Bihar Board Class 10 · Maths · Matric exam 2027
Bihar Board Class 10 Maths Model Paper 2023
The board’s 2023 model paper — all 138 questions as printed. Each question names its chapter, where that chapter’s most-asked questions are.
Section A · Objective questions (100)
In the form p/q (where p, q are integers, q!=0), 0.41 bar can be expressed as
- (A) 41/100
- (B) 41/90
- (C) 41/99
- (D) 41/9
Which of the following is an irrational number ?
- (A) sqrt(9)
- (B) sqrt(8)
- (C) sqrt(16)
- (D) sqrt(25)
The zero of the polynomial 6x - 21 is
- (A) 7/2
- (B) -7/2
- (C) 7/3
- (D) -7/3
Degree of the polynomial x^2 - 10x + 25 is
- (A) 1
- (B) 2
- (C) 0
- (D) 25
Which of the following is a quadratic equation ?
- (A) x^2 + 5sqrt(x) + 3 = 7
- (B) (x - 1)^3 = x^2
- (C) (x + 1)^2 = 2(x + 1)
- (D) (x + 1)^3 = 9
If m and n are prime numbers then the H.C.F. of m^3n^2 and m^2n^2 is
- (A) m^3n^2
- (B) m^2n
- (C) m^2n^2
- (D) mn
If the L.C.M. of a and 18 is 36 and the H.C.F. of a and 18 is 2 then a =
- (A) 3
- (B) 4
- (C) 2
- (D) 1
The product of two irrational numbers 2 + sqrt(3) and 2 - sqrt(3) is a/an
- (A) rational number
- (B) irrational number
- (C) either A or B
- (D) none of these
The zeroes of the polynomial x^2 + x - 20 are
- (A) -4, -5
- (B) 5, 4
- (C) -5, 4
- (D) 5, -4
If alpha, beta are zeros of any quadratic polynomial p(x) = 2x^2 + 5x - 12 then alpha + beta =
- (A) 5/2
- (B) -5/2
- (C) 5/12
- (D) 12/5
If alpha, beta are the zeroes of the polynomial x^2 + 4x + 1 then 1/alpha + 1/beta =
- (A) 2
- (B) 4
- (C) -4
- (D) 1
If the sum and product of roots of the equation Kx^2 + 6x + 9K = 0 are equal then the value of K is
- (A) 3/2
- (B) -2/3
- (C) 2/3
- (D) -3/2
If x = 1 is a root of both the equations x^2 + x + a = 0 and bx^2 + bx + 3 = 0 then ab =
- (A) 4
- (B) 3
- (C) -3
- (D) -4
If 2^(x+y) = 2^(x-y) = 2^(3/2) then the value of y is
- (A) 3/2
- (B) 1/2
- (C) 0
- (D) none of these
cos theta * cosec theta =
- (A) tan theta
- (B) cot theta
- (C) sec theta
- (D) cos theta
If zeroes of the polynomial x^2 + ax - b be reciprocal of each other then b =
- (A) 2
- (B) 1
- (C) -1
- (D) a
If equation ax^2 + bx + c = 0 has equal roots then the value of each root is
- (A) -b/a
- (B) -b/2a
- (C) b/2a
- (D) a/b
Which of the following are the roots of (x + 2) (x - 2) = 5 ?
- (A) 2, -2
- (B) 3, -3
- (C) 4, -4
- (D) 5, -5
If 6 and -2 be the zeroes of a quadratic polynomial p(x) = x^2 + bx + c then the values of b and c are respectively
- (A) 4, 12
- (B) -4, 12
- (C) 4, -12
- (D) -4, -12
Which of the following is a linear polynomial ?
- (A) 2x - 9sqrt(5)
- (B) x + 1/x
- (C) 5 - 1/z^2
- (D) z^2 - 1/z
The maximum number of zeroes of a cubic polynomial is
- (A) 1
- (B) 2
- (C) 3
- (D) 4
The sum of the zeroes of the polynomial m^3 - 3m^2 - m + 3 is
- (A) -3
- (B) 2
- (C) 3
- (D) 4
The discriminant of the equation bx^2+bx+c=0 is
- (A) b^2 - 4ac
- (B) c^2 - 4bc
- (C) b^2 - 4bc
- (D) c^2 - 4b^2
If x-y = 2 and x+y = 10 then
- (A) x = 4, y = 2
- (B) x = 5, y = 3
- (C) x = 6, y = 4
- (D) x = 7, y = 5
If 5, a, b and 11 are in A. P. then the value of a+b is
- (A) 7
- (B) 9
- (C) 11
- (D) 16
The common difference of the A. P. : 1/p, (1-p)/p, (1-2p)/p, ... is
- (A) p
- (B) -p
- (C) -1
- (D) 1
The 30th term of the A. P. : 10, 7, 4, ... is
- (A) 97
- (B) 77
- (C) -77
- (D) -97
Which term of A. P. : 5, 8, 11, ... is 383 ?
