Bihar Board Class 10 · Maths · Matric exam 2027
Bihar Board Class 10 Maths Question Paper 2021
The board’s 2021 question paper, set H — 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.
Section A · Objective questions (100)
If the midpoint of the line segment joining the points (6, -5) and (-2, 11) is (2, p), then the value of p will be
- (A) 2
- (B) 3 ✓
- (C) -3
- (D) None of these
Answer (the board’s key): (B)
The ordinate of the point (-6, -7) is
- (A) -6
- (B) 7
- (C) -7 ✓
- (D) 6
Answer (the board’s key): (C)
The equation of x-axis is
- (A) x = 0
- (B) y = 0 ✓
- (C) x = y
- (D) None of these
Answer (the board’s key): (B)
The value of sec^2 10° - cot^2 80° is
- (A) 1 ✓
- (B) 0
- (C) 3/2
- (D) 1/2
Answer (the board’s key): (A)
If the coordinates of the vertices of a triangle are (-1, 0), (5, -2) and (8, 2), then its centroid will be
- (A) (0, 4)
- (B) (4, 0) ✓
- (C) (0, -4)
- (D) (-4, 0)
Answer (the board’s key): (B)
If the radius of a circle is 7 cm, then the perimeter of its semicircle will be
- (A) 36 cm ✓
- (B) 63 cm
- (C) 37 cm
- (D) None of these
Answer (the board’s key): (A)
How many tangents can be drawn to a circle from an external point?
- (A) 1
- (B) 2 ✓
- (C) 3
- (D) 4
Answer (the board’s key): (B)
The value of 1 + cot^2 theta is
- (A) cot^2 theta
- (B) sin^2 theta
- (C) cosec^2 theta ✓
- (D) cos^2 theta
Answer (the board’s key): (C)
The value of tan 10° . tan 15° . tan 75° . tan 80° will be
- (A) sqrt(3)
- (B) 1/sqrt(3)
- (C) -1
- (D) 1 ✓
Answer (the board’s key): (D)
If tan theta = sqrt(3) then sec theta will be
- (A) 2/sqrt(3)
- (B) sqrt(3)/2
- (C) 1/2
- (D) 2 ✓
Answer (the board’s key): (D)
If alpha and beta are the roots of the quadratic equation ax^2 + bx = c, (a != 0), then the value of alpha + beta will be
- (A) -b/a ✓
- (B) b/a
- (C) -a/b
- (D) -c/a
Answer (the board’s key): (A)
If one root of the equation 3x^2 - 10x + 3 = 0 is 1/3, then the other root will be
- (A) -1/3
- (B) 1/3
- (C) -3
- (D) 3 ✓
Answer (the board’s key): (D)
If the roots of the quadratic equation 9x^2 + kx + 4 = 0 are real and equal, then the value of k will be
- (A) 2 or 0
- (B) -2 or 0
- (C) 2 or -2 ✓
- (D) Only 0
Answer (the board’s key): (C)
The product of the roots of the quadratic equation 4y^2 + 4y + 1 = 0 will be
- (A) 1/4 ✓
- (B) -1/4
- (C) 1
- (D) None of these
Answer (the board’s key): (A)
If x + 2, 3x and 4x + 1 are in arithmetic progression, then the value of x will be
- (A) 1
- (B) 2
- (C) 3 ✓
- (D) 4
Answer (the board’s key): (C)
The common difference of the arithmetic progression 0, -4, -8, -12, ... is
- (A) 0
- (B) 4
- (C) -4 ✓
- (D) 2
Answer (the board’s key): (C)
The 12th term of the arithmetic progression 14, 9, 4, -1, -6, ... is
- (A) 41
- (B) -41 ✓
- (C) 40
- (D) 36
Answer (the board’s key): (B)
If the first term of an arithmetic progression is 13 and the common difference is -4, then the sum of its first 10 terms will be
- (A) 50
- (B) -50 ✓
- (C) 30
- (D) -30
Answer (the board’s key): (B)
Which of the following is the point on the y-axis whose distance from the point (4, 2) is 5?
