CBSE Class 10 Mathematics 2023
315 questions · 680 marks · 314 with the board’s own marking-scheme answer, 1 with no official answer on the site.
Marks by chapter
Counted from the 315 questions listed below, as the 2023 papers print them. CBSE allocates marks per unit, not per chapter — this is what was observed, not an official CBSE weightage. Every year compared →
| Chapter | Marks | Share |
|---|---|---|
| Arithmetic Progressions28 questions | 78 | 11.5% |
| Circles34 questions | 70 | 10.3% |
| Introduction to Trigonometry36 questions | 68 | 10.0% |
| Some Applications of Trigonometry14 questions | 58 | 8.5% |
| Triangles21 questions | 55 | 8.1% |
| Coordinate Geometry32 questions | 53 | 7.8% |
| Real Numbers24 questions | 50 | 7.4% |
| Statistics18 questions | 47 | 6.9% |
| Surface Areas and Volumes16 questions | 46 | 6.8% |
| Polynomials26 questions | 36 | 5.3% |
| Probability25 questions | 36 | 5.3% |
| Quadratic Equations16 questions | 31 | 4.6% |
| Areas Related to Circles10 questions | 29 | 4.3% |
| Pair of Linear Equations in Two Variables14 questions | 22 | 3.2% |
315 questions from 2023
- Q1The distance between the points (0,5) and (-3, 1) is:1 mk
- Q1If 'n' is a natural number, then which of the following numbers end with zero?1 mk
- Q1Which of the following quadratic equations has sum of its roots as 4?1 mk
- Q1((1- tan^2 30^°)/(1+ tan^2 30^°)) is equal to:1 mk
- Q1The ratio of HCF to LCM of the least composite number and the least prime number is:1 mk
- Q1The graph of y=p(x) is given, for a polynomial p(x). The number of zeroes of p(x) from the graph is1 mk
- Q1If p^2=32/50, then p is a/an1 mk
- Q1Which of the following is true for all values of θ(0^° ≤ θ ≤ 90^°)?1 mk
- Q1If one zero of the polynomial x^2-3 kx+4 k be twice the other, then the value of k is:1 mk
- Q1The number of polynomials having zeroes -3 and 5 is:1 mk
- Q2The pair of equations ax+2 y=9 and 3 x+by=18 represent parallel lines, where a, b are integers, if:1 mk
- Q2If k+2,4 k-6 and 3 k-2 are three consecutive terms of an A.P., then the value of k is:1 mk
- Q2In a lottery, there are 5 prizes and 20 blanks. The probability of getting a prize is:1 mk
- Q2The value of k for which the pair of equations k x=y+2 and 6 x=2 y+3 has infinitely many solutions,1 mk
- Q2The roots of the equation x^2+3 x-10=0 are:1 mk
- Q2The distance of the point (- 6, 8) from x -axis is1 mk
- Q2What is the length of the arc of the sector of a circle with radius 14 cm and of central angle 90°?1 mk
- Q2The ratio in which the x-axis divides the line segment joining the points (-2, 3) and (6, -7) is:1 mk
- Q2In the given figure, TA is a tangent to the circle with centre O such that OT=4 cm, ∠ OTA=30^°, then length of TA is:1 mk
- Q2If tan θ=x/y, then cos θ is equal to1 mk
- Q3The next term of the A.P.: √6, √24, √54 is:1 mk
- Q3The common difference of the A.P. whose n^th term is given by a_n=3 n+7, is:1 mk
- Q3If Δ ABC ∼ Δ PQR with ∠ A=32^° and ∠ R=65^°, then the measure of ∠ B is:1 mk
- Q3In ABC, PQ ∥ BC. If PB=6 cm, AP=4 cm, AQ=8 cm, find the length of AC.1 mk
- Q3The 13^th term from the end of the A.P.: 20,13,6,-1, …..,-148 is:1 mk
- Q3If 2 x+3 y=15 and 3 x+2 y=25, then the value of x-y is:1 mk
- Q3The zeroes of the polynomial 3 x^2+11 x-4 are:1 mk
- Q3If p-1, p+1 and 2 p+3 are in A.P., then the value of p is1 mk
- Q3The number of quadratic polynomials having zeroes -5 and -3 is1 mk
- Q3The area of metal sheet required to make a closed hollow cylinder of height 2.4 m and base radius 0.7 m, is1 mk
- Q4If the mean and the mode of a distribution are 15 and 18 respectively, then the median of the distribution is:1 mk
- Q4The distance of the point (-1,7) from x -axis is:1 mk
- Q4In the given figure, DE ∥ BC. The value of x is:1 mk
- Q4If 'p' and 'q' are natural numbers and 'p' is the multiple of 'q', then what is the HCF of 'p' and 'q'?1 mk
- Q4In what ratio, does x -axis divide the line segment joining the points A(3, 6) and B(-12, -3)?1 mk
- Q4The point of intersection of the line represented by 3 x-y=3 and y-axis is given by1 mk
- Q4In the given figure, ∠ A=∠ C, AB=6 cm, AP=12 cm, CP=4 cm. Then length of CD is:1 mk
- Q4Two dice are rolled together. What is the probability of getting a sum greater than 10?1 mk
- Q5A quadratic equation whose roots are (2+√3) and (2-√3) is:1 mk
