Important Questions / Chapterwise Assignment
Summative Assessment-1 CLASS - X
MATHEMATICS
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REAL NUMBERS CLASS X Questions From CBSE Examination Papers 1.
Can the number 6n, n being a natural number, end with the digit 5? Give reasons.
18. Show that any positive odd integer is of the form 6q+ 1, , 6q+ 3 or 6q+ 5 where q is a positive integer.
2.
Use Euclid’s division lemma to show that square of any positive integer is either of form 3m or 3m+ 1 for some integer m.
19. Prove that (5 -
3 ) is an irrational number.
3.
Find the L.C.M. of 120 and 70 by fundamental theorem of Arithmetic.
20. Show that any positive odd integer is of the form 4q + 1 or 4q + 3, where q is a positive integer. 5 2 21. Prove that is an irrational number. 3
4.
In the given factor tree, find the numbers m, n :
22. Prove that
m
2 is an irrational number.
23. Show that the square of any positive odd integer is of the form 8m + 1, for some integer m. 210
24. Prove that
7 is an irrational number.
25. Use Euclid’s division algorithm to fnd the HCF of 10224 and 9648. 26. Prove that 3 5 - 2 is an irrational number.
n 35
27. Prove that 2 3 - 4 is an irrational number.
7 5.
Without actually performing the long division, state whether the following number has a terminating decimal expansion or non terminating recurring 543 decimal expansion . 225
6.
Use Euclid’s division algorithm to fnd HCF of 870 and 225.
7. 8. 9.
Check whether 6n can end with the digit 0, for any natural number n. Explain why 11 × 13 × 15 × 17 + 17 is a composite number. Show that every positive even integer is of the form 2 q and that every positive odd integer is of the form 2q + 1, where q is some integer.
10.
Check whether 15 n can end with digit zero for any natural number n.
11.
Find the LCM of 336 and 54 by prime factorisation method.
12.
Find the LCM and HCF of 120 and 144 by fundamental theorem of arithmetic.
13.
Use Euclid’s Lemma to show that square of any positive integer is of form 4 m or 4m+ 1 for some integer m.
14.
Using fundamental theorem of arithmetic, fnd the HCF of 26, 51 and 91.
15.
Find the LCM and HCF of 15, 18, 45 by the prime factorisation method.
16.
Prove that 2 + 3 2 is irrational.
17.
Prove that
7 2 is not a rational number. 5
28. A merchant has 120 litres of oil of one kind, 180 litres of another kind and 240 litres of third kind. He wants to sells the oil by flling, the three kinds of oil in tins of equal capacity. What should be the greatest capacity of such a tin? 1 29. Prove that is an irrational number. 2+ 3 30. Show that 9n can’t end with 2 for any integer n. 31. An army contingent of 616 members is to march behind an army band of 32 members in a parade. The two groups are to march in the same number of columns. What is the maximum number of columns in which they can march? 32. Find the HCF and LCM of 306 and 54. Verify that HCF × LCM = Product of the two numbers. 33. Use Euclid division lemma to show that cube of any positive integer is either of the form 9m, 9m + 1, or 9m + 8. 34. Show that the square of any positive integer cannot be of the form 5q + 2 or 5q + 3 for any integer q. 35. Show that any positive even integer is of the form 6m, 6m + 2 or 6m + 4. where m is some integer. 36. Show that 5n can’t end with the digit 2 for any natural number n. 37. Prove that
5 is an irrational number.
38. Show that any positive odd integer is of the form 6 p + 1, 6 p + 3 or 6p+5, where p is some integer. 1 39. Prove that is an irrational number. 2+ 3
POLYNOMIALS CLASS X Questions From CBSE Examination Papers 1. 2. 3.
4.
5.
Divide 6x3 + 13x2 + x – 2 by 2x + 1, and find the quotient and remainder. Divide x4 – 3x2 + 4x + 5 by x2 – x + 1, find quotient and remainder.
17.
3x 2 - 8 x + 4 3. 18.
Check whether x2 + 3x + 1 is a factor of 3x4 + 5x3 – 7x2 + 2x + 2.
a , ß are the roots of the quadratic polynomial p(x) = x2 – (k – 6) x + (2k + 1). Find the value of
19.
k, if a + ß= aß .
20.
a , ß are the roots of the quadratic polynomial p(x) = x2 – (k + 6)x + 2 (2k – 1). Find the value 1 of k, if a + ß= aß . 2 Find the zeroes of the polynomial
21.
