Bihar Board Class 10 · Maths · Matric exam 2027
Bihar Board Class 10 Maths Question Paper 2026
The board’s 2026 question paper, set I — 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)
Nature of the roots of the quadratic equation x^2 - 3x - 10 = 0 is
- (A) real and unequal ✓
- (B) real and equal
- (C) non-real
- (D) none of these
Answer (the board’s key): (A)
The value of the discriminant of the quadratic equation bx^2 + ax + c = 0, where b != 0 is
- (A) a^2 - 4bc ✓
- (B) b^2 - 4ac
- (C) c^2 - 4ab
- (D) b^2 + 4ac
Answer (the board’s key): (A)
If 3, x, y and 15 are in A.P. then x + y =
- (A) 9
- (B) 18 ✓
- (C) 27
- (D) 36
Answer (the board’s key): (B)
Common difference of the sequence 5, 9, 13, 17, 21, ... is
- (A) 5
- (B) -4
- (C) 4 ✓
- (D) -5
Answer (the board’s key): (C)
13th term of the sequence - 5, - 1, 3, 7, ... is
- (A) 35
- (B) 39
- (C) 43 ✓
- (D) 47
Answer (the board’s key): (C)
Sum of the first n natural numbers is
- (A) n(n+1)(2n+1)/6
- (B) n(n+1)/2 ✓
- (C) {n(n+1)/2}^2
- (D) n(n-1)/2
Answer (the board’s key): (B)
11 + 12 + 13 + ... + 45 =
- (A) 970
- (B) 980 ✓
- (C) 990
- (D) 1080
Answer (the board’s key): (B)
The next term of the A.P. sqrt(7), sqrt(28), sqrt(63), ... is
- (A) sqrt(70)
- (B) sqrt(84)
- (C) sqrt(98)
- (D) sqrt(112) ✓
Answer (the board’s key): (D)
How many terms of the A.P. 3, 7, 11, 15, .... will make the sum 406 ?
- (A) 10
- (B) 12
- (C) 14 ✓
- (D) 20
Answer (the board’s key): (C)
The sum of first n terms of an A.P. is (5n - n^2). The nth term of the A.P. is
- (A) 5 - 2n
- (B) 6 - 2n ✓
- (C) 2n - 5
- (D) 2n - 6
Answer (the board’s key): (B)
If the ratio of circumferences of two circles is 3 : 4 then the ratio of their areas is
- (A) 9 : 16 ✓
- (B) 16 : 9
- (C) 4 : 3
- (D) 3 : 4
Answer (the board’s key): (A)
The value of central angle of a sector is 60° and radius is 21 cm. Then the length of the arc is
- (A) 11 cm
- (B) 21 cm
- (C) 22 cm ✓
- (D) 44 cm
Answer (the board’s key): (C)
Three cubes having edge 6 cm are joined end to end to form a cuboid. Then its resultant volume is
- (A) 648 cm^3 ✓
- (B) 216 cm^3
- (C) 432 cm^3
- (D) 864 cm^3
Answer (the board’s key): (A)
If the surface areas of two spheres are in the ratio of 16 : 9 then the ratio of their volumes is
- (A) 27 : 64
- (B) 64 : 27 ✓
- (C) 4 : 3
- (D) 3 : 4
Answer (the board’s key): (B)
If the total surface area of a cube is 864 sq.cm then its volume is
- (A) 1728 cubic cm ✓
- (B) 3456 cubic cm
- (C) 864 cubic cm
- (D) 432 cubic cm
Answer (the board’s key): (A)
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) sqrt(xyz) ✓
- (C) x^2y^2z^2
- (D) x^3y^3z^3
Answer (the board’s key): (B)
The height of a cylinder is 14 cm and its curved surface area is 264 sq.cm ; then its volume is
- (A) 1848 cm^3
- (B) 1232 cm^3
- (C) 396 cm^3 ✓
- (D) 308 cm^3
Answer (the board’s key): (C)
Cumulative frequency table is required for the calculation of which of the following measures of central tendency ?
