Book

TUHI NIRANKAR

10TH CLASS

BOOK (Mathematics)

PAIR OF LINEAR EQUATIONS IN TWO VARIABLES

Q. 1 Find the value of k for which the following pair of linear equations has an infinite number of solutions : kx + 3y = k - 3, 12x + ky = k

Q. 2 Determine the value of k for which the following pair of linear equations has infinite number of solutions : (k - 3)x + 3y = k, kx + ky = 12

Q. 3 Solve graphically the following pair of linear equations : x + y = 3, 3x – 2y = 4

Q. 4 Determine the value of c for which the following pair of linear equations has no solutions : cx + 3y = 3, 12x + cy = 6

Q. 5 Find the value of k for which the pair of linear equations 2x + 5y = 3, (k + 1)x + 2(k + 2)y = 2k

Q. 6 For what value of k, the following pair of linear equations has (i) a unique solution (ii) no solution kx + 2y = 5, 3x – 4y = 10,

Q. 7 Solve the system of equations graphically : 2(x - 1) = y, x + 3y = 15 Also, find the co-ordinates of the points where the lines meet the axis of y.

Q. 8 For what value of k, will the following pair of linear equations have no solutions : 3x + y = 1, (2k - 1)x + (k - 1)y = 2k + 1

Q. 9 Determine graphically the co-ordinates of the vertices of a triangle, the equations of whose sides are y = x, y = 2x, x + y = 6.

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TUHI NIRANKAR

Q. 10 Find the value of p and q for which the following pair of linear equations has infinite number of solutions : 2x + 3y = 7, (p + q)x +(2p – q)y = 21

Q. 11 Solve graphically for x and y : 2x + y = 8, x - y = 1. Shade the region between the two lines and the y-axis.

Q. 12 Determine the value of k so that the following pair of linear equations has no solution : (3k + 1)x + 3y - 2 = 0, (k 2 + 1)x + (k - 2)y - 5 = 0

Q. 13 For what values of a and b, the following pair of linear equations has an infinite number of solutions ; 2x + 3y = 7, (a - b)x + (a + b)y = 3a + b – 2.

Q. 14 Draw the graph of 2x + y = 6 and 2x - y + 2 = 0. Shade the region bounded by these lines and the x-axis . Find the area of the shaded region.

Q. 15 Find the value of k for which the following pair of linear equations has infinite number of solutions : x + (k + 1)y = 5, (k + 1)x +9y = 8k -1.

Q. 16 Solve the following system of linear equations graphically : 2x – 3y = 1, 3x – 4y = 1. Does the point (3, 2) lie on any of the lines ? Write its equation.

Q. 17 Solve for x and y : (i) 4 5y = 7, 3 4y = 5 x x (ii) a b = 0, ab 2 a 2 b = a 2 + b 2 , x ≠ 0, y ≠ 0 x y x y

Q. 18 Solve the following systems of linear equations : (i) 2(ax – by) + (a + 4b) = 0, 2(bx + ay) + (b – 4a) = 0 (ii) 6(ax + by) = 3a + 2b, 6(bx – ay) = 3b -2a

Q. 19 Solve the following system of linear equations graphically : 4x – 5y -20 = 0, 3x + 5y -15 = 0 Determine the vertices of the triabngle formed by the lines representing the above equations and the y-axis.

Q. 20 Solve for x and y : (i) 4 3y = 14, 3 – 4y = 23 x x (ii) bx + ay = a 2 + b 2 , x + y = 2ab a b (iii) 2 + 3 = 13, 5 – 4 = -2, x = 0, y = 0 x y x y (iv) ax + by -a + b = 0, bx - ay - a - b = 0

Q. 21 Solve for x and y : x + y = 2, ax – by = a 2 – b 2 a b

Q. 22 Solve for x and y : 2a + 3b + 1 = 0, 3a – b – 4 = 0 x y x y



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TUHI NIRANKAR

Q. 23 Solve for x and y : 3a – 2b + 5 = 0, a + 3b – 2 = 0 x y x y

Q. 24 Solve the following system of equations graphically : x + 2y = 5, 2x – 3y = -4 Also find the points where the lines meet the x-axis.

Q. 25 Solve the following system of equations graphically : 2x – y = 4, 3y – x = 3 Find the points where the lines meet the y-axis.

Q. 26 Solve the following system of equations graphically : 3x – y = 3, x – 2y = -4 Shade the area of the region bounded by the lines and the x-axis.Q. 27 Solve for x and y : 47x + 31y = 63 31x + 47y = 15

Q. 28 Solve for x and y : ax – by = a + b b a ax – by = 2ab

Q. 29 Draw the graphs of the following equations : 2x – y + 6 = 0, 4x + 5y – 16 = 0. Also determine the co-ordinates of the vertices of the triangle formed by these lines and the x-axis.

Q. 30 Solve the following equations for x and y : mx – ny = m 2 + n 2 x + y = 2m

Q. 31 Solve the following system of equations graphically : 3x – 2y – 1 = 0 2x – 3y + 6 = 0 Shade the region bounded by the lines and x-axis.

Q. 32 Solve the following system of equations graphically : 3x + 2y - 4 = 0 2x – 3y – 7 = 0 Shade the region bounded by the lines and x-axis.

Q. 33 Solve the following system of equations graphically : 4x – y = 4 3x + 2y = 14 Shade the region bounded by the lines and y-axis.

Q. 34 Solve the following system of equations for x and y : (i) x + 6 = 6, 3x – 8 = 5 y y (ii) x + 1 y – 1 = 8, x – 1 y + 1 = 9



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TUHI NIRANKAR

2 3 3 2 (iii) 8x – 9y = 6xy, 10x + 6y = 19xy (iv) 4x y = 8, x + 3y = - 5 3 3 2 4 2 (v) 31x + 23y = 39, 23x + 31y = 15 (vi) 2 + 3 = 13, 5 – 4 = -2, x ≠ 0, y ≠ 0 x y x y

Q. 35 Solve the following system of equations graphically : (i) 2x + 3y = 2, x – 2y = 8 (ii) 2x – 3y = 1, 4x – 3y + 1 = 0 (iii) x + 2y + 2 = 0, 3x + 2y – 2 = 0

Aim of life is to know God and to serve mankind.(Nirankari Baba Hardev Singh Ji Maharaj)

QUADRATIC EQUATIONSQ. 1 Find the value of k so that the quadratic equation x 2 -2(1 + 3k)x + 7(3+ 2k) = 0 has equal roots.

Q. 2 If - 4 is a root of the quadratic equation x 2 + px – 4 = 0 and the quadratic equation x 2 + px + k = 0 has equal roots, find the value of k.

Q. 3 Solve for x : 5x 2 – 2x – 2 = 0

Q. 4 One root of the equation 2x 2 – 8x -m = 0 is 5. Find the value of m and the other root. 2

Q. 5 Solve for z : 7z 2 + 8z + 2 = 0.

Q. 6 Solve for x : x + 1 x – 2 = 3, x ≠ 1, - 2. x – 1 x + 2

Q. 7 Find the value of α such that the quadratic equation ( α – 12)x 2 + 2(α – 12)x + 2 = 0. has equal roots.

Q. 8 Solve for z : z 2 + 1z – 3 = 0. 2

Q. 9 Solve for x : abx 2 + (b 2 – ac)x – bc = 0.

Q. 10 If the equation (1 + m 2 )x 2 + 2mcx + (c 2 – a 2 ) = 0 has equal roots, prove that c 2 = a 2 (1 + m 2 ).

Q. 11 Find the value of c such that the equation 4x 2 -2(c + 1)x + (c + 4) = 0 has real and equal roots.

Q. 12 Solve for x : 2x + 1 + 3x + 9 = 0. x – 3 2x + 3 (x – 3)(2x + 3)



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TUHI NIRANKAR

Q. 13 Solve for y : y 2 – 6y + 2 = 0.

Q. 14 Solve for x : x 2 + 1 – 1 = 0 2

Q. 15 If the roots of the equation (a – b)x 2 + (b – c)x + (c – a) = 0 are equal, prove that 2a = b + c.

Q. 16 Solve for x : 3x 2 – 2 2x – 2 3 = 0.

Q. 17 Divide 12 into two parts such that their product is 32.

Q. 18 Solve for z : z 2 – 6z + 6 = 0.

Q. 19 Determine the value(s) of p for which the quadratic equation 2x 2 + 3x + p = 0 has real roots.

Q. 20 Determine the value(s) of p for which the quadratic equation 4x 2 – 3px + 9 = 0 has real roots.

Q. 21 Find the values of k so that (x – 3) is a factor of the polynomial k 2 x 2 – kx - 2.

Q. 22 Solve for y : 7 y 2 – 6y – 13 7 = 0. Q. 23 Solve the following equations for x :

(i) 4x 2 – 2(a 2 + b 2 )x + a 2 b 2 = 0. (ii) 9x 2 – 9(a + b)x + (2a 2 + 5ab + 2b 2 ) = 0 (iii) 4x 2 – 4a 2 x + (a 4 – b 4 ) = 0.

