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Learn MoreCBSE Class 10 - NCERT Full Text Book Video Solutions
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CBSE Class 10 - NCERT Full Text Book Video Solutions
CBSE Class 10 video solutions for all exercises, including optional are presented in a simple and comprehensive way.
Created byLakshmi Narayana
Last updated 1/2023
English
What you'll learn
- Real Numbers
- Polynomials
- Pair of Linear Equations in Two Variables
- Quadratic Equations
- Arithmetic Progressions
- Triangles
- Coordinate Geometry
- Introduction to Trigonometry
- Some Applications of Trigonometry
- Circles
- Constructions
- Areas Related to Circles
- Surface Areas and Volumes
- Statistics
- Probability
- At the end of the course, students will not only have learned about the topics in detail but also be able to solve various problems based on them.
Explore related topics
Course content
15 sections • 520 lectures • 75h 14m total length
- Exercise 1 - Q 1 (i)7:00
Use Euclid’s division algorithm to find the HCF of :
(i) 135 and 225
- Exercise 1 - Q 1 (ii)2:54
Use Euclid’s division algorithm to find the HCF of :
(ii) 196 and 38220
- Exercise 1 - Q 1 (iii)5:39
Use Euclid’s division algorithm to find the HCF of :
(iii) 867 and 255
- Exercise 1 - Q 211:50
Show that any positive odd integer is of the form 6q + 1, or 6q + 3, or 6q + 5, where q is some integer.
- Exercise 1 - Q 37:08
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?
- Exercise 1 - Q 48:45
Use Euclid’s division lemma to show that the square of any positive integer is either of the form 3m or 3m + 1 for some integer m.
[Hint: Let x be any positive integer then it is of the form 3q, 3q + 1 or 3q + 2. Now square each of these and show that they can be rewritten in the form 3m or 3m + 1.]
- Exercise 1 - Q 59:12
Use Euclid’s division lemma to show that the cube of any positive integer is of the form 9m, 9m + 1 or 9m + 8.
- Exercise 2 - Q 110:30
- Exercise 2 - Q 2 (i)7:18
Find the LCM and HCF of the following pairs of integers and verify that
LCM × HCF = product of the two numbers.
(i) 26 and 91
- Exercise 2 - Q 2 (ii)7:03
Find the LCM and HCF of the following pairs of integers and verify that LCM × HCF = product of the two numbers.
(ii) 510 and 92
- Exercise 2 - Q 2 (iii)7:47
Find the LCM and HCF of the following pairs of integers and verify that LCM × HCF = product of the two numbers.
(iii) 336 and 54
- Exercise 2 - Q 3 (i)4:06
Find the LCM and HCF of the following integers by applying the prime factorization method.
(i) 12, 15 and 21
- Exercise 2 - Q 3 (ii)3:25
Find the LCM and HCF of the following integers by applying the prime factorization method.
(ii) 17, 23 and 29
- Exercise 2 - Q 3 (iii)3:16
Find the LCM and HCF of the following integers by applying the prime factorization method.
(iii) 8, 9 and 25
- Exercise 2 - Q 43:20
Given that HCF (306, 657) = 9, find LCM (306, 657).
- Exercise 2 - Q 55:38
Check whether 6n can end with the digit 0 for any natural number n.
- Exercise 2 - Q 67:26
Explain why 7 × 11 × 13 + 13 and 7 × 6 × 5 × 4 × 3 × 2 × 1 + 5 are composite numbers.
- Exercise 2 - Q 77:20
There is a circular path around a sports field. Sonia takes 18 minutes to drive one round of the field, while Ravi takes 12 minutes for the same. Suppose they both start at the same point and at the same time, and go in the same direction. After how many minutes will they meet again at the starting point?
- Exercise 3 - Q 112:39
Prove that √5 is irrational.
- Exercise 3 - Q 26:33
Prove that (3 + 2√5) is irrational.
- Exercise 3 - Q 3 (i)6:24
Prove that the following are irrationals.
(i) 1/√2
- Exercise 3 - Q 3 (ii)6:22
Prove that the following are irrationals
(ii) 7√5
- Exercise 3 - Q 3 (iii)5:52
Prove that the following are irrationals
(iii) 6 + √2
- Exercise 4 - Q 1 (i)3:21
Without actually performing the long division, state whether the following rational numbers will have a terminating decimal expansion or a non-terminating decimal expansion.
(i)13/3125
- Exercise 4 - Q 1 (ii)2:06
Without actually performing the long division,state whether the following rational numbers will have a terminating decimal expansion or a non-terminating decimal expansion.
(ii)17/8
- Exercise 4 - Q 1 (iii)2:55
Without actually performing the long division,state whether the following rational numbers will have a terminating decimal expansion or a non-terminating decimal expansion.
(iii)64/455
- Exercise 4 - Q 1 (iv)2:07
Without actually performing the long division,state whether the following rational numbers will have a terminating decimal expansion or a non-terminating decimal expansion.
iv)15/1600
- Exercise 4 - Q 1 (v)2:08
Without actually performing the long division,state whether the following rational numbers will have a terminating decimal expansion or a non-terminating decimal expansion.
v)29/343
- Exercise 4 - Q 1 (vi)1:25
Without actually performing the long division, state whether the following rational numbers will have a terminating decimal expansion or a non-terminating decimal expansion.
vi)23/(2^3×5^2 )
- Exercise 4 - Q 1 (vii)2:00
Without actually performing the long division,state whether the following rational numbers will have a terminating decimal expansion or a non-terminating decimal expansion.
(vii)29/(2^2×5^7×7^5 )
- Exercise 4 - Q 1 (viii)1:57
Without actually performing the long division,state whether the following rational numbers will have a terminating decimal expansion or a non-terminating decimal expansion.
(viii)6/15
- Exercise 4 - Q 1 (ix)1:55
Without actually performing the long division,state whether the following rational numbers will have a terminating decimal expansion or a non-terminating decimal expansion.
(ix)35/50
- Exercise 4 - Q 1 (x)2:31
Without actually performing the long division, state whether the following rational numbers will have a terminating decimal expansion or a non-terminating decimal expansion.
(x)77/210
- Exercise 4 - Q 2 (i)3:52
Write down the decimal expansions of given rational numbers,
(i) 13/3125
- Exercise 4 - Q 2 (ii)2:26
Write down the decimal expansions of given rational numbers,
(ii) 17/8
- Exercise 4 - Q 2 (iii)3:41
Write down the decimal expansions of given rational numbers,
(iii) 15/1600
- Exercise 4 - Q 2 (iv)2:00
Write down the decimal expansions of given rational numbers,
(iv) 23/(2^3 x 5^2)
- Exercise 4 - Q 2 (v)1:21
Write down the decimal expansions of given rational numbers,
(v) 6/15
- Exercise 4 - Q 2 (vi)1:00
Write down the decimal expansions of given rational numbers,
(vi) 35/50
- Exercise 4 - Q 3 (i)4:49
The following real numbers have decimal expansions as given below. In each case, decide whether they are rational or not. If, they are rational, and of the form what can you say about the prime factors of q?
(i) 43.123456789
- Exercise 4 - Q 3 (ii)1:51
The following real numbers have decimal expansions as given below. In each case, decide whether they are rational or not. If, they are rational, and of the form p/q what can you say about the prime factors of q?
(ii) 0.1201120012000120000...
- Exercise 4 - Q 3 (iii)9:26
The following real numbers have decimal expansions as given below. In each case, decide whether they are rational or not. If, they are rational, and of the form p/q what can you say about the prime factors of q?
(iii) 43.(12346789) ̅
- Exercise 1 - Q 14:39
The graphs of y = p(x) are given in Figure below, for some polynomials p(x). Find the number of zeroes of p(x), in each case.
- Exercise 2 - Q 1 (i)10:06
Find the zeroes of the following quadratic polynomials and verify the relationship between the zeros and the coefficients.
(i) x2 - 2x - 8
- Exercise 2 - Q 1 (ii)6:15
Find the zeroes of the following quadratic polynomials and verify the relationship between the zeros and the coefficients.
(ii) 4s2 - 4s + 1
- Exercise 2 - Q 1 (iii)8:23
Find the zeroes of the following quadratic polynomials and verify the relationship between the zeros and the coefficients.
(iii) 6x2 – 3 - 7x
- Exercise 2 - Q 1 (iv)7:24
Find the zeroes of the following quadratic polynomials and verify the relationship between the zeros and the coefficients.
(iv) 4u2 + 8u
- Exercise 2 - Q 1 (v)8:00
Find the zeroes of the following quadratic polynomials and verify the relationship between the zeros and the coefficients.
(v) t2 - 15
- Exercise 2 - Q 1 (vi)8:41
Find the zeroes of the following quadratic polynomials and verify the relationship between the zeros and the coefficients.
(vi) 3x2 – x - 4
- Exercise 2 - Q 2 (i)4:31
Find a quadratic polynomial each with the given numbers as the sum and product of its zeroes respectively
1/4 , -1
- Exercise 2 - Q 2 (ii)3:03
Find a quadratic polynomial each with the given numbers as the sum and product of its zeroes respectively
√2 , 1/3
- Exercise 2 - Q 2 (iii)2:40
Find a quadratic polynomial each with the given numbers as the sum and product of its zeroes respectively
0 , √5
- Exercise 2 - Q 2 (iv)1:59
Find a quadratic polynomial each with the given numbers as the sum and product of its zeroes respectively
1 , 1
- Exercise 2 - Q 2 (v)3:30
Find a quadratic polynomial each with the given numbers as the sum and product of its zeroes respectively
- 1/4 , 1/4
- Exercise 2 - Q 2 (vi)2:03
Find a quadratic polynomial each with the given numbers as the sum and product of its zeroes respectively
4 , 1
- Exercise 3 - Q 1 (i)8:16
Divide the polynomial p(x) by the polynomial g(x) and find the quotient and remainder in each of the following.
(i) P (x) = x3 – 3x2 + 5x – 3 , g(x) = x2 - 2
- Exercise 3 - Q 1 (ii)7:26
Divide the polynomial p(x) by the polynomial g(x) and find the quotient and remainder in each of the following.
(i) P (x) = x4 – 3x2 + 4x + 5 , g(x) = x2 + 1 - x
- Exercise 3 - Q 1 (iii)4:56
Divide the polynomial p(x) by the polynomial g(x) and find the quotient and remainder in each of the following.
P (x) = x4 – 5x + 6 , g(x) = 2 - x2
- Exercise 3 - Q 2 (i)6:41
Check whether the first polynomial is a factor of the second polynomial by dividing the second polynomial by the first polynomial.