Check whether x2 – x + 1 is a factor of x3 – 3x2 + 3x – 2. Find the zeroes of the quadratic polynomial x2 + 7x + 12 and verify the relationship between the zeroes and its coeffcients. Divide (2x2 + x – 20) by (x + 3) and verify division algorithm. If a and ß are the zeroes of x2 + 7x + 12, then 1 1 - 2aß . find the value of + a ß
22.
2
4 3 x + 5 x - 2 3. 6.
Find a quadratic polynomial whose zeroes are 3+ 5 and 3 -
7.
8.
9.
5.
What must be added to polynomial f(x) = x4 + 2x3 – 2x2 + x – 1 so that the resulting polynomial is exactly divisible by x2 + 2x – 3. Find a quadratic polynomial, the sum of whose zeroes is 7 and their product is 12. Hence find the zeroes of the polynomial. Find a quadratic polynomial whose zeroes are 2 and –6. Verify the relation betweeen the coeffcients and zeroes of the polynomial. 1 are the zeroes of the polynomial a 2 4x – 2 x + (k – 4), find the value of k.
10.
If a and
11.
Find the zeroes of the polynomial 100x2 – 81.
12.
13.
14.
15. 16.
Find the zeroes of the quadratic polynomial
2
23.
For what value of k, is –2 a zero of the polynomial 3x2 + 4x + 2k?
24.
For what value of k, is –3 a zero of the polynomial x2 + 11x + k? If a and ß are the zeroes of the polynomial 2y2 + 7y + 5, write the value of á + â + áâ.
25.
26.
For what value of k, is 3 a zero of the polynomial 2x2 + x + k?
27.
If the product of zeroes of the polynomial ax2 – 6x – 6 is 4, find the value of a.
28.
Find the quadratic polynomial, sum of whose zeroes is 8 and their product is 12. Hence, find the zeroes of the polynomial.
29.
If one zero of the polynomial (a2 + 9) x2 + 13x + 6a is reciprocal of the other, find the value of a. If á and â are zeroes of the quadratic polynomial x2 – 6x + a; find the value of a if 3a + 2ß= 20.
3
Divide the polynomial p(x) = 3x – x – 3x + 5 by g(x) = x – 1 – x2 and find its quotient and remainder. Can (x + 3) be the remainder on the division of a polynomial p(x) by (2x – 5)? Justify your answer. Can (x – 3) be the remainder on division of a polynomial p(x) by (3 x + 2)? Justify your answer. Find the zeroes of the polynomial 2x2 – 7x + 3 and hence find the sum of product of its zeroes. It being given that 1 is one of the zeros of the polynomial 7x – x3 – 6. Find its other zeros.
30. 31.
Divide (6 + 19x + x2 – 6x3) by (2 + 5x – 3x2) and verify the division algorithm.
32.
If á, â, ã are zeroes of the polynomial 6x3 + 3x2 – 5x + 1, then find the value of á –1 + â–1 + ã–1.
33.
If the zeroes of the polynomial x3 – 3x2 + x + 1 are a – b, a and a + b, find the values of a and b.
34.
On dividing x3 – 3x2 + x + 2 by a polynomial g(x), the quotient and remainder were x – 2 and - 2 x + 4 respectively. Find g(x).
35.
If á, â are zeroes of the polynomial x2 – 2x – 8, then form a quadratic polynomial whose zeroes are 2á and 2â.
36.
If á, â are the zeroes of the polynomial 6y2– 7y + 2 + 2, find a quadratic polynomial whose zeroes are 1 1 and . a ß If á, â are zeroes of the polynomial x2 – 4x + 3, then form a quadratic polynomial whose zeroes are 3a and 3b . Obtain all zeroes of f(x) = x4 – 3x3 – x2 + 9x – 6 if two of its zeroes are (- 3 ) and 3 .
37.
38. 39.
Check whether the polynomial g(x) = x3 – 3x + 1 is the x5 - 4 x3 + x2 + 3x + 1 factor of polynomial p ( x) =
52.
If the polynomial x 4 + 2 x 3 + 8 x 2 + 12 x + 18 is divided by another polynomial x 2 + 5, the remainder comes out to be px + q. Find the values of p and q.
53.
Find all the zeroes of the polynomial x3 + 3x2 – 2x –6, if two of its zeroes are - 2 and 2.
54.
