Important Key Points of Chapter Quadratic Equations (CBSE Class 10)

 

Important Key Points of Chapter Pair of Linear Equations (CBSE Class 10)

Important Key Points of Chapter Polynomial (CBSE Class 10)

 

 Important Key Points of Chapter Polynomial 

(CBSE Class 10)

Important Key Points of Chapter Real Number (CBSE Class 10)

 Important Key Points of Chapter Real Number 

(CBSE Class 10)

UP Board Class 10 Examination Paper (2023) PDF

UP Board Class 10 Examination Paper 

(Session 2022 - 23)

                        This Paper is also available in PDF. 

                    Click the link and download the paper.

https://drive.google.com/file/d/1YQq62NX9oS6QZ3JCUr0Hde2HQCExzFsE/view?usp=share_link

Self Evaluation Test (Practice Paper) Class - X

Self Evaluation Test (Practice Paper)

Class - X

 1. Find the H.C.F of 1656 and 4025 by Euclid’s division algorithm.

2. Prove that 4-3√2 is an irrational.

3. The HCF of 65 and 117 is expressible in the form of 65m-117. Find the value of m. Also, find the LCM of 65 and 117 using prime factorization method.

4. For any positive integer n, prove that n3 -n is divisible by 6.

5. Find the quadratic polynomial whose zeroes are 3 and -4 respectively.

6. If a and b are the zeroes of a quadratic polynomial such that a+b = 24 and

a-b = 8. Find the quadratic polynomial having a and b as its zeroes. Verify the relationship between the zeroes and coefficients of the polynomial.

7. If the polynomial f(x) = 3x4 -9x3 +x2 +15x+k is completely divisible by     3x² -5, find the value of k and hence the other two zeroes of the polynomial.

8. For which value of k will the following pair of linear equation is inconsistent? 3x+y=1;        (2k-1)x+(k-1)y=2k+1.

10. Solve the following pair of equations graphically: 2x+3y=12; x-y-1=0. Shade the region between the two lines represented by the above equations and the x-axis.

11. Raghav scored 70 marks in a test, getting 4 marks for each right answer and losing 1 marks for each wrong answer. Had 5 marks been awarded for each correct answer and 2 marks been deducted for each wrong answer, then Raghav would have scored 80 marks. How many questions were there in the test? Which values would have Raghav violated if he resorted to unfair means.

12. Find the values of k for which the quadratic equation 9x2-3kx+k=0 has real roots.

https://miteshtutorialinfo.blogspot.com/2023/09/blog-post.html

13. A motor boat whose speed is 24 km/h in still water takes 1 hour more to go 32 km upstream than to return downstream to the same spot. Find the speed of the stream.

14. The sum of the squares of two consecutive multiples of 7 is 637. Find the multiples.

15. If the number x-2, 4x-1 and 5x+2 are in A.P. Find the value of x.

16. The angles of a triangle are in A.P. The greatest angle is twice the least . Find all angles of the triangle.

17. Which term of the A.P : 121, 117, 113, … is its first negative term?

18. The 17th term of an A.P is 5 more than twice its 8th term. If the 11th term of the A.P is 43, then find its nth term.

19. State and prove Basic Proportionality Theorem.

20. State and prove Pythagorus Theorem.

21. The perpendicular from A on side BC of a △ABC intersects BC at D, such that

DB = 3CD. Prove that 2AB2 = 2AC2 + BC2.

22. Prove that the points (3,0), (6,4) and (-1,3) are vertices of right angled isosceles triangle.

23. The x-coordinate of a point P is twice its y-coordinate. If P is equidistant from Q(2,-5) and R(-3,6), find the coordinates of P.

24. Find the ratio in which the line 2x + y = 4 divides the join of A (2,-2) and B (3,7). Also, find the coordinates of the point of their intersection.

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25. Find the points on the x-axis which is equidistant from (2,-5) and (-2,9).

26. If sin 3 A= cos (A-26○) , where 3A is an acute angle, find the value of A?

27. Evaluate: sin 25○ cos65○ + cos25○ sin 65○

28. If x= a sin A + b cos A and y= a cos A – b sin A, then prove that x2 + y2 = a2+ b2.

29. What happens to value of cosA when A increases from 0○-90○?

30. If the height and the length of the shadow of a man are the same, then find the angle of elevation of the Sun.

31. An observer 1.5m tall is 20.5 m away from a tower of 22m high. Determine the angle of elevation of the tower from the eye of the observer.

