REMAINDER THEOREM AND FACTOR THEOREM
1) (x+1) Is a factor of
a) x³+ 4x²+ 5x +2 b) x³- 4x²- 5x +2 c) x³- 3x²- 7x -1 d) 2x³- 4x²- 5x +1
2) On dividing mx³+ 9x²+ 4x -10 by (x +3), the remainder is 5. The value of m is
a) -1 b) 2 c) -2 d) 3
3) When x⁴+1 is divided (x +1), the remainder is
a) 0 b) 2 c) 1 d) -1
4) if a polynomial p(x) is divided by (x + k), then the remainder is
a) p(k) b) p(-k) c) kp(x) d) p(x)+ k
5) The remainder when x⁴- x³+ x² - x +1 is divided by (x -1) is
a) 0 b) 1 c) -1 d) 2
6) if (x -2) is a factor x²+ mx - 2, then the value of m is
a) 1 b) -1 c) 0 d) -2
7) when f(x) is divided by (ax - b), then the remainder is:
a) f(a) b) f(ab) c) f(-b/a) d) f(b/a)
8) when a polynomial f(x) is divided by (3x +4), the remainder is
a) f(3/4) b) f(4/3) c) f(-3/4) d) f(-4/3)
9) if (x -2) is a factor of x² - 4x +m, then the value of m is
a) -4 b) 4 c) 0 d) -1
10) If (x -1) is a factor of x³+ 2x²- x +k, then the value of k is
a) 1 b) 2 c) -2 d) -1
11) if (2x -1) is a factor of f(x)= 2x²+ px -5, then the value of p is
a) 10 b) 9 c) 8 d) 5
12) If (x- 2) is a factor of f(x)= x³+ kx²- 5x -6, then the value of k is
a) 2 b) 1 c) -1 d) -2
13) Given f(x)= 3x²- 5x +p. If (x -2) is a factor of f(x), then the value of p is
a) -2 b) 2 c) -1 d) 1
14) If (x -1) and (x +2) are factors of f(x)= x³+ 10x²+ ax +b, then the value of b is
a) 10 b) 12 c) 18 d) -18
15) When the polynomials f(x)= px³+ 3x²-3 and g(x)= 2x³-5x + p are divided by x - 4 , they leave the same reminder. The value of p is
a) -1 b) 0 c) 1 d) 2
16) Given f(x)= x³+ (kx+8)x +k. If f(x) is divided by x + 1 , the remainder is
a) 2k- 9 b) 2k+9 c) 2k d) 0
17) If (x -3) is a factor of f(x)= x² - kx + 12, then the other factor of f(x) is
a) x- 4 b) x -1 c) x - 2 d) x+ 4
18) (x +2) and (x -3) are factors of f(x)= x³+ ax +b , then the value of f(-3) is
a) 1 b) 2 c) -1 c) 0
19) Given f(x)= (3km+2)x³+ (k -1). If f(x) is divided by (2x+1), the remainder is
a) f(1/2) b) f(-1/2) c) f(2) d) f(-2)
20) On dividing x²- 4x +m by x-2, the remainder is -1. The value of m is
a) 1 b) 2 c) -2 d) 3
SHORT QUESTIONS
1) Using reminder theory, find the value of k if on the dividing 2x³+ 3x² - kx +5 by (x -2), leaves a remainder 7.
2) What must be subtracted from 16x³- 8x²+ 4x +7 so that the resulting expression has 2x + 1 as a factor ?
3) When divided by x -3, the polynomials p(x)= x³ - px²+ x + 6 and Q(x)= 2x³- x²-(p+3)x -6 leave the same remainder. Find the value of p. Also find the remainder .
4) Find the value of k, if x - 2 is a factor of x³+ 2x²- kx + 10. Hence, determine whether x + 5 is also a factor.
5) Find a, if the two polynomials ax³+ 3x²-9 and 2x³+ 4x+ a leave the same remainder when divided by x + 3. also find the remainder.
6) if p(x)= 2x³+ 3x² - ax - b is divided by x - 1, the remainder is 6, then find the value of a + b.
7) Find the value of p if x³+ px +2p -2 is exactly divisible by x + 1.
LONG QUESTION
1) If x + 2 and x+ 3 are factors of x³+ ax +b, find the value of a and b.
2) Use factor theorem to factorise 6x³+ 17x²+ 4x -12 completely.
3) If x- 2 is a factor of 2x³ - x²- px - 2,
a) find the value of o
b) with the value of p, Factorize the above expression completely .
4) Given that (x +2) and (x +3) are factors of 2x³+ ax²+ 7x - b, determine the value of a and b.
5) Using reminder theorem, factorise completely the following polynomials: 3x³+ 2x²- 19x +6
6) if x - 2 is a factor of the expression, 2x³+ ax²+ bx -14 and when the expression is divided by x-3, it leaves a remainder 52, find the value of a and b.
7) Using remainder theorem and factor theorem, factorise the following polynomial : x³+ 10x²- 37x +26.
TRIGO IDENTITY
Prove the following
T-1
1) cosx/(1- sinx) + cosx/(1 + sinx) = 2secx.
2) cosx/(coseccx+1) + cosx/(coseccx - 1)= 2 tanx.
3) tanx + cotx = secx cosecx
4) 1/(1+ sinx) + 1/(1- sinx)= 2 sec²x.
5) (1- sinx)/(1+ sinx)= (secx - tanx)²
6) (1- cosx)/(1+ cosx)= (cotx - cosx)²
7) (1+ tan²x). cotx/cosec²x = tanx.
8) (tanx + sinx)/(tanx - sinx)= (secx +1)/(secx -1).
9) (sinx - 2 sin³x)/(2cos³x - cosx)= tanx.
10) sin⁴x - cos⁴x = sin²x - cos²x.
11) sin³x. + cos³x= (sinx + cosx)(1- sinx cosx)
12) cos²x(1+ tan²x)= sin²x(1 +cot²x)= 1.
13) sinx/(1+ cosx) + (1+ cosx)/sinx = 2 cosecx.
14) (cosecx - sinx)(secx - cosx)(tanx + cotx)= 1
15) (cotx + tan y)/(cot y + tanx)= cotx tany
16) cotx/(cosecx. +1) + (cosecx +1)/cotx = 2 secx
17) sin²x + cos⁴x = cos²x + sin⁴x.
18) (cosecx - sinx)(secx - cosx)= 1/(tanx+ cotx).
19) sec⁴x(1- sin⁴x) - 2 tan²x = 1.
20) (1+ tanx tan y)² + (tanx - tan y)²= sec²x sec²y
21) cos⁶x + sin⁶x = 1- 3 cos²x sin²x.
22) 1/(cosx + sinx -1) + 1/(cosx + sinx +1)= cosecx + secx.
23) 1/(cosecx - cotx) - 1/sinx = 1/sinx - 1/(cosecx + cotx).
24) (1+ cosx)/(1- cosx) = cosecx + cotx.
25) (cotx + cosecx -1)/(cotx - cosecx +1)= (1+ cosx)/sinx.
26) sec²x + cosec²x = sec²x coaec²x
27) √{(1- sinx)/(1+ sinx)}= secx - tanx.
28) sec⁴x - sec²x = tan²x + tan⁴x.
29) cot⁴x + cot²x = cosec⁴x - cosec²x.
30) sinx/(1- cotx) + cosx/(1- tanx)= sinx + cosx.
31) (tanx + secx -1)/(tanx - secx +1) = (1+ sinx)/cosx.
T-2
1) If tanA=√2, find the value of (sec²A - cosec²A)/(sec²A + cosec²A).
2) Prove: i) sin²15° + sin²75°= 1
ii) tan²20° - cot²70° = sin²40° - cos²50°.
iii) sin²35 + sin²55° + tan²25° - cosec²65° = 0
iv) secx/cosec(90°- x) + cotx/tan(90- x)= 2.
v) (sin47°/cos43°)² + (sin43°/cos47°)² - 4 cos²45° = 0
vi) sin²20° + sin²70 - tan²45= 0
vii) If cosx = cos60° cot45° cos30° + sin60° tan45° sin30° find x.
ix) sin64°/cos26° - sec40°/cosec50° = 0
x) Evaluate: (cos²10° + cos²80°)/(sin²69° + sin²21°)
xi) Evaluate: cos(90°- A) cosecA - tan(90° - A) tanA.
T-3
1) Find the value of x in the following
a) sin2x = cos3x
b) tan5x = cot3x.
2) If tanx = √2, find the value of (sec²x - cosec²x)/(sec²x + cosec²x).
3) If secx + tanx = m and secx - tanx = n, show that mn= 1.
4) If x= r sinA cosB, y= r sinA sinB, z= r cosA, then show that x²+ y²+ z²= r².
5) x= asecA cosB, y= secA sinB , z= c tanA then show that x²/a² + y²/b² - z²/c² = 1.
6) If 2sinA - 1= 0, show that sin3A = 3 sinA - 4 sin³A.