Q. 24 Using quadratic formula, solve the following equations for x : (i) p 2 x 2 + (p 2 – q 2 )x – q2 = 0. (ii) x 2 – 2ax + (a 2 – b 2 ) = 0. (iii) x 2 – 4ax + 4a 2 – b 2 = 0

Q. 25 Solve the following equations for x : (i) 2 (2x – 1) - 3(x + 3) = 5, x = -3, 1

( x + 3) (2x – 1) 2 (ii) 2 (2x + 3) -25 (x -3) = 5, x = 3, -3

( x – 3) (2x + 3) 2(iii) (4x - 3) – 10(2x + 1) = 3, x = -1, 3

(2x +1) (4x – 3) 2 4

Q. 26 Solve for x : 1 = 1 + 1 + 1, a = 0, b = 0, x = 0. a+b+x a b x

Q. 27 Solve for x : a 2 b 2 x 2 + b 2 x -a 2 x – 1 = 0.

Q. 28 Solve for x : x – 1 + x – 3 = 1, x = 2, 4. x – 2 x – 4 3

Q. 29 Using quadratic formula, solve for x : (i) 9x 2 – 3(a + b)x + ab = 0 (ii) 9x 2 – 3(a 2 + b 2 )x + a 2 b 2 = 0.

Q. 30 Solve for x : 12abx 2 – (9a 2 – 8b 2 )x – 6ab = 0.



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TUHI NIRANKAR

Q. 31 Solve for x : 4 3x 2 + 5x – 2 3 = 0.

ARITHMETIC PROGRESSIONSQ. 1 The 7 th term of an A.P. is 32 and its 13 th term is 62. Find the A.P. Q. 2 The 7 th term of an A.P. is -4 and its 13 th term is -16. Find the A.P.

Q. 3 The 8 th term of an A.P. is -23 and tis 12 th term is -39. Find the A.P.

Q. 4 Find the sum of first 25 terms of an A.P. whose n th term is given by t n = 2 – 3n.

Q. 5 Which term of the Arithmetic Progression 3, 10, 17, .......................... will be 84 more than its 13 th term.

Q. 6 The n th term (t n) of an Arithmetic progression is given by tn = 5n – 3. Find the sum of its first 20 terms.

Q. 7 If the sum of first n terms of an A.P. is given by S n = 3n 2 + 2n, find the nth term of the A.P.

Q. 8 If m times the m th term of an A.P. is equal to n times its n th term, find its (m + n)th term.

Q. 9 How many terms of the A.P. 3, 5, 7, .......................... must be taken so that the sum is 120?

Q. 10 Find the sum of first 15 terms of an A.P. whose nth term is 9 – 5n.

Q. 11 which term of the A.P. 5, 9, 13, ............... is 81? Also find the sum 5 + 9 + 13 + .............. + 81.

Q. 12 Find the 10 th term from end of the A.P. 4, 9, 14 ......, 254.

Q. 13 Find the number of terms of the A.P. 54, 51, 48, ........... so that their sum is 513.

Q. 14 If the n th term of A.P. is (2n + 1), find the sum of first n terms.

Q. 15 Find the sixth term from end of the A.P. 17, 14, 11, ................, -40.

Q. 16 Find the sum of all two digit odd positive numbers.

Q. 17 The 8 th term of an A.P. is zero. Prove that its 38 th term is triple of its 18 th term.

Q. 18 Find the sum of all multiples of 9 lying between 300 and 700.

Q. 19 Find the sum of first 21 terms of the A.P. is whose 2 nd term is 8 and 4 th term is 14.

Q. 20 The sum of first n terms of an A.P. is given by n 2 + 5n. Find the 25 th term of the progression.

Q. 21 Find the sum of first 51 terms of the A.P. whose 2 nd term is 2 and 4 th term is 8.

Q. 22 The sum of first n terms of the A.P. is given by S n = 3n 2 – 4n. Determine the A.P. and its 12 th term.

Q. 23 The sum of three numbers in A.P. is 27 and their product is 405. Find the numbers.

Q. 24 The sum of three numbers in A.P. is 3 and product is -35. Find the numbers.

Q. 25 The 6 th term of an A.P. is -10 and the 10 th term is -26. Determine the 15 th term of the A.P.

Q. 26 Find the sum of all the two digit natural numbers which are divisible by 4.

Q. 27 Find the sum of all natural numbers between 100 and 200 which are divisible by 4.

Q. 28 Find the sum of all the natural numbers less than 100 which are divisible by 6.













































TUHI NIRANKAR

Q. 29 The 10 th term of an A.P. 52 and the 17 th term is 20 more than the 13 th term. Find the A.P.

Q. 30 The 8 th term of an A.P. is 31 and the 15 th term is 16 more than the 11 th term, find the A.P.

Q. 31 For what value of n is the n th term of the following two A.P.'s the same :

(i) 1, 7, 13, 19................ (ii) 69, 68, 67, ......?

Q. 32 Find the sum of all 3 digit natural numbers which are divisible by 13.

Q. 33 Find the sum of the following

25 + 28 +31 + ........ + 100

Q. 34 Using A.P., find the sum of all 3 digit natural numbers which are divisible by 7.

Q. 35 In an A.P., the sum of its first n terms is n 2 + 2n. Find its 18 th term.

Q. 36 The first term, common difference and last term of an A.P. are 12, 6, and 252 respectively. Find the sum of all

the terms of this AP.

Q. 37 In an A.P., the sum of its first n term is 6n – n 2 . Find its 25 th term.

Q. 38 How many terms are there in an A.P. whose first term is -14, common difference is 4 and the sum of terms is 40?

Q. 39 The third term of an A.P. is 3 and 11 th term is -21. Find its first term and common difference.

Q. 40 In an A.P., the sum of its first n terms is 3n 2 + n. Find its 22 nd term.

Q. 41 Which term of the A.P. 72, 68, 64, 60, ......... is 0?

Q. 42 How many terms of the A.P. 17, 15, 13, 11, ............... must be added to get the sum 72?

Q. 43 Find the sum of first 25 terms of an A.P. whose n th term is 1 – 4n.

Q. 44 Which term of the A.P. 3, 15, 27, 39, ....................... will be 132 more than its 60 th term?

Q. 45 Find the sum of first 32 terms of an A.P. whose n th term is 5 -2n.

Everyone must be true in his thoughts, words and deeds.(Nirankari Baba Hardev Singh Ji Maharaj)





































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TUHI NIRANKAR

TRIGONOMETRY

Q. 1 If cos ө + sin ө = 2 cos ө, show that cos ө – sin ө = 2 sin ө. Q. 2 show that: 1 + 1 1 + 1 = 1 tan 2 ө cot 2 ө sin 2 ө – sin 4 ө Q. 3 In the figure given below, ABCD is a rectabgle with AD = 12 cm and DC = 20 cm. Line segment DE is drawn making an angle of 30 º with AD, intersecting AB in E. find the lengths of DE and AE. A E B

D 20 cm C

Q. 4 Prove that : sec 4 ө – sec 2 ө = tan 4 ө + tan 2 ө Q. 5 Show that : (1 + cotө – cosecө)(1 + tanө + secө) = 2 Q. 6 If sinө + cosө = p and secө + cosecө = q, show that q(p 2 – 1) = 2p Q. 7 If cos α = m and cos α = n, show that (m 2 + n 2 )cos 2 β = n 2 cos β sin β Q. 8 Show that : 1 _ 1 = 1 _ 1 secx – tanx cosx cosx secx + tanx Q. 9 Prove that : 1 + sinө _ cosө = 2secө cosө 1+ sinөQ. 10 Prove that : (cosө + secө) 2 + (sinө + cosecө) 2 = 7 tan 2 ө + cot 2 ө Q. 11 Prove that : 1 – sinө cosө = 2secө cosө 1 – sinөQ. 12 Without using trigonometric tables, evaluate : (i) cos67º - tan40º - sin90º sin23º cot50º(ii) cos 2 90º + cos 2 70º sin 2 59º + sin 2 31º (iii) sec70ºsin20º – cos20ºcosec70º