(i) t2 – 3, 2t4 + 3t3 – 2t2 – 9t - 12
- Exercise 3 - Q 2 (ii)6:03
Check whether the first polynomial is a factor of the second polynomial by dividing the second polynomial by the first polynomial.
(ii) x2 + 3x +1, 3x4 + 5x3 - 7x2 + 2x + 2
- Exercise 3 - Q 2 (iii)4:32
Check whether the first polynomial is a factor of the second polynomial by dividing the second polynomial by the first polynomial.
(iii) x3 - 3x +1, x5 - 4x3 + x2 + 3x + 1
- Exercise 3 - Q 314:05
Obtain all other zeroes of 3x4 + 6x3 - 2x2 – 10x - 5 if two of its zeroes are √(5/3) and -√(5/3) .
- Exercise 3 - Q 48:09
On dividing x3 – 3x2 + x + 2 by a polynomial g(x), the quotient and remainder were x – 2 and -2x + 4 respectively. Find g(x).
- Exercise 3 - Q 57:23
Give examples of polynomials p(x), g(x), q(x) and r(x), which satisfy the division algorithm and
(i) deg p(x) = deg q(x)
(ii) deg q(x) = deg r(x)
(iii) deg r(x) = 0
- Exercise 4 - Q 1 (i)14:07
Verify that the numbers given alongside of the cubic polynomials below are their zeroes. Also verify the relationship between the zeroes and the coefficients in each case:
(i) 2x3 + x2 – 5x + 2; 1/2, 1, -2
- Exercise 4 - Q 1 (ii)9:53
Verify that the numbers given alongside of the cubic polynomials below are their zeroes. Also verify the relationship between the zeroes and the coefficients in each case:
(ii) x3 – 4x2 + 5x – 2; 2, 1, 1
- Exercise 4 - Q 24:32
Find a cubic polynomial with the sum, sum of the product of its zeroes taken two at a time, and the product of its zeroes as 2, –7, –14 respectively.
- Exercise 4 - Q 35:42
If the zeroes of the polynomial x3 – 3x2 + x + 1 are a – b, a, a + b, find a and b.
- Exercise 4 - Q 412:49
If two zeroes of the polynomial x4 – 6x3 – 26x2 + 138x – 35 are 2 ± √3, find other zeroes.
- Exercise 1 - Q 118:51
Aftab tells his daughter, "Seven years ago, I was seven times as old as you were then. Also, three years from now, I shall be three times as old as you will be." (Isn't this interesting?) Represent this situation algebraically and graphically.
- Exercise 1 - Q 215:40
The coach of a cricket team buys 3 bats and 6 balls for ₹3900. Later, she buys another bat and 3 more balls of the same kind for ₹1300. Represent this situation algebraically and graphically.
- Exercise 1 - Q 314:54
The cost of 2 kg of apples and 1 kg of grapes on a day was found to be ₹160. After a month, the cost of 4 kg of apples and 2 kg of grapes is ₹300. Represent the situation algebraically and geometrically.
- Exercise 2 - Q 1 (i)13:02
Form the pair of linear equations in the following problems, and find their solutions graphically.
(i). 10 students of class X took part in a mathematics quiz. If the number of girls is 4 more than the number of boys, find the number of boys and girls who took part in the quiz.
- Exercise 2 - Q 1 (ii)11:03
Form the pair of linear equations in the following problems, and find their solutions graphically.
(ii). 5 pencils and 7 pens together cost ₹ 50, whereas 7 pencils and 5 pens together cost ₹ 46. Find the cost of one pencil and that of one pen.
- Exercise 2 - Q 211:57
On comparing the ratios a_1/a_2 = b_1/b_2 and c_1/c_2 find out whether the lines representing the following pairs of linear equations intersect at a point, are parallel or coincide:
(i) 5x − 4y + 8 = 0;7x + 6y - 9 = 0
(ii) 9x + 3y + 12 = 0; 18x + 6y + 24 = 0
(iii) 6x − 3y + 10 = 0; 2x – y + 9 = 0
- Exercise 2 - Q 315:58
On comparing the ratios a_1/a_2 = b_1/b_2 and c_1/c_2 , find out whether the following pair of linear equations are consistent, or inconsistent.
(i) 3x + 2y = 5, 2x − 3y = 7
(ii) 2x − 3y = 8, 4x − 6y = 9
(iii). 3/2x + 5/3y = 7, 9x − 10y = 14
(iv). 5x − 3y = 11, −10x + 6y = −22
(v). 4/3x + 2y = 8, 2x + 3y = 12
- Exercise 2 - Q 4 (i)9:20
Which of the following pairs of linear equations are consistent/inconsistent? If consistent, obtain the solution graphically:
(i). x + y = 5, 2x + 2y = 10
- Exercise 2 - Q 4 (ii)2:24
Which of the following pairs of linear equations are consistent/inconsistent? If consistent, obtain the solution graphically:
(ii). x – y = 8, 3x − 3y = 16
- Exercise 2 - Q 4 (iii)11:03
Which of the following pairs of linear equations are consistent/inconsistent? If consistent, obtain the solution graphically:
(iii). 2x + y - 6 = 0, 4x − 2y - 4 = 0
- Exercise 2 - Q 4 (iv)3:12
Which of the following pairs of linear equations are consistent/inconsistent? If consistent, obtain the solution graphically:
(iv). 2x − 2y – 2 = 0, 4x − 4y – 5 = 0
- Exercise 2 - Q 56:12
Half the perimeter of a rectangle garden, whose length is 4 m more than its width, is 36 m. find the dimensions of the garden.
- Exercise 2 - Q 612:08
Given the linear equation (2x + 3y – 8 = 0), write another linear equation in two variables such that the geometrical representation of the pair so formed is:
(i). Intersecting lines
(ii). Parallel lines
(iii). Coincident lines
- Exercise 2 - Q 79:19
Draw the graphs of the equations x – y + 1 = 0 and 3x + 2y – 12 = 0. Determine the coordinates of the vertices of the triangle formed by these lines and the x-axis, and shade the triangular region.
- Exercise 3 - Q 1 (i)4:16
Solve the following pair of linear equations by the substitution method.
(i). x + y = 14
x – y = 4
- Exercise 3 - Q 1 (ii)4:26
Solve the following pair of linear equations by the substitution method.
(ii). s – t = 3
s/3 + t/2 = 6
- Exercise 3 - Q 1 (iii)3:11
Solve the following pair of linear equations by the substitution method.
(iii). 3x – y = 3
9x − 3y = 9
- Exercise 3 - Q 1 (iv)6:17
Solve the following pair of linear equations by the substitution method.
(iv) 0.2x + 0.3y = 1.3
0.4x + 0.5y = 2.3
- Exercise 3 - Q 1 (v)3:41
Solve the following pair of linear equations by the substitution method.
(v). √2x + √3y = 0
√3x - √8y = 0
- Exercise 3 - Q 1 (vi)6:46
Solve the following pair of linear equations by the substitution method.
(vi). 3x/2 - 5y/3 = -2
x/3 + y/2 = 13/6
- Exercise 3 - Q 26:25
Solve 2x + 3y = 11 and 2x − 4y = −24 and hence find the value of 'm' for which
y = mx + 3.
- Exercise 3 - Q 3 (i)6:17
Form a pair of linear equations for the following problems and find their solution by substitution method.
(i) The difference between two numbers is 26 and one number is three times the other. Find them.
- Exercise 3 - Q 3 (ii)6:46
Form a pair of linear equations for the following problems and find their solution by substitution method.
(ii) The larger of two supplementary angles exceeds the smaller by 18 degrees. Find them.
- Exercise 3 - Q 3 (iii)9:37
Form a pair of linear equations for the following problems and find their solution by substitution method.
(iii). The coach of a cricket team buys 7 bats and 6 balls for ₹ 3800. Later, she buys 3 bats and 5 balls for ₹ 1750. Find the cost of each bat and each ball.
- Exercise 3 - Q 3 (iv)9:33
Form a pair of linear equations for the following problems and find their solution by substitution method.
(iv). The taxi charges in a city consist of a fixed charge together with the charge for the distance covered. For a distance of 10 km, the charge paid is ₹ 105 and for a journey of 15 km, the charge paid is ₹ 155. What are the fixed charges and the charge per km? How much does a person have to pay for travelling a distance of 25 km?
- Exercise 3 - Q 3 (v)9:02
Form a pair of linear equations for the following problems and find their solution by substitution method.
(v). A fraction becomes 9/11 if 2 is added to both the numerator and the denominator. If, 3 is added to both the numerator and denominator it becomes5/6. Find the fraction?
- Exercise 3 - Q 3 (vi)8:58
Form a pair of linear equations for the following problems and find their solution by substitution method.
(vi). Five years hence, the age of Jacob will be three times that of his son. Five years ago, Jacob's age was seven times that of his son. What are their present ages?
- Exercise 4 - Q 1 (i)11:30
Solve the following pair of linear equations by the elimination method and the substitution method:
(i) x + y = 5, 2x – 3y = 4
- Exercise 4 - Q 1 (ii)8:23
Solve the following pair of linear equations by the elimination method and the substitution method:
(ii) 3x + 4y = 10, 2x – 2y = 2
- Exercise 4 - Q 1 (iii)10:33
Solve the following pair of linear equations by the elimination method and the substitution method:
iii) 3x − 5y – 4 = 0, 9x = 2y + 7
- Exercise 4 - Q 1 (iv)8:25
Solve the following pair of linear equations by the elimination method and the substitution method:
(iv). x/2 + 2y/3 = -1, x - y/3 = 3
- Exercise 4 - Q 2 (i)9:37
Form the pair of linear equations in the following problems, and find their solutions (if they exist) by the elimination method:
If we add 1 to the numerator and subtract 1 from the denominator, a fraction reduces to 1. It becomes 1/2 if we only add 1 to the denominator. What is the fraction?
- Exercise 4 - Q 2 (ii)7:35
Form the pair of linear equations in the following problems, and find their solutions (if they exist) by the elimination method:
(i) Five years ago, Nuri was thrice as old as sonu. Ten years later, Nuri will be twice as old as sonu. How old are Nuri and Sonu?
- Exercise 4 - Q 2 (iii)8:52
Form the pair of linear equations in the following problems, and find their solutions (if they exist) by the elimination method:
(i) The sum of the digits of a two–digit number is 9. Also, nine times this number is twice the number obtained by reversing the order of the digits. Find the number.