7) Solve for x:
a) 4 cos²x -1= 0
b) 4 sin²x -1=0.
c) 4 sin²(2x)= 3
d) sin²2x - (1/2) sin2x = 0
RATIO AND PROPORTION
1) If x², 4 and 9 are in continued proportion, then the value of x is:
a) 2/3 b) 4/3 c) 3/4 d) 16/9
2) if a: b = 5 : 3, then (5a + 8b): (6a - 7b) is equals to:
a) 40 : 9 b) 9 : 49 c) 49 : 9 d) 25: 9
3) The fourth proportional to 7, 13 and 35 is:
a) 65 b) 62 c) 52 d) 50
4) The third proportional to 9 and 15 is:
a) 10 b) 15 c) 18 d) 25
5) if (7m + 2n)/(7m - 2n)= 5/3 then m: n is:
a) 7 : 8 b) 2:7 c) 8 : 7 d) 1:8
6) The mean proportion between 28 and 63 is:
a) 42 b) 45 c) 36 d) 32
7) The fourth proportional to 3,12, 15 is:
a) 40 b) 45 c) 60 d) 62
8) If x : y= 2: 3 then (3x +2y)/(2x + 5y) is:
a) 12/19 b) 19/12 c) 12/13 d) 19/21
9) The mean proportion between x - y and x³ - x²y is :
a) x(x + y) b) x²(x - y) c) x²(x + y) d) x(x - y)
10) If a :b = 2: 3 and b: c= 4:5, then a: c is
a) 12 :15 b) 15:7 c) 15:8 d) 8:15
11) If a/b = c/d, then (a+ c)/(b+ d) is equal to
a) a/b b) c/d c) both a and b d) neither a nor b
12) The fourth proportional to 3, 6 and 4.5 is
a) 10 b) 9 c) 8.5 d) 6
13) The third proportional to x - y and x²- y² is
a) (x + y)(x²- y²) b) (x - y)(x² +y²) c) (x² + y²)(x²- y²) d) (x + y)(x- y)
14) if a/b : c/d, then each ratio is equals to :
a) a+ b : c+ d b) a+ c : b + d c) a+ d: b + c d) a- b: c - d
15) the mean proportion between a²b and 1/b is
a) a b) a² c) ab d) √(ab)
16) If 2x = 3y and 4y= 5z, then 8x/z is equals to
a) 5 b) 15 c) 10 d) 8
17) Two numbers are in the ratio 1 :4. If the mean proportion between them is 28 and third proportional to them is 224, then the smaller number is
a) 12 b) 14 c) 16 d) 21
18) If three quantities a, b, c are in continued proportion, then the mean proportion between:
a) a and c is b b) b and c is a c) c and a is b d) all the above are true
19) x, y, z are in continue proportion, then x/z is equals to
a) y²/z² b) y²/x² c) x²y² d) x/y²
20) If (4a + 9b)/(4c + 9d) = (4a - 9b)/(4c - 9d), then a: b =
a) c: d b) d: c c) c: d + c d) d: c - d
SHORT QUESTIONS
1) If (3a + 2b) : (5a + 3b)= 18: 29, find a, b.
2) What least number must be added to each of the numbers 2, 5, 18 and 33, so that the resulting numbers are proportional ?
3) If b is the mean proportion between a and c, show that b(a+ c) is the mean proportion between (a²+ b²) and (b²+ c²).
4) If a: b = c : d, then prove that (a+ b): (c + d)= √(a²+ b²): √(c²+ d²).
5) If x/(b - c) = y/(c - a)= z/(a - b), then show that ax+ by + cz =0.
6) If a, b, c, d are in continued proportion, prove that (b + c)(b + d)= (c + a)(c + d).
7) it a/b = c/d, then, show that (3a - 5b)/(3a + 5b)= (3c - 5d)/(3c + 5d).
LONG ANSWER
1) If b is the mean proportion between a and c, show (a²+ b²+ c²)/(a⁻² + b⁻² + c⁻²) = b⁴.
2) if a, b, c are in continued proportion, show that (2a²+ 7b²- 5ab)/(2b²+ 7c²- 5bc)= a/c.
3) If a, b, c, d are in continued proportion, show that
√(ab) + √(bc) - √(cd)= √{(a+ b - c)(b + c - d)}.
4) If a: b= c : d, show that (a²+ c²+ ac): (a²+ c²- ac)= (b²+ d²+ bd):(b²+ d²- bd)
5) If ax = by = cz, then show that x²/yz + y²/zx + z²/xy = bc/a²+ ca/b²+ ab/c².
6) If x= {√(a+ 3b) +√(a - 3b)}/{√(a+ 3b) - √(a - 3b)}, show that 3bx²- 2ax + 3b = 0
7) If p= 4xy/(x + y), show that (p+ 2x)/(p- 2x) + (p+ 2y)/(p - 2y)= 2.
8) If If {√(x+ 15) +√(x - 6)}/{√(x+ 15)- √(x - 6)}= 7/3, find the value of x.
9) If 11 = {√(6x)+√(3x +7)}/{√(6x)- √(3x+7)}, then find x
CLASS- X
REVISION - 1
1) Girija buys the following electrical goods:
i) 10 bulbs costing Rs 15.80 each. GST 2.5%
ii) 8 fans costing Rs 920 each, GST 10%
iii) 16 switches costing Rs 8.50 each, GST 7%.
iv) 70 metres of wires costing Rs 4.40 per metres.
Find the payment paid by Girija to the shopkeeper.
2) Hari buys a TV set for Rs 13300 and gets 6% rebate on it. He has to pay a GST of 10%. Find the total amount he has to pay.
3) The listed price of a refrigerator is Rs 15000. The customer is allowed a rebate of 50/3%. If the customer pays Rs13570 for it, calculate the GST percent.
4) Leena wants to buy an electric iron which is listed Rs 1308. She has to pay a GST of 9%. The shopkeeper reduces the price so that Leena has to pay Rs 1308 inclusive GST. Find the reduced price.
5) Sita deposited Rs 400 per month in a bank in recurring deposit. On maturity she gets Rs 97854.40. find the period of which she has deposited.
6) Which is better investment 8% Rs 100 shares at Rs 20 premium or 6% Rs 100 shares at Rs 20 discount.
7) A company declares a semi-annual dividend of 5%. Sanjay owns 25 shares of par value Rs 12.50 each. How much annual dividend must be receive?
8) A company having a capital of Rs 450000 declares a dividend of 4% semi-annually.
a) what is the annual income of a stock-holder owing 135 shares at par of Rs 10?
b) What is the total amount of dividend paid annually by the company?
9) Vinay owns 150 Rs 25 shares of a company which declares a dividend of 12%. What is Vinay's dividend income? If he sells the shares at Rs 40 and invests the proceeds in 7% stock (par value Rs 100) at Rs 80, what is the change in his dividend income?
10) Mr. Gupta purchased 360 Rs 50 shares at Rs 20 premium. The company declares an annual dividend of 12%.
a) Find his dividend income from the shares.
b) Find his total investment in the shares.
11) A man invests Ra 1426 in 5% stock at Rs 115. He sells this stock at Rs 125 and invests the proceeds in 3% stock at 93. Find the change in his income.
12) Solve: (11- 2x)/(9- 3x)≥ 5/8, x ∈ R, c < 3.
13) Solve: 8/3 ≤ x + 1/3 < 10/3, x ∈ R. Hence represent the solution on a number line.
14) A={x : -1< x ≤ 5, x ∈ R}
B={x : -4≤ x < 3, x ∈ R}
Represent a) A ∩ B b) A' ∩ B on different number lines, where universal set is R.
15) Solve the equation 3x²- x -7=0 and give your answer correct to two decimal places.
16) Find the roots of x²- 6x +2, using formula method.
17) Find the roots of √3 x² - 9x + 6√3, using formula method.
18) Solve: (4x²-1) - 3(2x +1) + x(2x +1)= 0.
19) Solve: x² - 1/x² = (29/10) (x - 1/x). x> 0.
20) Solve: √(x +15)= x +3, x ∈ N.
21) Solve: √{x(x -3)}= √10. State the sum of the roots.
22) Solve: √(6x -5) - √(3x -2)= 2.
23) A two digit number is four times the sum and two times the product of its digits. Find the number.
24) A says to B, 'I am twice as old as you were when I was as old as you are '. If the product of their ages is 588. Find their present ages.
25) A number consists of two digits such that the square of the digit in the ten's place exceeds the digit in the unit's place by 11. If the number is five times the sum of the digits, find the number.
26) Ten years ago, the sum of the ages of two sons was half of their father's age. The ratio of the present ages of the two sons is 4:3 and the sum of the present ages of all the three is 117 years. Find the present ages of the father & each of the two sons.
27) A party of students arranged an excursion costing Rs 540, whose amount was to be shared equally by all of them. But later, it was found that three of the students, could not pay, though they had joined the excursion. As a result, the rest of the students had each to pay Rs 2 more. Find the total number of students in the party.
28) In a certain examination, the those number of candidates passed was four times the number of those who failed. If the number of candidates that appeared had been 35 less, and the number that failed had been 9 more, the number passing would have been twice the number failing. Find the total number of candidates that appeared at the examination.
29) In a certain year, the incomes of Nitesh and Mitesh are in the ratio 5:6; their expenses are in the ratio 6:7, and their savings are in ratio 4:5. If the sum of their expenses during that year is Rs 3900, find the income of each.
30) The length of a rectangle exceeds its breadth by 5m. If the breadth were doubled and the length reduced by 9m, the area of the rectangle would have increased by 140 sq.m. Find its original area.
31) If the usual speed of a train is reduced by 4km per hour, it takes one hour more to cover a distance of 360 km then it usually does. Find the usual speed of the train in km per hour.
32) The area of a right angled triangle is 60 sq.units. Determine its base and the altitude if the latter exceeds the former by 7 units.