2x 3+ x2 – 6x – 3, if two of its zeroes are - 3 and 3. 55.
56.
57.
41.
Find the zeroes of 4 3 x 2 + 5 x - 2 3 and verify the relation between the zeroes and coeffcients of the polynomial.
58.
42.
If á, â are the zeroes of the polynomial 25p2 –15p + 2 find a quadratic polynomial whose zeroes are 1 1 and . 2a 2ß
43.
Divide 3x2 – x3 – 3x + 5 by x – 1 – x2 and verify the division algorithm.
44.
If á, â are the zeroes of the polynomial 21y2 – y -2 find a quadratic polynomial whose zeroes are 2á and 2â.
45.
Find the zeroes of 3 2 x 2 + 13 x + 6 2 and verify the relation between the zeroes and coeffcients of the polynomial.
46.
Find the zeroes of 4 5 x 2 + 17 x + 3 5 and verify the relation between the zeroes and coeffcients of the polynomial.
47.
48.
49. 50. 51.
If the polynomial 6 x4 + 8x3 + 17x2 + 21x + 7 is divided by another polynomial 3x2 + 4x + 1, the remainder comes out to be (ax + b), find a and b. 4
– 3x2 + 6x – 2, if - 2 and 2 are the zeroes of the given polynomial. If the remainder on division of x3 + 2x2 + kx + 3 by x – 3 is 21, find the quotient and the value of k. Hence, fnd the zeroes of the cubic polynomial x3 + 2x2 + kx – 18. If two zeroes of p(x) = x4 – 6x3 – 26x2 + 138x – 35
59.
If the polynomial x4 – 6x3 + 16x2 – 25x + 10 is divided by another polynomial x2 – 2x + k, the remainder comes out to be (x + a), find the values of k and a.
60.
Find all the zeroes of the polynomial 2x 4 + 7x3 – 19x2 – 14x + 30, if two of its zeros are
61.
62.
2 , - 2. Find other zeroes of the polynomial x4 + x3 – 9x2 – 3x + 18, if it is given that two of its zeroes are 3 and- 3. Divide 2x4 – 9x3 + 5x2 + 3x – 8 by x2 – 4x + 1 and verify the division algorithm.
63.
Divide 30x4 + 11x3 – 82x2 – 12x + 48 by (3x2 + 2x – 4) and verify the result by division algorithm.
64.
Find all zeroes of the polynomial 4x4– 20 x3 + 23x2 + 5x – 6, if two of its zeroes are 2 and 3. Find all the zeroes of the polynomial 2x4 – 10x3 + 5x2 + 15x – 12, if it is given that two of its zeroes 3 3 are and. 2 2
65.
Find all the zeroes of the polynomial 2x – 3x – 5x2 + 9x – 3, it being given that two of its zeros
What must be added to the polynomial P(x) =5x4 + 6x3 – 13x2 + 7 x – 44 so that the resulting polynomial is exactly divisible by the polynomial Q(x) = x2 + 4x + 3 and the degree of the polynomial to be added must be less than degree of the polynomial Q(x).
Find the other zeroes of the polynomial 2x4 – 3x3
are 2 ± 3 , find the other zeroes.
3
are 3 and - 3. Obtain all the zeroes of x4 – 7x3 + 17x2 – 17x + 6, if two of its zeroes are 1 and 2. Find all other zeroes of the polynomial p(x) = 2 x3 + 3x2 – 11x – 6, if one of its zero is –3.
What must be added to the polynomial f(x) = x4 + 2x3 – 2x2 + x – 1 so that the resulting polynomial is exactly divisible by x2 + 2x – 3?
Find the zeroes of the quadratic polynomial 6x2 – 3 – 7x, and verify the relationship between the zeroes and the coefficients.
40.
Find all the zeroes of the polynomial
66.
Find all the zeroes of the polynomial x4 + x3 – 34x2– 4x + 120, if two of its zeroes are 2 and –2.
67.
If the polynomial 6x4 + 8x3 – 5x2 + ax + b is exactly divisible by the polynomial 2x2 – 5, then find the values of a and b.
Linear Equation in Two Varibale CLASS X Questions From CBSE Examination Papers 1.
2.
3.
4.
5.