- Exercise 4 - Q 2 (iv)7:25
Form the pair of linear equations in the following problems, and find their solutions (if they exist) by the elimination method:
(i) Meena went to a bank to withdraw ₹ 2000. She asked the cashier to give her ₹ 50 and ₹ 100 notes only. Meena got 25 notes in all. Find how many notes of ₹ 50 and ₹ 100 she received.
- Exercise 4 - Q 2 (v)8:09
Form the pair of linear equations in the following problems, and find their solutions (if they exist) by the elimination method:
A lending library has a fixed charge for the first three days and an additional charge for each day thereafter. Saritha paid ₹27 for a book kept for seven days, while Susy paid ₹21 for the book she kept for five days. Find the fixed charge and the charge for each extra day
- Exercise 5 - Q 1 (i)6:57
Which of the following pairs of linear equations has unique solution, no solution, or infinitely many solutions? In case there is a unique solution, find it by using cross multiplication method.
(i). x – 3y – 3 = 0; 3x – 9y – 2 = 0
- Exercise 5 - Q 1 (ii)7:29
Which of the following pairs of linear equations has unique solution, no solution, or infinitely many solutions? In case there is a unique solution, find it by using cross multiplication method.
(ii). 2x + y = 5; 3x + 2y = 8
- Exercise 5 - Q 1 (iii)4:09
Which of the following pairs of linear equations has unique solution, no solution, or infinitely many solutions? In case there is a unique solution, find it by using cross multiplication method.
(iii). 3x − 5y = 20; 6x − 10y = 40
- Exercise 5 - Q 1 (iv)8:06
Which of the following pairs of linear equations has unique solution, no solution, or infinitely many solutions? In case there is a unique solution, find it by using cross multiplication method.
(iv). x − 3y – 7 = 0; 3x − 3y – 15 = 0
- Exercise 5 - Q 2 (i)11:00
(i) For which values of a and b does the following pair of linear equations have an infinite number of solutions?
2x + 3y = 7, (a − b) x + (a + b) y = 3a + b – 2
- Exercise 5 - Q 2 (ii)5:30
(ii) For which value of k will the following pair of linear equations have no solution? 3x + y = 1, (2k − 1) x + (k − 1) y = 2k + 1
- Exercise 5 - Q 311:42
Solve the following pair of linear equations by the substitution and cross- multiplication methods:
8x + 5y = 9
3x + 2y = 4
- Exercise 5 - Q 4 (i)10:24
Form the pair of linear equations in the following problems and find their solutions (if they exist) by any algebraic method:
(i) 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 ₹1000 as hostel charges whereas a student B, who takes food for 26 days, pays ₹1180 as hostel charges. Find the fixed charges and the cost of food per day.
- Exercise 5 - Q 4 (ii)8:24
Form the pair of linear equations in the following problems and find their solutions (if they exist) by any algebraic method:
A fraction becomes 1/3 when 1 is subtracted from the numerator and it becomes 1/4 when 8 is added to its denominator. Find the fraction.
- Exercise 5 - Q 4 (iii)9:18
Form the pair of linear equations in the following problems and find their solutions (if they exist) by any algebraic method:
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?
- Exercise 5 - Q 4 (iv)9:55
Form the pair of linear equations in the following problems and find their solutions (if they exist) by any algebraic method:
(i) 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?
- Exercise 5 - Q 4 (v)10:39
Form the pair of linear equations in the following problems and find their solutions (if they exist) by any algebraic method:
The area of a rectangle gets reduced by 9 square units, if its length is reduced by 5 units and breadth is increased by 3 units. If we increase the length by 3 units and the breadth by 2 units, the area increases by 67 square units. Find the dimensions of the rectangle.
- Exercise 6 - Q 1 (i)12:45
Solve the following pairs of equations by reducing them to a pair of linear equations:
(i). 1/2x + 1/3y = 2; 1/3x + 1/2y = 13/6
- Exercise 6 - Q 1 (ii)8:29
Solve the following pairs of equations by reducing them to a pair of linear equations:
(ii). 2/√x + 3/√y = 2; 4/√x - 9/√y = -1
- Exercise 6 - Q 1 (iii)10:29
Solve the following pairs of equations by reducing them to a pair of linear equations:
(iii). 4/x + 3y = 14; 3/x - 4y = 23
- Exercise 6 - Q 1 (iv)9:04
Solve the following pairs of equations by reducing them to a pair of linear equations:
(iv). 5/(x - 1) + 1/(y - 2) = 2; 6/(x - 1) - 3/(y - 2) = 1
- Exercise 6 - Q 1 (v)10:29
Solve the following pairs of equations by reducing them to a pair of linear equations:
v). (7x - 2y)/xy = 5; (8x + 7y)/xy = 15
- Exercise 6 - Q 1 (vi)9:27
Solve the following pairs of equations by reducing them to a pair of linear equations:
(vi). 6x + 3y = 6xy; 2x + 4y = 5xy
- Exercise 6 - Q 1 (vii)9:13
Solve the following pairs of equations by reducing them to a pair of linear equations:
(vii). 10/(x + y) + 1/(x - y) = 4; 15/(x + y) - 5/(x - y) = -2
- Exercise 6 - Q 1 (viii)9:32
Solve the following pairs of equations by reducing them to a pair of linear equations:
(viii). 1/(3x + y) + 1/(3x - y) = 3/4; 1/(2(3x + y)) - 1/(2(3x – y)) = - 1/8
- Exercise 6 - Q 2 (i)13:23
Formulate the following problems as a part of equations, and hence find their solutions.
(i). Ritu can row downstream 20 km in 2 hours, and upstream 4 km in 2 hours. Find her speed of rowing in still water and the speed of the current.
- Exercise 6 - Q 2 (ii)12:09
Formulate the following problems as a part of equations, and hence find their solutions.
(ii). 2 women and 5 men can together finish an embroidery work in 4 days, while 3 women and 6 men can finish it in 3 days. Find the time taken by 1 woman alone to finish the work, and also that taken by 1 man alone.
- Exercise 6 - Q 2 (iii)12:13
Formulate the following problems as a part of equations, and hence find their solutions.
(iii). Roohi travels 300 km to her home partly by train and partly by bus. She takes 4 hours if she travels 60 km by train and the remaining by bus. If she travels 100 km by train and the remaining by bus, she takes 10 minutes longer. Find the speed of the train and the bus separately
- Exercise 7 - Q 113:43
The age of two friends Ani and Biju differ by 3 years. Ani’s father Dharam is twice as old as Ani and Biju is twice as old as his sister Cathy. The ages of Cathy and Dharam differ by 30 years. Find the ages of Ani and Biju.
- Exercise 7 - Q 210:51
One says, “Give me a hundred, friend! I shall then become twice as rich as you.” The other replies, “If you give me ten, I shall be six times as rich as you.” Tell me what is the amount of their (respective) capital? [From the Bijaganita of Bhaskara II]
[Hint: x + 100 = 2(y – 100), y + 10 = 6(x – 10)].
- Exercise 7 - Q 312:30
A train covered a certain distance at a uniform speed. If the train would have been 10 km/h faster, it would have taken 2 hours less than the scheduled time. And, if the train were slower by 10 km/h, it would have taken 3 hours more than the scheduled time. Find the distance covered by the train.
- Exercise 7 - Q 414:09
The students of a class are made to stand in rows. If 3 students are extra in a row, there would be 1 row less. If 3 students are less in a row, there would be 2 rows more. Find the number of students in the class.
- Exercise 7 - Q 510:14
In a ∆ABC, ∠c = 3 ∠B = 2(∠A + ∠B). Find three angles.
- Exercise 7 - Q 69:29
Draw the graphs of the equations 5x – y = 5 and 3x – y = 3. Determine the co- ordinate of the vertices of the triangle formed by these lines and the y - axis.
- Exercise 7 - Q 7 (i)6:34
Solve the following pair of linear equations:
(i). px + py = p – q; qx – py = p + q
- Exercise 7 - Q 7 (ii)8:12
Solve the following pair of linear equations:
(ii). ax + by = c; bx + ay = 1 + c
- Exercise 7 - Q 7 (iii)6:24
Solve the following pair of linear equations:
(iii). x/a - y/b = 0; ax + by = a2 + b2
- Exercise 7 - Q 7 (iv)7:17
Solve the following pair of linear equations:
(iv). (a - b)x + (a + b)y = a2 – 2ab – b2; (a + b)(x + y) = a2 + b2
- Exercise 7 - Q 7 (v)8:29
Solve the following pair of linear equations:
(v). 152x – 378y = -74; -378x + 152y = -604
- Exercise 7 - Q 811:30
ABCD is a cyclic quadrilateral (see figure). Find the angles of the cyclic quadrilateral.
- Exercise 1 - Q 120:19
Check whether the following are Quadratic Equations.
(i) (x + 1)2= 2 (x − 3)
(ii) x2 - 2x = (−2) (3 − x)
(iii) (x − 2) (x + 1) = (x − 1) (x + 3)
(iv) (x − 3) (2x + 1) = x (x + 5)
(v) (2x − 1) (x − 3) = (x + 5) (x − 1)
(vi) x2 + 3x +1 = (x - 2)2
(vii) (x + 2)3 = 2x(x2 - 1)
(viii) x3 - 4x2 - x + 1 = (x - 2)3
- Exercise 1 - Q 2 (i)4:22
Represent the following situations in the form of Quadratic Equations:
(i) The area of rectangular plot is 528m2. The length of the plot (in metres) is one more than twice its breadth. We need to find the length and breadth of the plot.
- Exercise 1 - Q 2 (ii)2:40
Represent the following situations in the form of Quadratic Equations:
(ii) The product of two consecutive positive integers is 306. We need to find the integers.
- Exercise 1 - Q 2 (iii)3:52
Represent the following situations in the form of Quadratic Equations:
(iii) Rohan's mother is 26 years older than him. The product of their ages (in years) after 3 years will be 360. We would like to find Rohan's present age.
- Exercise 1 - Q 2 (iv)9:14
Represent the following situations in the form of Quadratic Equations:
(iv) A train travels a distance of 480 km at uniform speed. If, the speed had been 8km/h less, then it would have taken 3 hours more to cover the same distance. We need to find speed of the train.
- Exercise 2 - Q 120:54
Find the roots of the following quadratic equations by factorization:
(i) x2 – 3x – 10 = 0 (ii) 2x2 + x – 6 = 0
(iii) √2x2 + 7x + 5√2 = 0
(iv) 2x2 - x + 1/8 = 0 (v) 100x2 - 20x + 1 = 0
- Exercise 2 - Q 2 (i)12:11
Represent the following situations mathematically:
(i) John and Jivanti together have 45 marbles. Both of them lost 5 marbles each, and the product of the number of marbles they now have is 124. We would like to find out how many marbles they had to start with.