QUADRATIC EQUATION
1) If x²= 3x, then
a) x= 0 b) x=0 or x=3 c) x=3 d) x=0 and x= 3
2) If 3x²+ 8= 10x, then
a) x= 2 or 4/3 b) x=2 or x=3 c) x=3 or 4/3 d) x=1 and x= 1/3
3) The quadratic equation whose solution set is {-2, 3} is
a) x²- 3x -6=0 b) x²- x + 6=0 c) x² +x -6=0 d) x²- x -6=0
4) One of the roots of 2x²- 7x +6=0, is
a) -2 b) 2 c) - 3/2 d) 4
5) The discriminant of the quadratic equation 2x²- x +3 =0 is:
a) 24 b) 25 c) -23 d) -20
6) On solving x² + 4x - 21=0, we get:
a) x= -3, or -7 b) x=7 or -3 c) x = 7 or 3 d) x = -7 or 3
7) The roots of the quadratic equation 2x² + x - 1 =0 are
a) 1/2 or -1 b) -1/2 or 1 c) -1/2 or -1 d) 1/2 or 1
8) The discriminant of x²- 4x - 7 =0 is:
a) 44 b) √44 c) 12 d) -12
9) The quadratic equation whose roots are -1, -5 is:
a) x² +6x +5 =0 b) x²- 6x +5 =0 c) x²+ 6x - 5=0 d) x²- 6x - 5=0
10) For the equation 3x²- 4x - 2 =0, the roots are:
a) real and equal b) real and unequal c) imaginary d) both (a) and (b)
11) For the quadratic equation 2x²- 4x + 1 =0, the discriminant is:
a) 0 b) +ve c) -ve d) none
12) For the quadratic equation 2x²- 3x + 1 =0, the discriminant is:
a) 1 b) -1 c) 0 d) 9
13) For the quadratic equation 9x²+ 6x +1 =0, the discriminant is :
a) +ve b) -ve c) 0 d) imaginary
14) For the quadratic equation 2x² + ax - a²=0, the sum of the roots is:
a) a/2 b) -a/2 c) 2a d) -a
15) Given a quadratic equation mx²+ 8x -2 =0, m≠ 0. For this quadratic equation the value of discriminant is:
a) 64- 8m b) √(64+ 8m) c) 64+ 8m d) √(8m - 64)
16) For the quadratic equation 3x²+ 7x + 8 =0, the roots are:
a) real and distinct b) real and equal c) imaginary d) both (a) and (b)
17) The roots of the quadratic equation 2x²- kx + k =0 are equal . If k ∈N, then k is
a) 0 b) 8 c) 6 d) -6
18) Given the quadratic equation x² + 2√2x +1 =0. The roots of the quadratic equation are:
a) (-√2±1)/2 b) (√2±1)/2 c) (√2±1) d) (-√2±1)
19) The quadric equations (m +1)x² + 2(m +3)x +(m +8) =0. has equal roots. The value of m is :
a) 1/3 b) 1/4 c) 3 d) -3
20) The quadratic equation x² + m(2x + m -1)+ 2 =0 has equal roots . The value of m is
a) 1 b) 2 c) -2 d) 0
SHORT ANSWER TYPE QUESTIONS
1) Find the value of k for which the following equation has equal roots. x² + 4kx + (k²- k+2) =0.
2) Find the value of p for which the equation px² - 5x +p =0. has real and equal roots.
3) For what values of m the equation 2x² + mx +2 =0 has real roots ?
4) Find the values of p for which the equation px² + 2x +1 = 0 has distinct real roots.
5) Show that the equation 2(p²+ q²)x² + 2(p+ q)x +1 =0 has no real roots when p≠ q.
6) Solve the equation x² - 4x - 8=0 for x. Give your answer correct to 3 significant figures.
7) Solve 3x² + 11x +10 =0, when x belongs to I by factorization.
8) Solve : √(2x +9)= 13 - x by factorization.
9) Solve the equation 2x² + √5 x -5 =0 using formula, write your answer correct two decimal places.
10) Solve for x and give your answer correct to 2 decimal places.
(x +1)/(2x +5) = (x +3)/(3x +4).
Long Answer Type Questions
1) For what values of k will be the following quadratic equation (k +1)x² - 4kx + 9 = 0 have real and equal roots. Solve the equation.
2) Solve: √(2x² -2x + 21) =2x -3.
3) Solve: 2{x/(x +1)}² - {5/(x +1)}+ 2 = 0, x≠ 1.
4) If a. b c are rational, prove that the roots of the equation (b - c)x² + (c - a)x +(a - b) =0 are also rational.
5) The difference of the squares of two natural number is 84. The square of the larger number is 25 times the smaller number. Find the numbers .
6) The sum of the ages of Vivek and his younger brother Amit is 47 years . The product of their ages in years is 550. Find their ages.
7) An aeroplane travelled a distance of 400 km at an average speed of x km/hr. On the return journey the speed was increased by 40 kmph. Write down an expression for the time for:
a) the onward journey. b) the return journey
If the return journey took 30 minutes less than the onward journey write an equation in x and find the value of x.
8) The hotel bill for a number of people for overnight stay is Rs14400. If there were 4 more people, the bill each person had to pay would have reduced by Rs600. Find the number people staying overnight .
9) A fruitseller bought x apples for Rs1200.
a) Write the cost price of each apple in terms of x.
b) If 10 apples were rotten and he sold rest at Rs3 more than the cost price of each, write the selling price of (x -10) apples.
c) If he made a profit of Rs60 in this transaction , form an equation in x and solve it to evaluate x.
LINEAR INEQUATIONS
1) If 2x - 7 < 4, where x is a natural number less than 8, than the solution set is:
a) {0,1,2,3,4} b) {1,2,3,4,5} c) {1,2,3,4,5,6} d) {0,1, 2,3,4,5,6}
2) If - x ≥ -3 then:
a) x≤ -3 b) x ≥ 3 c) x = 3 d) x ≤ 3
3) if 2 + 4 x< 2 x - 5 ≤ 3x ∈Z, then the solution set is :
a) {5,4} b) {- 5,-4} c) {- 5, -4, -3} d) {- 4,-3,-2,-1}
4) if 2≤2x - 3 ≤ 5, x∈ R, then the solution set is:
a) {2.5≤ x ≤ 4, x∈ R} b) {2≤ x ≤ 5, x∈ R} c) {3≤ x ≤ 5, x∈ R} d) {2< x < 4, x∈ R}
5) If a ≥ b, then:
a) a - c ≤ b - c b) a - c≥ b - c c) a - c = b - c d) a - c > b - c
6) If - x≥ 5 and - ax ≥ 5a, then :
a) a > 0 b) a < 0 c) both a and b d) neither a nor b
7) If x+1≥ 13 - 5x, x ∈{1,2,3,4.....10}, then the solution set is:
a) {1,2,3,4,5,6} b) {6,7,8,9,10} c) {7,8,9,10} d) {6,7,8.....}
8) If 7 - 5x ≥ 3x -1, then the solution set, when x ∈ W is:
a) {0,1} b) {0} c) {1} d) {0,1,2}
9) Given a >0, b >0, c >0 and d <0, then a < b implies :
a) a+ d> b + d b) a - d < b - d c) a - d > b - d d) a + d = b + d
10) Given 2x - 5 ≤ 5x +4 < 11. If x ∈ I, the solution set is:
a) {-2,-1,0,1} b) {-3,-2,-1,0,1} c) {-3,-2,-1,0} d) {-2,-1,0,1}
11) For the inequation -12< 3 - 4x ≤ 11, x ∈ N,
the solution set on the number line can be shown as:
a) 5 b) 4 c) 3 d) 2
13) If 2x - 5 ≤ 5x + 4 < 11, x ∈ I, then the solution set can be represented as:
a) -2 b) -1 c) 0 d) 1
15) If -9(x -7)≥ 45 - 21x > x +1, x ∈ R, then the solution set is:
a) {-3/2≤ x < 2, x ∈R}. b) {-3/2 < x < 2, x ∈R} c) {-2/3 ≤ x ≤ 1, x ∈R} d) {-1/3 ≤ x ≤ 2, x ∈R}
16) If 2x - 5 ≤ 5x + 4 < 11, x ∈ I, then the smallest whole number for x is:
a) 0 b) 1 c) -3 d) 2
17) If 5 - 3x < 11, x ∈ R, then the solution set is:
a) {x> -2, x∈R} b) {x≥ -2, x∈R} c) {x< 2, x∈R} d) {x< -2, x∈R}
18) Given 3x -1 ≤ x +5, If x ∈N, then the solution set is:
a) {1,2,3} b) {1,2,3,4} c) {1,2} d) {0,1,2,3}
19) If 8 < 5(x +1) -2 ≤ 18, x ∈R, then the smallest integer value of x is:
a) 1 b) 0 c) -1 d) 2
20) Given a >0, b >0, c >0 and d <0. Then a > b implies:
a) ad >bd b) ad = bd c) ad < bd d) none
SHORT ANSWER TYPE QUESTIONS
1) Find the value of x, which satisfies the inquation -2≤ 1/2 - 2x/3 ≤ 11/6, x ∈N.
Graph the solution set on the number line.
2) Solve the following inequation, write the solution set and represent it on the number line.
-3(x -7)≥ 15 - 7x > (x +1)/3, x ∈R
3) Solve the following inequation, write down the solution set and represent it on the real number line:
-2+ 10x ≤ 13x +10 < 24+ 10x, x ∈ Z
4) Solve the following inequation and write down the solution set:
11x - 4 < 15x +4 ≤ 13x + 14, x ∈ W
Represent the solution on a real number line.
5) Solve the given inequation and graph the solution set on the number line:
2y - 3 < y +1 ≤ 4y +7, y ∈ R
6) Solve the following inequation and represent the solution set on the number line:
2x -5 ≤ 5x +4 < 11, x ∈I
7) Solve the following inequation and write the solution set:
13x -5 < 15x +4 < 7x +12, x∈R
LONG ANSWER TYPE
1) Solve: - 31/3 < 5y/3 + 3 ≤ y/2 + 16/3, y ∈R
Graph the solution set on the number line.