Solve for x and y : (a – b)x + (a + b)y = a2 – 2ab – b2 (a + b)(x + y) = a2 + b2 The sum of the digits of a two digit number is 12. The number obtained by interchanging the two digits exceeds the given number by 18. Find the number. The taxi charges in a city consists of a fixed charge together with the charge for the distance covered. For a distance of 10 km, the charge paid is Rs. 105 and for journey of 15 km, the charge paid is Rs. 155. What are the fixed charges and the charges per km? The monthly incomes of A and B are in the ratio of 5 : 4 and their monthly expenditures are in the ratio of 7 : 5. If each saves Rs. 3000 per month, find the monthly income of each. A part of monthly hostel charges is fixed and the remaining depends on the number of days one has taken food in the mess. When a student A takes food for 20 days, she has to pay Rs. 1000 as hostel charges whereas a student B, who takes food 26 days, pays Rs. 1180 as hostel charges. Find the fixed charges and the cost of the food per day.
6.
The sum of a two-digit number and the number obtained by reversing the digits is 66. If the digits of the number differ by 2, fnd the number.
7.
Nine times a two-digit number is the same as twice the number obtained by interchanging the digits of the number. If one digit of the number exceeds the other number by 7, find the number.
8.
9.
10. 11.
The ratio of incomes of two persons is 9 : 7 and the ratio of their expenditures is 4 : 3. If each of them manages to save Rs. 2000 per month, find their monthly incomes. Solve for x and y : 5 1 6 3 ; += 2 += 1 x- 1 y- 2 x- 1 y- 2 x y = 2, ax - by = a 2 - b2 . Solve for x and y : + a b Solve for x and y : mx – ny = m 2 + n 2 ; x – y = 2n.
12. 13.
14.
15.
16.
17.
18.
19.
20.
21.
Solve for u and v by changing into linear equations 2(3u – v) = 5uv; 2(u + 3v) = 5uv. Solve the following system of linear equations by cross multiplication method : 2( ax – by) + (a + 4b) = 0 2( bx + ay) + (b – 4a) = 0 For what values of a and b does the following pair of linear equations have an infinite number of solutions : 2x + 3y = 7; a(x + y) – b(x – y) = 3a + b – 2. If 4 times the area of a smaller square is subtracted from the area of a larger square, the result is 144 m 2. The sum of the areas of the two squares is 464 m2. Determine the sides of the two squares. Half the perimeter of a rectangular garden, whose length is 4 m more than its breadth is 36 m. Find the dimensions of the garden. Yash scored 40 marks in a test, getting 3 marks for each right answer and losing 1 mark for each wrong answer. Had 4 marks been awarded for each correct answer and 2 marks been deducted for each incorrect answer, then Yash would have scored 50 marks. How many questions were there in the test? A man travels 370 km partly by train and partly by car. If he covers 250 km by train and rest by car, it takes him 4 hours. But if he travels 130 km by train and rest by car, he takes 18 minutes longer. Find the speed of the train and that of the car. Six years hence a man’s age will be three times his son’s age and three years ago, he was nine times as old as his son. Find their present ages. A boat goes 24 km upstream and 28 km downstream in 6 hours. It goes 30 km upstream and 21 km downstream in 6 hours 30 minutes. Find the speed of the boat in still water. A person travels 600 km partly by train and partly by car. If he covers 400 km by train and the rest by car, it takes 6 hours 30 minutes. But if he travels 200 km by train and the rest by car, he takes half an hour longer. Find the speed of the car and that of the train.
22.
23.
24.
25.
26.
27.
28.
29. 30. 31. 32.
33. 34. 35.
36.
37.
The age of a father is equal to sum of the ages of his 6 children. After 15 years, twice the age of the father will be the sum of ages of his children. Find the age of the father. The auto fare for the first kilometer is fixed and is different from the rate per km for the remaining distance. A man pays Rs. 57 for the distance of 16 km and Rs. 92 for a distance of 26 km. Find the auto fare for the first kilometer and for each successive kilometer. A lending library has a fixed charge for first three days and an additional charge for each day there after. Bhavya paid Rs. 27 for a book kept for seven days, while Vrinda paid Rs. 21 for a book kept for fve days. Find the fixed charge and the charge for each extra day. Father’s age is 3 times the sum of ages of his two children. After 5 years his age will be twice the sum of ages of the two children. Find the age of father. Places A and B are 100 km apart on a highway. One car starts from A and another from B at the same time. If the cars travel in the same direction at different speeds, they meet in 5 hours. If they travel towards each other, they meet in 1 hour. What are the speeds of the two cars ? Rekha’s mother is five times as old as her daughter Rekha. Five years later, Rekha’s mother will be three times as old as her daughter Rekha. Find the present age of Rekha and her mother’s age. Two numbers are in the ratio 5 : 6. If 8 is subtracted from each of the numbers, the ratio becomes 4 : 5. Find the numbers. b a Solve : x + 2ab y= a2 + b2 ; x + y= a b Solve : ax + by – a + b= 0; bx – ay – a – b = 0 3
38.