(iv) sin47º 2 + cos43º 2 - 2cos 2 45º cos43º sin47º(v) cos75º + sin12º - cos18º sin15º cos78º sin72º(vi) sin80º + sin59ºsec31º



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TUHI NIRANKAR

cos10º Q. 13 Evaluate : (sec 2 ө – 1)(1 – cosec 2 ө) Q. 14 Prove : sec 2 ө + cosec 2 ө = sec 2 ө cosec 2 ө Q. 15 If sin ө + cos ө = 2sin(90º - ө), find cot ө. Q. 16 Show that : 2sec 2 ө – sec 4 ө – 2cosec 2 ө + cosec 4 ө = cot 4 ө – tan 4 ө . Q. 17 Show that : cosA + sinA = sinA + cosA 1 – tanA 1 – cotAQ. 18 Show that : sin ө = 2 + sin ө cot ө + cosec ө cot ө - cosec өQ. 19 Show that : (cosec ө - sin ө)(sec ө - cos ө) = 1 tan ө + cot өQ. 20 Show that : sin ө - 2sin 3 ө = tan ө 2cos 3 ө - cos ө Q. 21 If tan ө - sin ө = m and tan ө - sin ө = n, show that m 2 - n 2 = 4 mn. Q. 22 Prove : (sin ө + sec ө) 2 + (cos ө + cosec ө) 2 = (1 + sec ө cosec ө) 2 Q. 23 Evaluate : cosec(65º + ө) – sec(25º - ө) – tan(55º - ө) + cot(35º + ө) Q. 24 Without using trigonometric tables, find the value of :

(i) tan7ºtan23ºtan60ºtan67ºtan83º (ii) tan50º + sec50º + cos40ºcosec50º

cot40º + cosec40º(iii) sin50º + cosec40º – 4cos50ºcosec40º

cos40º sec50ºQ. 25 If 3tan ө = 3sin ө, find the value of sin 2 ө - cos 2 ө. Q. 26 If cosec ө = 13, find the value of 2sin ө - 3cos ө 12 4sin ө - 9cos өQ. 27 If 5tan ө = 4, find the value of 2sin ө - 3cos ө 4sin ө - 9cos ө Q. 28 If sec ө = x + 1 , prove that sec ө + tan ө = 2x or 1 4x 2xQ. 29 Prove that : tan ө + sec ө – 1 = 1 + sin ө tan ө - sec ө + 1 cos өQ. 30 without using trigonometric tables, evaluate :

(i) sec 2 10º – cot 2 80º + sin15ºcos75º + cos15ºsin75º cos өsin(90º- ө) + sin өcos(90º - ө)

(ii) cos70º + cos55ºcosec35º sin20º tan5ºtan25ºtan45ºtan65ºtan85º (iii) cos(40º + ө) - sin(50º - ө) + cos 2 40º + cos 2 50º sin 2 40º + sin 2 50º Q. 31 If x = a sec ө + b tan ө and y = a tan ө + b sec ө, prove that x 2 – y 2 = a 2 - b 2 Q. 32 Prove that : cosA _ sin 2 A = sinA + cosA 1 – tanA cosA – sinA

Q. 33 Prove that : 1 + cos ө - sin 2 ө = cot ө sin ө(1 + cos ө)Q. 34 Without using trigonometric tables, evaluate the following : (i) cos58º + sin22º _ cos38ºcosec52º sin32º cos68º tan18ºtan35ºtan60ºtan72ºtan55º(ii) tan20º 2 cot20º 2 + 2tan15º tan37º tan53vtan65º tan75º cosec70º sec70º (iii) cos67º _ tan40º - cos0º + tan15º tan25º tan60º tan65º tan75º sin23º cot50º (iv) cos 2 20º + cos 2 70º - tan45º + tan13ºtan23ºtan30ºtan67ºtan77º



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TUHI NIRANKAR

sin 2 25º + sin 2 65º

Q. 35 If sec ө + tan ө = p, prove that sin ө = p 2 – 1 p 2 + 1 Q. 36 Prove that : tan ө cot ө = 1 + tan ө + cot ө 1 - cot ө 1 - tan өQ. 37 Without using trigonometric tables, evaluate the following : (i) sin68º – 2cot15º - 3tan45ºtan20ºtan40ºtan50ºtan70º cos22º 5tan75º 5(ii) 3tan25ºtan40ºtan50ºtan65º – 1 2 4(cos 2 29º + cos 2 61º) Q. 38 Prove that tan 2 A – tan 2 B = sin 2 A – sin 2 B cos 2 Acos 2 B

Q. 39 Find the value of : (i) -tan ө cot(90 º – ө) + secө cosec(90 º -ө) + sin 2 35 º + sin 2 55 º tan10 º tan20 º tan30 º tan70 º tan80 º

(ii) sec 2 54º – cot 2 36º + 2sin 2 38º sec 2 52º – sin 2 45º cosec 2 57º – tan 2 33º

Q. 40 Prove that : sin ө + cosө + sinө – cosө = 2sec 2 ө sinө – cosө sinө + cosө tan 2 ө – 1 Q. 41 Prove that : 1 - 1 = 1 - 1 cosecө – cotө sinө sinө cotө + cosecөQ. 42 Without using trigonometric tables, evaluate : (i) sec 2 (90 º – ө) – cot 2 ө - 2cos 2 60 º tan 2 28 º tan 2 62 º 2(sin 2 25 º + sin 2 65 º ) 3(sec 2 43 º – cot 2 47 º )

(ii) cosec 2 (90 º - ө ) - tan 2 ө - 2tan 2 30ºsec 2 52ºsin 2 38º 2(cos 2 48º + cos 2 42º) cosec 2 70º – tan 2 20º Q. 43 Without using trigonometric tables, evaluate the following : cos 2 20º+ cos 2 70º + 2cosec 2 58º – 2cot58ºtan32º - 4tan13ºtan37ºtan45ºtan53ºtan77º sec 2 50º – cot 2 40º

Q. 44. prove that sec ө - 1 + sec ө + 1 = 2cosec ө sec ө +1 sec ө - 1Q. 45 Without using trigonometric tables, evaluate the following :

(i) tan7 º.tan23º.tan60º.tan67º.tan83º + cot54º + sin20º.sec70º – 2 tan36º

(ii) 2(sec 2 35º – cot 2 55º) - cos28º. cosec62º tan18º.tan36º.tan30º.tan54º.tan72º

(iii) 3cos55º - 4(cos70º.cosec20º) 7sin35º 7(tan5º.tan25º.tan45º.tan65º.tan85º)

Q. 46 Show that : sec 4 ө(1 - sin 4 ө) - 2tan 2 ө = 1.



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TUHI NIRANKAR

● Divine Knowledge (Brahm Gyan) is the only way to peace, unity and universal brotherhood.

● Everyone must be true in his thoughts, words and deeds. —Nirankari Baba Hardev Singh Ji Maharaj