- (A) 124
- (B) 125
- (C) 126
- (D) 127
Which of the following is equal to sqrt(3)/2 ?
- (A) sin30^0
- (B) sin60^0
- (C) tan30^0
- (D) cos60^0
If tan(theta) =1 then what will be the value of (theta) ?
- (A) 30^0
- (B) 60^0
- (C) 45^0
- (D) 90^0
The value of sec^2(44^0) - cot^2(46^0) will be
- (A) 0
- (B) 1
- (C) -1
- (D) 2
2sec^2(53^0) - 2cot^2(37^0) =
- (A) 0
- (B) 1
- (C) -2
- (D) 2
The value of cosec^2(35^0) - tan^2(55^0) is
- (A) 0
- (B) 1
- (C) -1
- (D) 2
If sin(theta) = cos(theta) then the value of (theta) is
- (A) 0^0
- (B) 30^0
- (C) 45^0
- (D) 60^0
tan(theta) x cosec(theta) x sec(theta) =
- (A) 1
- (B) cos^2(theta)
- (C) sec^2(theta)
- (D) sin^2(theta)
If 4tan(theta) = 3 then the value of (cos^2(theta) - sin^2(theta)) is
- (A) 4/25
- (B) 7/25
- (C) 1
- (D) -7/25
tan23^0 / cot67^0 + cos12^0 / sin78^0 =
- (A) 0
- (B) 1
- (C) 2
- (D) 1/2
The number of possible outcomes in tossing two coins is
- (A) 2
- (B) 4
- (C) 9
- (D) none of these
In throwing a die, the probability of not getting 6 is
- (A) 1/6
- (B) 1/2
- (C) 5/6
- (D) 2/3
If P(E) = 0.07 then P(E') is equal to
- (A) 0.93
- (B) 0.08
- (C) 0.07
- (D) 0.91
A bag contains 4 white, 6 red, 7 black and 3 blue balls. One ball is taken out at random. The probability that it is white is
- (A) 1/20
- (B) 2/5
- (C) 3/20
- (D) 1/5
If sin(theta) = a/b then cos(theta) =
- (A) b/sqrt(b^2-a^2)
- (B) sqrt(b^2-a^2)/b
- (C) a/sqrt(b^2-a^2)
- (D) b/a
ABC is a right angled triangle in which <C=90^0 then the value of cos(A+B) is
- (A) 0
- (B) 1
- (C) 1/2
- (D) sqrt(3)/2
The value of (cosec(theta) - cot(theta))^2 will be
- (A) (1+cos(theta))/(1-cos(theta))
- (B) (1-cos(theta))/(1+cos(theta))
- (C) (1+sin(theta))/(1-sin(theta))
- (D) (1-sin(theta))/(1+sin(theta))
The value of cos60^0.cos30^0 - sin60^0.sin30^0 is
- (A) 0
- (B) 1
- (C) -1
- (D) 2
The angle of elevation of a tower at a point 100m away from the tower is 60^0 then the height of the tower is
- (A) 100sqrt(3) m
- (B) 100/sqrt(3) m
- (C) 50sqrt(3) m
- (D) 200/sqrt(3) m
If alpha = beta = 45^0 then the value of sin(alpha + beta) is
- (A) 1/sqrt(2)
- (B) 1
- (C) 0
- (D) 1/2
If sqrt(2) = 2sinA then the value of A is
- (A) 0^0
- (B) 30^0
- (C) 45^0
- (D) 60^0
The value of tan26^0 / cot64^0 is
- (A) 0
- (B) 1
- (C) -1
- (D) 2
The circumference of a circle of diameter 42 cm will be
- (A) 14 cm
- (B) 21 cm
- (C) 132 cm
- (D) 122 cm
If the ratio of the radii of two circles is 2:3 then the ratio of their areas will be
- (A) 3:2
- (B) 2:3
- (C) 4:9
- (D) 9:4
If the circumference of a circle is 8pi then its area is
- (A) 8pi
- (B) 16pi
- (C) 4pi
- (D) 32pi
The angle constructed by the minute hand of a clock in completing 35 minutes is
- (A) 45^0
- (B) 60^0
- (C) 90^0
- (D) 210^0
In which quadrant is the point (2, -5) located ?