- (A) (-1, 0)
- (B) (0, -1) ✓
- (C) (2, -5)
- (D) (3, 5)
Answer (the board’s key): (B)
If R (5, 6) is the midpoint of the line segment AB joining the points A (6, 5) and B (4, y), then the value of y is
- (A) 5
- (B) 7 ✓
- (C) 12
- (D) 6
Answer (the board’s key): (B)
The median of 7, 3, 5, 8, 6, 10, 9, 12 is
- (A) 7
- (B) 7.5 ✓
- (C) 8
- (D) None of these
Answer (the board’s key): (B)
What is the mode of 9, 3, 4, 7, 2, 9, 6, 7, 9, 9?
- (A) 2
- (B) 3
- (C) 7
- (D) 9 ✓
Answer (the board’s key): (D)
Which of the following is a prime number?
- (A) 27
- (B) 35
- (C) 13 ✓
- (D) 56
Answer (the board’s key): (C)
If the mean of a frequency distribution is 8.9 and the median is 9, then the mode will be
- (A) 7.2
- (B) 8.2
- (C) 9.2 ✓
- (D) 10.2
Answer (the board’s key): (C)
The arithmetic mean of 19 and 25 is
- (A) 21
- (B) 22 ✓
- (C) 23
- (D) 24
Answer (the board’s key): (B)
For any event E, P(E) + P(not E) = ?
- (A) 0
- (B) 1 ✓
- (C) 1/2
- (D) -1
Answer (the board’s key): (B)
The probability of getting 2 heads in the toss of two coins will be
- (A) 1
- (B) 3/4
- (C) 1/2
- (D) 1/4 ✓
Answer (the board’s key): (D)
If the height of a vertical pillar is sqrt(3) times the length of its shadow on the ground, then the angle of elevation of the sun will be
- (A) 30°
- (B) 45°
- (C) 60° ✓
- (D) 75°
Answer (the board’s key): (C)
If a and b are prime numbers, then LCM(a, b) will be
- (A) a
- (B) b
- (C) ab ✓
- (D) a/b
Answer (the board’s key): (C)
Which of the following cannot be the probability of an event?
- (A) 1/3
- (B) 0.3
- (C) 33%
- (D) 7/6 ✓
Answer (the board’s key): (D)
If the coordinates of the ends of a diameter of a circle are (2, 3) and (-6, 7), then the coordinates of its centre will be
- (A) (-2, 5) ✓
- (B) (2, -5)
- (C) (2, 5)
- (D) (3, 5)
Answer (the board’s key): (A)
In triangle ABC, P and Q are points on sides AB and AC respectively such that PQ || BC. If AP = 3 cm, PB = 2 cm, AQ = 6 cm and QC = x cm, then the value of x will be
- (A) 2 cm
- (B) 4 cm ✓
- (C) 8 cm
- (D) 10 cm
Answer (the board’s key): (B)
The ratio of the areas of two similar triangles is 9:4, then the ratio of their corresponding heights will be
- (A) 2:3
- (B) 3:2 ✓
- (C) 4:9
- (D) 9:4
Answer (the board’s key): (B)
In triangle PQR, if PQ^2 = PR^2 + RQ^2, then which angle of the triangle will be a right angle?
- (A) <P
- (B) <Q
- (C) <R ✓
- (D) None of these
Answer (the board’s key): (C)
A line intersecting a circle at two distinct points is called
- (A) Chord
- (B) Secant ✓
- (C) Tangent
- (D) None of these
Answer (the board’s key): (B)
If chord AB subtends an angle of 60° at the centre of a circle, then the angle between the tangents drawn at points A and B will be
- (A) 30°
- (B) 60°
- (C) 90°
- (D) 120° ✓
Answer (the board’s key): (D)
How many tangents can be drawn on a circle?
- (A) 1
- (B) 2
- (C) 3
- (D) Infinite ✓
Answer (the board’s key): (D)
sqrt(1 + tan^2 A) = ? (0° <= A <= 90°)
- (A) cot A
- (B) sec A ✓
- (C) cos A
- (D) sin A
Answer (the board’s key): (B)
cosec^2 57° - tan^2 33° = ?
- (A) 0
- (B) 1 ✓
- (C) -1
- (D) 2
Answer (the board’s key): (B)
cos 48° - sin 42° = ?