2) Solve the inequation and represent the solution set on the number line:
-3 + x ≤ 8x/3 + 2 ≤ 14/3 + 2x, x∈ I
3) Solve the following inequation and represent the solution set on the number line:
- 3 < 1/2 - 2x/3 ≤ 5/6 , x ∈ R
4) Solve the following inequation and represent the solution set on the number line:
4x - 19 < 3x/5 - 2 ≤ -2/5 , x∈ R
5) Solve the inequation, write the solution set and represent it on the number line:
-x/3 ≤ x/2 - 4/3 < 1/6, x ∈ R
6) Find the value of x which satisfy the inequation:
-17/6 < 1/2 - 2x/3 ≤ 2, x ∈W.
Graph the solution set on the number line:
7) Solve the following inequation, write the solution set and represent it on the number line:
2x -1 ≥ x + (7 -x)/3 > 2, x ∈ R.
SHARES AND DIVIDEND
1) The total amount of money needed to run the company is called:
a) shares b) capital c) dividend d) principal
2) The whole capital to run a company is divided into small units, called:
a) Shares b) Share holders c) face value d) dividend
3) The annual profit distributed among share holders is called:
a) Nominal value b) Market value c) Dividend d) Face value
4) The value of a share printed on the share certificate is called its :
a) Nominal value b) Market value c) Discount d) Below par
5) The shares of different companies can the bought or sold in the market through stock exchange. The price at which the share is share is sold or purchased is called its:
a) Face value b) Market value c) Par value d) Nominal value
6) A share is said to be____, if its market value is the same as its face value.
a) Premium b) Discount c) Par c) nominal value
7) A share is said to be at premium, if its market value is ____then its face value.
a) More b) Less c) Same d) Equal
8) A share is said to be____, if its market value is less than its face value.
a) At par b) Above par c) Below par d) Premium
9) The face value of a share:
a) Change every year b) Changes from time to time
c) Akways remains the same d) none of these
10) Dividend is always paid on the____ of a share.
a) Market value b) Face value c) Investment d) Dividend
11) The market value of a share:
a) Never changes b) Changes from time to time c) Changes every month d) none
12) Number of shares hold by a person=
a) Total Nominal value/Face value of one share.
b) Total market value/ Face value of one share.
c) Dividend/market value of one share.
d) (Dividend/Investment) x 100
13) Dividend =
a) Number of shares x Nominal value
b) Number of shares x Market value
c) (Face value x Number of shares x rate of dividend)/100
d) None of these
14) Rate of return on investment=
a) Investment/Dividend
b) Dividend/Investment
c) (Dividend/Investment) x 100
d) (Investment/Dividend) x 100
15) Investment/Sale proceeds =
a) Number of shares x market value.
b) Number of shares x Nominal value
c) Face value x Number of shares x Rate of Dividend
d) Dividend/Investment
16) Annual Income of a share holder=
a) Number of shares x Face value
b) Number of shares x Rate of dividend x Face value of 1 share
c) Number of shares x Market value x Face value
d) Market value x Nominal value x 100
17) If a share of Rs100 is selling at Rs125, then it is said to be selling at a ___ of Rs25.
a) Discount b) Premium c) At par d) Below par
18) if a share of Rs125 is selling at Rs96, Then it is said to be selling at Rs 29___.
a) Below par b) At par c) Above par d) Premium
19) if Kabir invests Rs10320 on Rs100 shares at a discount of Rs14, then the number of shares he buys is:
a) 110 b) 120 c) 150 d) 100
20) Shahrukh has some shares of Rs50 of a company paying 15% dividend. If his annual income is Rs 3000, then the number of shares he processes is
a) 400 b) 600 c) 800 d) 200
21) If Kiran invests Rs19250 on Rs50 shares at a premium of 20%, then the number of shares she buys is:
a) 640 b) 160 c) 320 d) 240
22) Varun possesses 600 shares of Rs25 of a company. If the company announces a dividend of 8%, then his annual income is:
a) Rs600 b) Rs1200 c) Rs480 d) Rs120
23) A man invests Rs24000 on Rs60 shares at a discount of 20%. If the dividend declared by the company is 10%, then his annual income is
a) Rs2880 b) Rs1500 c) Rs3000 d) none of these
24) Rs25 shares of a company are selling at Rs20. If the company is paying a dividend of 12%, then the rate of return is:
a) 10% b) 18% c) 15% d) 12%
25) Rs 40 shares of a company are selling at 25% premium. If Mr. Wasim wants to buy 280 shares of the company, then the investment required by him is:
a) Rs14000 b) Rs16800 c) Rs8400 d) Rs10000
SHORT ANSWERS:
1) Find the % return on investment in:
a) 6% Rs100 shares at Rs120.
b) 25/3% Rs100 share at Rs150.
2) Find the price of 5% Rs100 shares when an investor gets a dividend of Rs65 by investing Rs1430.
3) At what price should a 25/4% Rs 100 shares be quoted when money is worth 5% ?
4) A person invested Rs1284 in 9/2% Rs100 shares . His dividend was Rs54. Calculate the M.V of the shares.
5) A company pays 7% dividend . For how much should a man buy a Rs100 shares for getting 8% on his investment?
6) Which share is more profitable: 4% Rs100 share at Rs80 or 9/2% Rs100 share at Rs88 ?
7) A part of Rs3020 is invested in 6% Rs100 shares at Rs97 and the rest in 12% Rs100 share at Rs108. If both bring the same dividend, find the sum invested in the shares selling a) at discount b) above par c) the total dividend.
8) A person bought 360 ten-rupee shares paying 12% dividend at par and sold them at Rs21. The proceeds were invested in five-rupees shares paying 9/2% at Rs 3.5 per share. Find
a) sale proceeds.
b) the number of five-rupees shares bought.
c) the percentage in income.
9) A person invests Rs4368 in hundred-rupee share at Rs91. Shares worth Rs24000 face value are sold at Rs95 and the rest at Rs85. Find
a) the number of shares bought Rs91?
b) the number of shares sold at Rs85.
c) the loss or gain in the deal.
10) Rs8000 and Rs10000 were invested in Rs100 shares giving dividends 12% and 8% respectively. The dividends are collected and all the shares are sold at a loss of 2% and 3% respectively on the investment. Find
a) the dividend collected.
b) the total sale proceeds.
c) gain% on the whole transaction.
LONG ANSWER
1) A man has a choice to invest in hundred-rupees shares of two firms at Rs120 each or at Rs132 each. The first firm pays a dividend of 5% per annum and the second firm pays a dividend of 6% per annum. How much more will his annual return be if he invests Rs 26400 with the firm from which he gets a better return on his investment ?
2) A man bought 1000 shares each of face value Rs5 at Rs7 per share . At the end of the year, the company from which he bought the shares declared a dividend of 8%. Calculate
a) the amount of money invested by the man.
b) the percentage return on his outlay. (Correct to one decimal place).
3) Ajay owns 560 share of a company. The face value of each share is Rs25. The company declares a dividend of 9%. Calculate
a) the dividend that Ajay will get.
b) the rate of interest on his investment, if Ajay had paid Rs30 for each share.
4) A company with 4000 shares of nominal value of Rs110 each declares an annual dividend of 15%. Calculate
a) the total amount of dividend paid by the company.
b) the annual income of Shah Rukh who holds 88 shares of the company.
c) If he received only 10% on his investment, find the price Shah Rukh paid for each share.
5) Amit Kumar invests Rs36000 in buying Rs100 shares at Rs20 premium. The dividend is 15% per annum. Find:
a) the number of shares he buys .
b) his yearly dividend.
c) the percentage return on his investment.
6) Vivek invested Rs45000 in 8% Rs10 shares at Rs15. He sells the shares when the price rise to Rs30 and invests the proceeds in 12% Rs100 shares at Rs125. Calculate
a) the sale proceeds.
b) the number of Rs125 shares he buys.
c) the change in his annual income from dividend.