- (A) Mean
- (B) Median ✓
- (C) Mode
- (D) None of these
Answer (the board’s key): (B)
Median of the first 10 prime numbers is
- (A) 11
- (B) 12 ✓
- (C) 13
- (D) 17
Answer (the board’s key): (B)
In symmetrical frequency distribution table it holds
- (A) Mean > Mode > Median
- (B) Mean < Mode < Median
- (C) Mean = Mode = Median ✓
- (D) Mode = (Mean + Median) / 2
Answer (the board’s key): (C)
According to Euclid's division lemma for any two given positive integers a and b, there exist unique integers q and r satisfying the equation a = bq + r where
- (A) 0 <= b < r
- (B) 0 <= r < b ✓
- (C) 0 >= b > r
- (D) 0 >= r >= b
Answer (the board’s key): (B)
According to Euclid's division algorithm the HCF of 867 and 255 is
- (A) 13
- (B) 15
- (C) 17
- (D) 51 ✓
Answer (the board’s key): (D)
Product of prime factors of 7429 is
- (A) 17 x 19 x 23 ✓
- (B) 13 x 17 x 19
- (C) 13 x 19 x 23
- (D) 17 x 19 x 29
Answer (the board’s key): (A)
If c and d are the LCM and HCF of two numbers a and b respectively then
- (A) a + b = c + d
- (B) a - b = c - d
- (C) ab = cd ✓
- (D) ad = bc
Answer (the board’s key): (C)
Which of the following numbers is a rational number ?
- (A) sqrt(2)
- (B) sqrt(3)
- (C) sqrt(4) ✓
- (D) sqrt(5)
Answer (the board’s key): (C)
The number whose generalised form is p/q where p and q are integers and q!=0 is said to be
- (A) even number
- (B) odd number
- (C) rational number ✓
- (D) irrational number
Answer (the board’s key): (C)
The number whose decimal expansion is neither terminating nor repeating is said to be
- (A) irrational number ✓
- (B) rational number
- (C) prime number
- (D) integer
Answer (the board’s key): (A)
(3 - 2sqrt(5))^2 =
- (A) 29 - 12sqrt(5) ✓
- (B) 29 + 12sqrt(5)
- (C) 29
- (D) - 11
Answer (the board’s key): (A)
Factors of the polynomial x^2 - 4x + 3 is
- (A) (x - 1)(x + 3)
- (B) (x + 1)(x - 3)
- (C) (x - 1)(x - 3) ✓
- (D) (x + 1)(x + 3)
Answer (the board’s key): (C)
Zeroes of the polynomial x^2 + 7x + 10 are
- (A) - 2, - 5 ✓
- (B) 2, 5
- (C) - 2, 5
- (D) 2, - 5
Answer (the board’s key): (A)
If the lengths of diagonals of a rhombus are 10 cm and 24 cm respectively then each side of the rhombus is
- (A) 5 cm
- (B) 12 cm
- (C) 13 cm ✓
- (D) 17 cm
Answer (the board’s key): (C)
How many tangents can be drawn to a circle from an external point ?
- (A) One
- (B) Two ✓
- (C) Infinite
- (D) None of these
Answer (the board’s key): (B)
If the lengths of radius of a circle and the tangent drawn from an external point to the circle are 7 cm and 24 cm respectively then the distance of the external point from the centre of the circle is
- (A) 18 cm
- (B) 32 cm
- (C) 25 cm ✓
- (D) 12 cm
Answer (the board’s key): (C)
If the lengths of chord and radius of a circle are 16 cm and 17 cm respectively then perpendicular distance of the chord from the centre is
- (A) 8 cm
- (B) 15 cm ✓
- (C) 14 cm
- (D) 13 cm
Answer (the board’s key): (B)
The sum ( in right angle ) of pair of opposite angles of a cyclic quadrilateral is
- (A) 1/2
- (B) 2/3
- (C) 3/4
- (D) 2 ✓
Answer (the board’s key): (D)
Which of the following pairs of lines in a circle cannot be parallel?
- (A) Two chords
- (B) A chord and a tangent
- (C) Two tangents
- (D) Two diameters ✓
Answer (the board’s key): (D)
If the angle between any two radii of a circle is 130° then the angle between the tangents at the ends of the radii is
- (A) 65°
- (B) 40°
- (C) 50° ✓
- (D) 90°
Answer (the board’s key): (C)
If the diagonals of the parallelogram are equal and bisect each other at right angle then the quadrilateral is
- (A) rectangle
- (B) square ✓
- (C) trapezium
- (D) rhombus
Answer (the board’s key): (B)
What relation holds between the diagonals of rhombus?