- (A) first
- (B) second
- (C) third
- (D) fourth
The distance of the point (7, 3) from y-axis is
- (A) 3
- (B) 7
- (C) 0
- (D) 10
What is the distance of the point (-5, 5) from the origin ?
- (A) 5
- (B) 5sqrt(2)
- (C) 10
- (D) 10sqrt(2)
The distance between the points P(2, 3) and Q(4, 1) is
- (A) 2
- (B) 2sqrt(2)
- (C) 4
- (D) 4sqrt(2)
cosec^2theta x tan^2theta x cos^2theta =
- (A) 1
- (B) -1
- (C) 0
- (D) 2
The ratio of the total surface area of a sphere and that of a hemisphere of the same radius is
- (A) 4:9
- (B) 2:1
- (C) 3:2
- (D) 4:3
If a point A is situated on the x-axis at a distance of 5 units on the right side of y-axis then the co-ordinates of A will be
- (A) (0, 5)
- (B) (-5, 0)
- (C) (5, 0)
- (D) (0, -5)
The sum of 1/2 + 1/3 + 1/6 + ... up to 9 terms is
- (A) 3/2
- (B) 1/2
- (C) 3/4
- (D) none of these
cos^2A(1 + tan^2A) =
- (A) sin^2A
- (B) cotA
- (C) 1
- (D) cosec^2A
If (x, 6) is the midpoint of the line segment joining the point A(6, 5) and B(4, y) then the value of y is
- (A) 5
- (B) 6
- (C) 8
- (D) 7
The co-ordinates of the vertices of a triangle are (4, 6) (0, 4) and (5, 5). The co-ordinates of its centroid will be
- (A) (5, 2)
- (B) (3, 5)
- (C) (1, 3)
- (D) (4, 4)
If the edge of a cube is doubled then the ratio of total surface area of the new cube to the total surface area of the old cube is
- (A) 2:1
- (B) 4:1
- (C) 6:1
- (D) 8:1
If the co-ordinates of the vertices of a triangle are (a, 0), (0, 0) and (0, b) then the area of the triangle is
- (A) ab
- (B) 1/2ab
- (C) 1/2b^2
- (D) 1/2a^2b^2
If triangle ABC and triangle DEF are similar and AB = 1.2 cm, DE = 1.4 cm then the ratio of areas of triangle ABC and triangle DEF is
- (A) 16:25
- (B) 49:81
If ΔABC and ΔDEF are similar and perimeter of ΔABC = 30 cm, perimeter of ΔDEF = 20 cm and AB = 12 cm then DE is equal to
- (A) 12 cm
- (B) 8 cm
- (C) 16 cm
- (D) 10 cm
PQ and PR are two tangents to a circle from an external point P. If PQ = 6 cm then PR is equal to
- (A) 12 cm
- (B) 3 cm
- (C) 6 cm
- (D) 5 cm
If the diameter of a circle is 14cm then the area of the circle is
- (A) 616 cm^2
- (B) 221 cm^2
- (C) 154 cm^2
- (D) 77 cm^2
If the diameter of a circle is 14cm then the circumference of the circle is
- (A) 7 cm
- (B) 22 cm
- (C) 44 cm
- (D) 66 cm
The sum of the circumference and diameter of a circle is 29 cm. The diameter of the circle is
- (A) 3.5 cm
- (B) 7 cm
- (C) 14 cm
- (D) 21 cm
The area of the left part after cutting a circle of diameter 6 cm from a circle of radius 8 cm is
- (A) 28 cm^2
- (B) 2pi cm^2
- (C) 55pi cm^2
- (D) 14pi cm^2
If the area of an equilateral triangle is 4sqrt(3)cm^2 then its each side will be
- (A) 9 cm
- (B) 4sqrt(3) cm
- (C) 4 cm
- (D) 2sqrt(3) cm
If sin3A = cos(A - 10^0) and 3A is acute angle then angle A =
- (A) 20^0
- (B) 25^0
- (C) 30^0
- (D) 35^0
In triangle ABC, AC^2 = AB^2 + BC^2 then angle B =
- (A) 90^0
- (B) 60^0
- (C) 45^0
- (D) none of these