COORDINATE GEOMETRYQ. 1 Show that the points A(5, 6), B(1, 5), C(2, 1) and D(6, 2) are the vertices of a sqaure.Q. 2 Determine the ratio in which the point P(m, 6) divides the join of A(-4, 3) and B(2, 8). Also find the value of m.Q. 3 A(3, 2) and B(-2, 1) are two vertices of a triangle ABC, whose centroid G has coordinates 5, - 1 . Find

the coordinates of the third vertex C of the triangle. 3 3 Q. 4 Show that the points A(2, -2), B(14, 10), C(11, 13) and D(-1, 1) are the vertices of a rectangle.Q. 5 Prove that the coordinates of the centroid of a ABC, with vertices A(x1, y1), B(x2, y2) and C(x3, y3) are

given by x1 + x2 + x3, y1 + y2 + y3

3 3Q. 6 Determine the ratio in which the point (-6, a) divides the join of A(-3, -1) and B(-8, 9). Also find the value of a.Q. 7 Find the point on the x-axis which is equidistant from the points (-2, 5) and (2, -3).Q. 8 The coordinates of the mid-point of the joining the points (3p, 4) and (-2, 2q) are (5, p). Find the values of p

and q.Q. 9 Two vertices of a triangle are (1, 2) and (3, 5). If the centroid of the triangle is at the origin, find the coordinates

of the third vertex.Q. 10 If 'a' is the length of one of the sides of an equilateral triangle ABC, base BC lies on x-axis and vertex B is at

the origin, find the coordinates of the vertices of the triangle ABC. Q. 11 The coordinates of the mid-point of the line joining the points (2p + 2, 3) and (4, 2q + 1) are (2p, 2q). Find the

values of p and q.Q. 12 Find the ratio in which the line segment joining the points (6, 4) and (1, -7) is divided by x-axis.Q. 13 Find the value of m for which the points with coordinates (3, 5), (m, 6) and 1, 15 are collinear. 2 2Q. 14 Find the value of k for which the points with coordinates (2, 5), k, 11 and (4, 6) are collinear. 2 Q. 15 Find the value of x such that PQ = QR where the coordinates of P, Q and R are (6, -1), (1, 3) and (x, 8)

respectively.Q. 16 Find a point on x-axis which is equidistant from the points (7, 6) and (-3, 4).Q. 17 The line segment joining the points (3, -4) and (1, 2) is trisected a the points P and Q. If the coordinates of P

and Q are (p, -2) and 5, q respectively, find the values of p and q. 3Q. 18 Prove that point (0, 0), (5, 5) and -5, 5) are the vertices of a right isosceles triangle.Q. 19 If the point P (x, y) is equidistant from the points A(5, 1) and B(-1, 5). Prove that 3x = 2y.Q. 20 The line joining the points (2, 1) and (5, -8) is trisected at the points P and Q. If the point P lies on the line

2x – y + k = 0, find the value of k.Q. 21 Find the coordinates of the point equidistant from the points A(1, 2), B(3, -4) and C(5, -6).Q. 22 Prove that the points A(-4, -1), B(-2, -4), C(4, 0) and D(2, 3) are the vertices of a rectangle.Q. 23 Find the coordinates of the points which divide the line-segment joining the points (-4, 0) and (0, 6) in three























































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TUHI NIRANKAR

equal parts.Q. 24 Find the coordinates of the points which divide the line-segment joining the points (-2, 0) and (0, 8) in four

equal parts.Q. 25 Find the coordinates of the point equidistant from the points A(-2, -3), B(-1, 0) and C(7, -6).Q. 26 Two vertices of a triangle are given by A(2, 3) and B(-2, 1) and its centroid is G 1, 2 , find the coordinates

of the third vertex C of the ABC. 3Q. 27 Find the coordinates of the point equidistant from the three given points A (5, 1) B(-3, -7) and C(7, -1).Q. 28 Show that the points A(1, 2), B(5, 4), C(3, 8) and D(-1, 6) are the vertices of a square.Q. 29 Find the value of p for which the points (-1, 3), (2, p) and (5, -1) are collinear.Q. 30 Find the value of p for which the points (-5, 1) (1, p) and (4, -2) are collinear.Q. 31 Show that the points A(3, -1), B(5, -1) and C(3, -3) are the vertices of a right angle isosceles triangle.Q. 32 If the points (10, 5) (8, 4) and (6, 6) are the mid-points of the sides of a triangle, find its vertices.Q. 33 Prove that the points (3, 0) (6, 4) and (-1, 3) are the vertices of a right angled triangle. Also, prove that these

are the vertices of an isosceles triangle.Q. 34 In what ratio is the line segment joining the points (-2, -3) are the and (3, 7) divided by the y-axis? Also, find

the coordinates of the point of division.Q. 35 If A(5, -1), B(-3, -2) and C(-1, 8) are the vertices of a triangle ABC, find the length of median through A and

coordinates of the centroid.Q. 36 Three consecutive vertices of a parallelogram are (-2, -1), (1, 0) and (4, 3). Find the coordinates of the fourth

vertex.Q. 37 If the point C(-1, 2) divides the lien segment AB in the ratio 3:4, where the coordinates of A are (2, 5), find

the coordinates of B.Q. 38 Show that the points (1, 1), (-2, 7) and (3, -3) are collinear.Q. 39 Find the ratio in which C(p, 1) divides the join of A(-4, 4) and B(6, -1) and hence find the value of p.Q. 40 Show that the points (7, 10), (-2, 5) and 3, -4) are the vertices of an isosceles right triangle.Q. 41 In what ratio does the line x – y -2 = 0 divides the line segment joining (3, -1) are (8, 9)?Q. 42 Find the ratio in which the pint (-3, k) divides the line segment joining the point (-5, -4) and (-2, 3). Hence

find the value of k.

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If God answers your prayer, he is increasing your faith.

If he delays, He is increasing your patience.














































TUHI NIRANKAR

If he does not answer, he has definitely something better for you .

STATISTICSQ. 1 Find the value of p if the mean of the following distribution is 7.5 :

X 3 5 7 9 11 13ƒ 6 8 15 9 8 4

Q. 2 The mean of the following frequency distribution is 62.8 and the sum of all the frequencies is 50. Compute the missing frequencies ƒ1 and ƒ2

Classes 0-20 20-40 40-60 60-80 80-100

100-120 Total

Frequencies 5 ƒ1 10 ƒ2 7 8 50

Q. 3 The mean of the following frequency distribution is 57.6 and the sum of the observations is 50. Find the missing frequencies ƒ1 and ƒ2 :

Classes 0-20 20-40 40-60 60-80 80-100 100-120Frequencies 7 ƒ1 12 ƒ2 8 5

Q. 4 The following table gives the distribution of expenditure of different families on education. Find the mean expenditure on education of a family.

Expenditure (in Rs.) Number of families1000-15001500-20002000-25002500-30003000-35003500-40004000-45004500-5000

244033283012167

Q. 5 The following table gives the marks scored by 80 students in a class-test :
























































































TUHI NIRANKAR

Marks 0-50 50-100 100-150 150-200 200-250 250-300Number of students 8 12 20 25 10 5

Q. 6 Find the mean of the following distribution : Class Number of students4-88-1212-1616-2020-2424-2828-3232-36

21215251812133

Q. 7 If the mean of the following data is 18.75 find the value of p :

X1 10 15 P 25 30ƒ1 5 10 7 8 2

Q. 8 Find the mean of the following data : Class 0-100 100-200 200-300 300-400 400-500

Frequency 6 9 15 12 8

Q. 9 Find the mean of the following data : Class 0-10 10-20 20-30 30-40 40-50

Frequency 18 17 22 24 19

Q. 10 The Arithmetic Mean of the following frequency distribution is 53. Find the value of p. Class 0-20 20-40 40-60 60-80 80-100

Frequency 12 15 32 9 13

Q. 11 The Arithmetic mean of the following frequency distribution is 52.5. Find the value of p. Class 0-20 20-40 40-60 60-80 80-100

Frequency 15 22 37 P 21

Q. 12 The Arithmetic mean of the following frequency distribution is 25. Determine the value of p. Class 0-10 10-20 20-30 30-40 40-50


Q. 13 The Arithmetic mean of the following frequency distribution is 47. Determine the value of p. Class 0-20 20-40 40-60 60-80 80-100


Q. 14 If the mean of the following distribution is 27, find the value of p.












































































































































TUHI NIRANKAR

Class 0-10 10-20 20-30 30-40 40-50Frequency 8 P 12 13 10

Q. 15 If the mean of the following distribution is 54, find the value of p. Class 0-20 20-40 40-60 60-80 80-100

Frequency 7 P 10 9 13

Q. 16 The mean of the following frequency distribution is 62.8. Find the missing frequency x. Class 0-20 20-40 40-60 60-80 80-100 100-120

Frequency 5 8 X 12 7 8Q. 17 Find the mean of the following distribution :

Class 0-10 10-20 20-30 30-40 40-50Frequency 8 12 10 11 9

Q. 18 Find the mean of the following data : Class 0-10 10-20 20-30 30-40 40-50 50-60 60-70 70-80

Frequency 12 7 16 12 11 7 19 16

Q. 19 The mean of the following frequency distribution is 50 and the sum of all the frequencies is 120. find the values of p and q :

Class Intervals Frequency0-2020-4040-6060-80080-100

17p32q19

Q. 20 The mean of the following frequency distribution is 62.8 and the sum of all the frequencies is 50. Find the values of ƒ 1 and ƒ2 :

Class 0-20 20-40 40-60 60-80 80-100 100-120Frequency 5 ƒ1 10 ƒ2 7 8

Q. 21 The arithmetic mean of the given frequency distribution is Rs. 472 and the total number of employees is 100. Find the values of p and q :

Daily income (in Rs.) Number of employees200-300300-400400-500500-600600-700700-800800-900

5p24q964

Q. 22 The following table shows the number of patients in different wards of a hospital on a particular day : Age (in years) 10 5 30 40 60 4p + 30















































































































































TUHI NIRANKAR

No. of patients 5 16 21 P + 8 15 10If the average of a patient is 35.5 years, find the value of p.