Draw the graphs of 2 x + y = 6 and 2x – y + 2 = 0. Shade the region bounded by these lines and x-axis. Find the area of the shaded region.
39.
Solve the following system of equations graphically and from the graph, find the points where these lines intersect the y-axis : x – 2y = 2, 3x + 5y = 17.
40.
Solve the following system of equations graphically and find the vertices of the triangle formed by these lines and the x-axis : 4 x – 3y + 4 = 0, 4x + 3y – 20 = 0
41.
Solve graphically the following system of equations : x + 2y = 5, 2x – 3y = –4. Also, find the points where the lines meet the x-axis.
42.
Draw the graph of the pair of equations 2 x + y = 4 and 2x – y = 4. Write the vertices of the triangle formed by these lines and the y-axis. Also shade this triangle.
43.
Draw the graphs of the equations 4x – y – 8 = 0 and 2x – 3y + 6 = 0. Shade the region between two lines and x-axis. Also, find the co-ordinates of the vertices of the triangle formed by these lines and the x-axis. Solve the following system of equations graphically and find the vertices of the triangle bounded by these lines and the x-axis. 2x – 3y – 4 = 0, x – y – 1 = 0. Solve the following system of equations graphically and from the graph, find the points where the lines intersect x-axis. 2x – y = 2, 4x – y = 8. Solve graphically the pair of linear equations : 3 x + y – 3 = 0; 2x – y + 8 = 0 Write the co-ordinates of the vertices of the triangle formed by these lines with x-axis. A sailor goes 8 km downstream in 40 minutes and returns in 1 hour. Find the speed of the sailor in still water and the speed of current. Draw the graphs of equations 3x + 2y = 14 and 4x – y = 4. Shade the region between these lines and y-axis. Also, find the co-ordinates of the triangle formed by these lines with y-axis. Check graphically whether the pair of linear equations 4x – y – 8 = 0 and 2x – 3y + 6 = 0 is consistent. Also, find the vertices of the triangle formed by these lines with the x-axis.
44.
45.
46.
2
If (x + 3) is a factor of x + ax – bx + 6 and a + b = 7, find the values of a and b. If (x + 1) is a factor of 2x3 + ax2 + 2bx + 1, then find the values of a and b given that 2a + 3b = 4. bx ay Solve : + a+ b= 0; bx - ay + 2ab = 0 a b x y Solve : + + a - b; ax + by = a3 + b3 a b Find the values of a and b for which the following system of linear equations has infinite number of solutions : 2 x + 3y = 7; (a + b + 1) x + (a + 2b + 2) y = 4(a +b) + 1 For what value of k, will the system of equations x + 2y = 5, 3x + ky + 15 = 0 has (i) a unique solution, (ii) no solution. The sum of the numerator and denominator of a fraction is 4 more than twice the numerator. If the numerator and denominator are increased by 3, they are in the ratio 2 : 3. Determine the fraction.
47.
48.
49.
50.
A boat goes 24 km upstream and 28 km downstream in 6 hours. It goes 30 km upstream and 21 km 1 downstream in 6 hours. Find the speed of boat 2 in still water and also the speed of the stream.
51.
52.
53.
54.
8 men and 12 boys can finish a piece of work in 10 days while 6 men and 8 boys can finish it in 14 days. Find the time taken by one man alone and that by one boy alone to finish the work. Solve graphically : x + y + 1 = 0, 3x + 2y = 12 (i) Find the solution from the graph.
55.
Solve graphically 4x – y = 4 and 4x + y = 12. Shade the triangular region formed by these lines and x-axis. Also, write the coordinate of the vertices of the triangle formed by these lines and x-axis.
56.
(ii) Shade the triangular region formed by the lines and the x-axis.
Solve for x and y : 6( ax + by ) = 3 a +2 b, 6(bx – ay) = 3b – 2a.
57.
Solve for x and y : (a – b)x + ( a + b )y = a2 – 2 ab – b2, (a + b)(x + y) = a2 + b2
58.