- (A) equal
- (B) intersect at right angle
- (C) bisect at right angle ✓
- (D) none of these
Answer (the board’s key): (C)
The angle subtended at the centre of a circle is ∠AOB and ∠ACB where C lies on major arc at the circumference. Then ∠ACB =
- (A) ∠AOB
- (B) 2∠AOB
- (C) ∠AOB/2 ✓
- (D) None of these
Answer (the board’s key): (C)
If the mean and mode of a frequency distribution are 28 and 16 respectively then median is
- (A) 22
- (B) 23
- (C) 24 ✓
- (D) 25
Answer (the board’s key): (C)
The average of 24 numbers is zero. Out of them maximum how many numbers may be greater than zero ?
- (A) 0
- (B) 1
- (C) 12
- (D) 23 ✓
Answer (the board’s key): (D)
The probability of a sure event is
- (A) 0
- (B) less than 1
- (C) more than 1
- (D) 1 ✓
Answer (the board’s key): (D)
A die is thrown once ; then the probability that it shows a prime number is
- (A) 1/2 ✓
- (B) 1/3
- (C) 0
- (D) 1
Answer (the board’s key): (A)
Which of the following options cannot be the probability of an event ?
- (A) 1.2 ✓
- (B) 3/4
- (C) 20%
- (D) 0.3
Answer (the board’s key): (A)
If p(E) denotes the probability of an event E then
- (A) p(E)<0
- (B) p(E)>1
- (C) 0 <= p(E) <= 1 ✓
- (D) -1 <= p(E) <= 1
Answer (the board’s key): (C)
The probability of an impossible event is
- (A) less than 1
- (B) greater than 1
- (C) 1
- (D) 0 ✓
Answer (the board’s key): (D)
A die is thrown once ; then the probability that it shows an odd number is
- (A) 0
- (B) 1
- (C) 1/2 ✓
- (D) 1/3
Answer (the board’s key): (C)
A number is selected at random from the numbers 1 to 30. Then the probability that the selected number is a prime number, is
- (A) 11/30
- (B) 1/3 ✓
- (C) 3/10
- (D) 2/3
Answer (the board’s key): (B)
If the probability of winning a game is 0.4 then the probability of losing it is
- (A) 0.96
- (B) 0.6 ✓
- (C) 1/0.4
- (D) 1/0.6
Answer (the board’s key): (B)
Sum of the zeroes of the polynomial 6x^2 - 7x - 3 is
- (A) -7/6
- (B) 7/6 ✓
- (C) -1/2
- (D) 1/2
Answer (the board’s key): (B)
Product of the zeroes of the polynomial 2x^2 + 3x + 1 is
- (A) -3/2
- (B) 3/2
- (C) -1/2
- (D) 1/2 ✓
Answer (the board’s key): (D)
Polynomial having zeroes -3 and 2 is
- (A) x^2 - x - 6
- (B) x^2 + x - 6 ✓
- (C) x^2 - x + 6
- (D) x^2 + x + 6
Answer (the board’s key): (B)
Graphical representation of the pair of linear equations x - y = 0, 3x + 4y - 20 = 0 is
- (A) intersecting lines ✓
- (B) coincident lines
- (C) parallel lines
- (D) none of these
Answer (the board’s key): (A)
If a1/a2 = b1/b2 = c1/c2 in the pair of linear equations a1x + b1y + c1 = 0, a2x + b2y + c2 = 0, then their solution is
- (A) unique solution
- (B) infinitely many solutions ✓
- (C) no solution
- (D) none of these
Answer (the board’s key): (B)
If the graphical representation of pair of linear equations a1x + b1y + c1 = 0, a2x + b2y + c2 = 0 are parallel lines then
- (A) a1/a2 != b1/b2
- (B) a1/a2 = b1/b2 = c1/c2
- (C) a1/a2 = b1/b2 != c1/c2 ✓
- (D) none of these
Answer (the board’s key): (C)
If the sum and difference of two numbers are 14 and 4 respectively then their product is
- (A) 45 ✓
- (B) 48
- (C) 54
- (D) 36
Answer (the board’s key): (A)
If the speed of a boat in still water is 8 km/hr and the speed of the stream is 3 km/hr then relative speed of the boat in upstream is
- (A) 11 km/hr
- (B) 5 km/hr ✓
- (C) 8 km/hr
- (D) 3 km/hr
Answer (the board’s key): (B)
ax^2 + bx + c = 0 is said to be a quadratic equation in variable x where a, b & c are real numbers then
- (A) a != 0 ✓
- (B) b != 0
- (C) c != 0
- (D) a = 0
Answer (the board’s key): (A)
Discriminant of the quadratic equation 5x^2 - 6x - 2 = 0 is
- (A) 66
- (B) 76 ✓
- (C) 73
- (D) 72
Answer (the board’s key): (B)
Distance between the points (9, 3) and (15, 11) is