Q. 23 Find the median for the following data : Wages per day (in rupees) 38 45 48 55 62 65

No. of workers 14 8 7 10 6 2

Q. 24 Find the median for the following data : Weight (in nearest kg.) 15 18 20 23 25 27 28 29

No. of children 6 7 10 14 13 8 3 2

Q. 25 Marks obtained by 70 students are given below :

Marks 20 70 50 60 75 90 40No. of students 8 12 18 6 9 5 12

Calculate the median marks.Q. 26 Calculate the mean, the median and the mode of the following distribution :

No. of Goals 0 1 2 3 4 5No. of matches 2 4 7 6 8 3

Q. 27 At a shooting competition, the scores of a competitor were as given below :

(i) What was his modal score? (ii) What was his median score? (iii) What was his total score? (iv) What was his mean score?

Q. 28 Find the median for the following data :

Marks Obtained 2.5 - 3.5 3.5 - 4.5 4.5 - 5.5 5.5 - 6.5 6.5 - 75 7.5 - 8.5No. of students 7 31 33 17 11 1

Q. 29 The following table gives the distribution of the life time of 400 neon lamps :

Life time (in hours) Number of lamps1500-20002000-25002500-30003000-35003500-40004000-45004500-5000

14566086746248

Find the median life time of lamp.

Q. 30 Find the median income for the following data :































































































































TUHI NIRANKAR

Monthly income(in rupees)

600-700 700-800 800-900 900-1000 1000-1100 1100-1200 1200-1300

No. of employees 40 68 86 120 90 40 26

Q. 31 Weights of 40 eggs were recorded as given below :

Weight (in grams) 85-89 90-94 95-99 100-104 105-109No. of eggs 10 12 12 4 2

Q. 32 The marks of 200 students in a test were recorded as follows :

Marks % 10-19 20-29 30-39 40-49 50-59 60-69 70-79 80-89No. of students 7 11 20 46 57 37 15 7

Find the median marks.

Q. 33 The marks obtained by 53 students, out of 100 in a certain examination are given below : Marks Number of students0-1010-2020-3030-4040-5050-6060-7070-8080-9090-100

5343347978

Find the median marks.

Q. 34 A survey regarding the heights (in cm) of 51 girls of class X of a school was conducted and the following data was obtained :

Height (in cm) Number of girlsLess than 140Less than 145Less than 150Less than 155Less than 160Less than 165

41129404651

Find the median height.

Q. 35 A life insurance agent found the following data for distribution of ages of 100 policy holders. Calculate the median age, if policies are only given to persons having age 18 years onwards but less than 60 years.












































































































TUHI NIRANKAR

Age (in years) Number of policy holdersBelow 20Below 25Below 30Below 35Below 40Below 45Below 50Below 55Below 60

26244578899298100

Q. 36 100 surnames were randomly picked up from a local telephone directory and the frequency distribution of the number of letters in the English alphabets in the surnames was obtained as follows :

Number of letters 1-4 4-7 7-10 10-13 13-16 16-19No. of surnames 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.

Q. 37 The median of the following data is 525. Find the value of x and y, if the total frequency is 100. Class Interval Frequency

0-100100-200200-300300-400400-500500-600600-700700-800800-900900-1000

25x121720y974

Q. 38 The mean of the following frequency distribution is 50 and the sum of all the frequencies is 120. Find the value of p and q :

Class Interval Frequency0 – 2020 – 4040 – 6060 – 8080 – 100

17p32q19

Q. 39 The mean of the following distribution is 62.8 and the sum of all the frequencies is 50. Find the values of ƒ 1

and ƒ 2 :Classes 0-20 20-40 40-60 60-80 80-100 100-120

Frequency 5 ƒ1 10 ƒ2 7 8

Q. 40 The arithmetic mean of the view frequency distribution is Rs. 472 and the total number of employees is 100. Find the value of p and q :












































































































TUHI NIRANKAR

Daily income (in Rs.) Number of employees200 – 300300 – 400400 – 500500 – 600600 – 700700 – 800800 - 900

5p24q964

Q. 41 The following table shows the number of patients in different wards of a hospital on a particular day :

Age (in years) 10 15 30 40 60 4p + 30No. of patients 5 16 21 p + 8 15 10

If the average age of a patient is 35.5 years, find the value of p.

Q. 42 The wickets taken by a bowler in 10 cricket matches are as follows :

2 6 4 5 0 2 1 3 2 3 Find the mode of the data.


















































TUHI NIRANKAR

PROBABILITYQ. 1 A bag contains 5 black, 7 red and 3 white balls. A ball is drawn from the bag at random. Find the probability that the ball drawn is (i) red (ii) black or white (iii) not black

Q. 2 A bag contains 7 black, 5 red and 2 white balls. A ball is drawn from the bag at random. Find the probability that the ball drawn is (i) red (ii) black or white (iii) not black.

Q. 3 A bag contains 6 red, 5 black and 4 white balls. A ball is drawn from the bag at random. Find the probability that the ball drawn is (i) white (ii) red (iii) not black (iv) red or white

Q. 4 A card is drawn at random from a pack of 52 playing cards. Find the probability that the card drawn is neither an ace nor a king.

Q. 5 A card is drawn at random from a pack of 52 playing cards. Find the probability that the card drawn is neither a red card nor a black king.

Q. 6 Out of 400 bulbs in a box 15 bulbs cards. One bulb is taken out at random from the box. Find the probability that the drawn bulb is not defective.

Q. 7 Find the probability of getting 53 Fridays in a leap year.

Q. 8 A card is drawn at random from well shuffled pack of 52 cards. Find the probability that the card drawn is neither a red card nor a queen.

Q. 9 Find the probability that a number selected from number 1 to 25 is not a prime number when each of given numbers is equally likely to be selected.




































TUHI NIRANKAR

Q. 10 There are 30 cards, of same size, in a bag on which numbers 1 to 30 are written. One card is taken out of the bag at random. Find the probability that the number on the selected card is not divisible by 3.

Q. 11 A bag contains 5 red, 8 white and 7 black balls. A ball is drawn from the bag at random. Find the probability that the drawn ball is (i) red or white (ii) not black (iii) neither white nor black.

Q. 12 A box contains 20 cards, numbered from 1 to 25. A card is drawn from the box at random. Find the probability that the number on the card is : (i) even (ii) prime (iii) multiple of 3.

Q. 13 A box contains 25 cards, numbered from 1 to 25. A card is drawn from the box at random. Find the probability that the ball is : (i) even (ii) prime (iii) multiple of 6.

Q. 14 A bag contains 6 red, 8 white balls, 5 green balls and 3 black balls. One ball is drawn at random from the bag. Find the probability that the ball is :

(i) white ii) red or black (iii) not green (iv) neither white nor black.

Q. 15 A com contains 29 balls bearing numbers 1, 2, 3, 4, ..... 20. A ball is drawn at random from the box. What is the probability that the number on the ball is (i) an odd number (ii) divisible by 2 or 3 (iii) prime number (iv) not divisible by 10 ?

Q. 16 A bag contains 5 white balls, 7 red balls, 4 black balls and 2 blue balls. One ball is drawn at random from the bag. What is the probability that the ball drawn is : (i) white or blue (ii) red or black (iii) not white (iv) neither white nor black ?

Q. 17 A card is drawn at random from a well shuffled deck of playing cards. Find the probability that eh card drawn is : (i) a king or jack (ii) a non-ace (iii) a red card (iv) neither a king nor queen.

Q. 18 A card drawn at random from a well shuffled deck of playing cards. Find the probability that the card is : (i) a card of spade or an ace (ii) a red king (iii) neither a king nor a queen (iv) either a king or a queen.