- Q5The sum of zeroes of the polynomial √2 x^2-17 are given as:1 mk
- Q5If α, β are zeroes of a polynomial p(x)=2 x^2-x-1 then α^2+β^2 is equal to1 mk
- Q5The circumferences of two circles are in the ratio 4: 5. What is the ratio of their radii?1 mk
- Q5What is the area of a semi-circle of diameter 'd'?1 mk
- Q5The distribution below gives the marks obtained by 80 students on a test: The modal class of this distribution is:1 mk
- Q5If x=0 · 3, is a root of the equation x^2-0 · 9 k=0, then k is equal to:1 mk
- Q5In the given figure, PQ is tangent to the circle centred at O. If ∠ AOB=95^°, then the measure of ∠ ABQ will be1 mk
- Q5The coordinates of the vertex A of a rectangle ABCD whose three vertices are given as B(0,0), C(3,0) and D(0,4) are:1 mk
- Q6If 2 tan A=3, then the value of (4 sin A+3 cos A)/(4 sin A-3 cos A) is1 mk
- Q6The empirical relation between the mode, median and mean of a distribution is:1 mk
- Q6If α and β are the zeroes of the polynomial x^2-1, then the value of (α+β) is1 mk
- Q6The LCM of smallest 2-digit number and smallest composite number is1 mk
- Q6If the area of the base of a cone is 51 cm^2 and its volume is 85 cm^3, then the vertical height of the cone is given…1 mk
- Q6If the pair of equations 3 x-y+8=0 and 6 x-ry+16=0 represent coincident lines, then the value of 'r' is:1 mk
- Q6One card is drawn at random from a well shuffled pack of 52 playing cards. The probability that the drawn card is a…1 mk
- Q6If tan θ=5/12, then the value of (sin θ+ cos θ)/(sin θ- cos θ) is:1 mk
- Q7The pair of linear equations 2 x=5 y+6 and 15 y=6 x-18 represents two lines which are:1 mk
- Q7The end-points of a diameter of a circle are (2,4) and (-3, -1). The radius of the circle is1 mk
- Q7PQ is tangent to a circle centered at O. If the radius of the circle is 5 cm, then the length of the tangent PQ is:1 mk
- Q7The next term of the A. P.: √7, √28, √63 is:1 mk
- Q7The distance between the points P(-11/3, 5) and Q(-2/3, 5) is:1 mk
- Q7(cos^2 θ)/(sin^2 θ)-1/(sin^2 θ), in simplified form, is:1 mk
- Q7The distance of the point (-4,3) from y -axis is1 mk
- Q7If α, β are the zeroes of a polynomial p(x)=x^2+x-1, then 1/(α)+1/(β) equals to1 mk
- Q7A bag contains 100 cards numbered 1 to 100. A card is drawn at random from the bag. What is the probability that the…1 mk
- Q8If PQR ∼ ABC; PQ=6 cm, AB=8 cm and the perimeter of ABC is 36 cm, then the perimeter of PQR is1 mk
- Q8If α, β are zeroes of the polynomial x^2-1, then value of (α+β) is:1 mk
- Q8Which of the following is a quadratic polynomial with zeroes 5/3 and 0?1 mk
- Q8( sec^2 θ-1)(cosec^2 θ-1) is equal to:1 mk
- Q8The pair of equations x=a and y=b graphically represents lines which are:1 mk
- Q8In the given figure, AB=BC=10 cm. If AC=7 cm, then the length of BP is:1 mk
- Q8The graph of y=p(x) is given in the adjoining figure. Zeroes of the polynomial p (x) are1 mk
- Q8If one zero of the polynomial x^2+3 x+k is 2, then the value of k.1 mk
- Q8The least positive value of k, for which the quadratic equation 2 x^2+k x-4=0 has rational roots, is1 mk
- Q9If a pole 6 m high casts a shadow 2 √3 m long on the ground, then sun's elevation is:1 mk
- Q9If sec θ- tan θ=1/3, then the value of ( sec θ+ tan θ) is:1 mk
- Q9The point of intersection of the line represented by 3 x-y=3 and the y-axis is given by1 mk
- Q9If one zero of the polynomial 6 x^2+37 x-(k-2) is reciprocal of the other, then what is the value of k?1 mk
- Q9If the quadratic equation a x^2+b x+c=0 has two real and equal roots, then 'c' is equal to1 mk
- Q9Water in a river which is 3 m deep and 40 m wide is flowing at the rate of 2 km / h. How much water will fall into the…1 mk
- Q9[3/4 tan^2 30^°- sec^2 45^°+ sin^2 60^°] is equal to1 mk
- Q10What is the total surface area of a solid hemisphere of diameter 'd'?1 mk
- Q10If the mean and the median of a data are 12 and 15 respectively, then its mode is:1 mk
- Q10In the given figure, DE ∥ BC. If AD=3 cm, AB=7 cm and EC=3 cm, then the length of AE is1 mk
- Q10sec θ when expressed in terms of cot θ, is equal to:1 mk
- Q10A quadratic equation whose roots are (3-√2) and (3+√2) is:1 mk
- Q10Curved surface area of a cylinder of height 5 cm is 94.2 cm^2. Radius of the cylinder is (Take π=3.14 )1 mk