Solve :
59.
Solve : 1 12 1 7 4 + = ; + = 2, 2( 2 x + 3 y ) 7(3 x + 2 y ) 2 (2 x + 3 y ) 3x + 2y
Solve graphically; x – y = 1, 2x + y = 8. Shade the region bounded by these lines and y-axis. Also find its area. Solve graphically the pair of linear equations : x – y = –1 and 2x + y – 10 = 0. Also find the area of the region bounded by these lines and x-axis.
44 30 55 40 += 10; += 13 x+ y x-y x+ y x-y
where (2x + 3y) ≠0 and (3 x + 2 y )≠ 0
TRIGONOMETRY CLASS X Questions From CBSE Examination Papers p2 - 1 = sin ?. p2 + 1
20.
If sec è + tan è = p, show that
Prove that : cosec2 (90° – è) – tan2 è = cos2(90° - q )+ cos2 è.
21.
If a sin è + b cos è = c, then prove that, 2 b2 - c2 . a cos è – b sin è = a +
4.
If 2 cos è – sin è = x and cos è – 3 sin è = y, prove that 2x2 + y2 – 2xy = 5.
22.
Prove that
5.
Without using trigonometric tables, evaluate the following :
1.
If sin è + cos è = p and sec è + cosec è = q then prove that q(p2 – 1) = 2p.
2.
Prove that : cos4 è – cos2 è = sin4 – sin2 è.
3.
6. 7. 8.
9. 10. 11.
12. 13. 14.
15.
16. 17.
18.
sin A + cos A sin A - cos A 2 . + = sin A - cos A sin A + cos A sin 2 A - cos 2 A
cos 2 20° + cos 2 70° 23. + 2 c osec 2 58° - 2 cot 58 ° tan 32° cos 2 50° + cos 2 40° - 4 taan 13° tan 37° tan 45° tan 53° tan 77° 24. sin A cos A Prove that : + = 1 sec A + tan A - 1 cosec A + cot A - 1 25. sin ? - cos ? sin ? + cos ? 2 Prove that : + = sin ? + cos ? sin ? - cos ? 2 sin 2 ? - 1 26. Prove that : 27. 2 1 2 1 +=cot 4 ? tan 4 ? 2 4 2 4 cos ? cos ? sin ? sin ? 28. Prove that : (1 + tan A tan B)2 + (tan A – tan B )2 = sec2 A.sec2 B. If tan è + sin è = m and tan è – sin è = n, prove that m2 – n2 = 4 mn . Prove that : 1 1 1 1 =cosec A - cot A sin A sin A cosec A + cot A 2 4 sin ? - 2 sin ? 2 1. = Prove that : sec ? 2 cos 4 ? - cos 2 ? 8 . If sec è – tanè = 4, then prove that cos è = 17
29.
30.
Find the value of sin2 5° + sin2 10° + sin2 80° + sin2 85°.
31.
1+ sec A sin 2 A Prove that = = . 1 - cos A sec A
32.
Prove that :
tan ? + 1+ sec ? 1 = . tan ? + 1 - sec ? sec ? - tan ?
1 If sec ? = x+ , then prove that 4x 1 sec ? += tan ? 2 x or . 2x Prove that
tan ? cot ? tan ? + cot ?. + = 1+ 1 - cot ? 1 - tan ?
33.
Prove that
cos A 1 - sin A + = 2 sec A. cos A 1 - sin A
If x = r sin A cos C, y = r sin A sin C, z = r cos A prove that r2 = x 2 + y2 + z2 . 2 . If tan A = 2 - 1, show that sin A cos A = 4 Prove that
cot A + cosec A - 1 1 + cos A = . cot A - cosec A + 1 sin A
Prove that sin6 è +
cos 6
è = 1- 3 sin2 è
cos 2 è.
Evaluate : 1 2 - ?) + sin(50° + ?) - cos(40° cot 30° 4 3 tan 45° tan 20° tan 40° tan 50° tan 70° + 5 sin 2 63° + sin 2 27° + 2 cos 17° + cos 2 73° Prove that : (cosec A – sin A)(sec A – cos A) (tan A + cot A) = 1. 1 1 1 1 =. Prove that : sec ? - tan ? cos ? cos ? sec ? + tan ? tan ? + sec ?sin ? 1 1+ Prove that = tan ? - sec ? + cos ? 1 Prove that
cos A sin A sin A + cos A + = 1 - tan A 1 - cot A
Prove that 2(sin
6
è + cos6 è) – 3(sin4 è + cos4 è) + 1 = 0
34.