Q. 19 A bag contains 4 white balls, 6 red balls, 7 black balls and 3 blue balls. One ball is drawn at random from the bag. Find the probability that the ball drawn is : (i) white (ii) not black (iii) neither white nor black (iv) red or white

Q. 20 A box contains 19 balls bearing numbers 1, 2, 3, ..., 19. A ball is drawn at random from the box. Find the probability that the number on the ball is :

(i) a prime number (ii) divisible by 3 or 5 (iii) neither divisible by 5 nor by 10 (iv) an even number

Q. 21 Find the probability that a number selected at random from the numbers 1, 2, 3,..., 35 is a (i) prime number (ii) multiple of 7 (iii) multiple of 3 or 5

Q. 22 From a pack of 52 cards, a black jack, a red queen and two black kings fell down. A card was then drawn from the pack at random. Find the probability that the selected card is a (i) black card (ii) king (iii) red queen.

Q. 23 Tickets numbers 3, 5, 7, 9,..., 29 are placed in a box and mixed thoroughly. One ticket is drawn at random




















































TUHI NIRANKAR

from the box. Find the probability that the number on the ticket is (i) a prime number (ii) a number less than 16 (iii) a number divisible by 3

Q. 24 A box contains 5 red balls, 4 green balls and 7 white balls. A balls is drawn at random from the box. Find the probability that the ball drawn is (i) white (ii) neither red nor white

Q. 25 All the three face cards of spades are removed from a well-shuffled pack of 52 cards. A card is then drawn at random from the remaining pack. Find the probability of getting (i) a black face card (ii) a queen

Q. 26 A box contains 7 red balls, 8 green balls and 5 white balls. A ball is drawn at random from the box. Find the probability that the ball drawn is (i) white (ii) neither red nor white.

Q. 27 Cards marked with numbers 3, 4, 5,..., 50 are placed in a box and mixed thoroughly. One card is drawn at random from the box. Find the probability that the number on the card is (i) divisible by 7 (ii) a number which is a perfect square.

Q. 28 Cards marked with numbers 13, 14, 15,..., 60 are placed in a box and mixed thoroughly. One card is drawn at random from the box. Find the probability that the number on the card is (i) divisible by 5 (ii) a number which is a perfect square.

Q. 29 A bag contains 6 red balls and some blue balls. If the probability of drawing a blue ball is twice that of a red ball, find the number of blue balls in the bag.

Q. 30 A bag contains 5 red balls and some blue balls. If the probability of drawing a blue ball is four times that of a red balls, find the number of blue balls in the bags.

ANSWERS :1. (i) 7 (ii) 8 (iii) 2 2. (i) 1 (ii) 2 (iii) 8 3. (i) 4 (ii) 2 (iii) 2 (iv) 2 15 15 3 3 3 15 15 5 3 3

4. (i) 11 5. 6 6. 77 7. 2 8. 6 9. 16 10. 2 11. (i) 13 (ii) 13 (iii) 1 13 13 80 7 13 25 3 20 20 4 12. (i) 1 (ii) 2 (iii) 3 13. (i) 12 (ii) 9 (iii) 4 14. (i) 4 (ii) 9 (iii) 17 (iv) 1 2 5 10 25 25 25 11 22 22 2 15. (i) 1 (ii) 13 (iii) 2 (iv) 9 2 20 5 10 16. (i) 7 (ii) 11 (iii) 13 (iv) 1 18 18 18 2 17. (i) 2 (ii) 12 (iii) 1 (iv) 11 13 13 2 13 18. (i) 4 (ii) 1 (iii) 11 (iv) 2 13 26 13 13 19. (i) 1 (ii) 13 (iii) 9 (iv) 1 5 20 20 2 20. (i) 8 (ii) 8 (iii) 16 (iv) 9 19 19 19 19
























































TUHI NIRANKAR

21. (i) 11 (ii) 1 (iii) 16 22. (i) 23 (ii) 1 (iii) 1 23. (i) 9 (ii) 1 (iii) 5 35 7 35 48 24 48 14 2 14 24. (i) 7 (ii) 1 25. (i) 3 (ii) 3 (iii) 23 26. (i) 1 (ii) 2 27. (i) 7 (ii)1 16 4 49 49 49 4 5 48 8 28. (i) 5 (ii) 1 29. 12 30. 20. 24 12

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Humility is the key to peace and harmony.(Nirankari Baba Hardev Singh Ji Maharaj)

APPLICATION TO TRIGNOMATRYQ. 1 The angle of elevation of a jet plane from a point A on the ground is 60 º . After a flight of 15 seconds, the angle of elevation changes to 30 º . If the jet plane is flying at a constant height of 1500 3 m, find the speed of the jet plane.

Q. 2 Determine the height of a mountain if the elevation of its top at an unknown distance from the base is 45 º and at a distance 10km further off from the mountain, along the same line, the angle of elevation is 30º. (Use tan 30º = 0.5774)

Q. 3 The angles of elevation of the top of a rock from the top and the foot of a 100 m high tower are 30º and 45º respectively. Find the height of the rock.

Q. 4 A man on the deck of a ship is 16m above water level. He observes that the angle of elevation of the top of a cliff is 45º and the angle of depression of the base is 30º. Calculate the distance of the cliff from the ship and the height of the cliff.

Q. 5 From the top of a cliff 50 m high, the angles of depression of the top and bottom of a tower are observed to be 30º and 45º respectively. Find the height of the tower.

Q. 6 An aeroplance, when 3000 m high, passes vertically above another aeroplane at an instant when the angles of elevation of the two aeroplanes from the same point on the ground are 60º and 45º respectively. Find the vertical distance between the aeroplanes at that instant.

Q. 7 The angle of elevation of the top of a hill at the foot of a tower is 60º and the angle of elevation of the top fo the tower from the foot of the hill is 30º. If the tower is 50 m high, what is the height of the hill?

Q. 8 A pole 5 m high is fixed on the top of a tower. The angle of elevation of the top of the pole observed from a point A on the ground is 60º and the angle of depression of the point A from the top of the tower is 45º. Find the height of the tower.













































( )

TUHI NIRANKAR

Q. 9 An observed from the top of a 150 m tall lighthouse, the angles of depression of two ships approaching it are 30º and 45º. If one ship is directly behind the other, find the distance between the two ships.

Q. 10 A tower is 50 m high. Its shadow is x meters shorter when the sun's altitude is 45º than when it is 30º. Find x correct to the nearest cm.

Q. 11 A 7 meters long flagstaff is fixed on the top of a tower on the horizontal plane. From a point on the ground, the angles of elevation of the top and bottom of the flagship are 60º and 45º respectively. Find the height of the tower correct to one place of decimal.

Q. 12 From the top of a building 100 m high, the angle of elevation of the top of a tower in found to be 60º and the angle of depression of the base of the tower is 45º. Find the height of the tower and its distance on the ground from the building.

Q. 13 A tree is broken by wind. The top stuck the ground at an angle of 30º and at a distance of 30 meters from the root. Find the whole height of the tree.

Q. 14 Two ships are sailing in the sea on the either side of the lighthouse, the angles of depression of two ships as observed from the top of the lighthouse are 60º and 45º respectively. If the distance between the ships is

200 3 + 1 metres, find the height of the lighthouse. 3

Q. 15 The angle of elevation of the top of a tower from a point A on the ground is 30º. On moving a distance of 20 metres towards the foot of the tower to a point B the angle of elevation increases to 60º. Find the height of the tower and the distance of the tower from the point A.

Q. 16 The angle of elevation of a cloud from a point 60 m above the lake is 30º and the angle of depression of the reflection of the cloud in the lake is 60º. Find the height of cloud.

Q. 17 Two men on either side of a cliff 80m high observe the angles of elevation of the top of the cliff to be 30º and 60º respectively. Find the distance between the two men.

Q. 18 From a window 60 metres high above the ground of a house in a street the angles of elevation and depression of the top and the foot of another house on opposite side the street are 60º and 45º respectively. Show that the height of the opposite house 60 (1 + 3) metres.

Q. 19 If the angle of elevation of a cloud from a point h metres above a lake is α and the angle of depression of its reflection in the lake is β, prove that the height of the cloud is h (tanβ + tanα ) tanβ – tanα Q. 20 The angles of elevation and depression of the top and bottom of a lighthouse from the top of a building, 60 m high, are 30º and 60º respectively. Find :

(i) the difference between the heights of the lighthouse and the building. (ii) Distance between the lighthouse and the building.