- Q11The distribution below gives the marks obtained by 80 students on a test: The modal class of this distribution is:1 mk
- Q11If three coins are tossed simultaneously, what is the probability of getting at most one tail?1 mk
- Q11For the following distribution: Class Class Frequency Frequency 10 15 12 20 9 The sum of lower limits of median class…1 mk
- Q11A bag contains 5 pink, 8 blue and 7 yellow balls. One ball is drawn at random from the bag. What is the probability of…1 mk
- Q11In the given figure, AB is a tangent to the circle centered at O. If OA=6 cm and ∠ OAB=30^°, then the radius of the…1 mk
- Q11Two dice are thrown together. The probability of getting the difference of numbers on their upper faces equals to 3 is:1 mk
- Q11In a single throw of two dice, the probability of getting 12 as a product of two numbers obtained is:1 mk
- Q11If a, b, c form an A.P. with common difference d, then the value of a - 2b - c is equal to1 mk
- Q12In the given figure, DE ∥ BC. If AD=2 units, DB=AE=3 units and EC=x units, then the value of x is:1 mk
- Q12The length of tangent drawn to a circle of radius 9 cm from a point 41 cm from the centre is:1 mk
- Q12The curved surface area of a cone having height 24 cm and radius 7 cm, is1 mk
- Q12If ' α ' and ' β ' are the zeroes of the polynomial ax^2-5 x+c and α+β=α β=10, then:1 mk
- Q12In the given figure, ABC ∼ QPR. If AC=6 cm, BC=5 cm, QR=3 cm and PR=x; then the value of x is:1 mk
- Q12The volume of a right circular cone whose area of the base is 156 cm^2 and the vertical height is 8 cm, is1 mk
- Q12((2 tan 30^°)/(1+ tan^2 30^°)) is equal to:1 mk
- Q13The hour-hand of a clock is 6 cm long. The angle swept by it between 7:20 a.m. and 7:55 a.m. is:1 mk
- Q133 chairs and 1 table cost ₹ 900; whereas 5 chairs and 3 tables cost ₹ 2,100. If the cost of 1 chair is ₹ x and the cost…1 mk
- Q13The distance between the points (0,2 √5) and (-2 √5, 0) is1 mk
- Q13The distance of the point (-6,8) from origin is:1 mk
- Q13The solution of the pair of equations x+y=a+b and ax-by=a^2-b^2 is:1 mk
- Q13In the given figure, O is the centre of the circle and PQ is the chord. If the tangent PR at P makes an angle of 50^°…1 mk
- Q13In Δ ABC and Δ DEF, AB/DE=BC/FD. Which of the following makes the two triangles similar?1 mk
- Q14The 11^th term from the end of the A.P.: 10,7,4, … ….,-62 is:1 mk
- Q14In the given figure, PA and PB are tangents from external point P to a circle with centre C and Q is any point on the…1 mk
- Q14The zeroes of the polynomial p(x)=x^2+4 x+3 are given by:1 mk
- Q14In the given figure, PQ is a tangent to the circle with centre O. If ∠ OPQ=x, ∠ POQ=y, then x+y is:1 mk
- Q14A bag contains 5 red balls and n green balls. If the probability of drawing a green ball is three times that of a red…1 mk
- Q14Which of the following is a quadratic polynomial having zeroes -2/3 and 2/3?1 mk
- Q15In the given figure, TA is a tangent to the circle with centre O such that OT=4 cm, ∠ OTA=30^°, then length of TA is:1 mk
- Q15If the value of each observation of a statistical data is increased by 3, then the mean of the data1 mk
- Q15A card is drawn at random from a well shuffled deck of 52 playing cards. The probability of getting a face card is1 mk
- Q15Two coins are tossed together. The probability of getting at least one tail is:1 mk
- Q15For the following distribution: The modal class is:1 mk
- Q15If end points of a diameter of a circle are (-5, 4) and (1, 0), then the radius of the circle is:1 mk
- Q15In the given figure, the quadrilateral PQRS circumscribes a circle. Here PA+CS is equal to:1 mk
- Q16In ABC, PQ ∥ BC. If PB=6 cm, AP=4 cm, AQ=8 cm, find the length of AC.1 mk
- Q16If θ is an acute angle of a right angled triangle, then which of the following equation is not true?1 mk
- Q16A box contains 90 discs, numbered from 1 to 90. If one disc is drawn at random from the box, the probability that it…1 mk
- Q16The number of polynomials having zeroes -1 and 2 is:1 mk
- Q16If ' p ' is a root of the quadratic equation x^2-(p+q) x+k=0, then the value of 'k' is1 mk
- Q16In the given figure, PT is a tangent at T to the circle with centre O. If ∠ TPO=25^°, then x is equal to:1 mk
- Q16Probability of happening of an event is denoted by p and probability of non-happening of the event is denoted by q.…1 mk