BANKING
1) Sujata has a recurring deposit account in a bank for 2 years at 6% p.a, simple interest. If she gets Rs1200 as interest at the time of maturity, then her monthly installment is :
a) Rs500 b) Rs600 c) Rs700 d) Rs800
2) A man deposited Rs1000 per month in a recurring deposit account for 3 years at 8% p.a, The maturity value is:
a) Rs440000 b) Rs 40000 c) Rs40440 d) Rs44444
3) Abhinav deposited Rs200 per month for 3 years in a bank's recurring deposit account. If the bank pays intrest at the rate of 11% p.a., then the interest at the time of maturity is:
a) Rs1210 b) Rs1221 c) Rs1415 d) Rs1521
4) Kamal has a recurring deposit account in a bank for 2 years at 6% p.a., S. I. if he gets Rs1350 as interest, then the monthly installment is:
a) Rs1000 b) Rs900 c) Rs850 d) Rs 800
5) if Kavita deposited Rs900 per month for 2 years and 6 months in a recurring deposit account, then the total money she deposited in the account is:
a) Rs 26000 b) Rs27000 c) Rs28000 d) Rs30000
6) A man deposited RsX per month for Y years in a recurring deposit account. If at the time a maturity he got RsZ as interest, then the total maturity amount is:
a) Rs(12XY + Z) b) Rs 12 X Y Z c) Rs( X Y + 12Z) d) Rs XYZ/12
7) Karan deposited Rs2000 per month for 3 years in a recurring deposit account . if the rate of interest is 11% p.a., the amount he gets on maturity is:
a) Rs72000 b) Rs80000 c) Rs82210 d) Rs 84210
8) A man deposited Rs2000 per month in a recurring deposit account for 18 months at 8% p.a, The interest he will get at the time of maturity is :
a) Rs2280 b) Rs2350 c) Rs2400 d) Rs2410
9) Mr Awasthi has a 4 year time deposit account and deposits Rs 650 per month. If he received Rs5096, as interest at the time of maturity, the rate of interest is:
a) 8% p.a b) 8.5%p.a c) 9%p.a d) 10%p.a
10) Sharda deposited Rs150 per month in a bank for 8 months under the recurring deposit scheme. If the rate of interest is 8% p.a., then intrest earned at the time of maturity is:
a) Rs50 b) Rs40 c) Rs36 d) Rs30
11) Ravi has a 4 year cumulative time deposit account and deposits Rs650 per month. If he receives Rs36296 at the time of maturity , the rate of interest was :
a) 10% p.a b) 9%p.a c) 8%p.a d) 7.5%p.a
12) Piyush has a recurring deposit account for 2 years at 10%p.a. If he receives Rs1900 as intrest , the monthly installment paid by him is:
a) Rs700 b) Rs750 c) Rs760 d) Rs800
13) A man deposited Rs1500 every month in a bank for 8 months under the recurring deposit scheme. If the rate of interest is 8%p.a, than the interest earned at the time of maturity is:
a)Rs 400 b) Rs350 c) Rs360 d) Rs320
14) Kamal deposits Rs4000 per month in a recurring deposit account for 3/2 years at 6%p.a. The intrest he will receive at the time of maturity is:
a)Rs 3420 b) Rs3240 c) Rs3150 d) Rs3110
15) Garima deposited Rs500 per month in a recurring deposit account for 3 years. If the rate of interest is 4%p.a, then the amount she gets at the time in maturity is:
a) Rs1110 b) Rs19110 c) Rs 20150 d) Rs21110
16) Amrita deposited Rs1600 per months in a recurring deposit account for 2 years. If the rate of interest is 8% p.a, then the interest earned by her at the time of maturity is:
a) Rs 3200 b) Rs3450 c) Rs3550 d) Rs3600
17) Manisha deposited Rs500 per month in a recurring deposit account for 2 years. If the bank pays interest at 8%p.a, then the interest she gets at the time of maturity is:
a) Rs 1000 b) Rs1100 c) Rs 1200 d) Rs1500
18) Ajay deposited Rs2400 per month for 18 months in a bank's recurring deposit account. If the bank pays interest at 6%p.a, the interest he gates at the time of maturity is:
a) Rs 1550 b) Rs18000 c) Rs2052 d) Rs3250
19) Sameer has a recurring deposit account in a bank for 3 years at 4% simple interest. If he gets Rs4440 as interest at the time of maturity, then the monthly installment is:
a) Rs2000 b) Rs26000 c) Rs3000 d) Rs3200
20) Dinesh has a recurring deposit account, which pays interest at 5%p.a. if he pays Rs 2500 per month for 2 years, then the interest he will get at the time and maturity is
a) Rs 3000 b) Rs3025 c) 3200 d) Rs3125
Short Answer Type Questions:
1) Rekha opened a recurring deposit account for 20 months. The rate of interest is 9% per annum and Rekha receives Rs441 as interest at the time of maturity. Find the amount Rekha deposited each month.
2) Mr Sonu has a recurring deposit account and deposited Rs750 per month for 2 years. If he gets Rs19125 at the time of maturity, find the rate of interest.
3) Kabeer opened a recurring deposit account in a bank and deposited Rs300 per month for 2 years. If he received Rs7725 at the time of maturity, find the rate of interest per annum.
4) Ram deposits Rs1000 per month in a recurring deposit account for 3 years at 8% per annum intrest. Find the maturity value.
5) Reema deposited Rs200 per month for 3 years in a bank's recurring deposit account. If the bank pays interest at the rate of 11% p.a., find the amount she gets on maturity.
6) Harishankar opened a recurring deposit account in a bank and deposited Rs800 per month for 18 month. If he received Rs15084 at the time of maturity, find the rate of interest per annum.
Long Answer Type Questions
1) Mohan has a recurring account in a bank for 2 years at 6% simple interest. If he gets Rs1200 as intrest at the time of maturity, find
a) the monthly installment.
b) the amount of maturity.
2) Priyanka has a recurring deposit account of Rs1000 per month at 10% per month. If she gets Rs5550 as interest at the time of maturity, find the total for which the account was held.
3) Manish opened a recurring deposit account in a bank. He deposited Rs2500 per month for two years. At the time of maturity, he got Rs67500. Find
a) the total interest earned by Manish.
b) the rate of interest per annum.
4) Mr Gupta has a recurring deposit account in a bank. He deposits Rs2500 per month for 2 years. If he gets Rs66250 at the time of maturity, find
a) the interest paid by the bank.
b) the rate of interest.
5) Mr Garg deposits a certain sum of money each month in a recurring deposit account of a bank. If the rate of interest is 8% p.a and Mrs Garg gets Rs 8088 from the bank after 36 months, find the value of his monthly installment.
6) Vandana has a recurring time deposit account of Rs340 per month at 6% p.a. If she gets Rs7157 at the time of maturity, find the total time for which the account was held.
Paper-1
Full Marks -20
Answer the following questions:
1) a) A man has a recurring deposit account in a bank for 7/2 years. If the rate of interest is 12% per annum and the men gets Rs 10206 in maturity, find the value of monthly installments. (3)
b) Find the values of x which satisfies the inequation: (3)
-2≤ 1/2 - 2x/3 ≤ 11/6, x belongs to N. Graph the solution on the number line.
c) Find the values of x, y, z if the matrix
x+2 6 = -5 y²+y
3 5z 3 -20 (4)
2) a) If matrix
A= 1 -2 & B= 3 2
2 -1 -2 1
Find 2B - A². (3)
b) The three equation of straight lines are : 3x + 2y= 1, x - y= 2, 3x+ y= 1 .
Prove that these three lines are concurrent or not. (3)
Or
Prove that: cosec⁶A - cot⁶A= 3cot² A cosec² A +1
c) i) The line 4x - 3y + 1 = 0 meet the x-axis at A. Write down the coordinates of A.
ii) Determine the equation of the line passing through A and perpendicular to 4x - 3y +12= 0 (2+2)
1) if 7th and 13th terms of an AP be 34 and 64 respectively, then its 18th term is
a) 87 b) 88 c) 89 d) 90
2) If the sum of p terms of an AP is q and the sum of q terms is p, then the sum of p+ q terms will be
a) 0 b) p- q c) p+ q d) -(p+ q)
3) If the sum of n terms of an AP be 3n²- n and its common difference is 6, then its first term is
a) 2 b) 3 c) 1 d) 4
4) Sum of all two digit numbers which when divided by 4 yield unity as remainder is
a) 1200 b) 1210 c) 1250 d) none
7) The first and last terms of an AP are 1 and 11. If the sum of its terms is 36, then the number of terms will be
a) 5 b) 6 c) 7 d) 8
8) If the sum of n terms of an AP is 3n² + 5n then which of its terms is 164 ?
a) 26th b) 27th c) 28th d) none
9) If the sum of n terms of an AP is 2n² + 5n, then its nth term is
a) 4n -3 b) 3n -4 c) 4n + 3 d) 3n +4
13) If four numbers in AP are such that their sum is 50 and the greatest number is 4 times the least, then the numbers are
a) 5,10,15,20 b) 4,10,16,22 c) 3,7,11,15 d) none
18) If the first, second and last term of an AP are a, b and 2a respectively, then its sum is
a) ab/2(b - a)
b) ab/(b - a)
c) 3ab/2(b - a) d) none
23) If the first term of an AP is 2 and common difference is 4, then the sum of its 40 terms is
a) 3200 b) 1600 c) 200 d) 2800
24) The number of terms of the AP 3,7,11,15,.....to be taken so that the sum is 406 is
a) 5 b) 10 c) 12 d) 14 e) 20
1c 2d 3a 4b 5b 6b 7b 8b 9c 10c 11a 12d 13a 14d 15b 16b 17d 18c 19a 20c 21b 22b 23a 24d 25a
ARITHMETIC PROGRESSION (1)
2) The sum of the first 5 terms of an AP is 20, find its third term.
3) The 10th term an AP is 44 and the common difference is 4, find the first term of the series.
4) The 15th term of an AP and the 32nd term is 67, find the first term and the common difference.
5) The first two terms of an AP are 5 and 8 and the last term is 80, find the number of terms.
6) 3K + 1, 7k and 10K + 8 are in AP, what will be the common difference.
7) The third term of an AP is twice that of its first term and the fourth term is 15, find the common difference of the AP.
8) Find one arithmetic mean between -5 and 15.
9) Find the arithmetic mean between (x -y)² and (x + y)².
10) 2, x, 8, y are in AP, find the values of x and y.
11) In an AP the first term is (-5), the last term is 25 and the number of terms is 10. What is the sum of the series ?
12) A polygon has 25 sides , the lengths of which are in AP. If the perimeter of the polygon be 1100cm and the length of the largest side is 10 times that of the smallest, find the length of the smallest side.
13) The sum of how many terms of the AP 21 + 18 + 15 + 12 ....will be zero?
14) find the sum of the first 100 natural numbers.
15) If each term of a series in AP be multiply by 3. Would the series so obtained be again in AP?