- (A) 8 units
- (B) 10 units ✓
- (C) 12 units
- (D) 15 units
Answer (the board’s key): (B)
Perimeter of the triangle having vertices (0, 7), (0, 0) and (24, 0) is
- (A) 28 units
- (B) 42 units
- (C) 48 units
- (D) 56 units ✓
Answer (the board’s key): (D)
Distance of the point (11, 60) from the origin is
- (A) 51 units
- (B) 53 units
- (C) 59 units
- (D) 61 units ✓
Answer (the board’s key): (D)
If the points (1, 2), (0, 0) and (a, b) are collinear then
- (A) a = b
- (B) a = 2b
- (C) 2a = b ✓
- (D) a = -b
Answer (the board’s key): (C)
Area of the triangle formed having vertices (3, 0), (7, 0) and (8, 4) is
- (A) 16 cm^2
- (B) 24 cm^2
- (C) 8 cm^2 ✓
- (D) 32 cm^2
Answer (the board’s key): (C)
Perimeter of semicircle is
- (A) pi r
- (B) 2pi r
- (C) pi r + 2r ✓
- (D) pi r + r
Answer (the board’s key): (C)
Circumference of the circle having diameter 28 cm is
- (A) 22 cm
- (B) 44 cm
- (C) 88 cm ✓
- (D) 176 cm
Answer (the board’s key): (C)
The perimeter of a sector of a circle of radius 14 cm is 50 cm ; then central angle of the sector is
- (A) 36°
- (B) 45°
- (C) 72°
- (D) 90° ✓
Answer (the board’s key): (D)
A wire is looped in the form of a circle of radius 28 cm. It is rebent into a square form, then the length of its side is
- (A) 11 cm
- (B) 22 cm
- (C) 44 cm ✓
- (D) 88 cm
Answer (the board’s key): (C)
The area of a circle is the same as the area of a square. Then the ratio of their perimeters is
- (A) 2 : pi
- (B) pi : 2
- (C) sqrt(pi) : 2 ✓
- (D) 2 : sqrt(pi)
Answer (the board’s key): (C)
All the triangles of which of the following types are similar ?
- (A) Scalene triangle
- (B) Isosceles triangle
- (C) Equilateral triangle ✓
- (D) None of these
Answer (the board’s key): (C)
Two polygons having the same number of sides are similar. Then the relation among their corresponding sides is
- (A) equal
- (B) directly proportional ✓
- (C) inversely proportional
- (D) none of these
Answer (the board’s key): (B)
Ratio of the areas of two similar triangles is 4 : 9 ; then the ratio of their corresponding sides is
- (A) 3 : 2
- (B) 9 : 4
- (C) 4 : 9
- (D) 2 : 3 ✓
Answer (the board’s key): (D)
If delta ABC ~ delta DEF and AB/DE = 3/5 then FD/CA =
- (A) 9 : 25
- (B) 25 : 9
- (C) 3 : 5
- (D) 5 : 3 ✓
Answer (the board’s key): (D)
delta ABC ~ delta PQR and their perimeters are 32 cm and 24 cm respectively. If AB = 10 cm then PQ =
- (A) 7 cm
- (B) 40/3 cm
- (C) 15/2 cm ✓
- (D) 15 cm
Answer (the board’s key): (C)
Length of the altitude of an equilateral triangle having side 2a cm is
- (A) a*sqrt(3)/2 cm
- (B) a*sqrt(2)/3 cm
- (C) a*sqrt(2) cm
- (D) a*sqrt(3) cm ✓
Answer (the board’s key): (D)
Which of the following options is not a Pythagorean triplet ?
- (A) 12, 35, 37
- (B) 28, 45, 53
- (C) 16, 63, 65
- (D) 17, 61, 69 ✓
Answer (the board’s key): (D)
Area of the equilateral triangle having height 6*sqrt(3) cm is
- (A) 27*sqrt(3) cm^2
- (B) 36*sqrt(3) cm^2 ✓
- (C) 54*sqrt(3) cm^2
- (D) 72*sqrt(3) cm^2
Answer (the board’s key): (B)
If the bisector of an angle of a triangle bisects the opposite side then the triangle is
- (A) scalene triangle
- (B) equilateral triangle
- (C) isosceles triangle ✓
- (D) right angled triangle
Answer (the board’s key): (C)
The structure formed with the line segments joining the mid-points of the adjacent sides of a quadrilateral is
- (A) parallelogram ✓
- (B) rectangle
- (C) square
- (D) rhombus
Answer (the board’s key): (A)
If sin theta = 45/53 then cos theta =
- (A) 53/28
- (B) 28/53 ✓
- (C) 28/45
- (D) 45/28
Answer (the board’s key): (B)
sin 29° / cos 61° =
- (A) 90
- (B) 0
- (C) 1 ✓
- (D) 32
Answer (the board’s key): (C)
Which of the following options is not equal to unity ?