Prove :
sin ? sin ? = 2+ (cot ? + cosec ?) (cot ? - cosec ?)
35.
Prove :
cosec ? + cot ? = (cosec ? + cot ? ) 2 cosec ? + cot ? = 1+ 2 cot 2 ? + 2 cosec ?cot ?
STATISTICS CLASS X Questions From CBSE Examination Papers 1.
In the figure the value of the median of the data using the graph of less than ogive and more than ogive is :
2.
(a) 5 (b) 40 (c) 80 (d) 15 If mode = 80 and mean = 110, then the median is :
3.
(a) 110 (b) 120 (c) 100 (d) 90 The lower limit of the modal class of the following data is : C.I. Frequency
4. 5. 6.
7.
8.
9. 10.
11.
0–10 5
10–20 8
20–30 13
30–40 7
40–50 6
(a) 10 (b) 30 (c) 20 (d) 50 The mean of the following data is : 45, 35, 20, 30, 15, 25, 40 : (a) 15 (b) 25 (c) 35 (d) 30 The mean and median of a data are 14 and 15 respectively. The value of mode is : (a) 16 (b) 17 (c) 13 (d) 18 For a given data with 50 observations the ‘less than ogive’ and the ‘more then ogive’ intersect at (15.5, 20). The median of the data is : (a) 4.5 (b) 20 (c) 50 (d) 15.5 The empirical relationship among the Median, Mode and Mean of a data is : (a) mode = 3 median + 2 mean (b) mode = 3 median – 2 mean (c) mode = 3 mean – 2 median (d) mode = 3 mean + 2 median For a symmetrical distribution, which is correct? (a) Mean > Mode > Median (b) Mean < Mode < Median Mean + Median (c) Mode = (d) Mean = Median = Mode 2 Which of the following is not a measure of central tendency ? (a) Mean (b) Median (c) Range (d) Mode The class mark of a class interval is : (a) Lower limit + Upper limit (b) Upper limit – Lower limit 1 1 (c) (Lower limit + Upper limit) (d) (Lower limit + Upper limit) 2 4 If mode of a data is 45, mean is 27, then median is : (a) 30 (b) 27 (c) 23
(d) None of these
12.
For the following distribution : marks
Below 10
Below 20
Below 30
Below 40
Below 50
Below 60
3
12
27
57
75
80
No. of students
15.
The modal class is : (a) 10–20 (b) 20–30 (c) 30–40 (d) 50–60 For a given data with 60 observations the ‘less than ogive’ and ‘more than ogive’ intersect at (66.5, 30). The median of the data is : (a) 66.5 (b) 30 (c) 60 (d) 36.5 The abscissa of the point of intersection of the less than type and of the more than type cumulative frequency curves of a grouped data gives its : (a) mean (b) median (c) mode (d) all the three above A data has 25 observations (arranged in descending order). Which observation represents the med?ian
16.
(a) 12th (b) 13th (c) 14th (d) 15th If mode of the following data is 7, then value of k in 2, 4, 6, 7, 5, 6, 10, 6, 7, 2k + 1, 9, 7, 13 i:s
13.
14.
17. 18.
19. 20. 21.
(a) 3 The mean and median (a) 13 The upper limit of the
(b) 7 (c) 4 (d) 2 of a data are 14 and 16 respectively. The value of mode is : (b) 16 (c) 18 (d) 20 median class of the following distribution is :
Class 0–5 6–11 12–17 18–23 Frequency 13 10 15 8 (a) 17 (b) 17.5 (c) 18 (d) 18.5 The measures of central tendency which can’t be found graphically is : (a) mean (b) median (c) mode (d) none of these The measure of central tendency which takes into account all data items is : (a) mode (b) mean (c) median (d) none of these
Convert the following data to a less than type distribution and draw its ogive. Also fnd the median from the graph. Class Interval
100–120
120–140
140–160
160–180
180–200
12
14
8
6
10
Frequency 22.
Convert the following data into a more than type distribution and draw its ogive. Also fnd the median of the data from the graph. Class Interval
100–120
120–140
140–160
160–180
180–200
12
14
8
6
10
Frequency 23.
Draw ‘more than ogive’ for the following frequency distribution and hence obtain the median. Class Interval Frequency
24.
25.