- (A) 1
- (B) 0 ✓
- (C) 2
- (D) -1
Answer (the board’s key): (B)
The ratio of the sum of the roots and the product of the roots of the equation 7x^2 - 12x + 18 = 0 will be
- (A) 7:12
- (B) 7:18
- (C) 3:2
- (D) 2:3 ✓
Answer (the board’s key): (D)
If the roots of the quadratic equation ax^2 + bx + c = 0 are equal, then the value of c will be
- (A) b/2a
- (B) b/2a
- (C) -b^2/4a
- (D) b^2/4a ✓
Answer (the board’s key): (D)
What kind of straight line will the graph of y = 10 be?
- (A) Parallel to x-axis ✓
- (B) Parallel to y-axis
- (C) Passing through origin
- (D) None of these
Answer (the board’s key): (A)
Which of the following points lies in the third quadrant?
- (A) (4, 6)
- (B) (-4, -6) ✓
- (C) (4, -6)
- (D) (-4, 6)
Answer (the board’s key): (B)
If 'a' is the first term and 'd' is the common difference of an arithmetic progression, then the sum of its first 'n' terms will be
- (A) a + (n - 1)d
- (B) a + nd
- (C) n/2 [2a + (n - 1)d] ✓
- (D) n/2 [2a + nd]
Answer (the board’s key): (C)
The 10th term of the arithmetic progression 2, 7, 12, ... is
- (A) 50
- (B) 40
- (C) 47 ✓
- (D) 53
Answer (the board’s key): (C)
The n-th term of the arithmetic progression 14, 9, 4, -1, -6, ... will be
- (A) (19 + 5n)
- (B) (19 - 5n) ✓
- (C) (19 + n)
- (D) None of these
Answer (the board’s key): (B)
The sum of the first 10 terms of the arithmetic progression 2, 7, 12, ... will be
- (A) 245 ✓
- (B) 240
- (C) 244
- (D) 254
Answer (the board’s key): (A)
The values of x in the quadratic equation ax^2 + bx + c = 0 (a != 0) will be
- (A) (-b ± sqrt(b^2 + 4ac)) / 2a
- (B) (-b ± sqrt(4ac - b^2)) / 2a
- (C) (-b ± sqrt(b^2 - 4ac)) / 2a ✓
- (D) b ± sqrt(4ac)
Answer (the board’s key): (C)
The distance between the points (9, 3) and (15, 11) is
- (A) 11
- (B) 10 ✓
- (C) 13
- (D) 12
Answer (the board’s key): (B)
If the circumference of a circular field is 4pi meters, then its radius will be
- (A) 1 meter
- (B) 2 meters ✓
- (C) pi meters
- (D) None of these
Answer (the board’s key): (B)
The total perimeter of a semicircle with radius 6 cm will be
- (A) (pi + 1) 6 cm
- (B) (pi + 2) 6 cm ✓
- (C) (pi + 3) 6 cm
- (D) 6pi cm
Answer (the board’s key): (B)
What will be the length of the arc of a circle with radius 21 cm that subtends an angle of 60° at the centre of the circle?