- Exercise 2 - Q 2 (ii)9:13
(ii) A cottage industry produces a certain number of toys in a day. The cost of production of each toy (in rupees) was found to be 55 minus the number of toys produced in a day. On a particular day, the total cost of production was ₹ 750. We would like to find out the number of toys produced on that day.
- Exercise 2 - Q 36:14
Find two numbers whose sum is 27 and product is 182.
- Exercise 2 - Q 48:00
Find two consecutive positive integers, sum of whose squares is 365.
- Exercise 2 - Q 58:37
The altitude of a right triangle is 7 cm less than its base. If the hypotenuse is 13 cm, find the other two sides.
- Exercise 2 - Q 610:55
A cottage industry produces a certain number of pottery articles in a day. It was observed on a particular day that cost of production of each article (in rupees) was 3 more than twice the number of articles produced on that day. If, the total cost of production on that day was ₹90, find the number of articles produced and the cost of each article.
- Exercise 3 - Q 1 (i)13:41
Find the roots of the following quadratic equations if they exist by the method of completing square.
(i) 2x2 – 7x + 3 = 0
- Exercise 3 - Q 1 (ii)7:11
Find the roots of the following quadratic equations if they exist by the method of completing square.
(ii) 2x2 + x – 4 = 0
- Exercise 3 - Q 1 (iii)5:40
Find the roots of the following quadratic equations if they exist by the method of completing square.
(iii) 4x2 + 4√3x + 3 = 0
- Exercise 3 - Q 1 (iv)4:44
Find the roots of the following quadratic equations if they exist by the method of completing square.
(vi) 2x2 + x + 4 = 0
- Exercise 3 - Q 214:52
Find the roots of the following quadratic equations if they exist by the method of completing square.
(i) 2x2 – 7x + 3 = 0
(ii) 2x2 + x – 4 = 0
(iii) 4x2 + 4√3x + 3 = 0
(vi) 2x2 + x + 4 = 0
- Exercise 3 - Q 312:01
Find the roots of the following equations:
(i) x - 1/x = 3, x ≠ 0
(ii) 1/(x+4) - 1/(x-7) = 11/30, x ≠ -4,7
- Exercise 3 - Q 49:05
The sum of reciprocals of Rehman's ages (in years) 3 years ago and 5 years from now is1/3. Find his present age.
- Exercise 3 - Q 510:20
In a class test, the sum of Shefali's marks in Mathematics and English is 30. Had she got 2 marks more in Mathematics and 3 marks less in English, the product of their marks would have been 210. Find her marks in the two subjects.
- Exercise 3 - Q 611:03
The diagonal of a rectangular field is 60 meters more than the shorter side. If, the longer side is 30 meters more than the shorter side, find the sides of the field.
- Exercise 3 - Q 710:26
The difference of squares of two numbers is 180. The square of the smaller number is 8 times the larger number. Find the two numbers.
- Exercise 3 - Q 811:08
A train travels 360 km at a uniform speed. If, the speed had been 5 km/hr more, it would have taken 1 hour less for the same journey. Find the speed of the train.
- Exercise 3 - Q 916:10
Two water taps together can fill a tank in 93/8hours. The tap of larger diameter takes 10 hours less than the smaller one to fill the tank separately. Find the time in which each tap can separately fill the tank.
- Exercise 3 - Q 1013:16
An express train takes 1 hour less than a passenger train to travel 132 km between Mysore and Bangalore (without taking into consideration the time they stop at intermediate stations). If, the average speed of the express train is 11 km/h more than that of the passenger train, find the average speed of two trains.
- Exercise 3 - Q 1112:13
Sum of areas of two squares is 468 m2. If, the difference of their perimeters is 24 meters, find the sides of the two squares.
- Exercise 4 - Q 113:46
Find the nature of the roots of the following quadratic equations. If the real roots exist, find them.
(i) 2x2 - 3x + 5 = 0
(ii) 3x2 - 4√3x + 4 = 0
(iii) 2x2 - 6x + 3 = 0
- Exercise 4 - Q 27:17
Find the value of k for each of the following quadratic equations, so that they have two equal roots.
(i) 2x2 + kx + 3 = 0
(ii) kx (x − 2) + 6 = 0
- Exercise 4 - Q 35:56
Is it possible to design a rectangular mango grove whose length is twice its breadth, and the area is 800 m2? If so, find its length and breadth.
- Exercise 4 - Q 410:00
Is the following situation possible? If so, determine their present ages. The sum of the ages of two friends is 20 years. Four years ago, the product of their ages in years was 48.
- Exercise 4 - Q 59:36
Is it possible to design a rectangular park of perimeter 80 m and area 400 m2? If so, find its length and breadth.
- Exercise 1 - Q 113:05
In which of the following situations, does the list of numbers involved make an arithmetic progression, and why?
The tax if are after each km when the fare is ₹15 for the first km and ₹8 for each additional km.
The amount of air present in a cylinder when a vacuum pump removes 1/4of the air remaining in the cylinder at a time.
The cost of digging a well after every meter of digging, when it costs ₹150 for the first meter and rises by ₹50 for each subsequent meter.
The amount of money in the account every year, when ₹ 10,000 is depositedat compoundInterestat
8%perannum.
- Exercise 1 - Q 210:25
Write first four terms of the AP, when the first term a and common difference d are given as follows:
a = 10, d = 10
a=-2,d = 0
a= 4,d =-3
a =-1,d = 1/2
a =-1.25, d =-0.25
- Exercise 1 - Q 35:28
For the following APs, write the first term and the common difference.
(i). 3,1,–1,–3, ….…
(ii). –5, –1, 3, 7, ……..
(iii). 1/3, 5/3, 9/3, 13/3…….
(iv). 0.6, 1.7, 2.8, 3.9 ...
- Exercise 1 - Q 444:00
Which of the following are APs? If they form an AP, find the common difference d and write three more terms.
2, 4, 8, 16...
2, 5/2, 3, 7/2 ….
−1.2, −3.2, −5.2, −7.2...
−10, −6, −2, 2, ...
3, 3+√2, 3+2√2, 3+3√2, …..
0.2,0.22,0.222,0.2222...
0, −4, −8, −12...
-1/2,-1/2, - 1/2, - 1/2…..
1, 3, 9, 27...
a, 2a, 3a, 4a...
a, a2, a3, a4…….
√2,√8,√18,√32….
√3,√6,√9,√12…
12, 32, 52, 72……
12, 52, 72, 73…..
- Exercise 2 - Q 18:47
Find the missing variable from a, d, n and an, where a is the first term, d is the common difference and an is the nth term of AP.
- Exercise 2 - Q 23:35
Choose the correct choice in the following and justify:
30th term of the AP: 10, 7, 4,... is
97
77
–77
–87
11th term of the AP:−3, −12, 2,... is
28
22
–38
- 481/2
- Exercise 2 - Q 314:59
In the following AP's find the missing terms:
(i) 2, ___, 26
(ii) ___, 13, ___, 3
(iii) 5, ___, ___, 91/2
(iv) –4, ___, ___, ___, ___, 6
(v) ___, 38, ___, ___, ___, –22
- Exercise 2 - Q 43:27
Which term of the AP: 3, 8, 13, 18, . . . , is 78?
- Exercise 2 - Q 58:11
Find the number of terms in each of the following APs:
(i) 7, 13, 19, , 205 (ii) 18, 151/2, 13, ….., −47
- Exercise 2 - Q 65:31
Check whether – 150 is a term of the AP: 11, 8, 5, 2....
- Exercise 2 - Q 78:46
Find the 31st term of an AP whose 11th term is 38 and the 16th term is 73.
- Exercise 2 - Q 87:50
An AP consists of 50 terms of which 3rd term is 12 and the last term is 106. Find the 29th term.
- Exercise 2 - Q 97:17
If the 3rd and the 9th terms of an AP are 4 and – 8 respectively, which term of this AP is zero?
- Exercise 2 - Q 103:05
The 17th term of an AP exceeds its 10th term by 7. Find the common difference.
- Exercise 2 - Q 117:49
Which term of the AP: 3, 15, 27, 39,…. will be 132 more than its 54th term?
- Exercise 2 - Q 129:37
Two APs have the same common difference. The difference between their 100th terms is100, what is the difference between their 1000th terms?
- Exercise 2 - Q 139:56
How many three-digit numbers are divisible by 7?
- Exercise 2 - Q 145:21
How many multiples of 4 lie between 10 and 250?
- Exercise 2 - Q 157:09
For what value of n, are the nth terms of two APs: 63, 65, 67,... and 3, 10, 17,.... equal?
- Exercise 2 - Q 168:59
Determine the AP whose third term is 16 and the 7th term exceeds the 5th term by 12.
- Exercise 2 - Q 176:53
Find the 20th term from the last term of the AP: 3, 8, 13,...., 253.
- Exercise 2 - Q 1812:53
The sum of the 4th and 8th terms of an AP is 24 and the sum of the 6th and 10th terms is 44. Find the first three terms of the AP.
- Exercise 2 - Q 198:23
Subba Rao started work in 1995 at an annual salary of ₹ 5000 and received an increment of ₹ 200 each year. In which year did his income reach ₹ 7000?
- Exercise 2 - Q 208:09
Ramkali saved ₹ 5 in the first week of a year and then increased her weekly savings by ₹ 1.75. If in the nth week, her weekly savings become ₹ 20.75, find n.
- Exercise 3 - Q 112:30
Find the sum of the following AP's.
(i) 2, 7, 12, ..., to 10 terms. (ii) –37, –33, –29, ..., to 12 terms.
(iii) 0.6, 1.7, 2.8, ..., to 100 terms. (iv) 1/15, 1/12, 1/(10 ), …, to 11 terms.
- Exercise 3 - Q 214:51
Find the sums given below:
(i) 7 + 101/2 + 14 +……+ 84
(ii) 34 + 32 + 30 + … + 10
(iii) –5 + (–8) + (–11) + ... + (–230)
- Exercise 3 - Q 346:35
In an AP
(i) Given a = 5, d = 3, an = 50, find n and Sn.
(ii) Given a = 7, a13 = 35, find d and S13.
(iii) Given a12 = 37, d = 3, find a and S12.
(iv) Given a3 = 15, S10 = 125, find d and a10.
(v) Given d = 5, S9 = 75, find a and a9.
(vi) Given a = 2, d = 8, Sn = 90, find n and an.
(vii) Given a = 8, Sn = 210, find n and d.
(viii) Given an = 4, d = 2, Sn = -14, find n and a.
(ix) Given a = 3, n = 8, S = 192, find d.