16) Let Sₙ denote the sum of first n terms of an AP. If S₂ₙ = 3Sₙ then find the values of S₃ₙ/Sₙ.
1c 2d 3a 4b 5b 6b 7b 8b 9c 10c 11a 12d 13a 14d 15b 16b 17d 18c 19a 20c 21b 22b 23a 24d 25a
20/8/26
EQUATION OF STRAIGHT LINE
SAP-1
1) Find the gradient of the line joining the pair of points:
a) (0,3),(4,5)
b) (1/2,3/2),(5/2,7/2)
c) (√3+ 1,2),(√3+3, 4)
2) Find the slope of a line through each pair of the following points :
a) (2,3),(3,4)
b) (-5,-3) and (4,3)
3) Find the inclination of the line joining the pair of points :
a) (1,2),(2,3)
b) (2,-4),(3,-2)
4) If A(4,-3), B(6,5) and C(5,1) are 3 points, find the slope of AB and BC. Hence show that the points are collinear.
5) Show that the following points are collinear:
A(5.-2),(4,-1) and C(1,2). Hence find the inclination of the line AC .
6) If (-5,a),(3,6) and (7,8) are collinear , find a.
7) Find a, if the points (-2,3),(3,4) are collinear.
8) if the points (a,1),(1,2) and (0, b+1) are collinear, show that 1/a + 1/b = 1.
9) If 2y - p²x = 3 and 2y - 4px +1=0 are parallel, find the value of p.
10) If 3(k -1)y - 6x=2 and 4y - 8x +10= 0 are parallel, find the value of k.
11) If (p+1)x + y = 3, and 3y - (p -1)x =4 are perpendicular, find the value of p.
12) If y+ (2p+1)x +3=9 and 8y - (2p -1)x= 5 are perpendicular, find the value of p.
13) ABCD is a trapezium in which AD||BC and BC is parallel to the x-axis. If angle ABC=60° and AB= AD= DC,
calculate
a) the slop of AB
b) the slope of DC
c) the slope of diagonal BD
d) the slope of diagonal AC
14) The side BC of an equilateral ∆ ABC is parallel to the x-axis, as shown in the diagr.
Calculate
a) the slope of BC
b) The slope of AB
c) the slope of AC
15) The side of AB of a square ABCD is parallel to the y-axis, as shown in the adjoining diagram, 
calculate
a) the slope of AD
b) the slope of diagonal BD
c) the slope of diagonal AC.
4) 4, 4 5) 135 6) 2 7) 8 9) 4 10) 2 11) ±2 12) ±3/2 13) √3,-√3,1/√3-1/√3 14) 0,√3-√3 15) 0,1,-1
SAP-2
1) State the slope (m) and y-intercept (c) of the line
a) y= x+1
b) 3y= 6x -2
c) 3x + 2y -4=0
2) Find the slope of a line which is parallel to
a) 2y - x= 2
b) 2x - 3y/4= 2.
3) Find the Slope of a line which is perpendicular to
a) 3y - x = 5.
b) x - 2y/3= 3
4) Find the equation of the line with y-intercept of 3, and a slope of 2.
5) find the equation of the line with y-intercept of 4 and a slope of - 3.
6) Find the equation of the line with y-intercept of 1/2 and a slope of 2.
7) Find the equation of the line with y-intercept of 5, and a slope of -2/3.
8) Given , y- intercept of 2 and an inclination of 45° with the positive direction of the x-axis.
9) Find the equation of the line having y-intercept of 4 and which is equally inclined to the axes, in the second quadrant.
10) Find the equation the line through (0,2) parallel to y= 3x+2
11) find the equation the line through (-0,3) and parallel to 2y = x -2.
12) find the equation of the line (0,2) and perpendicular to
a) y= 2x -3
b) y= x/3+ 2.
13) find the equation of the line through (0,2) and (2,0).
14) find the equation the line through (3,0) and (0,3).
15) find the equation of the straight line through (2,3) and (3,4) y-intercept of 5 units .
16) find the slope of the line joining the points (3,4) and (0,16). Hence, or otherwise, write down the equation of this line.
17) The equation of the line is y= 3x -5. Write down the slope of this line and the intercept made by it on the y-axis. hence or otherwise, write down the equation of a line which is parallel to this line and which passes through the point (0, 5).
18) A(2,1), B(5,3), C(-1,3) are the vertices of the triangle ABC. Find
a) the equation of the median AD .
b) the equation of the altitude BE.
c) the equation of the altitude CF.
19) i) find the equation of the line through (-4, 8) and parallel to x-axis .
ii) find the equation of the line through (3,5) and perpendicular to the x-axis.
20) A(1,3) and C(6,8) at the opposite vertices of a square ABCD. Find the equation of the diagonal BD .
21) Find the equation of the perpendicular bisector of the line segment joining (3,2) and (7,6).
22) A(-1,4) and B(5,-2) are two points. Find the equation of the Perpendicular bisector of AB.
23) A straight line cuts off, on the axea of co-ordinates , positive intercepts whose sum is 7.
if the line passes through the point (-3,8), find its equation.
24) A straight line cuts off, on the coordinate axes, positive intercepts whose sum is 7. If the line passes through the point (-8,9), find its equation.
25) The coordinate of the vertex A of a square ABCD are (1,2) and the equation of the diagonal BD is x+ 2y = 10. Find the equations of the other diagonal and the coordinates of the centre of the square.
26) A, B are the points (0,6) and (10,0). O is the origin, OM is a median and OP an altitude of triangle AOB. Find the equations of OM and OP.
27) find the equation of a straight line passing through the point P(3,4)
such that the portion between the axes is divided by P in the ratio 2:3.
28) Find the equation of a line, which has the y-intercept 5 and is parallel to the line 4x - 6y = 9. Find the coordinates of the point, where it cuts the x-axis.
29) Match the equation A, B, C, D with the lines L₁, L₂, L₃, L₄, whose graphs are roughly drawn in the adjoining diagram.
A≡ y= 4x
B ≡ y- 3x + 4 =0
C≡ 2x +3y = 6
D≡ y= 3.
30) The midpoint the line segment AB shown in the diagram is P(6,-3).
Write down the co-ordinate of A and B. If OB= OC and Q divides AC in the ratio 2:1, find the coordinates of Q
19/8/26
x²- 7x +3=0.
Give your answer correct to two decimal places. (2)
b) If A= x 3
y 3
If A²= 3I, where I is the identity Matrix of order 2. Find x and y. (2)
c) Using ruler and compass construct a triangle ABC where AB= 3cm, BC= 4cm and angle ABC= 90°. Hence construct a circle circumscribing the triangle ABC. Measure and write down the radius of the circle. (2(
2)a) Use factor theorem to factorise 6x³+ 17x²+ 4x - 12 completely. (3)
b) Solve the following inequation and represent the solution set on the number line.
3x/5 + 2 < x +4 ≤ x/2 +5, x ∈R. (2)
c) Draw a Histogram for the given data, using graph paper:
Wages(Rs) No of people
3000-4000 4
4000-5000 9
5000-6000 18
6000-7000 6
7000-8000 7
8000-9000 2
9000-10000 4
Estimate the mode from the graph. (3)
3)a) In the figure given below, O is the centre of the circle and AB is a diameter.
If AC = BD and angle AOC=72°. Find :
i) angle ABC
ii) angle BAD
iii) angle ABD. (3)
b) Prove : sinA/(1+ cotA) - cosA/(1+ tanA)= sinA - cosA. (3)
c) In what ratio is the line joining P(5,3) and Q(-5,3) divided by the y-axis ? Also find the co-ordinate of the point of intersection. (2)
4) a) A solid spherical ball of radius 6cm is melted and recast into identical spherical marbles. Find the radius of each marble. (3)
b) Each of the letters of the word AUTHORIZES is written on identical circular discs and put in a bag. They are well shuffled . If a discount is drawn at random from the bag, what is the probability that the letter is:
i) a vowel. (3(
ii) one of the first 9 letters of the English alphabet which appears in the given word.
iii) one of the last 9 letters of the English alphabet which appears in the given word ?
c) Mr Bedi visit the market and buys the following articles:
Medicines cost Rs950, GST @ 5%
A pair of shoes costing Rs 3000, GST @18%
A laptop bag costing Rs1000 with a discount of 30%, GST @ 18%
i) Calculate the total of GST paid.
ii) The total bill amount GST paid by Mr Bedi. (3)
SECTION - B
Attempt any four questions from this Section:
5) A company with 500 shares of nominal value Rs120 declares an annual dividend of 15%, Calculate :
i) the total amount of dividend paid by the company.
ii) annual income of Mr. Sharma who holds 80 shares of the company.
If the return percent of Mr.Sharma from shares is 10%, find the market value of each share. (3)
b) The mean of the following data 16, Calculate the value of a:
Marks: 5 10 15 20 25
F: 3 7 a 9 6 (3(
c) The 4th, 6th and the last term of a geometric progression are 10, 40 and 640 respectively. If the common ratio is positive, find the first term, common ratio and the number of terms of the series. (3)
6) a) If A= 3 0 B= -4 2
5 1 & 1 0
find A²- 2AB + B² (3)
b) In the given figure AB= 9cm, PA= 7.5 cm and PC= 5cm.