- (A) cosec^2 theta - cot^2 theta
- (B) sec^2 theta - tan^2 theta
- (C) cos^2 theta - sin^2 theta ✓
- (D) cos^2 theta + sin^2 theta
Answer (the board’s key): (C)
If the length of the shadow of a tree is equal to its height then angle of elevation of the sun is
- (A) 30°
- (B) 45° ✓
- (C) 60°
- (D) 75°
Answer (the board’s key): (B)
The value of cos theta in the form of sin theta is
- (A) sqrt(1 - sin^2 theta) ✓
- (B) sqrt(1 + sin^2 theta)
- (C) sqrt(sin^2 theta - 1)
- (D) none of these
Answer (the board’s key): (A)
If sec theta = cosec theta then theta =
- (A) 90°
- (B) 45° ✓
- (C) 30°
- (D) 75°
Answer (the board’s key): (B)
sin^2 30° + cos^2 60° - cot^2 45° =
- (A) 1/2
- (B) -1/2 ✓
- (C) 1
- (D) 0
Answer (the board’s key): (B)
If sqrt(3) tan theta = 1 then theta =
- (A) 30° ✓
- (B) 45°
- (C) 60°
- (D) 75°
Answer (the board’s key): (A)
If cot A + 1/cot A = 2 then tan^2 A + 1/tan^2 A =
- (A) 0
- (B) 1
- (C) 2 ✓
- (D) 4
Answer (the board’s key): (C)
sin^4 theta - cos^4 theta =
- (A) cos^2theta - sin^2theta
- (B) 2cos^2theta - 1
- (C) 1 + 2cos^2theta
- (D) 2sin^2theta - 1 ✓
Answer (the board’s key): (D)
If sin theta + sin^2 theta = 1 then cos^2 theta + cos^4 theta =
- (A) 0
- (B) 1 ✓
- (C) 2
- (D) 4
Answer (the board’s key): (B)
(1 + tan^2 A) / (1 + cot^2 A) =
- (A) sec^2 A
- (B) cosec^2 A
- (C) tan^2 A ✓
- (D) cot^2 A
Answer (the board’s key): (C)
If tan theta + cot theta = 5 then tan^2 theta + cot^2 theta =
- (A) 23 ✓
- (B) 24
- (C) 25
- (D) 27
Answer (the board’s key): (A)
Assuming centre as the mid-point of hypotenuse in a right angled triangle, a circle is drawn, then the centre is called
- (A) centroid
- (B) circumcentre ✓
- (C) orthocentre
- (D) incentre
Answer (the board’s key): (B)
A 15 m long ladder is resting at the top of a vertical wall forming an angle 60° with the wall ; then the height of the wall is
- (A) 15/2 m ✓
- (B) 15 m
- (C) 30 m
- (D) 15/4 m
Answer (the board’s key): (A)
Lengths of the sides of a triangle are 13 cm, 14 cm and 15 cm respectively, then its area is
- (A) 42 cm^2
- (B) 48 cm^2
- (C) 84 cm^2 ✓
- (D) 96 cm^2
Answer (the board’s key): (C)
Area of four walls of a room whose length, breadth and height are l, b and h units respectively, is
- (A) (l + b) h sq.unit
- (B) lbh sq.unit
- (C) 2(l + b)h sq.unit ✓
- (D) 4(l + b)h sq.unit
Answer (the board’s key): (C)
Area of an isosceles triangle each of which equal sides is 13 cm and having base 24 cm is
- (A) 30 cm^2
- (B) 60 cm^2 ✓
- (C) 90 cm^2
- (D) 120 cm^2
Answer (the board’s key): (B)
If the area of an equilateral triangle is 36sqrt(3) cm^2 then its perimeter is
- (A) 24 cm
- (B) 27 cm
- (C) 36 cm ✓
- (D) 48 cm
Answer (the board’s key): (C)
The length of the diagonal of a square is 10sqrt(2) cm; then its area is
- (A) 200 cm^2
- (B) 100 cm^2 ✓
- (C) 150 cm^2
- (D) 100sqrt(2) cm^2
Answer (the board’s key): (B)
Section B · Written questions (38)
Find the zeroes of the quadratic polynomial 3x^2 - x - 4.