- Q16In the given figure, AC and AB are tangents to a circle centered at O. If ∠ COD=120^°, then ∠ BAO is equal to:1 mk
- Q16If α and β are the zeroes of the quadratic polynomial p(x)=x^2-ax-b, then the value of α^2+β^2 is:1 mk
- Q16In a survey, it is found that every fifth person has a vehicle. The probability of a person NOT having a vehicle, is1 mk
- Q17[5/8 sec^2 60^°- tan^2 60^°+ cos^2 45^°] is equal to1 mk
- Q17A girl calculates that the probability of her winning the first prize in a lottery is 0.08. If 6000 tickets are sold,…1 mk
- Q17If α, β are the zeroes of the polynomial p(x)=4 x^2-3 x-7, then (1/(α)+1/(β)) is equal to:1 mk
- Q17The coordinates of the point where the line 2 y=4 x+5 crosses x -axis is1 mk
- Q17Which of the following numbers cannot be the probability of happening of an event?1 mk
- Q17If the zeroes of the quadratic polynomial x^2+(a+1) x+b are 2 and -3, then1 mk
- Q17The area of the triangle formed by the line x/a+y/b=1 with the coordinate axes is:1 mk
- Q17Cards bearing numbers 3 to 20 are placed in a bag and mixed thoroughly. A card is taken out of the bag at random. What…1 mk
- Q17In the given figure, PQ ∥ AC. If BP=4 cm, AP=2.4 cm and BQ=5 cm, then length of BC is:1 mk
- Q18The points (-4,0),(4,0) and (0,3) are the vertices of a:1 mk
- Q18If the sum of the first n terms of an A.P be 3 n^2+n and its common difference is 6, then its first term is1 mk
- Q18A card is drawn at random from a well-shuffled pack of 52 cards. The probability that the card drawn is not an ace is:1 mk
- Q18( cos^4 A- sin^4 A) on simplification, gives1 mk
- Q18The common difference of the A.P. whose n^th term is given by a_n=5 n-7 is:1 mk
- Q18In a group of 20 people, 5 can't swim. If one person is selected at random, then the probability that he/she can swim,…1 mk
- Q18The condition for the system of linear equations a x+by=c; l x+my=n to have a unique solution is1 mk
- Q18If every term of the statistical data consisting of n terms is decreased by 2, then the mean of the data:1 mk
- Q18In the given figure, AB ∥ PQ. If AB=6 cm, PQ=2 cm and OB=3 cm, then the length of OP is:1 mk
- Q19Statement A (Assertion): If 5+√7 is a root of a quadratic equation with rational co-efficients, then its other root is…1 mk
- Q19Assertion: A tangent to a circle is perpendicular to the radius through the point of contact. Reason (R): The lengths…1 mk
- Q19Assertion: Point P(0, 2) is the point of intersection of y-axis with the line 3 x+2 y=4. Reason (R): The distance of…1 mk
- Q19Assertion: If the points A(4, 3) and B(x, 5) lie on a circle with centre O(2,3), then the value of x is 2. Reason (R):…1 mk
- Q19Assertion: The probability that a leap year has 53 Sundays is 2/7. Reason (R): The probability that a non-leap year has…1 mk
- Q20Assertion: The perimeter of ABC is a rational number. Reason (R): The sum of the squares of two rational numbers is…1 mk
- Q20Assertion (A): The number 5^n cannot end with the digit 0, where n is a natural number. Reason (R): Prime factorisation…1 mk
- Q20Assertion: If PA and PB are tangents drawn from an external point P to a circle with centre O, then the quadrilateral…1 mk
- Q20Statement A (Assertion): For 0<θ ≤ 90^°, cosec θ- cot θ and cosec θ+ cot θ are reciprocal of each other. Statement R…1 mk
- Q20Assertion: a, b, c are in A.P. if and only if 2 b=a+c. Reason (R): The sum of first n odd natural numbers is n^2.1 mk
- Q20Assertion (A): The polynomial p(x)=x^2+3 x+3 has two real zeroes. Reason (R): A quadratic polynomial can have at most…1 mk
- Q21Find the greatest 3-digit number which is divisible by 18, 24 and 36.2 mk
- Q21Find the ratio in which the y -axis divides the line segment joining the points (5, -6) and (-1, -4).2 mk
- Q21Find the least number which when divided by 12, 16 and 24 leaves remainder 7 in each case.2 mk
- Q21If sin θ+ sin^2 θ=1, then prove that cos^2 θ+ cos^4 θ=1. OR If tan θ=1/(√7), then show that (cosec^2 θ- sec^2…2 mk
- Q21Solve the pair of equations x=3 and y=-4 graphically. OR Using graphical method, find whether following system of…2 mk
- Q21Evaluate: 5/(cot^2 30^°)+1/(sin^2 60^°)- cot^2 45^°+2 sin^2 90^° OR If θ is an acute angle and sin θ= cos θ, find the…2 mk
- Q21Two numbers are in the ratio 2: 3 and their LCM is 180. What is the HCF of these numbers?2 mk
- Q21Show that 6^n can not end with digit 0 for any natural number 'n'. OR Find the HCF and LCM of 72 and 120.2 mk