From a point Q, the length of the tangent to a circle is 24 cm and the distance of Q from the centre is 25 cm. The radius of circle is
- (A) 7 cm
- (B) 12 cm
- (C) 15 cm
- (D) 24 cm
A coin is tossed. The probability of getting head is
- (A) 0
- (B) 1/2
- (C) 3/4
- (D) 1
The probability that Saumya passes in an examination is 4/5. The probability that Saumya fails is
- (A) 2/5
- (B) 1/5
- (C) 4/5
- (D) 3/5
(2cos60^0 / sin23^0) - (tan40^0 / cot50^0) - cos0^0 =
- (A) 1
- (B) -1
- (C) 2
- (D) 0
The median of 2, 5, 8, 3, 8 is
- (A) 3
- (B) 2
- (C) 5
- (D) 8
The mode of 2, 5, 8, 3, 8, 3, 2, 3, 3 is
- (A) 3
- (B) 5
- (C) 8
- (D) 2
The class mark of the class interval 10-20 is
- (A) 10
- (B) 15
- (C) 20
- (D) 30
The length of the class intervals of the classes 5-10, 10-15, 15-20 .. is
- (A) 5
- (B) 10
- (C) 15
- (D) 20
(secA + tanA)(1 - sinA) =
- (A) cosA
- (B) sinA
- (C) cosecA
- (D) secA
The mean of five numbers is 18. If one number is removed then the mean is 16. The removed number is
- (A) 16
- (B) 18
- (C) 24
- (D) 26
Mode is usually a variable value whose frequency is
- (A) lowest
- (B) highest
- (C) sum of total frequencies
- (D) none of these
The following data has been written in ascending order : 24, 27, 28, 31, 34, x, 37, 40, 42, 45. If the median is 35 then the value of x is
- (A) 35
- (B) 34.5
- (C) 35.5
- (D) 36
(sin^2A / cos^2A) + 1 =
- (A) sin^2A
- (B) sec^2A
- (C) tan^2A
- (D) cot^2A
The total surface area of a right circular cylinder is
- (A) 2pirh
- (B) pir^2h
- (C) 2pir(r + h)
- (D) 1/3 pir^2h
A cube of side 5cm is cut into cubes of side 1cm. The number of cubes formed is
- (A) 5
- (B) 50
- (C) 125
- (D) 625
The ratio of square of the radii of two cylinders is 2:5 and the ratio of their heights is 5:2. The ratio of their volumes will be
- (A) 1:1
- (B) 2:5
- (C) 5:2
- (D) 2:3
(tan60^0 - tan30^0) / (1 + tan60^0.tan30^0) =
- (A) sqrt(3)
- (B) 1/sqrt(3)
- (C) 1
- (D) 2
If the distance between points (4, P) and (1, 0) is 5 then P is equal to
- (A) +- 4
- (B) +- 5
- (C) +- 3
- (D) 1
The radius of the base of a cone is 7 cm and its height is 15 cm then its volume is
- (A) 665 cm^3
- (B) 770cm^3
- (C) 880 cm^3
- (D) 429 cm^3
If the points A(1, 2), O(0, 0) and C(a, b) are collinear then
- (A) a = b
- (B) a = 2b
- (C) 2a = b
- (D) a + b = 0
The area of a triangle (in square units) with vertices (5, 0), (8, 0) and (8, 4) is
- (A) 6
- (B) 12
- (C) 16
- (D) 20
If the point (5, a) lies on the line 2x - 3y = 5 then the value of a is
- (A) 5/3
- (B) 3/5
- (C) -5/3
- (D) -3/5
If acos(theta) + bsin(theta) = 4 and asin(theta) - bcos(theta) = 3 then a^2 + b^2 =
- (A) 7
- (B) 16
- (C) 25
- (D) 36
A letter is chosen at random from the word “mobile”. The probability that the letter chosen is a vowel is
- (A) 1/3
- (B) 1/2
- (C) 1/6
- (D) 3/7
Section B · Written questions (38)
Find the HCF and LCM of 126 and 156 by prime factorization method.
If p(x) = 2x^2+3x-5 then find the values of p(3) and p(-3).