Medha Ma’am
AskMedha’s AI teacher
Model answer · zeroes of quadratic polynomial
- 1.We have p(x) = 3x^2 - x - 4. By the method of splitting the middle term, 3x^2 - x - 4 = 3x^2 - 4x + 3x - 4 = x(3x - 4) + 1(3x - 4)1 mark
- 2.So, 3x^2 - x - 4 = (3x - 4)(x + 1). Hence, the value of 3x^2 - x - 4 is zero when either 3x - 4 = 0 or x + 1 = 0, i.e., when x = 4/3 or x = -1. So, the zeroes of 3x^2 - x - 4 are 4/3 and -1.1 mark
- Write the splitting step showing 3x^2 - 4x + 3x - 4 before factoring.
- State the factors as (3x - 4)(x + 1) = 0 explicitly.
- End with both zeroes written as x = 4/3 and x = -1.
Marks split from a structure derived from past papers (checked twice); not Bihar Board’s official scheme
If x^2 + 2x - 3 is divided by x - 2 then find the remainder.
Medha Ma’am
AskMedha’s AI teacher
Model answer · Remainder Theorem
- 1.Given polynomial p(x) = x^2 + 2x - 3 is divided by x - 2. According to the Remainder Theorem, if a polynomial p(x) is divided by (x - a), the remainder is p(a). Here the divisor is (x - 2), so a = 2, and the remainder will be p(2).1 mark
- 2.The value of p(2) = (2)^2 + 2(2) - 3 = 4 + 4 - 3 = 5 So the remainder when x^2 + 2x - 3 is divided by x - 2 is 5.1 mark
- Begin by stating the Remainder Theorem and identifying that the remainder equals p(2).
- Show the substitution step clearly: p(2) = (2)^2 + 2(2) - 3.
- Write the final calculated value 5 as the remainder with the working 4 + 4 - 3.
Marks split from a structure derived from past papers (checked twice); not Bihar Board’s official scheme
Find the values of x and y of the pair of linear equations 3x + 2y = 12, 5x - 2y = 4 by cross-multiplication method.
Medha Ma’am
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Model answer · pair of linear equations, cross-multiplication method
- 1.The given pair of linear equations 3x + 2y = 12 and 5x - 2y = 4 is written in the standard form a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0, i.e. 3x + 2y - 12 = 0 and 5x - 2y - 4 = 0. Here a1 = 3, b1 = 2, c1 = -12, a2 = 5, b2 = -2, c2 = -4.0.5 marks
- 2.To solve this pair of linear equations by cross-multiplication method, we use the formula x/(b1c2 - b2c1) = y/(c1a2 - c2a1) = 1/(a1b2 - a2b1).0.5 marks
- 3.Substituting the values a1 = 3, b1 = 2, c1 = -12, a2 = 5, b2 = -2, c2 = -4 in the formula, we get x/((2)(-4) - (-2)(-12)) = y/((-12)(5) - (-4)(3)) = 1/((3)(-2) - (5)(2)) i.e. x/(-8 - 24) = y/(-60 + 12) = 1/(-6 - 10).0.5 marks
- 4.Simplifying x/(-32) = y/(-48) = 1/(-16), we get x = -32/-16 = 2 and y = -48/-16 = 3. Hence, x = 2 and y = 3 is the solution of the given pair of linear equations.0.5 marks
- Write the equations in the form a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0 before substituting values.
- State the formula x/(b1c2 - b2c1) = y/(c1a2 - c2a1) = 1/(a1b2 - a2b1) exactly as given.
- Show the substitution step with the values of a1, b1, c1, a2, b2, c2 before simplifying to get x = 2 and y = 3.
Marks split from a structure derived from past papers (checked twice); not Bihar Board’s official scheme
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How the model answer begins
1. Let the two sides of the right-angled triangle other than the hypotenuse be x and y, where x is the longer side and y is the shorter side. Since the difference of the lengths of these two sides is 7 cm, we get x - y =…
The full answer from Medha Ma’am →If the radius of a circle is 21 cm and central angle of the sector is 120 degree then find the area of the sector.
How the model answer begins
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How the model answer begins
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The full answer from Medha Ma’am →Total how many terms are there in the A.P. 6, 10, 14, 18, ..., 174 ?