(ix) Given l = 28, S = 144, and there are total of 9 terms. Find a.
- Exercise 3 - Q 410:29
How many terms of the AP: 9, 17, 25,... must be taken to give a sum of 636?
- Exercise 3 - Q 55:10
The first term of an AP is 5, the last term is 45 and the sum is 400. Find the number of terms and the common difference.
- Exercise 3 - Q 66:24
The first and the last terms of an AP are 17 and 350 respectively. If, the common difference is 9, how many terms are there and what is their sum?
- Exercise 3 - Q 75:36
Find the sum of first 22 terms of an AP in which d = 7 and 22nd term is 149.
- Exercise 3 - Q 86:08
Find the sum of first 51 terms of an AP whose second and third terms are 14 and 18 respectively.
- Exercise 3 - Q 99:33
If the sum of first 7 terms of an AP is 49 and that of 17 terms is 289, find the sum of first n terms.
- Exercise 3 - Q 108:18
Show that a1, a2,……, an,…. form an AP where an is defined as below:
(i) an = 3 + 4n
(ii) an = 9 - 5n
Also find the sum of the first 15 terms in each case.
- Exercise 3 - Q 1114:48
If the sum of the first n terms of an AP is (4n – n2), what is the first term (that is S1)? What is the sum of first two terms? What is the second term? Similarly, find the 3rd, the 10th and the nth terms.
- Exercise 3 - Q 123:56
Find the sum of the first 40 positive integers divisible by 6.
- Exercise 3 - Q 133:22
Find the sum of the first 15 multiples of 8.
- Exercise 3 - Q 144:35
Find the sum of the odd numbers between 0 and 50.
- Exercise 3 - Q 156:07
A contract on construction job specifies a penalty for delay of completion beyond a certain date as follows: ₹ 200 for the first day, ₹ 250 for the second day, ₹ 300 for the third day, etc., the penalty for each succeeding day being
₹ 50 more than for the preceding day. How much money the contractor has to pay as penalty, if he has delayed the work by 30 days?
- Exercise 3 - Q 168:47
A sum of ₹ 700 is to be used to give seven cash prizes to students of a school for their overall academic performance. If, each prize is ₹ 20 less than its preceding term, find the value of each of the prizes.
- Exercise 3 - Q 176:21
In a school, students thought of planting trees in and around the school to reduce air pollution. It was decided that the number of trees, that each section of each class will plant, will be the same as the class, in which they are studying, e.g., a section of Class I will plant 1 tree, a section of class II will plant two trees and so on till Class XII. There are three sections of each class. How many trees will be planted by the students?
- Exercise 3 - Q 189:50
A spiral is made up of successive semicircles, with centers alternatively at A and B, starting with center at A, of radii 0.5 cm, 1.0 cm, 1.5 cm, 2.0 cm, ... What is the total length of such a spiral made up of thirteen consecutive semicircles. (Take π = 22/7)
[Hint: Length of successive semicircles is l1, l2, l3, l4,.... with centers at A, B, A, B, . . ., respectively.]
- Exercise 3 - Q 1915:35
200 logs are stacked in the following manner: 20 logs in the bottom row, 19 in the next row, 18 in the row next to it and soon. In how many rows are the 200 logs placed and how many logs are in the top row?
- Exercise 3 - Q 2010:15
In a potato race, a bucket is placed at the starting point, which is 5 meters from the first potato, and the other potatoes are placed 3 meters apart in a straight line. There are ten potatoes in the line.
A competitor starts from the bucket, picks up the nearest potato, runs back with it, drops it in the bucket, runs back to pick up the next potato, runs to the bucket to drop it in, and she continues in the same way until all the potatoes are in the bucket. What is the total distance the competitor has to run?
[Hint: To pick up the first potato and the second potato, the total distance (in meters) run by a competitor is 2 × 5 + 2 × (5 + 3)]
- Exercise 4 - Q 19:03
Which term of the AP: 121, 117, 113,. . ., Is its first negative term?
[Hint : Find n for an< 0]
- Exercise 4 - Q 214:42
The sum of the third and the seventh terms of an AP is 6 and their product is 8. Find the sum of first sixteen terms of the AP.
- Exercise 4 - Q 39:20
A ladder has rungs 25 cm apart (see figure). The rungs decrease uniformly in length from 45 cm, at the bottom to 25 cm at the top. If the top and the bottom rungs are 21/2 m apart, what is the length of the wood required for the rungs?
[Hint: Number of rungs = 250/25 + 1]
- Exercise 4 - Q 413:51
The houses of a row are numbered consecutively from 1 to 49. Show that there is a value of x such that the sum of the numbers of the houses preceding the house numbered x is equal to the sum of the numbers of the houses following it. Find this value of x.
[Hint: Sx – 1 = S49 – Sx]
- Exercise 4 - Q 511:03
A small terrace at a football ground comprises of 15 steps each of which is 50 m long and built of solid concrete. Each step has a rise of 1/4 m and at read of 1/2 m (see figure). Calculate the total volume of concrete required to build the terrace.
[Hint : Volume of concrete required to build the first step = 1/4 × 1/2 × 50 m3]
- Exercise 1 - Q 14:16
Fill in the blanks using the correct word given in brackets :
(i) All circles are_______. (congruent, similar)
(ii) All squares are __________. (similar, congruent)
(iii) All _______ triangles are similar. (isosceles, equilateral)
(iv) Two polygons of the same number of sides are similar, if (a) their corresponding angles are _________ and (b) their corresponding sides are _________. (equal, proportional)
- Exercise 1 - Q 22:44
Give two different examples of pair of
(i) Similar figures. (ii) Non-similar figures.
- Exercise 1 - Q 32:01
State whether the following quadrilaterals are similar or not:
- Exercise 2 - Q 16:12
In figure, (i) and (ii), DE || BC. Find EC in (i) and AD in (ii).
- Exercise 2 - Q 28:21
E and F are points on the sides PQ and PR respectively of a ∆PQR. For each of the following cases, state whether EF || QR :
(i) PE = 3.9 cm, EQ = 3 cm, PF = 3.6 cm and FR = 2.4 cm
(ii) PE = 4 cm, QE = 4.5 cm, PF = 8 cm and RF = 9 cm
(iii) PQ = 1.28 cm, PR = 2.56 cm, PE = 0.18 cm and PF = 0.36 cm
- Exercise 2 - Q 35:02
In figure. 6.18, if LM || CB and LN || CD, prove that
(AM )/AB = (AN )/AD
- Exercise 2 - Q 46:07
In figure, DE || AC and DF || AE. Prove that (BF )/FE = (BE )/EC
- Exercise 2 - Q 57:49
In figure, DE || OQ and DF || OR. Show that EF || QR.
- Exercise 2 - Q 66:24
In figure, A, B and C are points on OP, OQ and OR respectively such that AB || PQ and AC || PR. Show that BC || QR.
- Exercise 2 - Q 75:37
Using Theorem 6.1, prove that a line drawn through the mid-point of one side of a triangle parallel to another side bisects the third side. (Recall that you have proved it in Class IX).
- Exercise 2 - Q 85:59
Using Theorem 6.2, prove that the line joining the mid-points of any two sides of a triangle is parallel to the third side. (Recall that you have done it in Class IX).
- Exercise 2 - Q 99:54
ABCD is a trapezium in which AB || DC and its diagonals intersect each other at the point O. Show that (AO )/BO = (CO )/DO
- Exercise 2 - Q 1012:09
The diagonals of a quadrilateral ABCD intersect each other at the point O such that (AO )/BO = (CO )/DO Show that ABCD is a trapezium.
- Exercise 3 - Q 115:09
State which pairs of triangles in figure are similar. Write the similarity criterion used by you for answering the question and also write the pairs of similar triangles in the symbolic form :
- Exercise 3 - Q 28:31
In figure, ∆ODC ~ ∆OBA, ∠BOC = 1250 and ∠CDO = 700. Find ∠DOC, ∠DCO and ∠OAB.
- Exercise 3 - Q 37:23
Diagonals AC and BD of a trapezium ABCD with AB || DC intersect each other at the point O. Using a similarity criterion for two triangles, show that
(OA )/OC = (OB )/OD
- Exercise 3 - Q 48:27
In figure, (QR )/QS = (QT )/PR and ∠1 = ∠2. Show that ∆PQS ~ ∆TQR.
- Exercise 3 - Q 53:56
S and T are points on sides PR and QR of ∆PQR such that ∠P = ∠RTS. Show that ∆RPQ ~ ∆RTS.
- Exercise 3 - Q 66:44
In figure, if ∆ABE = ∆ACD, show that ∆ADE ~ ∆ABC.
- Exercise 3 - Q 714:15
In figure, altitudes AD and CE of ∆ABC intersect each other at the point P. Show that:
(i) ∆AEP ~ ∆CDP
(ii) ∆ABD ~ ∆CBE
(iii) ∆AEP ~ ∆ADB
(iv) ∆PDC ~ ∆BEC
- Exercise 3 - Q 85:40
E is a point on the side AD produced of a parallelogram ABCD and BE intersects CD at F. Show that ∆ABE ~ ∆CFB.
- Exercise 3 - Q 96:41
In figure, ABC and AMP are two right triangles, right angled at B and M
respectively. Prove that:
(i) ∆ABC ~ ∆AMP
(ii) (CA )/PA = (BC )/MP
- Exercise 3 - Q 1014:37
CD and GH are respectively the bisectors of ∠ACB and ∠EGF such that D and H lie on sides AB and FE of ∆ABC and ∆EFG respectively. If ∆ABC ~ ∆FEG, show that:
(i) (CD )/GH = (AC )/FG
(ii) ∆DCB ~ ∆HGE
(iii) ∆DCA ~ ∆HGF
- Exercise 3 - Q 115:12
In figure, E is a point on side CB produced of an isosceles triangle ABC with AB = AC. If AD⊥BC and EF⊥AC, prove that ∆ABD ~ ∆ECF.
- Exercise 3 - Q 129:34
Sides AB and BC and median AD of a triangle ABC are respectively proportional to sides PQ and QR and median PM of ∆PQR (see figure). Show that ∆ABC ~ ∆PQR.
- Exercise 3 - Q 136:07
D is a point on the side BC of a triangle ABC such that ∠ADC = ∠BAC. Show that CA2 = CB. CD.
- Exercise 3 - Q 1419:11
Sides AB and AC and median AD of a triangle ABC are respectively proportional to sides PQ and PR and median PM of another triangle PQR.
Show that ∆ABC ~ ∆PQR.