- (A) 21 cm
- (B) 22 cm ✓
- (C) 23 cm
- (D) None of these
Answer (the board’s key): (B)
The ratio of the volumes of two cubes is 1 : 27, then the ratio of their surface areas will be
- (A) 1 : 3
- (B) 1 : 8
- (C) 1 : 9 ✓
- (D) 1 : 18
Answer (the board’s key): (C)
The ratio of the total surface area and the curved surface area of a cylinder whose base radius is 80 cm and height is 20 cm is
- (A) 2:1
- (B) 3:1
- (C) 4:1
- (D) 5:1 ✓
Answer (the board’s key): (D)
The median of 8, 7, 12, 15, 10, 9, 11 will be
- (A) 10 ✓
- (B) 11
- (C) 9
- (D) None of these
Answer (the board’s key): (A)
The mode of 70, 80, 50, 60, 70, 40, 50, 90, 50, 30 will be
- (A) 70
- (B) 50 ✓
- (C) 40
- (D) None of these
Answer (the board’s key): (B)
In grouped data, if x_i are the mid-points of class intervals, f_i are the corresponding frequencies and x_bar is the mean, then (sum(f_i * x_i) / sum(f_i)) =
- (A) x_bar ✓
- (B) 2x_bar
- (C) 3x_bar
- (D) None of these
Answer (the board’s key): (A)
If the probability of an event occurring is p, then the probability of it not occurring will be
- (A) (p - 1)
- (B) (1 - p) ✓
- (C) p
- (D) (1 - 1/p)
Answer (the board’s key): (B)
Two dice are thrown together. The probability of getting 5 on both dice will be
- (A) 1/2
- (B) 1/36 ✓
- (C) 1/6
- (D) None of these
Answer (the board’s key): (B)
If the mean of 4, 5, 9, 11 and x is 7, then the value of x is
- (A) 6 ✓
- (B) 4
- (C) 8
- (D) 7
Answer (the board’s key): (A)
The nature of the roots of the quadratic equation 4x^2 + 4x + 1 = 0 will be
- (A) Real and unequal
- (B) Real and equal ✓
- (C) No real roots
- (D) None of these
Answer (the board’s key): (B)
If 3alpha = 90°, then the value of sin alpha will be
- (A) 1/sqrt(3)
- (B) sqrt(3)/4
- (C) 1/2 ✓
- (D) sqrt(3)/2
Answer (the board’s key): (C)
sin(45° + theta) - cos(45° - theta) = ?
- (A) 2 sin theta
- (B) 2 cos theta
- (C) 0 ✓
- (D) 1
Answer (the board’s key): (C)
The solutions of the linear equation pair 5x + 2y = 16 and 7x - 4y = 2 are
- (A) x = 2, y = 3 ✓
- (B) x = 2, y = 1
- (C) x = 1, y = 3
- (D) x = 0, y = 3
Answer (the board’s key): (A)
If triangle ABC ~ triangle DEF and area(ABC) = 36 cm^2 and area(DEF) = 49 cm^2, then the ratio of the corresponding sides of the two triangles will be
- (A) 36 : 49
- (B) 6 : 7 ✓
- (C) 7 : 6
- (D) sqrt(6) : sqrt(7)
Answer (the board’s key): (B)
In triangle ABC, DE || BC such that AD/DB = 3/5. If AC = 5.6 cm, then AE = ?
- (A) 4.2 cm
- (B) 3.1 cm
- (C) 2.8 cm
- (D) 2.1 cm ✓
Answer (the board’s key): (D)
A chord of a circle with radius 10 cm makes a right angle at the center of the circle, the length of that chord will be
- (A) 5/sqrt(2) cm
- (B) 5sqrt(2) cm
- (C) 10sqrt(2) cm ✓
- (D) 10sqrt(3) cm
Answer (the board’s key): (C)
The value of cos 60° cos 30° - sin 60° sin 30° is
- (A) 1
- (B) 2
- (C) 0 ✓
- (D) -1
Answer (the board’s key): (C)
The number of powers of the variable in a linear equation is
- (A) 1 ✓
- (B) 2
- (C) 3
- (D) None of these
Answer (the board’s key): (A)
Which of the following is not irrational?
- (A) pi
- (B) sqrt(7)
- (C) sqrt(2)/sqrt(5)
- (D) 3sqrt(75)/sqrt(48) ✓
Answer (the board’s key): (D)
If alpha and beta are the zeros of the quadratic polynomial x^2 - 3x + 5, then the value of (alpha + beta) will be
- (A) 3 ✓
- (B) 5
- (C) -3
- (D) -5
Answer (the board’s key): (A)
The zeros of the quadratic polynomial x^2 + 1/6x - 2 will be
- (A) (-3, 4)
- (B) (-3/2, 4/3) ✓
- (C) (-4/3, 3/2)
- (D) None of these
Answer (the board’s key): (B)
If the product of two numbers is 2166 and their HCF is 19, then their LCM will be
- (A) 38
- (B) 57
- (C) 114 ✓
- (D) None of these
Answer (the board’s key): (C)
If (x + 1) is a factor of the polynomial p(x), then a zero of the polynomial p(x) will be
- (A) -1 ✓
- (B) 1
- (C) 0
- (D) None of these
Answer (the board’s key): (A)
The graph of the system of equations 6x - 2y + 9 = 0 and 3x - y + 12 = 0 are two straight lines which are
- (A) Coincident
- (B) Parallel ✓
- (C) Intersect at only one point
- (D) None of these
Answer (the board’s key): (B)
If a = (2^3 * 3 * 5) and b = (2^4 * 5 * 7), then LCM(a, b) will be
- (A) 40
- (B) 560
- (C) 1120
- (D) 1680 ✓
Answer (the board’s key): (D)
If the zeros of a quadratic polynomial are 5 and -3, then the quadratic polynomial will be
- (A) x^2 + 2x - 15
- (B) x^2 - 2x + 15
- (C) x^2 - 2x - 15 ✓
- (D) None of these
Answer (the board’s key): (C)
The common difference of the arithmetic progression -10, -6, -2, 2, ... is
- (A) -4
- (B) 4 ✓
- (C) 2
- (D) None of these
Answer (the board’s key): (B)
Which of the following is not a quadratic equation?