How the model answer begins
1. In the given AP 6, 10, 14, 18, ..., 174, the first term a = 6 and the common difference d = 10 - 6 = 4.
The full answer from Medha Ma’am →If the sum of three consecutive terms of an AP is 15 and product is 105 then find these terms.
How the model answer begins
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How the model answer begins
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The full answer from Medha Ma’am →Evaluate : (sec 39 degree / cosec 51 degree) + (sin 67 degree / cos 23 degree) - cot 45 degree - cos 0 degree.
How the model answer begins
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The full answer from Medha Ma’am →If 4tan theta = 3 then find the value of sin theta cos theta.
How the model answer begins
1. Given 4 tan theta = 3, we get tan theta = 3/4.
The full answer from Medha Ma’am →If the length of shadow of a pole is equal to sqrt(3) times of its height then find the angle of elevation of the sun.
How the model answer begins
1. Let the height of the pole be h and the length of its shadow be h√3.
The full answer from Medha Ma’am →A circle is inscribed in a triangle ABC touching the sides BC, CA and AB of the triangle at P, Q and R respectively. If AB = 10 cm, AQ = 7cm and CQ = 5 cm then find the length of BC.
How the model answer begins
1. The lengths of tangents drawn from an external point to a circle are equal. Since A is an external point with tangents AR and AQ touching the circle at R and Q, AR = AQ = 7 cm. Since AB = 10 cm and AR = 7 cm, BR = AB -…
The full answer from Medha Ma’am →If the lengths of a chord and radius of a circle are 24 cm and 13 cm respectively then find the perpendicular distance of the chord from the centre of the circle.
How the model answer begins
1. The perpendicular from the centre of a circle to a chord bisects the chord. Here the chord AB has length 24 cm, so the perpendicular from the centre O to the chord meets it at point M such that AM = MB = 12 cm.
The full answer from Medha Ma’am →Express sin theta and cos theta in the form of tan theta.
How the model answer begins
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How the model answer begins
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How the model answer begins
1. Let the two numbers be x and y. According to the given condition, the sum of the numbers is 137, so the equation obtained is x + y = 137. The difference of the numbers is 43, so the equation obtained is x - y = 43.
The full answer from Medha Ma’am →Find the sum and product of the roots of the quadratic equation x^2 - x + 3 = 0.
How the model answer begins
1. Comparing the quadratic equation x^2 - x + 3 = 0 with ax^2 + bx + c = 0, we get a = 1, b = -1, c = 3.
The full answer from Medha Ma’am →A metallic cuboid whose length, breadth and height are 100 cm, 80 cm and 64 cm respectively, is melted and recast into a cube. Then find the length of the edge of the cube.
How the model answer begins
1. When a metallic cuboid is melted and recast into a cube, the volume remains the same. The volume of the cuboid = length × breadth × height = 100 × 80 × 64 cubic cm = 512000 cubic cm.
The full answer from Medha Ma’am →Find the mean of first 10 prime numbers.
How the model answer begins
1. The first 10 prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23 and 29. Adding these numbers, we get 2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 = 129. So the sum of the first 10 prime numbers is 129.
The full answer from Medha Ma’am →What is the probability that an ordinary year has 53 Sundays ?
How the model answer begins
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How the model answer begins
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The full answer from Medha Ma’am →ΔABC ~ ΔPQR and their areas are 64 cm^2 and 121 cm^2 respectively. If QR = 15.4 cm then find the value of BC.
How the model answer begins
1. The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides. Since ΔABC ~ ΔPQR, their areas and corresponding sides BC and QR are related as Area(ΔABC)/Area(ΔPQR) =…
The full answer from Medha Ma’am →Find the height of an equilateral triangle of side 12 cm.
How the model answer begins
1. The height (altitude) of an equilateral triangle of side a is given by the formula (√3/2) × side.
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How the model answer begins
1. The required number must divide 245 and 1037 leaving remainder 5 in each case. So subtracting the remainder 5 from the given numbers, we get 245 - 5 = 240 and 1037 - 5 = 1032. The required largest number is the HCF of…
The full answer from Medha Ma’am →Without actual division, show that the rational number 77/210 is a non-terminating repeating decimal.
How the model answer begins
1. The rational number 77/210, when reduced to its simplest form by cancelling the common factor 7 from numerator and denominator, is 11/30.
The full answer from Medha Ma’am →Simplify : (2sqrt(27) + 3sqrt(12)) / (4sqrt(3)).