Chords AD and BC intersect at P.
i) prove that ∆PAB ~ ∆PCD
ii) Find the length of CD.
iii) Find the area ∆PAB: area of ∆PCD. (3)
c) From the top of a cliff, the angle of depression of the top and bottom of a tower are observed to be 45° and 60° respectively. If the height of the tower is 20m. Find:
i) the height of the cliff.
ii) the distance between the cliff and the tower. (3)
7)a) Find the value of p if the lines, 5x - 3y +2=0 and 6x - py +7=0 are perpendicular to each other. Hence find the equation of a line passing through (-2, -1) and parallel to 6x - py +7= 0. (3)
b) Using properties of proportion find x: y, given
(x²+ 2x)/(2x +4)= (y²+ 3y)/(3y +9). (3)
c) In the given figure
TP and TQ are two tangents to the circle with centre O, touching at A and C respectively. If Angle BCQ= 55 and angle BAP= 60, find:
u) Angle OBA and OBC.
ii) angle AOC.
iii) angle ATC. (3)
8)a) What must be added to polynomial 2x³- 3x²- 8x, so that it leaves a remainder 10 when divided by (2x +1). (3)
b) Mr. Sonu has a recurring deposit account and deposit Rs 750 per month for 2 years. If he gets Rs 19125 at the time of maturity, find the rate of interest. (3)
c) Use graph paper for this question.
Take 1cm= 1unit on both x and y axes.
i) plot the following points on your graph sheets:
A(-4,0), B(-3, 2), C(0, 4), D(4,1) and E(7,3).
ii) Reflect the points B, C, D and E on x-axis and name them as B', C', D' and E' respectively.
iii) join the points A, B, C, D, E, E', D', C', B' and A' in order.
iv) Name the closed figure formed. (3)
9)a) 40 student enter for a game of shot-put competition. The distance thrown (in metres) is recorded below:
Distance No of students
12-13 3
13-14 9
14-15 12
15-16 9
16-17 4
17-18 2
18-19 1
Use a graph to draw an ogive for the above distribution.
Use a scale of 2cm = 1m on one axis and 2cm = 5 students on the other axis.
Hence using your graph find:
i) the median.
ii) upper quartile
iii) number of students who cover a distance which is above 33/2 m. (3)
b) x= {√(2a +1) + √(2a -1)}/{√(2a +1) - √(2a -1)}, show that x² - 4ax +1=0. (3)
10) a) If the 6th term of an AP is equal to four times its first term and the sum of first 6 terms is 75, find the first term and the common difference. (3)
b) The difference of two natural number is 7 and their product is 450. Find the numbers. (3)
c) Use ruler and compass for this question.
Construct a circle of radius 4.5 cm
Draw a chord AB= 6cm.
i) Find the locus of the points of equidistant from A and B.
Mark the point where it meets the circle as D.
ii) Join AD and find the locus of points which are equidistant from AD and AB. Mark the point where it meets the circle as C.
iii) Join BC and CD. Measure and write down the length of side CD of the quadrilateral ABCD. (3)
11)a) A model of a high rise building is made to a scale 1:50.
i) If the height of the model is 0.8m, find the height of the actual building.
ii) if the floor area of a flat in the building is 20m², find the floor area of that in the model. (3)
b) From a solid wooden cylinder of height 28cm and diameter 6cm, two conical cavities are hollowed out. The diameter of the cones are also of 6cm and height 10.5cm (π= 22/7)
Find the volume of the remaining solid. (3(
c) prove: {(1- tanx)/(1- cotx)}²= tan²x (3)
Algebra - Test
1) Mr Desai opens a recurring deposit of Rs 2000 per month for 31 months paying simple interest of 12% p.a. Calculate the amount receives at the time of maturity.
1) Mr Desai opens a recurring deposit of Rs 2000 per month for 31 months paying simple interest of 12% p.a. Calculate the amount receives at the time of maturity.
2) Which is better investment ---12% shares at 90 or 15% at 120?
3) A man invests a sum of money in Rs 20 shares, paying 15% dividend, quoted at 20% premium . If his annual dividend is Rs 720, find
a) his total investment
b) the rate of interest on his investment.
4) A man bought 3200 shares of Rs 10 paying 15% per annum. He sold them when the price rose to Rs 28 and invested the proceeds in Rs fifty shares paying 11% per annum at 12% premium . Find the annual change in his income.
ARITHMETIC PROGRESSION
1) if 7th and 13th terms of an AP be 34 and 64 respectively, then its 18th term is
a) 87 b) 88 c) 89 d) 90
2) If the sum of p terms of an AP is q and the sum of q terms is p, then the sum of p+ q terms will be
a) 0 b) p- q c) p+ q d) -(p+ q)
3) If the sum of n terms of an AP be 3n²- n and its common difference is 6, then its first term is
a) 2 b) 3 c) 1 d) 4
4) Sum of all two digit numbers which when divided by 4 yield unity as remainder is
a) 1200 b) 1210 c) 1250 d) none
5)
6)
7) The first and last terms of an AP are 1 and 11. If the sum of its terms is 36, then the number of terms will be
a) 5 b) 6 c) 7 d) 8
8) If the sum of n terms of an AP is 3n² + 5n then which of its terms is 164 ?
a) 26th b) 27th c) 28th d) none
9) If the sum of n terms of an AP is 2n² + 5n, then its nth term is
a) 4n -3 b) 3n -4 c) 4n + 3 d) 3n +4
10)
11)
12)
13) If four numbers in AP are such that their sum is 50 and the greatest number is 4 times the least, then the numbers are
a) 5,10,15,20 b) 4,10,16,22 c) 3,7,11,15 d) none
14)
15)
16) The first and last term of an AP are a and l respectively. If S is the sum of all the terms of the AP and the common difference is given by (l² - a²)/{k - (l+ a)}, then k=
a) S b) 2S c) 3S d) none
17)
18) If the first, second and last term of an AP are a, b and 2a respectively, then its sum is
a) ab/2(b - a)
b) ab/(b - a)
c) 3ab/2(b - a) d) none
19)
20)
21)
22)
23) If the first term of an AP is 2 and common difference is 4, then the sum of its 40 terms is
a) 3200 b) 1600 c) 200 d) 2800
24) The number of terms of the AP 3,7,11,15,.....to be taken so that the sum is 406 is
a) 5 b) 10 c) 12 d) 14 e) 20
25)
PAPER - 4
1) a)
b) Solve: 21x²- 8x -4=0. (2)
c)
d) Find the coordinates of the image of (5, -4) after reflection in
i) x= 0
ii) y= 2. (2)
e) list the solution set of the following inequation and graph the solution set :
1/2+ 8x > 5x - 3/2, x belongs to Z. (2)
2) a) Calculate the ratio in which the line joining A(6,5) and B(4,-3) is divided by the line y= 2. (2)
b) In the figure, BC is parallel to DE.
Area of the triangle ABC= 25cm²,
Area of trapezium BCED= 24cm² and DE= 21 cm. Calculate the length of BC. (3)
c)
3) a) Calculate the mean, median and mode of the following numbers :
13, 11, 15, 13, 14, 15, 13, 17, 12, 16. (2)
b) Given A= 1 1
8 3
Evaluate A²- 3A. (3)
c)
4) a) Construct a triangle ABC with AB= 7cm, BC= 8cm and angle ABC=60°. Locate by constuction of the point P such that
i) P is equidistant from B and C.
ii) P is equidistant from AB and BC.
iii) measure and record the length of PB. (3)
b)
c) In the figure, I is the incentre of the circle. AI produced to meet the circle in D. Calculate
i) Angle DCB
ii) Angle IBC.
iii) angle BID
iv) angle BIC
Give angle BAC=50° and angle ABC=64°. (3)
5) a) Show that: √{(1- cosA)/(1+ cosA)}= sinA/(1+ cosA). (3)
b) In the figure, AB is a common tangent to two circles intersecting at C and D. Write down the measure of (angle ACB+ angle ADB). (2)
c)
6) a)
b) The surface area of a solid metallic sphere is 1256cm². It is melted and recast into solid right circular cones of radius 2.5cm and height 8cm. Calculate
i) the radius of the solid sphere.
ii) the number of cones recast (π=3.14). (3)
7)a) A dividend of 9% was declared on Rs 100 shares selling at a certain price. If the rate of return is 15/2%, find
i) the market value of the share.
ii) the amount to be invested to obtain an annual dividend of Rs 630. (3)
b) In the figure, AB and CD are the lines 2x - y+ 6=0 and x - 2y= 4 respectively.
i) write down the coordinates of A,B,C,D.
ii) prove that triangles OAB and ODC are similar.
iii) is figure ABCD cyclic ? (3)
8) a)
b)
c)
9) a)
b)
10)a) The following table shows the distribution of the heights of a group of factory workers:
Ht(cm) no of workers
140-145 6
145-150 12
150-155 18
155-160 20
160-165 13
165-170 8
170-175 6
i) Determine the cumulative frequencies
ii) write down the median height in cm. (3)
b) The hotel bill for a number of people for overnight stay is Rs 4800. If there were 4 more, the bill each person had to pay would have reduced by Rs 200. Find the number of people staying overnight. (3)
11) a) ABCD is a rhombus. The coordinates of A and C are (5,8) and (-1,2) respectively. Write down the equation of BD. (3)
b)
TEST PAPER - 24
1)a) The lines given by ax+ 4y= 8 and 2x+ 3y +2=0 are mutually perpendicular. Find the value of a. (2)
b) AB and CD are two chords of a circle intersecting at a point P outside the circle. Find if PC= 30cm, CD= 14cm and PA= 24cm. Determine AB. (2)
c)
d) Find the total surface area of an open pipe of length 50 cm, external diameter 20cm and internal diameter 6cm. (2)
e)
2)a)
b) Find the equation of the lines which passes through the point (-2,3) and are equally inclined to the co-ordinate axes. (3)
c) If P= 2 6 & Q= 3 x
3 9 y 2
Find x and y such that PQ= null matrix. (3)
3) a) Using remainder theorem, factorise the expression 3x³+ 10x²+ x - 6. (3)
b)
c) If (a²+ b²)+x²+ y²)= (ax + by)², show that a/x = b/y. (3)
4) a) If a= 4√6/(√2+ √3), find the value of
(a+ 2√2)/(a- 2√2) + (a+ 2√3)/(a- 2√3). (3)
b)
c) Mr. Ahuja invests equal sums of money in two companies offering 8% shares of Rs 100 at a premium of Rs 25 at 6% shares of Rs 100 at a discount of Rs 25. He finds that if with his total investment he has bought equal number of shares of each company, his income would have been Rs142 less. Find his total investment . (3)
5) a)
b)
6) a)
b) A recurring deposit account of Rs 1200 per month has a maturity value of Rs 12440.00. if the rate of interest is 8% and the interest is calculated at the end of every month, find the time of the recurring deposit account. (3)
7) a) i(plot the point A(3,5) and B(-2,4).