- Exercise 3 - Q 156:05
A vertical pole of length 6 m casts a shadow 4 m long on the ground and at the same time a tower casts a shadow 28 m long. Find the height of the tower.
- Exercise 3 - Q 169:46
If AD and PM are medians of triangles ABC and PQR, respectively where ∆ABC ~ ∆PQR, prove that (AB )/PQ = (AD )/PM
- Exercise 4 - Q 15:36
Let ∆ABC~∆DEF and their areas be, respectively, 64 cm2 and 121 cm2. If EF = 15.4 cm, find BC.
- Exercise 4 - Q 27:03
Diagonals of a trapezium ABCD with AB || DC intersect each other at the point O. If AB = 2 CD, find the ratio of the areas of triangles AOB and COD.
- Exercise 4 - Q 36:38
In figure, ABC and DBC are two triangles on the same base BC. If AD intersects BC at O, show that (ar (ABC) )/(ar (DBC)) = (AO )/DO
- Exercise 4 - Q 45:30
If the areas of two similar triangles are equal, prove that they are congruent.
- Exercise 4 - Q 57:48
D, E and F are respectively the mid-points of sides AB, BC and CA of ∆ABC. Find the ratio of the areas of ∆DEF and ∆ABC.
- Exercise 4 - Q 68:11
Prove that the ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding medians.
- Exercise 4 - Q 79:09
Prove that the area of an equilateral triangle described on one side of a square is equal to half the area of the equilateral triangle described on one of its diagonals.
- Exercise 4 - Q 83:39
ABC and BDE are two equilateral triangles such that D is the mid-point of BC. Ratio of the areas of triangles ABC and BDE is
(A) 2 : 1 (B) 1 : 2 (C) 4 : 1 (D) 1 : 4
- Exercise 4 - Q 91:58
Sides of two similar triangles are in the ratio 4 : 9. Areas of these triangles are in the ratio
(A) 2 : 3 (B) 4 : 9 (C) 81 : 16 (D) 16 : 81
- Exercise 5 - Q 19:59
Sides of triangles are given below. Determine which of them are right triangles. In case of a right triangle, write the length of its hypotenuse.
(i) 7 cm, 24 cm, 25 cm
(ii) 3 cm, 8 cm, 6 cm
(iii) 50 cm, 80 cm, 100 cm
(iv) 13 cm, 12 cm, 5 cm
- Exercise 5 - Q 26:39
PQR is a triangle right angled at P and M is a point on QR such that PM⊥QR. Show that PM2 = QM. MR.
- Exercise 5 - Q 39:47
In figure, ABD is a triangle right angled at A and AC⊥BD. Show that
(i) AB2 = BC. BD
(ii) AC2 = BC. DC
(iii) AD2 = BD. CD
- Exercise 5 - Q 43:27
ABC is an isosceles triangle right angled at C. Prove that AB2 = 2AC2.
- Exercise 5 - Q 55:52
ABC is an isosceles triangle with AC = BC. If AB2 = 2AC2, prove that ABC is a right triangle.
- Exercise 5 - Q 65:37
ABC is an equilateral triangle of side 2a. Find each of its altitudes.
- Exercise 5 - Q 77:23
Prove that the sum of the squares of the sides of a rhombus is equal to the sum of the squares of its diagonals.
- Exercise 5 - Q 813:31
In Fig. 6.54, O is a point in the interior of a triangle ABC, OD⊥BC, OE⊥AC and OF⊥AB. Show that
(i) OA2 + OB2 + OC2 – OD2 – OE2 – OF2 = AF2 + BD2 + CE2,
(ii) AF2 + BD2 + CE2 = AE2 + CD2 + BF2.
- Exercise 5 - Q 94:25
A ladder 10 m long reaches a window 8 m above the ground. Find the distance of the foot of the ladder from base of the wall.
- Exercise 5 - Q 106:03
A guy wire attached to a vertical pole of height 18 m is 24 m long and has a stake attached to the other end. How far from the base of the pole should the stake be driven so that the wire will be taut?
- Exercise 5 - Q 1111:18
An aeroplane leaves an airport and flies due north at a speed of 1000 km per hour. At the same time, another aeroplane leaves the same airport and flies due west at a speed of 1200 km per hour. How far apart will be the two planes after 11/2 hour?
- Exercise 5 - Q 125:17
Two poles of heights 6 m and 11 m stand on a plane ground. If the distance between the feet of the poles is 12 m, find the distance between their tops.
- Exercise 5 - Q 137:13
D and E are points on the sides CA and CB respectively of a triangle ABC right angled at C. Prove that AE2 + BD2 = AB2 + DE2.
- Exercise 5 - Q 147:59
The perpendicular from A on side BC of a ∆ABC intersects BC at D such that DB = 3 CD (see Fig. 6.55). Prove that 2AB2 = 2AC2 + BC2.
- Exercise 5 - Q 1513:28
In an equilateral triangle ABC, D is a point on side BC such that BD = 1/3BC. Prove that 9AD2 = 7AB2.
- Exercise 5 - Q 167:12
In an equilateral triangle, prove that three times the square of one side is equal to four times the square of one of its altitudes.
- Exercise 5 - Q 172:51
Tick the correct answer and justify: In ∆ABC, AB = 6√3 cm, AC = 12 cm and BC = 6 cm. The angle B is :
(A) 1200 (B) 600
(C) 900 (D) 450
- Exercise 6 - Q 18:45
In figure, PS is the bisector of ∠QPR of ∆PQR. Prove that QS/SR = PQ/PR
- Exercise 6 - Q 215:23
In figure, D is a point on hypotenuse AC of ∆ABC, such that BD⊥AC, DM⊥BC and DN⊥AB. Prove that :
(i) DM2 = DN. MC (ii) DN2 = DM. AN
- Exercise 6 - Q 36:25
In figure, ABC is a triangle in which ∠ABC > 90° and AD⊥CB produced. Prove that
AC2 = AB2 + BC2 + 2BC. BD.
- Exercise 6 - Q 45:20
In figure, ABC is a triangle in which ∠ABC < 90° and AD⊥BC. Prove that
AC2 = AB2 + BC2 – 2BC. BD.
- Exercise 6 - Q 58:44
In figure, AD is a median of a triangle ABC and AM ⊥ BC. Prove that:
(i) AC2 = AD2 + BC. DM + (BC/2)^2
(ii) AB2 = AD2 – BC. DM + (BC/2)^2
(iii) AC2 + AB2 = 2 AD2 + 1/2BC2
- Exercise 6 - Q 610:08
Prove that the sum of the squares of the diagonals of parallelogram is equal to the sum of the squares of its sides.
- Exercise 6 - Q 74:07
In figure, two chords AB and CD intersect each other at the point P. Prove that : (i) ∆APC ~ ∆DPB (ii) AP. PB = CP. DP
- Exercise 6 - Q 85:45
In figure, two chords AB and CD of a circle intersect each other at the point P (when produced) outside the circle. Prove that (i) ∆PAC ~ ∆PDB
(ii) PA. PB = PC. PD
- Exercise 6 - Q 99:26
In figure, D is a point on side BC of ∆ABC such that BD/CD = AB/AC Prove that AD is the bisector of ∠BAC.
- Exercise 6 - Q 108:55
Nazima is fly fishing in a stream. The tip of her fishing rod is 1.8 m above the surface of the water and the fly at the end of the string rests on the water 3.6 m away and 2.4 m from a point directly under the tip of the rod. Assuming that her string (from the tip of her rod to the fly) is taut, how much string does she have out (see Fig. 6.64)? If she pulls in the string at the rate of 5 cm per second, what will be the horizontal distance of the fly from her after 12 seconds?
- Exercise 1 - Q 110:46
Find the distance between the
following pairs of points:
(i) (2, 3), (4, 1)
(ii) (–5, 7), (–1, 3)
(iii) (a, b), (–a, –b)
- Exercise 1 - Q 25:36
Find the distance between the points (0, 0) and (36, 15). Also, find the distance betweentownsAandBiftownBislocatedat36kmeastand 15kmnorthoftownA.
- Exercise 1 - Q 39:33
Determine if the points (1, 5), (2, 3) and (–2, –11) are collinear
- Exercise 1 - Q 47:20
Check whether (5, –2), (6, 4) and (7, –2) are the vertices of an isosceles triangle.
- Exercise 1 - Q 515:45
In a classroom, 4 friends are seated at the points A, B, C and Das shown in figure. Champa and Chameli walk into the class and after observing for a few minutes Champa asks Chameli. "Don't you think ABCD is a square?"Chameli disagrees. Using distance formula, find which of them is correct.
- Exercise 1 - Q 642:29
Name the type of quadrilateral formed, if any, by the following points, and give reasons for your answer:
(i) (–1, –2), (1, 0), (–1, 2), (–3, 0)
(ii) (–3, 5), (3, 1), (0, 3), (–1, –4)
(iii) (4, 5), (7, 6), (4, 3), (1, 2)
- Exercise 1 - Q 711:24
Find the point on the x–axis which is equidistant from (2, –5) and (–2, 9)
- Exercise 1 - Q 88:19
Find the values of y for which the distance between the points P (2, –3) and Q (10, y) is 10 units.
- Exercise 1 - Q 913:26
If, Q (0, 1) is equidistant from P (5, –3) and R (x, 6), find the values of x. Also, find the distances QR and PR.
- Exercise 1 - Q 109:09
Find a relation between x and y such that the point (x, y) is equidistant from the point (3, 6) and (–3, 4).
- Exercise 2 - Q 15:06
Find the coordinates of the point which divides the line segment join of (–1, 7) and (4, –3) in the ratio 2:3.
- Exercise 2 - Q 210:35
Find the coordinates of the points of trisection of the line segment joining (4, –1) and (–2, –3).
- Exercise 2 - Q 312:47
To conduct sports day activities, in your rectangular shaped school ground ABCD, lines have been drawn with chalk powder at a distance of 1 m each. 100 flower pots have been placed at a distance of 1 m from each other along AD as shown in figure. Niharika runs 1/4 of the distance AD on the 2nd line and posts a green flag. Preet runs 1/5of the distance AD on the eighth line and posts a red flag. What is the distance between both the flags? If Rashmi has to post a blue flag exactly halfway between the line segment joining the two flags, where should she post her flag?
- Exercise 2 - Q 46:48
Find the ratio in which the line segment joining the points (–3, 10) and (6, –8) is divided by (–1, 6).
- Exercise 2 - Q 59:31
Find the ratio in which the line segment joining A (1, –5) and B(–4, 5) is divided by the x–axis. Also find the coordinates of the point of division.