How the model answer begins
1. We simplify 2√27 and 3√12 by breaking the numbers under the root into prime factors. Since 27 = 9 × 3, we get √27 = 3√3, so 2√27 = 2 × 3√3 = 6√3. Since 12 = 4 × 3, we get √12 = 2√3, so 3√12 = 3 × 2√3 = 6√3.
The full answer from Medha Ma’am →Prove that the points (1, 1), (-2, 7) and (3, -3) are collinear.
How the model answer begins
1. Points are collinear if the area of the triangle formed by them is zero. Let A(1, 1), B(-2, 7) and C(3, -3) be the given points. If the area of triangle ABC is zero, then A, B and C are collinear.
The full answer from Medha Ma’am →Find the area of the triangle formed having vertices (-5, 6), (3, 0) and (9, 8).
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)]. Here the vertices are A(-5, 6), B(3, 0) and C(9, 8),…
The full answer from Medha Ma’am →Points D and E are on the sides AB and AC respectively of triangle ABC such that DE || BC. If AD = 3.6 cm, AB = 10 cm and AE = 4.5 cm then find the values of EC and AC.
How the model answer begins
1. D and E are points on sides AB and AC of triangle ABC such that DE || BC. So, according to the Basic Proportionality Theorem (Thales Theorem), AD/DB = AE/EC.
The full answer from Medha Ma’am →Plot the graphs of the pair of linear equations 2x - 3y + 13 = 0 and 3x - 2y + 12 = 0 and solve.
How the model answer begins
1. The given pair of linear equations is 2x - 3y + 13 = 0 and 3x - 2y + 12 = 0. To represent these equations graphically, we need at least two solutions for each equation. For 2x - 3y + 13 = 0: When x = -5, 2(-5) - 3y + 13…
The full answer from Medha Ma’am →Quadratic equation ax^2 + bx + c = 0 where a, b & c are real numbers and a ≠ 0 then prove that x = (-b ± sqrt(b^2 - 4ac)) / 2a.
How the model answer begins
1. The given quadratic equation is ax² + bx + c = 0, where a, b and c are real numbers and a ≠ 0. Dividing throughout by a, we get x² + (b/a)x + (c/a) = 0.
The full answer from Medha Ma’am →State and prove Pythagoras' theorem.
How the model answer begins
1. Pythagoras Theorem states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
The full answer from Medha Ma’am →Prove that the ratio of the areas of two similar triangles is equal to the ratio of the squares of their corresponding sides.
How the model answer begins
1. Given two triangles ABC and PQR such that triangle ABC is similar to triangle PQR.
The full answer from Medha Ma’am →The angle of elevation of the top of a building from the foot of a tower is 30° and the angle of elevation of the top of the tower from the foot of the building is 60°. If the tower is 60 m high then find the height of the building.
How the model answer begins
1. Let the height of the tower AB be 60 m and the height of the building CD be h m, and let the distance between the foot of the tower and the foot of the building be x m. The angle of elevation of the top of the building…
The full answer from Medha Ma’am →If the points (-4, 8), (-3, -4), (0, -5) and (5, 6) are the vertices of a quadrilateral then find the area of the quadrilateral.
How the model answer begins
1. Let the vertices of the quadrilateral be A(–4, 8), B(–3, –4), C(0, –5) and D(5, 6), taken in order. Join B to D so that the quadrilateral ABCD is divided into two triangles ABD and BCD, which have no common area, so…
The full answer from Medha Ma’am →A circular disc of radius 6 cm is divided into three sectors with central angles 90°, 120° and 150° then what part of the whole disc is the sector with central angle 150° ? Also, calculate the areas of the three sectors.
How the model answer begins
1. The fraction of the whole disc represented by the sector with central angle 150° is found using the formula for area of a sector of a circle with radius r and angle with degree measure θ, which is πr²(θ/360°). Since the…
The full answer from Medha Ma’am →Find the median of the following data : | Marks obtained | less than 10 | less than 20 | less than 30 | less than 40 | less than 50 | less than 60 | less than 70 | less than 80 | less than 90 | less than 100 | | No. of students | 5 | 8 | 12 | 15 | 18 | 22 | 29 | 38 | 45 | 53 |
How the model answer begins
1. From the 'less than' cumulative frequencies, the class intervals and frequencies are: 0-10: 5, 10-20: 8 − 5 = 3, 20-30: 4, 30-40: 3, 40-50: 3, 50-60: 4, 60-70: 7, 70-80: 9, 80-90: 7, 90-100: 8.
The full answer from Medha Ma’am →
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