- Q21Prove that 2+√3 is an irrational number, given that √3 is an irrational number.2 mk
- Q21The line segment joining the points A(4,-5) and B(4,5) is divided by the point P such that AP: AB=2: 5. Find the…2 mk
- Q22If one zero of the polynomial p(x)=6 x^2+37 x-(k-2) is reciprocal of the other, then find the value of k.2 mk
- Q22If 4 cot^2 45^°- sec^2 60^°+ sin^2 60^°+p=3/4, then find the value of p. OR If cos A+ cos^2 A=1, then find the value of…2 mk
- Q22Find the ratio in which line y=x divides the line segment joining the points (6, -3) and (1, 6).2 mk
- Q22A line intersects y -axis and x -axis at point P and Q, respectively. If R(2,5) is the mid-point of line segment PQ,…2 mk
- Q22In the given figure, PT is a tangent to the circle centered at O. OC is perpendicular to chord AB. Prove that PA.…2 mk
- Q22Evaluate 2 sec^2 θ+3 cosec^2 θ-2 sin θ cos θ if θ=45^°. OR If sin θ- cos θ=0, then find the value of sin^4 θ+ cos^4 θ.2 mk
- Q22In the given figure, XZ is parallel to BC. AZ=3 cm, ZC=2 cm, BM=3 cm and MC=5 cm. Find the length of XY.2 mk
- Q22In the given figure, O is the centre of the circle. AB and AC are tangents drawn to the circle from point A. If ∠…2 mk
- Q23Find the length of the shadow on the ground of a pole of height 18 m when angle of elevation θ of the sun is such that…2 mk
- Q23Show that the points (-2, 3), (8,3) and (6,7) are the vertices of a right-angled triangle.2 mk
- Q23Find the sum and product of the roots of the quadratic equation 2 x^2-9 x+4=0. OR Find the discriminant of the…2 mk
- Q23If sin θ+ cos θ=√3, then find the value of sin θ · cos θ. OR If sin α=1/(√2) and cot β=√3, then find the value of cosec…2 mk
- Q23Solve the pair of equations x=5 and y=7 graphically. OR Using graphical method, find whether pair of equations x=0 and…2 mk
- Q23Show that 6^n can not end with digit 0 for any natural number 'n'. OR Find the LCM and HCF of 72 and 120.2 mk
- Q23If a cos θ+b sin θ=m and a sin θ-b cos θ=n, then prove that a^2+b^2=m^2+n^2. Prove that: √((sec A-1)/(sec A+1))+√((sec…2 mk
- Q23Using prime factorisation, find HCF and LCM of 96 and 120.2 mk
- Q24Find the ratio in which y-axis divides the line segment joining the points (5,-6) and (-1, -4).2 mk
- Q24If a fair coin is tossed twice, find the probability of getting 'atmost one head'.2 mk
- Q24Find the points on the x -axis, each of which is at a distance of 10 units from the point A(11, - 8).2 mk
- Q24Prove that the tangents drawn at the ends of a diameter of a circle are parallel.2 mk
- Q24In the given figure, PA is a tangent to the circle drawn from the external point P and PBC is the secant to the circle…2 mk
- Q24In the given figure, ABCD is a parallelogram. AE divides the line segment BD in the ratio 1: 2. If BE=1.5 cm, then find…2 mk
- Q24Find the greatest number which divides 85 and 72 leaving remainders 1 and 2 respectively.2 mk
- Q24The length of the shadow of a tower on the plane ground is √3 times the height of the tower. Find the angle of…2 mk
- Q25In the given figure, O is the centre of the circle. AB and AC are tangents drawn to the circle from point A. If ∠…2 mk
- Q25In the given figure, ABC is a triangle in which DE ∥ BC. If AD=x, DB=x-2, AE=x+2 and EC=x-1, then find the value of x.…2 mk
- Q25Prove that 4^n can never end with digit 0, where n is a natural number.2 mk
- Q25In the given figure, PQ is a chord of the circle centered at O. PT is a tangent to the circle at P. If ∠ QPT=55^°, then…2 mk
- Q25Prove that 6-√7 is irrational number, given that √7 is an irrational number.2 mk
- Q25A bag contains 4 red, 3 blue and 2 yellow balls. One ball is drawn at random from the bag. Find the probability that…2 mk
- Q25If a cos θ+b sin θ=m and a sin θ-b cos θ=n, then prove that a^2+b^2=m^2+n^2. OR Prove that: √((sec A-1)/(sec…2 mk
- Q25Evaluate (5 cos^2 60^°+4 sec^2 30^°- tan^2 45^°)/(sin^2 30^°+ cos^2 30^°) OR If A and B are acute angles such that sin…2 mk
- Q26Find the value of 'p' for which one root of the quadratic equation px^2-14 x+8=0 is 6 times the other.3 mk
- Q26Prove that: (sin θ)/(1+ cos θ)+(1+ cos θ)/(sin θ)=2 cosec θ3 mk
- Q26Prove that ( sin θ+ cos θ)( tan θ+ cot θ)= sec θ+cosec θ.3 mk
- Q26The centre of a circle is (2a, a -7). Find the values of 'a' if the circle passes through the point (11, -9). Radius of…3 mk
