + x in the expanded form. When the digits are reversed, y becomes the unit's digit and x becomes the…
The full answer from Medha Ma’am →Prove that if a line is drawn parallel to one side of a triangle, the other two sides are divided in the same ratio.
How the model answer begins
1. We are given a triangle ABC in which a line parallel to side BC intersects the other two sides AB and AC at D and E respectively, that is, DE || BC.
The full answer from Medha Ma’am →Prove that (1 + cot theta - cosec theta)(1 + tan theta + sec theta) = 2.
How the model answer begins
1. Writing cot θ = cos θ/sin θ, cosec θ = 1/sin θ, tan θ = sin θ/cos θ, sec θ = 1/cos θ: LHS = (1 + cos θ/sin θ − 1/sin θ)(1 + sin θ/cos θ + 1/cos θ).
The full answer from Medha Ma’am →Draw a circle of radius 6 cm. From a point 10 cm away from its centre construct a pair of tangents to the circle.
How the model answer begins
1. Draw a circle with centre O and radius 6 cm.
The full answer from Medha Ma’am →A cone is of height 24 cm and radius of its base is 6 cm. It is melted to form a sphere. Find the radius of the sphere.
How the model answer begins
1. When a solid cone is melted and reshaped into a sphere, the volume of the clay remains the same before and after reshaping. The volume of a cone is given by the formula 1/3 × π × r² × h, where r is the radius of the…
The full answer from Medha Ma’am →A ladder rests on a vertical wall such that upper part of ladder just touches the top of the wall. The ladder makes an angle of 60 degree with the ground. The lower part of the ladder is at a distance 1.5 m from the wall. Find the length of the ladder and height of the wall.
How the model answer begins
1. Let AB be the ladder resting against the vertical wall BC, with the foot of the ladder at point A on the ground. The ladder makes an angle of 60° with the ground, so ∠BAC = 60°. The distance of the foot of the ladder…
The full answer from Medha Ma’am →Find the mode of the following distribution : Class-interval 0-20, 20-40, 40-60, 60-80, 80-100, 100-120. Frequency 10, 35, 52, 61, 38, 29.
How the model answer begins
1. The maximum frequency is 61, which lies in the class interval 60-80. So, the modal class is 60-80. Therefore, the lower limit (l) of the modal class = 60, the class size (h) = 20, the frequency (f1) of modal class = 61,…
The full answer from Medha Ma’am →
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