24–29 11
5–10
10–15
15–20
20–25
25–30
30–35
35–40
2
12
2
4
3
4
3
Draw ‘less than ogive’ for the following frequency distribution and hence obtain the median. marks obtained
10–20
20–30
30–40
40–50
50–60
60–70
70–80
No. of students
3
4
3
3
4
7
9
If the median of the following data is 525. Find the values of x and y if the sum of the frequencies is 100. Class Interval Frequency Class Interval Frequency
0–100
100–200
200–300
300–400
400–500
2
5
x
12
17
500–600
600–700
700–800
800–900
900–1000
20
y
9
7
4
26.
Calculate the mode of the following frequency distribution table. marks
Above 25 52
Number of students 27.
Above 35 47
Above 45 37
Above 55 17
Above 65 8
Above 75 2
Above 85 0
During medical check up of 35 students of a class, their weights were recorded. Weight Less than 38 Less than 40 Less than 42 Less than 44 Less than 46 Less than 48 Less than 50 Less than 52
Number of students 0 3 5 9 14 28 32 35
Draw less than type ogive for the given data. Hence obtain the median weight from graph and verify the result by using formula. 28.
29.
Change the following data into less than type distribution and draw its ogive. Hence fnd the median of the data. marks obtained
30–39
40–49
50–59
60–69
70–79
80–89
90–99
No. of students
5
7
8
10
5
8
7
Draw less than and more than ogive for the following distribution and hence obtain the median. marks
30–40
40–50
50–60
60–70
70–80
80–90
90–100
14
6
10
20
30
8
12
No. of students 30.
The following distribution gives the annual profit earned by 30 shops of a shopping complex.
Profit (in Lakh Rs.) No. of shops
0–5
5–10
10–15
15–20
20–25
3
14
5
6
2
Change the above distribution to more than type distribution and draw its ogive. 31.
Following distribution shows the marks obtained by the class of 100 students. marks
10–20
20–30
30–40
40–50
50–60
60–70
10
15
30
32
8
5
No. of students
32.
Draw less than ogive for the above data. Find median graphically and verify the result by actual method. Find the median by drawing both ogives. Class Interval Frequency
33.
Frequency
70–80
80–90
90–100
5
9
12
6
0–10
10–20
20–30
30–40
40–50
50–60
Total
5
x
20
15
y
5
60
The mean of the following data is 50. Find the missing frequencies f1 and f2. C.I. No. of students
35.
60–70
3
If the median of the distribution given below is 28.5, fnd the values of x and y. Class Intervals
34.
50–60
0–20
20–40
40–60
60–80
80–100
Total
17
f1
32
f2
19
120
Find the mode, median and mean for the following data : marks obtained No. of students
25–35 7
35–45 31
45–55 33
55–65 17
65–75 11
75–85 1
36.
Draw a less than ogive for the following data : marks
Number of students
Less than 20
0
Less than 30
4
Less than 40
16
Less than 50
30
Less than 60
46
Less than 70
66
Less than 80
82
Less than 90
92
Less than 100
100
Find the median of the data from the graph and verify the result using the formula. 37.
The following table gives the distribution of expenditures of different families on education. Find the mean expenditure on education of a family. Expenditure (in Rs.) 1000–1500 1500–2000 2000–2500 2500–3000 3000–3500 3500–4000 4000–4500 4500–5000
38.
Number of families 24 40 33 28 30 22 16 7
100 surnames were randomly picked up from a local telephone directory and the frequency distribution of the number of letters in the English alphabet in the surnames was obtained as follows : Number of letters Number of surnames
1–4
4–7
7–10
10–13
13–16
16–19
6
30
40
16
4
4
Determine the median number of letters in the surnames. Find the mean number of letters in the surnames. Also, find the modal size of the surnames. 39.
Find the mean, mode and median of the following data : Classes Frequency
40.
0–10
10–20
20–30
30–40
40–50
50–60
60–70
5
10
18
30
20
12
5
The following table gives the daily income of 50 workers of a factory : Daily income (in Rs.)
100–120
120–140
No. of workers 12 14 Find the mean, median and mode of the above data. 41.
140–160
160–180
180–200
8
6
10
The median of the following data is 52.5. Find the values of x and y if the total frequency is 100. Class Interval 0–10 10–20 20–30 30–40 40–50 50–60 60–70 70–80 80–90 90–100
Frequency 2 5 x 12 17 20 y 9 7 4
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