- Q26Half of the difference between two numbers is 2. The sum of the greater number and twice the smaller number is 13. Find…3 mk
- Q26Find the mean of the following distribution:3 mk
- Q26Find the ratio in which the line segment joining the points A(6,3) and B(-2, -5) is divided by x -axis.3 mk
- Q26Find by prime factorisation the LCM of the numbers 18180 and 7575. Also, find the HCF of the two numbers. OR Three…3 mk
- Q26In the given figure, AB and CD are diameters of a circle with centre O perpendicular to each other. If OA=7 cm, find…3 mk
- Q26In the given figure, CD is the perpendicular bisector of AB. EF is perpendicular to CD. AE intersects CD at G. Prove…3 mk
- Q26Prove that √3 is an irrational number. OR The traffic lights at three different road crossings change after every 48…3 mk
- Q26How many terms are there in an A.P. whose first and fifth terms are - 14 and 2, respectively and the last term is 62.…3 mk
- Q26Prove that: (cos^2 θ)/(1- tan θ)+(sin^3 θ)/(sin θ- cos θ)=1+ sin θ cos θ3 mk
- Q27Prove that √5 is an irrational number.3 mk
- Q27A 2-digit number is seven times the sum of its digits. The number formed by reversing the digits is 18 less than the…3 mk
- Q27If sin θ+ cos θ=p and sec θ+cosec θ=q, then prove that q(p^2-1)=2 p.3 mk
- Q27If p^th term of an A.P. is q and q^th term is p, then prove that its n^th term is (p+q-n).3 mk
- Q27Prove that: 2( sin^6 θ+ cos^6 θ)-3( sin^4 θ+ cos^4 θ)+1=0.3 mk
- Q27A natural number, when increased by 12, equals 160 times its reciprocal. Find the number. OR If one root of the…3 mk
- Q27A fraction becomes 1/3 when 1 is subtracted from the numerator. It becomes 1/4 when 8 is added to the denominator. Find…3 mk
- Q27A chord of a circle of radius 14 cm subtends an angle of 60° at the centre. Find the area of the corresponding minor…3 mk
- Q27Prove that: (1/(cos θ)- cos θ)(1/(sin θ)- sin θ)=1/(tan θ+ cot θ).3 mk
- Q27From an external point, two tangents are drawn to a circle. Prove that the line joining the external point to the…3 mk
- Q27Prove that √5 is an irrational number.3 mk
- Q27Find the HCF and LCM of 26, 65 and 117, using prime factorisation. OR Prove that √2 is an irrational number.3 mk
- Q28If Q(0,1) is equidistant from P(5,-3) and R(x, 6), find the values of x.3 mk
- Q28Prove that the angle between the two tangents drawn from an external point to a circle is supplementary to the angle…3 mk
- Q28Prove that: (tan A)/(1+ sec A)-(tan A)/(1- sec A)=2 cosec A3 mk
- Q28In the given figure, CD is the perpendicular bisector of AB. EF is perpendicular to CD. AE intersects CD at G. Prove…3 mk
- Q28If (-5,3) and (5,3) are two vertices of an equilateral triangle, then find co- ordinates of the third vertex, given…3 mk
- Q28In the given figure, E is a point on the side CB produced of an isosceles triangle ABC with AB=AC. If AD ⊥ BC and EF ⊥…3 mk
- Q29Prove that (sin A-2 sin^3 A)/(2 cos^3 A- cos A)= tan A OR Prove that sec A(1- sin A)( sec A+ tan A)=1.3 mk
- Q29The sum of first 15 terms of an A.P. is 750 and its first term is 15. Find its 20^th term. OR Rohan repays his total…3 mk
- Q29How many terms are there in A.P. whose first and fifth term are - 14 and 2, respectively and the last term is 62. OR…3 mk
- Q29A car has two wipers which do not overlap. Each wiper has a blade of length 21 cm sweeping through an angle of 120°.…3 mk
- Q29Two people are 16 km apart on a straight road. They start walking at the same time. If they walk towards each other…3 mk
- Q29Two tangents TP and TQ are drawn to a circle with centre O from an external point T. Prove that ∠ PTQ=2 ∠ OPQ. OR In…3 mk
- Q29The sum of two numbers is 15. If the sum of their reciprocals is 3/10, find the two numbers. OR If α and β are roots of…3 mk
- Q30Prove that √3 is an irrational number.3 mk
- Q30If the system of linear equations 2 x+3 y=7 and 2 a x+(a+b) y=28 have infinite number of solutions, then find the…3 mk
- Q30In an A.P., the sum of the first n terms is given by S_n=6 n-n^2. Find its 30^th term.3 mk
- Q30Prove that: (tan θ+ sec θ-1)/(tan θ- sec θ+1)=(1+ sin θ)/(cos θ)3 mk
- Q30Two concentric circles are of radii 5 cm and 3 cm. Find the length of the chord of the larger circle which touches the…3 mk
- Q30Prove that: (tan θ)/(1- cot θ)+(cot θ)/(1- tan θ)=1+ sec θ cosec θ3 mk