Find a quadratic polynomial, the sum and product of whose zeroes are sqrt(2) and 1/3 respectively.
Find the discriminant of the quadratic equation 3sqrt(3)x^2 + 10x + sqrt(3) = 0.
Show that x = -3 is a solution of x^2 + 6x + 9 = 0.
Find the nature of the roots of the quadratic equation 2x^2 - 6x + 3 = 0
Which term of the A. P. : 18, 15 1/2, 13, ... is -47 ?
Find the values of K for which the equation 2x^2 - 10x + K = 0 has real roots.
Find the sum of the A. P. 25 + 28 + 31 + ... + 100.
For which value of K the following pair of linear equations have no solution ? 3x + y = 1, (2K - 1)x + (K - 1)y = 2K + 1
By comparing the ratios a1/a2, b1/b2 and c1/c2 find whether following pair is consistent or inconsistent : 2x - 3y = 8, 4x - 6y = 9
Find the zeroes of 6x^2 - 7x - 3 and verify the relation between zeroes and coefficients.
Find the relation between x and y such that the point (x, y) is equidistant from the points (7, 1) and (3, 5).
ABC is an isosceles triangle in which AC = BC. If AB^2 = 2AC^2 then prove that ABC is a right angled triangle.
Prove that the lengths of tangents drawn from an external point to a circle are equal.
If a tower 30m high casts a shadow 10sqrt(3) m long on the ground then find the angle of elevation of the sun.
The areas of two similar triangles ABC and DEF are 25 cm^2 and 49 cm^2 respectively. If EF = 9.8 cm, then find BC.
The length of a tangent drawn from a point A at distance 5 cm from the centre of the circle is 4 cm. Find the radius of the circle.
Find the area of a circle whose circumference is 66 cm.
If the perimeter of a semicircular protractor is 72 cm then find its diameter.
If A = 30^0 then prove that tan2A = (2tanA)/(1-tan^2A)
If A = 90^0 and B = 45^0 then prove that Sin(A - B) = sinAcosB - cosAsinB
Find the area of the triangle whose coordinates of the vertices are (1, -1), (-4, 6) and (-3, -5).
Find the solution of the equations x + y = 5 and 5x - 3y = 4 by substitution method.
Find the value of (cos45^0)/(sec30^0 + cosec30^0).
Divide 2x^2 + 3x + 1 by x+2 and verify division algorithm.
The volume of a right circular cylinder of length 14 cm is equal to the volume of a cube of side 11 cm. Find the radius of the cylinder.
Prove that (sin^3theta + cos^3theta)/(sin theta + cos theta) + sin theta.cos theta = 1
Show that points A(a, b+c), B(b, c+a) and C(c, a+b) are collinear.
The sum of the squares of two consecutive natural numbers is 421. Find the numbers.
Draw the graph of pair of equations 2(x+y) = 1 and 2y - 3x = 6 and solve them.
The sum of two numbers is 15. If the sum of their reciprocals is 3/10 then find the numbers.
Find the mean and median from the following frequency table | वर्ग | 30-40 | 40-50 | 50-60 | 60-70 | 70-80 | 80-90 | | बारंबारता | 5 | 22 | 63 | 74 | 30 | 6 |
A 1.2 m tall girl spots a balloon moving with the wind in a horizontal line at a height of 88.2 m from the ground. The angle of elevation of the balloon from the eyes of the girl at any instant is 60^0. After some time the angle of elevation reduces to 30^0. Find the distance travelled by the balloon during the interval.
Draw a line segment AB of length 8 cm. Taking A as centre draw a circle of radius 4 cm and taking B as centre, draw another circle of radius 3 cm. Construct tangents to each circle from the centre of the other circle.
A vessel is in the form of an inverted cone. Its height is 8 cm and the radius of its top, which is open, is 5 cm. It is filled with water up to the brim. When lead shots, each of which is a sphere of radius 0.5 cm, are dropped into the vessel, one fourth of the water flows out. Find the number of lead shots dropped in the vessel.
Prove that Sin^2 10^0 + sin^2 20^0 + sin^2 30^0 + sin^2 40^0 + sin^2 50^0 + sin^2 60^0 + sin^2 70^0 + sin^2 80^0 + sin^2 90^0 = 5.
Find the co-ordinates of the vertices of the triangle whose co-ordinates of the midpoints of the sides are (3, 3), (1, 4) and (2, 1).
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