- Exercise 2 - Q 67:38
If (1, 2), (4, y), (x, 6) and (3, 5) are the vertices of a parallelogram taken in order, find x and y.
- Exercise 2 - Q 77:11
Find the coordinates of a point A, where AB is the diameter of a circle whose centre is (2, –3) and B is (1, 4).
- Exercise 2 - Q 86:06
If A and B are (–2, –2) and (2, –4) respectively, find the coordinates of P such that AP = 3/7AB and P lies on the line segment AB.
- Exercise 2 - Q 911:40
Find the coordinates of the points which divides the line segment joining A (–2, 2) and B (2, 8) into four equal parts.
- Exercise 2 - Q 107:35
Find the area of a rhombus if its vertices are (3, 0), (4, 5), (–1, 4) and (–2, –1) taken in order.
[Hint: Area of a rhombus = 1/2(product of its diagonals)]
- Exercise 3 - Q 18:18
Findtheareaofthetrianglewhoseverticesare:
(i)(2,3),(–1,0),(2,–4)
(ii) (–5, –1), (3, –5), (5,2)
- Exercise 3 - Q 28:55
Ineachofthefollowingfindthevalueof'k',forwhichthepointsarecollinear.
(i)(7,–2),(5,1),(3,k)
(ii) (8, 1), (k, –4), (2,–5)
- Exercise 3 - Q 314:04
Find the area of the triangle formed by joining the mid–points of the sides of the triangle whose vertices are(0,–1),(2,1)and(0,3).Find the ratio of this area to the area of the given triangle.
- Exercise 3 - Q 49:36
Find the area of the quadrilateral whose vertices taken in order are (–4, –2), (–3,–5), (3, –2) and (2,3).
- Exercise 3 - Q 512:05
Weknowthatmedianofatriangledividesitintotwotrianglesofequalareas.Verify thisresultfor△ABCwhoseverticesareA(4,–6),B(3,–2)andC(5,2).
- Exercise 4 - Q 110:16
Determine the ratio in which the line 2x + y – 4 = 0 divides the line segment joining the points A (2, -2) and B (3, 7).
- Exercise 4 - Q 24:34
Find a relation between x and y if the points (x, y) , (1,2) and (7, 0) are collinear.
- Exercise 4 - Q 319:31
Find the centre of a circle passing through the points (6, -6), (3, -7) and (3,3).
- Exercise 4 - Q 417:49
The two opposite vertices of a square are (-1, 2) and (3, 2). Find the coordinates of the other two vertices.
- Exercise 4 - Q 514:33
The class X students of a secondary school in Krishinagar have been allotted a rectangular plot of land for their gardening activity. Saplings of Gulmohar are planted on the boundary at a distance of 1 m from each other. There is a triangular grassy lawn in the plot as shown in the figure. The students are to sow seeds of flowering plants on the remaining area of the plot.
Taking A as origin, find the coordinates of the vertices of the triangle.
What will be the coordinates of the vertices of ∆PQR if C is the origin? Also calculate the area of the triangle in these cases. What do you observe?
- Exercise 4 - Q 619:56
The vertices of a ∆ABC are A (4, 6), B (1, 5) and C (7, 2).
A line is drawn to intersect sides AB and AC at D and E respectively such that AD/AB=AE/AC=1/4. Calculate the area of
the ∆ADE and compare it with the area of ∆ABC.
- Exercise 4 - Q 722:55
Let A (4, 2), B (6, 5) and C (1, 4) be the vertices of ∆ABC.
The median from A meets BC at D. Find the coordinates of the point D.
Find the coordinates of the point P on AD such that
AP: PD = 2: 1.
Find the coordinates of points Q and R on medians BE and CF respectively such that BQ: QE = 2: 1 and
CR : RF = 2 : 1.
What do you observe?
(Note: The point which is common to all the three medians is called centroid and this point divides each median in the ratio 2: 1)
If A(x1, y1), B(x2, y2) and C (x3, y3) are the vertices of
∆ABC, find the coordinates of the centroid of the triangle.
- Exercise 4 - Q 824:54
ABCD is a rectangle formed by joining points A (-1, -1),
B (-1, 4), C (5, 4) and D(5, -1). P, Q, R and S are the
mid-points of AB, BC, CD and DA respectively. Is the quadrilateral PQRSs a square? a rectangle? Or a rhombus? Justify your answer.
- Exercise 1 - Q 19:42
In ∆ABC, right-angled at B, AB = 24 cm, BC = 7 cm. Determine :
(i) sin A, cos A
(ii) sin C, cos C
- Exercise 1 - Q 25:59
In Fig. 8.13, find tan P – cot R.
- Exercise 1 - Q 35:02
If sin A = calculate cos A and tan A.
- Exercise 1 - Q 46:29
Given 15 cot A = 8, find sin A and sec A.
- Exercise 1 - Q 56:53
Given secθ = 13/12 calculate all other trigonometric ratios.
- Exercise 1 - Q 64:50
If ∠A and ∠B are acute angles such that cos A = cos B, then show that ∠A = ∠ B.
- Exercise 1 - Q 711:43
If cot q = 7/8, evaluate : (i) ((1+sinθ)(1-sinθ))/((1+cosθ)(1-cosθ))
(ii) cot2 θ
- Exercise 1 - Q 89:40
If 3 cot A = 4, check whether (1-tan^2 A)/(1+tan^2 A) = cos2A – sin2A or not.
- Exercise 1 - Q 99:15
In triangle ABC, right-angled at B, if tan A = 1/√3 find the value of:
(i) sin A cos C + cos A sin C
(ii) cos A cos C – sin A sin C
- Exercise 1 - Q 107:45
In ∆PQR, right-angled at Q, PR + QR = 25 cm and PQ = 5 cm. Determine the values of
sin P, cos P and tan P.
- Exercise 1 - Q 113:58
State whether the following are true or false. Justify your answer.
(i) The value of tan A is always less than 1.
(ii) sec A = 12/5 for some value of angle A.
(iii) cos A is the abbreviation used for the cosecant of angle A.
(iv) cot A is the product of cot and A.
(v) sinθ = 1/2 for some angle θ.
- Exercise 2 - Q 115:46
Evaluate the following :
(i) sin 60° cos 30° + sin 30° cos 60° (ii) 2 tan2 45° + cos2 30° – sin2 60°
(iii) (cos〖45〗^0)/(sec30^0+cosec30^0 ) (iv) (s〖in30〗^0+tan45^0-cosec60^0)/(sec30^0+cos30^0+cot45^0 )
(v) (5cos^2 60^0+4sec^2 〖30〗^0-tan^2 〖45〗^0)/(sin^2 30^0+cos^2 30^0 )
- Exercise 2 - Q 26:38
Choose the correct option and justify your choice :
(i) (2tan30^0)/(1+tan^2 30^0 ) =
(A) sin 60° (B) cos 60° (C) tan 60° (D) sin 30°
(ii) (1-tan^2 〖45〗^0)/(1+tan^2 〖45〗^0 )
(A) tan 90° (B) 1 (C) sin 45° (D) 0
(iii) sin 2A = 2 sin A is true when A =
(A) 0° (B) 30° (C) 45° (D) 60°
(iv) (2tan30^0)/(1-tan^2 30^0 )
(A) cos 60° (B) sin 60° (C) tan 60° (D) sin 30°
- Exercise 2 - Q 35:51
If tan (A + B) = √3 and tan (A – B) = 1/√3; 0° < A + B ≤ 90°; A > B, find A and B.
- Exercise 2 - Q 44:34
State whether the following are true or false. Justify your answer.
(i) sin (A + B) = sin A + sin B.
(ii) The value of sinθ increases as θ increases.
(iii) The value of cosθ increases as θ increases.
(iv) sin θ = cos θ for all values of θ.
(v) cot A is not defined for A = 0°.
- Exercise 3 - Q 15:53
Evaluate :
(i) (sin18^0)/(cos72^0 ) (ii) (tan26^0)/(cot64^0 )
(iii) cos 48° – sin 42° (iv) cosec 31° – sec 59°
- Exercise 3 - Q 27:49
Show that :
(i) tan 48° tan 23° tan 42° tan 67° = 1
(ii) cos 38° cos 52° – sin 38° sin 52° = 0
- Exercise 3 - Q 33:49
If tan 2A = cot (A – 18°), where 2A is an acute angle, find the value of A.
- Exercise 3 - Q 42:17
If tan A = cot B, prove that A + B = 90°.
- Exercise 3 - Q 53:58
If sec 4A = cosec (A – 20°), where 4A is an acute angle, find the value of A.
- Exercise 3 - Q 64:55
If A, B and C are interior angles of a triangle ABC, then show that
sin((B+C)/2) = cosA/2
- Exercise 3 - Q 72:36
Express sin 67° + cos 75° in terms of trigonometric ratios of angles between 0° and 45°.
- Exercise 4 - Q 19:20
Express the trigonometric ratios sin A, sec A and tan A in terms of cot A.
- Exercise 4 - Q 27:56
Write all the other trigonometric ratios of ∠A in terms of sec A.
- Exercise 4 - Q 37:39
Evaluate :
(i) (sin^2 63^0+sin^2 27^0)/(cos^2 17^0+cos^2 73^0 )
(ii) sin 25° cos 65° + cos 25° sin 65°
- Exercise 4 - Q 412:07
Choose the correct option. Justify your choice.
(i) 9 sec2 A – 9 tan2 A =
(A) 1 (B) 9 (C) 8 (D) 0
(ii) (1 + tanθ + secθ) (1 + cotθ – cosecθ) =
(A) 0 (B) 1 (C) 2 (D) –1
(iii) (sec A + tan A) (1 – sin A) =
(A) sec A (B) sin A (C) cosec A (D) cos A
(iv) (1+tan^2 A)/(1+cot^2 A)
(A) sec2 A (B) –1 (C) cot2 A (D) tan2 A
- Exercise 4 - Q 5 (i)7:45
Prove the following identities, where the angles involved are acute angles for which the expressions are defined.
(i). (cosecθ - cotθ)2 = (1-cosθ)/(1+cosθ)
- Exercise 4 - Q 5 (ii)7:25
Prove the following identities, where the angles involved are acute angles for which the expressions are defined.
(ii). cosA/(1+sinA) + (1+sinA)/cosA = 2 secA
- Exercise 4 - Q 5 (iii)16:23
Prove the following identities, where the angles involved are acute angles for which the expressions are defined.
(iii). tanθ/(1-cotθ) + cotθ/(1-tanθ) = 1 + secθ cosecθ
- Exercise 4 - Q 5 (iv)6:26
Prove the following identities, where the angles involved are acute angles for which the expressions are defined.