- Q30Prove that (1+ sec A)/(sec A)=(sin^2 A)/(1- cos A).3 mk
- Q31Prove that (cosec A- sin A)( sec A- cos A)=1/(cot A+ tan A).3 mk
- Q31Jaya scored 40 marks in a test getting 3 marks for each correct answer and losing 1 mark for each incorrect answer. Had…3 mk
- Q31Find the common difference of an A.P. whose first term is 8, the last term is 65 and the sum of all its terms is 730.3 mk
- Q31Reeti prepares a Rakhi for her brother Ronit. The Rakhi consists of a rectangle of length 8 cm and breadth 6 cm…3 mk
- Q31A room is in the form of cylinder surmounted by a hemi-spherical dome. The base radius of hemisphere is one-half the…3 mk
- Q31Find the value of 'p' for which the quadratic equation px(x-2)+6=0 has two equal real roots.3 mk
- Q31In a circle of radius 21 cm, an arc subtends an angle of 60^° at the centre. Find the area of the sector formed by the…3 mk
- Q31Find the mean of the following frequency distribution:3 mk
- Q31In the given figure, O is the centre of the circle and QPR is a tangent to it at P. Prove that ∠ QAP+∠ APR=90^°.3 mk
- Q32A straight highway leads to the foot of a tower. A man standing on the top of the 75 m high tower observes two cars at…5 mk
- Q32In a PQR, N is a point on PR, such that QN ⊥ PR. If PN × NR=QN^2, prove that ∠ PQR=90^°. OR In the given figure, ABC…5 mk
- Q32A ladder set against a wall at an angle 45° to the ground. If the foot of the ladder is pulled away from the wall…5 mk
- Q32From a solid cylinder of height 20 cm and diameter 12 cm, a conical cavity of height 8 cm and radius 6 cm is hallowed…5 mk
- Q32If a line is drawn parallel to one side of a triangle to intersect the other two sides at distinct points, prove that…5 mk
- Q32The mode of the following frequency distribution is 55. Find the missing frequencies 'a' and 'b'.5 mk
- Q32How many terms of the arithmetic progression 45, 39, 33, must be taken so that their sum is 180? Explain the double…5 mk
- Q32Two tangents TP and TQ are drawn to a circle with centre O from an external point T. Prove that ∠ PTQ=2 ∠ OPQ. OR A…5 mk
- Q32The sum of first seven terms of an A.P. is 182. If its 4^th term and the 17^th term are in the ratio 1: 5, find the…5 mk
- Q32One observer estimates the angle of elevation to the basket of a hot air balloon to be 60°, while another observer 100…5 mk
- Q33Prerna saves ₹ 32 during the first month, ₹ 36 in the second month and ₹ 40 in the third month. If she continues to…5 mk
- Q33As observed from the top of a 75 m high lighthouse from the sea-level, the angles of depression of two ships are 30°…5 mk
- Q33The angle of elevation of the top of a tower 24 m high from the foot of another tower in the same plane is 60^°. The…5 mk
- Q33A wooden article was made by scooping out a hemisphere from each end of a solid cylinder, as shown in the figure. If…5 mk
- Q33A triangle ABC is drawn to circumscribe a circle of radius 4 cm such that the segments BD and DC are of lengths 10 cm…5 mk
- Q33A solid is in the shape of a right-circular cone surmounted on a hemisphere, the radius of each of them being 7 cm and…5 mk
- Q33D is a point on the side BC of a triangle ABC such that ∠ ADC=∠ BAC, prove that CA^2=CB. CD OR If AD and PM are medians…5 mk
- Q33Prove that a parallelogram circumscribing a circle is a rhombus. OR In the given figure, tangents PQ and PR are drawn…5 mk
- Q34Sides AB and AC and median AM of a ABC are proportional to sides DE and DF and Median DN of another DEF. Show that ABC…5 mk
- Q34A train travels at a certain average speed for a distance of 54 km and then travels a distance of 63 km at an average…5 mk
- Q34A student was asked to make a model shaped like a cylinder with two cones attached to its ends by using a thin…5 mk
- Q34A chord of a circle of radius 14 cm subtends an angle of 60° at the centre. Find the area of the corresponding minor…5 mk
- Q34The angle of elevation of the top of a vertical tower from a point P on the ground is 60^°. From another point Q, 10 m…5 mk
- Q34A student noted the number of cars passing through a spot on a road for 100 periods each of 3 minutes and summarised it…5 mk
- Q34The ratio of the 11^th term to the 18^th term of an A.P. is 2: 3. Find the ratio of the 5^th term to the 21 ^st term.…5 mk
- Q34PA, QB and RC are each perpendicular to AC. If AP=x, QB=z, RC=y, AB=a and BC=b, then prove that 1/x+1/y=1/z. OR In the…5 mk