Q. 21 The height of a house subtends a right angle at the opposite window. The angle of elevation of the window from the base of the house is 60º. If the width of the road is 6 m, find the height of the house.

Q. 22 If the angle of elevation of a cloud from a point h metres above a lake is α and the angle of depression of its reflection in the lake is β, prove that the distance of the cloud from the point of observation is 2h sec α tanβ – tanα





















































TUHI NIRANKAR

Q. 23 From an aeroplane vetically above a straight horizontal plane, the angles of depression of two consecutive kilometre stones on the opposite sides of the aeroplane are found to be α and β . Show that the heigh of the aeroplane is tanαtanβ tanα + tanβ Q. 24 A man standing on the deck of a ship, which is 10 m above water level, observes the angle of elevation of the top of a hill as 60º and angle of depression of the base of the hill as 30º. Find the distance of the hill from the ship and the height of the hill.

Q. 25 The angles of elevation of the top of a tower from two points P and Q at distancs of a and b respectively from the base and in the same straight line with it are complementary. Prove that the height of the tower is ab.

Q. 26 As observed from the top of a lighthouse, 100 m high above sea level, the angle of depression of a ship, sailing directly towards it, changes from 30º to 60º. Determine the distance travelled by the ship during the period of observation. (Use 3 = 1.732)

Q. 27 On a horizontal plane there is vertical towe rwith a flagpole on the tp fo the tower. At a pint 9 metres away from the foot of tht tower, the angles of elevation of the top and the bottom of the flagpole are 60º and 30º. Find the height of the tower and the flagpole mounted on it.Q. 28 Two pillars of equal height stand on either side of a roadway which is 150 m wide. From a point on the roadway between the pillars, the angles of elevation of the top of the pillars are 60º and 30º. Find the height of the pillars and the position of the point.

Q. 29 The angles of depression of the tp and the bottom of a building 50 m high as observed from the top of a tower are 30º and 60º respectively. Find the height of the tower and also the horizontal distance between the building and the tower.

Q. 30 The angle of elevation of the top of a tower as observed from a point on the ground is α and on moving 'a' metres towards the tower, the angle of elevation is β. Prove that the height of the tower is a tanα tanβ tanβ – tanα Q. 31 From a window x metres high above the ground in a street, the angles of elevation and depression of the top and the foot of the other house on the opposite side of the street are α and β respectively. Show that the height of the opposite house is x (1 + tanα cotβ) metres.

Q. 32 From a window 15 metres high above the ground in a street, the angles of elevation and depression of the top and foot of another house on the opposite side of the street are 30º and 45º respectively. Show that the height of the opposite house is 23.66 metres. (Take 3 = 1.732)

Q. 33 The angles of elevation of the top of a tower from a point on the same level as the foot of the tower is 30º. On advancing 150 metres towards the foot of the tower, the angle of elevation becomes 60º. Show that the height of the tower is 129.9 metres. (Use 3 = 1.732)

Q. 34 A pole 5 m high is fixed on the top of a tower. The angle of elevation of the top of the pole observed from a point A on the ground is 60º and the angle of depression of point A from the top of the tower is 45º. Find the height of the tower. Take 3 = 1.732)

Q. 35 A window in a building is at a height of 10 m from the ground. The angle of depression of a point P on the ground from the window is 30º. The angle of elevation of the top of the building from the point P is 60º. Find the height of the building.

Q. 36 The angle of elevation of the top of a tower from a point on the same level as the foot of the tower is 30º. On advancing 150 metres towards the foot of the tower, the angle of elevation becomes 60º. Find the height of the tower.

























































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Q. 37 A boy standing on a horizontal plane finds a bird flying at a distance of 100 m from him at an elevation of 30º. A girl standing on the roof of 20 metre high building finds the angle of elevation of the same bird to be 45º. Both the boy and the girl are on opposite sides of the bird. Find the distance of the bird from the girl.

Answers :1 720 km/hr 2 13.66 km 3 236.6 m 4 27.712 m,

43.712 m5 21.13 m 6 1268 m 7 150 m

8 6.83 m 9 109.8 m 10 3660 cm 11 9.56 m 12 273.2 m100 m

13 51.96 m 14 200 m

15 17.32 m,30 m

16 120 m 17 184.75 m 20 (i) 20 m,(ii)34.64 m

21 13.856 m

24 17.32 m ;40 m

26 115.46 m

27 5.193 m ;10.392 m

28 64.88 m ;37.5 m from the pillar whose elevation is 60 º

29 75 m ;43.3 m

34 6.83 m 35 30 m 36 75 3 37 30 2 m

Humility is the royal road to greatness and glory.

SURFACE AREAS AND VOLUMESTake = 22 , unless stated otherwise. 7Q. 1 A sphere of diameter 6 cm is dropped in a right circular cylindrical vessel partly filled with water. The diameter of the cylindrical vessel is 12 cm. If the sphere is completely submerged in water, by how much will the level of water rise in the cylindrical vessel?

Q. 2 The largest sphere is carved out of a cube of side 7 cm. Find the volume of the sphere.

Q. 3 A vessel, in the form of a hemispherical bowl, is full of water. Its contents are emptied in a right circular cylinder. The internal radii of the bowl and the cylinder are 3.5 cm and 7 cm respectively. Find the height to which water will rise in the cylinder.

Q. 4 An iron pillar has some part in the form of a right circular cylinder and the remaining in the form of a right circular cone. The radius of the base of each of the cone and cylinder is 8 cm. The cylindrical part is 240 cm high and the conical part is 36 cm high. Find the wight of the pillar if one cu cm of iron weighs 7.8 grams.

Q. 5 A solid wooden toy is in the shape of right circular cone mounted on a hemisphere. If the radius of the hemisphere is 4.2 cm and the total height of the toy is 10.2 cm, find the volume of the wooden toy.

Q. 6 A hemispherical bowl of internal diameter 36 cm contains a liquid. This liquid is to be filled in cylindrical bottles of radius 3 cm and height 6 cm. How many bottles are required to empty the bowl?

Q. 7 Marbles of diameter 1.4 cm are dropped into a cylindrical beaker of diameter 7 cm., containing some water. Find the number of marbles that should be dropped into the beaker so that the level rises by 5.6 cm.

Q. 8 The curved surface area of a right circular cone is 12320 cm 2. If the radius of the base is 56 cm, find its height.

Q. 9 The circumference of the edge of a hemispherical bowl is 132 cm. Find the capacity of the bowl.



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Q. 10 The volume of a vessel in the form of a right circular cylinder is 448 π cm 3 and its height is 7 cm. Find the radius of its base.

Q. 11 The height of a cylinder is 15 cm. The curved surface area is 660 cm 2 . Find the radius.

Q. 12 The sum of the radius of the base and the height of a solid cylinder is 37 cm. If the total surface area of the solid cylinder is 1628 cm 2 , find the volume of the cylinder.

Q. 13 A building is in the form of a cylinder surmounted by a hemispherical walled dome and contains 19 m 3 of 21 air. If the internal diameter of the building is equal to tis total height above the floor, find the height of the building.

Q. 14 A circus tent has cylindrical shape surmounted by a conical roof. The radius of the cylindrical base is 20 m. The heights of the cylindrical and conical portions are 4.2 m and 2.1 m respectively. Find the volume of tent.

Q. 15 50 circular plates, each of radius 7 cm and thickness 0.5 cm are place one above the another to form a solid right circular cylinder. Find the total surface area and the volume of the cylinder so formed.

Q. 16 If the radii of the circular ends of a conical bucket, which is 45 cm high, are 28 cm and 7 cm, find the capacity of the bucket.Q. 17 A cone is 8.4 cm high and the radius of its base is 2.1 cm. It is melted and recast into a sphere. Find the radius of the sphere.

Q. 18 A toy is in the form of a cone mounted on a hemisphere of diameter 7 cm. The total height of the toy is 14.5 cm. Find the volume and the total surface area of the toy.

Q. 19 A sphere of diameter 5 cm is dropped into a cylindrical vessel partly filled with water. The diameter of the base of the vessel is 10 cm. If the sphere is completely submerged, by how much will the level of water rise? A

Q. 20 In the adjoining figure, a cone of radius 10 cm is divided into two parts by drawing a plane through the mid-point of its axis, parallel to its base. E Compare the volumes of two parts.

C Q. 21 The volume of a right circular cylinder of height 7 cm is 567 π cm 3 . Find its curved surface area.