32. A man on the top of the vertical tower observes a car moving at a uniform speed towards him. If it takes 12 min for the angle of depression to change from 30○ to 45○ , how soon after this, the car will reach the tower?

33. Prove that the tangents drawn from an external point to a circle are equal in length.

34. From a point Q, 13cm away from the Centre of a circle, the length of tangent PQ to the circle is 12 cm .Find the radius of the circle?

35. A triangle ABC is drawn to circumscribe a circle of radius 4cm such that the segments BD and DC into which BC is divided by the point of contact D are of lengths 8cm and 6 cm respectively. Find the sides AB and AC?

36. Divide a line segment of length 7.6 cm in the ratio 2:7.

37. Draw a right triangle with sides of length 5cm and 4 cm making a right angle .Construct another triangle whose sides are 3/5 times the corresponding sides of the first triangle.

38. Draw a pair of tangents to a circle of radius 4cm which are inclined to each other at an angle of 60○.

39. If the area and circumference of a circle are numerically equal, then find the diameter of the circle.

40. A paper is in the form of a rectangle ABCD in which AB = 20 cm, BC = 14cm. A semi-circular portion with BC as diameter is cut off. Find the area of the remaining part.

41. Find the area of the minor segment of a circle of radius 14cm, when its central angle is 60○. Also find the area of the corresponding major segment.

42. Find the difference of the areas of a sector of angle 120○ and its corresponding major sector of a circle of radius 21 cm.

43. A cylindrical tub, whose diameter is 12 cm and height 15 cm is full of ice-cream. The whole ice cream is to be divided into 10 children in equal ice-cream cones, with conical base surmounted by hemispherical top. If the height of conical portion is twice the diameter of base, find the diameter of conical part of ice-cream cone.

44. Find the mass of silver cone of silver metal having base diameter 14 cm and vertical height 51 cm. the density of silver is 10 g/cm3.

45. Gayatri was making a mathematical model, in which she placed 4 cubes each of edge 20 cm one above another. Find the surface area of the resulting cuboid.

46. A bucket open at the top is in the form of a frustum of a cone with a capacity of 12308.8 cm3. The radii of the top and bottom circular ends are 20 cm and 12 cm respectively. Find the height of the bucket and the area of metal sheet used in making the bucket.

47. Find the mode if mean and median are 10.5 and 9.6 respectively.

48. The following table gives the daily income of 50 workers of a factory. Draw more than type ogive.

49. A box contains 20 cards numbered from 1 to 20. A card is drawn at random from the box. Find the probability that the no. drawn card is divisible by 2 or 3.

50. A no. x is numbered at random from the numbers 1, 2, 3 and 4. Another number y is selected at random from the numbers 1, 4, 9 and 16. Find the probability that product of x and y is less than 16.

51. A bag contains 18 balls out of which x balls are red. If 2 more        balls are put in the bag, the probability of drawing a red ball        will be 9/8 times the probability of not drawing red ball. Find

        the value of x.

Mathematics Basic Formulae Part - 1

                                     

         Mathematics Basic Formulae Part - 1

Algebraic Expansions:

1. (a + b)² = a² + 2ab + b²

2. (a - b)² = a² - 2ab + b²  

3. (a + b) (a - b) = a² - b²

4. (x + a)(x + b) = x² + (a + b)x + ab

5. (x + a)(x - b) = x² + (a - b)x - ab   

6. (x - a)(x + b) = x² + (b - a)x - ab 

7. (x - a)(x - b) = x² - (a + b)x + ab 

8. (a + b)³ = a³ + b³ + 3ab(a + b)

9. (a - b)³ = a³ - b³ - 3ab(a - b)  