- Q34A solid is in the shape of a right-circular cone surmounted on a hemisphere, the radius of each of them being 3.5 cm…5 mk
- Q34Solve the equation for x: 1+4+7+10+….+x=2875 mk
- Q35The angle of elevation of the top of a tower 30 m high from the foot of another tower in the same plane is 60^° and the…5 mk
- Q35In the given figure, ∠ ADC=∠ BCA; prove that Δ ACB ∼ Δ ADC. Hence find BD if AC=8 cm and AD=3 cm. OR If a line is drawn…5 mk
- Q35A horse is tied to a peg at one corner of a square shaped grass field of side 15 m by means of a 5 m long rope. Find…5 mk
- Q35Sides AB and BC and median AD of a triangle ABC are respectively proportional to sides PQ and QR and median PM of Δ…5 mk
- Q35250 apples of a box were weighed and the distribution of masses of the apples is given in the following table: (i) Find…5 mk
- Q35An aeroplane when flying at a height of 3000 m from the ground passes vertically above another aeroplane at an instant…5 mk
- Q35Find the sum of integers between 100 and 200 which are (i) divisible by 9 (ii) not divisible by 9. OR Solve the…5 mk
- Q35The ratio of the 11^th term to 17^th term of an A.P. is 3: 4. Find the ratio of 5^th term to 21^st term of the same…5 mk
- Q35The monthly expenditure on milk in 200 families of a Housing Society is given below: Find the value of x and also, find…5 mk
- Q35Two pillars are standing on either side of a 80 m wide road. Height of one pillar is 20 m more than the height of the…5 mk
- Q36A golf ball is spherical with about 300-500 dimples that help increase its velocity while in play. Golf balls are…4 mk
- Q36Two schools 'P' and 'Q' decided to award prizes to their students for two games of Hockey ₹ x per student and Cricket ₹…4 mk
- Q36While designing the school year book, a teacher asked the student that the length and width of a particular photo is…4 mk
- Q36A coaching institute of Mathematics conducts classes in two batches I and II and fees for rich and poor children are…4 mk
- Q36In a pool at an aquarium, a dolphin jumps out of the water travelling at 20 cm per second. Its height above water level…4 mk
- Q36In an annual day function of a school, the organizers wanted to give a cash prize along with a memento to their best…4 mk
- Q37India meteorological department observes seasonal and annual rainfall every year in different sub-divisions of our…4 mk
- Q37Radio towers are used for transmitting a range of communication services including radio and television. The tower will…4 mk
- Q37Jagdish has a field which is in the shape of a right angled triangle AQC. He wants to leave a space in the form of a…4 mk
- Q37While designing the school year book, a teacher asked the student that the length and width of a particular photo is…4 mk
- Q37The discus throw is an event in which an athlete attempts to throw a discus. The athlete spins anti-clockwise around…4 mk
- Q37Two schools 'P' and 'Q' decided to award prizes to their students for two games of Hockey ₹ x per student and Cricket ₹…4 mk
- Q37A middle school decided to run the following spinner game as a fund-raiser on Christmas Carnival. Making Purple: Spin…4 mk
- Q37In a coffee shop, coffee is served in two types of cups. One is cylindrical in shape with diameter 7 cm and height 14…4 mk
- Q38In a pool at an aquarium, a dolphin jumps out of the water travelling at 20 cm per second. Its height above water level…4 mk
- Q38The discus throw is an event in which an athlete attempts to throw a discus. The athlete spins anti-clockwise around…4 mk
- Q38Governing council of a local public development authority of Dehradun decided to build an adventurous playground on the…4 mk
- Q38Two schools 'P' and 'Q' decided to award prizes to their students for two games of Hockey ₹ x per student and Cricket ₹…4 mk
- Q38Computer-based learning (CBL) refers to any teaching methodology that makes use of computers for information…4 mk
- Q38"Eight Ball" is a game played on a pool table with 15 balls numbered 1 to 15 and a "cue ball" that is solid and white.…4 mk
- Q38In a pool at an aquarium, a dolphin jumps out of the water travelling at 20 cm per second. Its height above water level…4 mk
Other Mathematics years
Answers come from CBSE’s own marking schemes. How we source them.