Q. 22 If the radius of the base of a right circular cylinder is halved, keeping the height same, find the ratio of the volume of the reduced cylinder to that of the original cylinder.

Q. 23 Two right circular cones X and Y are made, X having three times the radius of Y and Y having half the volume of X. Calculate the ratio of heights of X and Y.

Q. 24 How many metres of cloth 5 m wide will be required to make a conical tent, the radius of whose base is 7 m and whose height is 24 cm?

Q. 25 The radii of the internal and external surfaces of a hollow spherical shell are 3 cm and 5 cm respectively. If its is melted and recast into a solid cylinder of height 2 cm, find the diameter and the curved surface area of the



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B

D

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3 cylinder.

Q. 26 A toy is in the form of a cone mounted on a hemisphere of radius 3.5 cm. The total height of the toy is 15.5 cm. Find the total surface area and the volume of the toy.

Q. 27 The radii of the ends of the frustum of a right circular cone are 5 meters and 8 meters and its lateral height is 5 m. Find the lateral surface area and the volume of the frustum. Take π = 3.142.

Q. 28 A conical vessel whose internal radius is 5 cm and height 24 cm is full of water. The water is empties intoa cylindrical vessel with internal radius 10 cm. Find the number of cones so formed.

Q. 29 A solid metallic sphere of diameter 21 cm is melted and recast into a number of smaller cones, each of diameter 7 cm and height 3 cm. Find the number of cones so formed.

Q. 30 A hemispherical bowl of internal diameter 30 cm is full of some liquid. This liquid is to be filled into cylindrical shaped bottles each of diameter 5 cm and height 6 cm. Find the number of bottles necessary to empty the bowl.

Q. 31 A circus tent is cylindrical to a height of 3 m and conical above it. If its base radius is 52.5 m and slant height of the conical portion is 53 m, find the area of the canvas needed to make the tent.

Q. 32 If the radii of the circular ends of a bucket, 45 cm high, are 28 cm and 7 cm, find the capacity and the total surface area of the bucket.

Q. 33 A hollow cone is cut by a plane parallel to the base and upper portion is removed. If the curved surface area of the remainder is 8 of the curved surface area of the whole cone, find the ratio of the line segments into which the altitude of the cone is divided by the plane.

Q. 34 A hemispherical bowl of internal radius 9 cm is full of liquid. The liquid is to be filled into cylindrical shaped small bottles each of diameter 3 cm and height 4 cm. How many bottles are needed to empty the bowl?

Q. 35 The base radius and height of a right circular solid cone are 2 cm and 8 cm respectively. It is melted and recast into spheres of diameter 2 cm. Find the number of spheres so formed.

Q. 36 A tent is in the shape of a right circular cylinder upto a height of 3 m and conical above it. The total height of the tent is 13.5 m and radius of the base is 14 m. Find the cost of the cloth required to make the tent at the rate of Rs. 80 per sq m.

Q. 37 The radii of the circular ends of a solid frustum of a cone are 33 cm and 27 cm and its slant height is 10 cm. Find its total surface area.

Q. 38 A tent is in the form of a cylinder of diameter 4.2 m and height 4 m, surmounted by a cone of equal base radius and height 2.8 m . Find the capacity of the tent and the cost of canvas for making the tent at Rs. 100 sq m.

Q. 39 The radius of the base and the height of a solid right circular cylinder are in the ratio 2:3 and its volume is 1617 cu cm. Find the total surface area of the cylinder.

Q. 40 A solid is in the form of a right circular cylinder with hemispherical ends. The total height of the solid is 58 cm and the diameter of the cylinder is 28 cm. Find the total surface area of the solid.

Q. 41 The rain water from a roof 22 m X 20 m drains into a cylindrical vessel having diameter of base 2 m and



















































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height 3.5. If the vessel is just full, find the rainfall in cm.

Q. 42 A bucket made up of a metal sheet is in the form of a frustum of a cone of height 16 cm with radii of its lower and upper ends as 8 cm and 20 cm respectively. Find the cost of the bucket if the cost of the metal sheet used is Rs. 15 per 100 cm 2 . (Use π = 3.14)

Q. 43 Water is flowing at the rate of 15 km per hour through a pipe of diameter 14 cm into a rectangular tank which is 50 m long and 44 m wide. Find the time in which the level of water in the tank will rise by 21 cm.

Q. 44 A bucket made up a metal sheet is in the form of a frustum of a cone. Its depth is 24 cm and the diameters of the top and the bottom are 30 cm and 10 cm respectively. Find the cost of milk which can completely fill the bucket at the rate of Rs. 20 per litre and the cost of the metal sheet used, if it costs Rs. 10 per cm 2 . (Use π = 3.14)

Q. 45 A solid cylinder of diameter 12 cm and height 15 cm is melted and recast into toys with the shape of a right circular cone mounted on a hemisphere of radius 3 cm. If the height of the toy s 12 cm, find the number of toys so formed.

Q. 46 A bucket is in the form of a frustum of a cone with a capacity of 12308.8 cm 3 of water. The radii of the top and bottom circular ends are 20 cm respectively. Find the height of the bucket and the area of the metal used in making it. (Use π = 3.14)

Q. 47 A toy in in the form of a cone mounted on a hemisphere of common base radius 7 cm. The total height of the toy is 31 cm. Find the total surface area of th e toy.

Q. 48 A sphere, of diameter 12 cm, is dropped into a right circular cylindrical vessel, partly filled with water If the sphere is completely submerged in water, the water level in the cylindrical vessel rises by 5 cm. Find the diameter 9 of the cylindrical vessel.

Q. 49 A solid right circular cone of diameter 14 cm and height 8 cm is melted to form a hollow sphere. If the external diameter of the sphere is 10 cm, find the internal diameter of the sphere.

Q. 50 A tent is in the form of a right circular cylinder surmounted by a cone. The diameter of the cylindrical portion is 24 m and the height of the cylindrical portion is 11 m while the vertex of the cone is 16 m above the ground. Find the area of the canvas required for making the tent.

Q. 51 A solid iron rectangular block of dimensions 4.4. X 2.6 m X 1 is cast into a hollow cylindrical pipe of internal radius 30 cm and thickness 5 cm. Find the length of the pipe.

Q. 52 Find the number of coins 1.5 cm in diameter and 0.2 cm thick, to be melted to form a right circular cylinder of height 10 cm and diameter 4.5 cm.

Q. 53 A toy is in the form of a cone mounted on a hemisphere with same radius. The diameter of the base of the conical portion is 7 cm and the total height of the toy is 14.5 cm. Find the volume of the toy.

Q. 54 A hemispherical bowl of internal diameter 36 cm is full of some liquid. This liquid is to be filled in cylindrical bottles of radius 3 cm and height 6 cm. Find the number of bottles needed to empty the bowl

Q. 55 Water flows out through a circular pipe whose internal radius is 1 cm, at the rate of 80 cm/sec into an empty cylindrical tank, the radius of whose base is 40 cm. By how much will the level of water rise in the tank in half an hour?



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TUHI NIRANKAR

ANSWERS :1 1 cm 2 179 2 cm

33 7 cm

124 395.37 kg 5 266.112 cm 3 6 72 7 150

8 42 cm 9 19404 cm 3 10 8 cm 11 7 cm 12 4620 cm 3 13 4 m 14 6160 m 3

15 1408 cm 2 ; 16 48510 cm 3 17 2.1 cm 18 231 cm 3 ; 203.94 cm 2

19 5 m6

20 1 : 7 21 396 cm 2

22 1 : 4 23 2 : 9 24 110 m 25 14 cm ; 352 cm2 3

26 214.5 cm 2 ; 243.cm 3

27 204.23 m 2 540.42 m 3

28 2 cm

29 126 30 60 31 9735 m 2 32 48510 cm 3 ; Rs. 5616.6 cm 2

33 1 : 2 34 54 35 8

36 Rs. 82720 37 7599.43 cm 2 38 68.376 m 3 ; Rs. 7590

39 770 cm 2 40 5104 cm 2 41 2.5 cm 42 Rs. 293.90

43 2 hours 44 Rs. 163 approx; Rs. 171 approx

45 12 46 15 cm; 2160.32 cm 2

47 858 cm 2 48 9 cm 49 6 cm

50 1320 m 2 51 112m 52 450 53 231 cm 3 54 72 55 90 cm










































































































































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