(iv) (1+secA)/secA = (sin^2 A)/(1-cosA)
- Exercise 4 - Q 5 (v)11:09
Prove the following identities, where the angles involved are acute angles for which the expressions are defined.
(v) (cosA-sinA+1)/(cosA+sinA-1) = cosec A + cot A, using the identity cosec2 A = 1 + cot2A.
- Exercise 4 - Q 5 (vi)7:19
Prove the following identities, where the angles involved are acute angles for which the expressions are defined.
(vi) √((1+sinA)/(1-sinA)) = secA + tanA
- Exercise 4 - Q 5 (vii)5:59
Prove the following identities, where the angles involved are acute angles for which the expressions are defined.
(vii) (sinθ-2sin^3 θ)/(2cos^3 θ-cosθ)
- Exercise 4 - Q 5 (viii)7:41
Prove the following identities, where the angles involved are acute angles for which the expressions are defined.
(viii) (sin A + cosec A)2 + (cos A + sec A)2 = 7 + tan2A + cot2A
- Exercise 4 - Q 5 (ix)8:17
Prove the following identities, where the angles involved are acute angles for which the expressions are defined.
(ix) (cosecA – sinA)(secA – cosA) = 1/(tanA+cotA)
- Exercise 4 - Q 5 (x)9:54
Prove the following identities, where the angles involved are acute angles for which the expressions are defined.
(x) ((1+tan^2 A)/(1+cot^2 A)) = ((1-tanA)/(1-cotA))^2= tan2A
- Exercise 1 - Q 14:25
A circus artist is climbing a 20 m long rope, which is tightly stretched and tied from the top of a vertical pole to the ground. Find the height of the pole, if the angle made by the rope with the ground level is 30°
(see Fig. 9.11).
- Exercise 1 - Q 28:18
A tree breaks due to storm and the broken part bends so that the top of the tree touches the ground making an angle 30° with it. The distance between the foot of the tree to the point where the top touches the ground is 8 m. Find the height of the tree.
- Exercise 1 - Q 39:06
A contractor plans to install two slides for the children to play in a park. For the children below the age of 5 years, she prefers to have a slide whose top is at a height of 1.5 m, and is inclined at an angle of 30° to the ground, whereas for elder children, she wants to have a steep slide at a height of 3m, and inclined at an angle of 60° to the ground. What should be the length of the slide in each case?
- Exercise 1 - Q 44:37
The angle of elevation of the top of a tower from a point on the ground, which is 30 m away from the foot of the tower, is 30°. Find the height of the tower.
- Exercise 1 - Q 55:15
A kite is flying at a height of 60 m above the ground. The string attached to the kite is temporarily tied to a point on the ground. The inclination of the string with the ground is 60°. Find the length of the string, assuming that there is no slack in the string.
- Exercise 1 - Q 611:23
A 1.5 m tall boy is standing at some distance from a 30 m tall building. The angle of elevation from his eyes to the top of the building increases from 30° to 60° as he walks towards the building. Find the distance he walked towards the building.
- Exercise 1 - Q 78:34
From a point on the ground, the angles of elevation of the bottom and the top of a transmission tower fixed at the top of a 20 m high building are 45° and 60° respectively. Find the height of the tower.
- Exercise 1 - Q 810:46
A statue, 1.6 m tall, stands on the top of a pedestal. From a point on the ground, the angle of elevation of the top of the statue is 60° and from the same point the angle of elevation of the top of the pedestal is 45°. Find the height of the pedestal.
- Exercise 1 - Q 98:00
The angle of elevation of the top of a building from the foot of the 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 50 m high, find the height of the building.
- Exercise 1 - Q 1011:50
Two poles of equal heights are standing opposite each other on either side of the road, which is 80 m wide. From a point between them on the road, the angles of elevation of the top of the poles are 60° and 30°, respectively. Find the height of the poles and the distances of the point from the poles.
- Exercise 1 - Q 1110:38
A TV tower stands vertically on a bank
of a canal. From a point on the other
bank directly opposite the tower, the
angle of elevation of the top of the
tower is 60°. From another point 20 m
away from this point on the line joing
this point to the foot of the tower, the
angle of elevation of the top of the tower is 30° (see Fig. 9.12). Find the height of the tower and the width of the canal.
- Exercise 1 - Q 128:19
- Exercise 1 - Q 1310:53
As observed from the top of a 75 m high lighthouse from the sea-level, the angles of depression of two ships are 30° and 45°. If one ship is exactly behind the other on the same side of the lighthouse, find the distance between the two ships.
- Exercise 1 - Q 1411:16
A straight highway leads to the foot of a tower. A man standing at the top of the tower observes a car at an angle of depression of 30°, which is approaching the foot of the tower with a uniform speed. Six seconds later, the angle of depression of the car is found to be 60°. Find the time taken by the car to reach the foot of the tower from this point.
- Exercise 1 - Q 11:20
How many tangents can a circle have?
- Exercise 1 - Q 22:32
Fill in the blanks :
(i) A tangent to a circle intersects it in ___________ point (s).
(ii) A line intersecting a circle in two points is called a _____________.
(iii) A circle can have ___________ parallel tangents at the most.
(iv) The common point of a tangent to a circle and the circle is called __________.
- Exercise 1 - Q 33:19
A tangent PQ at a point P of a circle of radius 5 cm meets a line through the centre O at a point Q so that OQ = 12 cm. Length PQ is :
(A) 12 cm (B) 13 cm (C) 8.5 cm (D) √119 cm.
- Exercise 1 - Q 41:44
Draw a circle and two lines parallel to a given line such that one is a tangent and the other, a secant to the circle.
- Exercise 2 - Q 14:41
From a point Q, the length of the tangent to a circle is 24 cm and the distance of Q from the centre is 25 cm. The radius of the circle is
(A) 7 cm (B) 12 cm
(C) 15 cm (D) 24.5 cm
- Exercise 2 - Q 23:17
In Fig. 10.11, if TP and TQ are the two tangents to a circle with centre O so that
∠POQ = 110°, then ∠PTQ is equal to
(A) 60° (B) 70° (C) 80° (D) 90°
- Exercise 2 - Q 39:10
If tangents PA and PB from a point P to a circle with centre O are inclined to each other at angle of 80°, then ∠POA is equal to
(A) 50° (B) 60° (C) 70° (D) 80°
- Exercise 2 - Q 46:03
Prove that the tangents drawn at the ends of a diameter of a circle are parallel.
- Exercise 2 - Q 57:16
Prove that the perpendicular at the point of contact to the tangent to a circle passes through the centre.
- Exercise 2 - Q 64:59
The length of a tangent from a point A at distance 5 cm from the centre of the circle is 4 cm. Find the radius of the circle.
- Exercise 2 - Q 77:03
Two concentric circles are of radii 5 cm and 3 cm. Find the length of the chord of the larger circle which touches the smaller circle.
- Exercise 2 - Q 87:17
A quadrilateral ABCD is drawn to circumscribe a circle (see Fig. 10.12). Prove that
AB + CD = AD + BC
- Exercise 2 - Q 910:15
In Fig. 10.13, XY and X’Y’ are two parallel tangents to a circle with centre O and
another tangent AB with point of contact C intersecting XY at A and X’Y’ at B. Prove that ∠AOB = 90°.
- Exercise 2 - Q 105:34
Prove that the angle between the two tangents drawn from an external point to a circle is supplementary to the angle subtended by the line-segment joining the points of contact at the centre.
- Exercise 2 - Q 116:57
Prove that the parallelogram circumscribing a circle is a rhombus.
- Exercise 2 - Q 1217:00
A triangle ABC is drawn to circumscribe a circle of radius 4 cm such that the segments BD and DC into which BC is divided by the point of contact D are of lengths 8 cm and 6 cm respectively (see Fig. 10.14). Find the sides AB and AC.
- Exercise 2 - Q 139:40
Prove that opposite sides of a quadrilateral circumscribing a circle subtend supplementary angles at the centre of the circle.
Requirements
- Basic elementary math knowledge.
- Each chapter includes prerequisite knowledge classes in which the child gains extensive knowledge and a thorough understanding of the chapters.
Description
This course is carefully designed to explain various areas of CBSE Class 10.
It has 520 lectures spanning more than 75 hours of on-demand videos that are divided into 15 sections, and each chapter is a section and further divided into simple sessions. The course is divided into a simplified day-by-day learning schedule.
Each topic is divided into simple sessions and explained extensively by solving multiple questions. Each session contains a detailed explanation of the concept.
This course will give you a firm understanding of the fundamentals and is designed in a way that a person with little or no previous knowledge can also understand it very well.
It covers 100% video solutions of various problems and situations.
Our design meets the real classroom experience by following classroom teaching practices. We have designed this course by keeping in mind all the needs of students and their desire to become masters in CBSE Class 10. This course is designed to benefit all levels of learners and will be the best gift for board-appearing students. Students love these easy methods and explanations. They enjoy learning math and never feel that math is troublesome.
Topics covered in the course:
Real Numbers
Polynomials
Pair of Linear Equations in Two Variables
Quadratic Equations
Arithmetic Progressions
Triangles
Coordinate Geometry
Introduction to Trigonometry
Some Applications of Trigonometry
Circles
Constructions
Areas Related to Circles
Surface Areas and Volumes
Statistics
Probability
With this course you'll also get:
Perfect your mathematical skills on CBSE Class 10 for better scores.
A Udemy Certificate of Completion is available for download.
Feel free to contact me with any questions or clarifications you might have.
I can't wait for you to get started on mastering the CBSE Class 10 Math.
I look forward to seeing you on the course! :)
Benefits of Taking this Course:
On completion of this course, one will have detailed knowledge of CBSE Class 10 and be able to easily solve all the problems, which can lead to scoring well in exams with the help of explanatory videos ensure complete concept understanding.
Who this course is for:
- This course has been designed for students of Grade 10th CBSE, SSC, SAT, ACT, GRE, and other board-appearing students.
- Students studying for the public or other competitive examinations as well as job aspirants.
- Home-school parents are looking for extra support with the fundamentals.
- Anyone interested in revising or learning the basics of mathematics should.
- Students in junior high and high school/secondary schools.
- Anyone who wants to proficient mathematics and the solving different real life situations as well.
- Anyone who wants to study math for fun after taking a break from school.
- It will also benefit schools who wish to run classes in the absence of a teacher and make learning fun for their students.
- It will also benefit teachers and schools who wish to improve their teaching skills and make learning fun for their students.
- For 11th and 9th grade students, this will help as a bridge course.
- These are the people whose jobs require them to solve basic daily math-related problems.