10. (x + y + z)² = x² + y² + z² + 2xy + 2yz + 2xz 

11. (x + y - z)² = x² + y² + z² + 2xy - 2yz - 2xz

12. (x - y + z)² = x² + y² + z² - 2xy - 2yz + 2xz 

13. (x - y - z)² = x² + y² + z² - 2xy + 2yz - 2xz 

14. x³ + y³ + z³ - 3xyz = (x + y + z)(x² + y² + z² - xy - yz -xz)

15. x² + y² = 1212 [(x + y)² + (x - y)²]

16. (x + a) (x + b) (x + c) = x³ + (a + b + c)x² 

+ (ab + bc + ca) x + abc

17.  x³ + y³ = (x + y) (x² - xy + y²)

18.  x³ - y³ = (x - y) (x² + xy + y²)

  

2D and 3D figures

Figure	Formula for surface area	Formula for volume
Rectangular Prism	SA = 2ab + 2bc + 2ca  sq. Units  where a, b,c are the sides of the cube.	V = abc  cubic units
Cylinder	SA = 2πrh  sq. units  TSA =  2πr(h+r)  sq. units  r = radius of the cylinder  h – height of the cylinder	V = πr2h  cubic units
Cube	SA = 6a2  sq. units  a = sides of the cube	V = a3   cubic units
Sphere	SA = 4πr2  sq. units  r = radius of the sphere	V = 4343 πr3   cubic units
Ellipsoid	SA = 4π[(apbp+apcp+bpcp3)]1p4π[(apbp+apcp+bpcp3)]1p  p = 1.6075  a,b,c are semi axis of ellipsoid	V = 4343  π r1,r2,r3   cubic units
Cone	CSA = πr1 sq. units	V = 1313  πr2h   cubic units
Pyramid	SA = a + 1212*p*1   p = perimeter of pyramid   l = slant height   a= area of the base of the pyramid	V = 1313*a*h    cubic units
Hemisphere	CSA = 2πr2  TSA = 3πr2  r = radius	V = 2323 πr3    cubic units
Triangle	SA = √s(s-a) (s-b) (s-c)  where s is the perimeter of the triangle  a, b, c are the sides of the triangle

Simple interest =  PTR / 100                      

Pythagorean Theorem:   a² + b² = c²

^{Rational Number}

§  n(n + l)(2n + 1) is always divisible by 6.

§  3²ⁿ leaves remainder = 1 when divided by 8 

§  n³ + (n + 1 )³ + (n + 2 )³ is always divisible by 9

§  10^2n + 1 + 1 is always divisible by 11

§  n(n²– 1) is always divisible by 6

§  n²+ n is always even

§  2³ⁿ-1 is always divisible by 7

§  15^2n-1 +l is always divisible by 16

§  n³ + 2n is always divisible by 3

§  3⁴ⁿ – 4 ³ⁿ is always divisible by 17

 

Angles

1.   Product of n consecutive numbers is always divisible by n!. 

2.   If n is a positive integer and p is a prime, then n^p – n is divisible by p.

3.   |x| = x if x ≥ 0 and |x| = – x if x ≤ 0.

4.   Minimum value of a².sec²Ɵ + b².cosec²Ɵ is (a + b)²; (0° < Ɵ < 90°) for eg. minimum value of 49 sec²Ɵ + 64.cosec²Ɵ is     (7 + 8)² = 225 

5.   Among all shapes with the same perimeter a circle has the largest area.

6.   If one diagonal of a quadrilateral bisects the other, then it also bisects the quadrilateral.

7.   Sum of all the angles of a convex quadrilateral = (n – 2)180°

8.   Number of diagonals in a convex quadrilateral = 0.5n(n – 3)

 

● Laws of Indices:

(i) aᵐ ∙ aⁿ = aᵐ + ⁿ 

(ii) aᵐ/aⁿ = aᵐ - ⁿ 

(iii) (aᵐ)ⁿ = aᵐⁿ 

(iv) a = 1 (a ≠ 0). 

(v) a-ⁿ = 1/aⁿ 

(vi) ⁿ√aᵐ = aᵐ/ⁿ 

(vii) (ab)ᵐ = aᵐ ∙ bⁿ. 

(viii) (a/b)ᵐ = aᵐ/bⁿ 

(ix) If aᵐ = bᵐ (m ≠ 0), then a = b. 

(x) If aᵐ = aⁿ then m = n.