- (A) 3x - x^2 = x^2 + 5
- (B) (x + 2)^2 = 2(x^2 - 5)
- (C) (sqrt(2)x + 3)^2 = 2x^2 + 6 ✓
- (D) (x - 1)^2 = 3x^2 + x - 2
Answer (the board’s key): (C)
If tan theta = 4/3, then (sin theta + cos theta) = ?
- (A) 7/3
- (B) 7/4
- (C) 7/5 ✓
- (D) 5/7
Answer (the board’s key): (C)
(cosec theta - cot theta)^2 = ?
- (A) (1 + cos theta) / (1 - cos theta)
- (B) (1 - cos theta) / (1 + cos theta) ✓
- (C) (1 + sin theta) / (1 - sin theta)
- (D) sin theta * cos theta
Answer (the board’s key): (B)
The length of the corresponding arc of a sector with angle theta in a circle of radius R will be
- (A) (2 * pi * R * theta) / 180
- (B) (2 * pi * R * theta) / 360 ✓
- (C) (pi * R^2 * theta) / 180
- (D) (pi * R^2 * theta) / 360
Answer (the board’s key): (B)
The value of sec 11° / cosec 79° will be
- (A) 1 ✓
- (B) 0
- (C) 2
- (D) -1
Answer (the board’s key): (A)
The diameter of a wheel is 40 cm. How many revolutions will it have to make to cover a distance of 176 m?
- (A) 140 ✓
- (B) 150
- (C) 160
- (D) 166
Answer (the board’s key): (A)
If the height of a cone is h and the radius of the base is r, then its volume will be
- (A) 1/3 * pi * r * h
- (B) 1/3 * pi * r^2 * h ✓
- (C) 1/3 * pi * r^3 * h
- (D) 1/3 * pi * r * h^2
Answer (the board’s key): (B)
If the diameter of a cylinder is 28 cm and its height is 20 cm, then its total surface area will be
- (A) 2993 cm^2
- (B) 2992 cm^2 ✓
- (C) 2292 cm^2
- (D) 2229 cm^2
Answer (the board’s key): (B)
If the radius of a sphere becomes 3 times, then its volume will become
- (A) 3 times
- (B) 6 times
- (C) 9 times
- (D) 27 times ✓
Answer (the board’s key): (D)
If the length of the diagonal of a cube is 6sqrt(3) cm, then its total surface area will be
- (A) 144 cm^2
- (B) 216 cm^2 ✓
- (C) 180 cm^2
- (D) 108 cm^2
Answer (the board’s key): (B)
If the curved surface area of a cone is 880 cm^2 and the radius is 14 cm, then its slant height will be
- (A) Not given
- (B) 20 cm ✓
- (C) 40 cm
- (D) 30 cm
Answer (the board’s key): (B)
The number 2.13 113 1113 11113 ... is
- (A) Integer
- (B) Rational number
- (C) Irrational number ✓
- (D) None of these
Answer (the board’s key): (C)
The HCF of co-prime numbers is
- (A) 2
- (B) 1 ✓
- (C) 3
- (D) None of these
Answer (the board’s key): (B)
If m is a positive integer, then the form of a positive odd integer will be
- (A) 4m + 2
- (B) 4m + 4
- (C) 4m + 1 ✓
- (D) None of these
Answer (the board’s key): (C)
The number of zeros of a linear polynomial is
- (A) 2
- (B) 1 ✓
- (C) 3
- (D) None of these
Answer (the board’s key): (B)
A zero of 2x^2 - 3x - 5 is
- (A) 1
- (B) -1 ✓
- (C) 0
- (D) None of these
Answer (the board’s key): (B)
If p(x) = q(x) * g(x) and degree of p(x) = 6 and degree of g(x) = 2, then the degree of p(x)/g(x) will be
- (A) 4 ✓
- (B) 6
- (C) 3
- (D) None of these
Answer (the board’s key): (A)
The sum of the zeros of the polynomial 6x^2 - 11x + 3 will be
- (A) 1/2
- (B) 6/11
- (C) 11/6 ✓
- (D) -11/6
Answer (the board’s key): (C)
How many solutions will the linear equation ax + by + c = 0 in two variables x, y have in the set of real numbers?
- (A) 1
- (B) 2
- (C) 3
- (D) None of these ✓
Answer (the board’s key): (D)
How many solutions will the pair of linear equations 2x + 3y = 5 and 4x + 6y = 15 have?
- (A) Unique solution
- (B) Three solutions
- (C) Infinite solutions
- (D) No solution ✓
Answer (the board’s key): (D)
If a pair of linear equations is consistent, then the lines represented by them are
- (A) Parallel
- (B) Always intersecting
- (C) Always coincident
- (D) Intersecting or coincident ✓
Answer (the board’s key): (D)
Section B · Written questions (38)
Find a quadratic polynomial whose sum of zeros is sqrt(2) and product of zeros is 1/3.
Medha Ma’am
AskMedha’s AI teacher
Model answer · quadratic polynomial from zeroes
- 1.A quadratic polynomial, the sum and product of whose zeroes are given, is found using the formula: quadratic polynomial = x² – (sum of zeroes)x + (product of zeroes).1 mark
- 2.Substituting the given values, sum of zeroes = √2 and product of zeroes = 1/3, in x² – (sum of zeroes)x + (product of zeroes), the polynomial obtained is x² – √2x + 1/3.1 mark
- Write the formula x² – (α+β)x + αβ first, using the words sum of zeroes and product of zeroes.
- Show the substitution step clearly with √2 and 1/3 before giving the final polynomial.
- End with the final answer x² – √2x + 1/3 written exactly in this form.
Marks split from a structure derived from past papers (checked twice); not Bihar Board’s official scheme
Write the solutions for the equation 3x + y = 11 in the set of natural numbers.
Medha Ma’am
AskMedha’s AI teacher
Model answer · solutions of linear equation in N
- 1.Substituting x = 1, 2, 3 in the equation 3x + y = 11, we get the corresponding values of y, since x and y are to be taken from the set of natural numbers N = {1, 2, 3, ...}. When x = 1, y = 11 - 3 = 8. When x = 2, y = 11 - 6 = 5. When x = 3, y = 11 - 9 = 2.1 mark
- 2.The solutions of the equation 3x + y = 11 in the set of natural numbers, written as ordered pairs (x, y), are (1, 8), (2, 5), and (3, 2).1 mark
- Mention that x and y belong to the set of natural numbers N before substituting values.
- Show each substitution step for x = 1, 2, 3 leading to y = 8, 5, 2.
- Write the final solutions as ordered pairs (1, 8), (2, 5), (3, 2).
Marks split from a structure derived from past papers (checked twice); not Bihar Board’s official scheme
For what value of k will the pair of linear equations kx + 3y - (k - 3) = 0 and 12x + ky - k = 0 have infinitely many solutions?
Medha Ma’am
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Model answer · value of k, infinitely many solutions
- 1.For infinitely many solutions a1/a2 = b1/b2 = c1/c2, i.e. k/12 = 3/k = (k − 3)/k.1 mark
- 2.k/12 = 3/k gives k^2 = 36, so k = ±6; 3/k = (k − 3)/k gives k − 3 = 3, so k = 6. Both hold only for k = 6.1 mark
Marks split from a structure derived from past papers (checked twice); not Bihar Board’s official scheme
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