ii) A' is the image of A when reflected in the x-axis . Write down the coordinates of A'b and plot it.
iii) B' is the image of B when reflected in the y-axis, followed by reflection in the origin, Write down the coordinates of B'.
iv) write down the geometrical name of the figure AA'BB'.
v) Name two invariant points under the reflection in the x-axis . (3)
b) The point P(5,-4) divides the line segment AB as shown in the figure in the ratio 2:5. Find the coordinates of points A and B. 3
8) a) The vertices of a ∆ ABC are (0,5), B(-1,-2) and C(11,7). Write down the equation of BC . Find
a) i) the equation of line through A and perpendicular to BC.
ii) the coordinates of the point P, where the perpendicular through A as obtained in (i) meets BC. (3)
b)
9) a) Find the mean of the given data:
C. I frequency
63-70 9
70-77 13
77-84 27
84-91 38
91-98 32
98-105 16
105-112 15 (3)
b)
10a)
b) A, B are the points (9,6) and (10,0). O is the origin, OM is a median and OP is an altitude of ∆ OAB. Find the equations of OM and OP. (3)
11) a) prove: (SinA + cosecA)² + (cosA + secA)²= 7+ tan²A + cot²A. (3)
b)
c) If A= -1 1
a b and A²= I, find a, b.
FACTOR THEOREM
Sap-2
1) Find the value of p if the polynomial f(x)= x³- 3x² - px + 24 is divisible by g(x)= x + 3. Hence find all the factors.
2) Find the value of q if the polynomial f(x)= 2x³ + qx² - 7x -12 is divisible by g(x)= x + 4. Hence find all the factors.
3) Find the value of q if the polynomial f(x)= x³ + 2x² - 13x + q is divisible by g(x)= x -2. Hence find all the factors.
4) Find the value of p if the polynomial f(x)= x³ - px² - x +3 is divisible by g(x)= x² -1. Hence find all the factors.
5) Find the value of p and q if the polynomial f(x)= px³ + qx² - 8x -12 is divisible by g(x)= x² - 4. Hence find all the factors.
6) Find the value of p and q if the polynomial f(x)= px³ + 6x² + qx + 6 is divisible by g(x)= x²+ 4x + 3. Hence find all the factors.
7) Find the value of p if the polynomial f(x)= x³ + px² - 16x +8 has g(x)= x -2 as one of the factors. Hence find the remaining two factors.
8) Find the value of q if the polynomial f(x)= 2x³ + qx² - x -15 has g(x)= 2x + 3 as one of the factors. Hence find all the factors.
9) Factorise 6x³ - 11x² - 3x +2. Completely.
Sap-1
1) Using factor theorem, find out whether polynomial g(x) is a factor of f(x) or not:
a) f(x)= 3x²- 2x - 8, g(x)= 3x +4.
b) f(x)= 9x²- 4a²+ 4ay - y² - 8, g(x)= 3x + 2a - y.
c) f(x)= x³- 3x² + 4x - 2, g(x)= x -1.
d) f(x)= x²(x - 14)+ 37x - 60, g(x)= x -2.
e) f(x)= x³+ x² - 2x - 30, g(x)= x -3.
2) Find the value of k when f(x)= x³- 3x² - x +k, g(x)= x +1 is a factor of f(x).
3) Find the values of p and q if g(x)= x +2 is a factor of f(x)= x³- px² + x + q and f(2)= 4.
4) Find the values of p and q if g(x)= x -4 is a factor of f(x)= 4x³- px² + qx -216 and f(-1)= -225.
5) Find the values of a and b if p(x)= x +2 is a factor of q(x)= ax³- bx² + 2(x -2) and q(2)= 20.
6) Find the value of a if x+ a is a factor of f(x)= x³ + a(x² +1) - 2x + 4.
7) Find the remainder, if f(x)= x³- 2x² + x - 3 is divided by g(x)= x +2.
8) Find the remainder, if f(x)= 2x³- 3x² -4x - 5 is divided by g(x)= 2x + 1.
9) If g(x)= 2x -3 is a factor of f(x)= 2x³- 9x² + x + p, find the value of p. Hence find all the factors.
10) If g(x)= 2x - 2 is a factor of f(x)= 3x³ + x² + px + 12, find the value of p. Hence find all the factors of f(x)
Sap-2 LINEAR INEQUATION- TEST
1) Solve the following inequations:
a) -5<-1-2x ≤ 3, x ∈N
b) -8< x - 7 ≤ 2 - 2x; x ∈ N
c) - 8/3 < - x/3 -1 ≤ -4/3; x ∈ R
d) - 17/5≤ x - 8/3 < 1/3 ; x ∈ R
e) -1≤ 2x -3 < 17 - 3x; x ∈N
f) -10≤ 2x -12 < 8 - 3x ; x∈ W
g) 2≤ x + 3 ≤ 14 - 2x ; x ∈ I
h) -20< x - 19 ≤ -2x - 8; x ∈ W
2) List one of the solution set of 1/8< m/n < 1/7, where m,n ∈ Z.
3) Given P={x: 8< 2x +2 ≤ 14, x ∈ R}
Q={ x: -5≤ -1+ 4x < 19, x ∈ I}
Represent P and Q on number lines. Write down the elements of P∩Q.
4) State for each of the following statements whether it is true or false:
a) If (x - a)(x - b)< 0, then x< a and x < b.
b) If a< 0 and b< 0, then (a+ b)²> 0.
c) If a and b are any two integers such that a> b, then a²> b².
d) If p= q+2, then p> q.
e) If a and b are two negative integers such that a< b, then 1/a < 1/b.
5) The diagram represents two inequations A and B on real number lines
i) Write down A and B in Rooster form.
ii) Represent A ∩B and A - B on two different number lines.
LINEAR INEQUATIONS
SAP-1
1) Solve and hence illustrate on the number line
a) 2x - 3< 5x -3≤ 12, x∈N. Hence .
b) 3(x -2)≥ 2x -3, x ∈R.
c) (x -2)/(2x +5) < 1/3, x ∈ R.
d) Given A={x: -8<5x +2≤ 17, x ∈ I}
B={x: -2≤7+3x < 17, x ∈ R}
Represent A and B on two different number lines. Write down the elements of A∩B.
e) 2x +5< 9; x ∈N.
f) 5x -20÷ 4; x ∈N
g) 5x ≤ 20, x ∈ W
h) 2p/3 + 1 > 3p -2, p∈ R.
2) The diagram represents two inequation A and B on real number lines
i) Write down A and B in set builder notation.
ii) Represent A ∩B and A ∩B' on two different number lines.
3) If the replacement set is {-2,-1,+1,+2,+4,+5,+9}, what is the solution set of each of the following mathematical sentences?
i) x+ 3/2> 5/2.
ii) x -4< -3.
iii) 2x -5≥ 10.
iv) 3y -2≤ 5/2.
4) Translate the following sentences into open sentences:
i) 5 more than 4 times a number.
ii) 2 less than half a number.
iii) Mother's age is 10 years greater than 3 times her daughter's age.
iv) Sum of a number and its reciprocal equals to 2.
v) The length of a rectangle of perimeter 8 is 3 times its width.
5) In the following graphs match each group of column A with one of the sets given in column B.
6) Write open mathematical sentences using, x for variable, whose graphs are the following:
7) Write open mathematical sentences using, x for variable, whose graphs would be
8) Figure shows the graph of the following lines:
Explain the meanings of the following ring, the solid rings and the arrowheads in the diagrams.
9) If the replacement set={-8,-7,....-1,0,1,2,....+8}.
List the solution set of the following:
i) {x: x> 6}
ii) {x: -2≤x≤0}
iii) {x: x< 8}
iv) {x: 0≤2x -3≤ 6}
v) {x: x²< 24< x³}
10) If x∈ {x: -5< x < +5 and x ∈ I} , find the truth sets of the following:
a) 7x > -10
b) 2(3x -5)< 6
c) 7x²÷3x > 4/3
d) x + 1/x = 2
e) 7x²+ 2 ≥ x(7x +2)
11) P is the solution set of 1/x > 3/4 and Q is the solution set of:
x(1- 1/x)≥ 5(x -1), where x ∈ W. Find the set P∩Q.
12) P is the solution set of 8x -1> 5x +2 and Q is the solution set of 7x -2≥ 3(x +6), where x ∈N. Find the set P∩Q
PAPER- 2
1) The price of a TV set inclusive GST of 9% is 40221. Find the marked price.
2) If x: y= 4:3, find (5x +8y): (6x - 7y).
3) Using the reminder theorem, find the remainder when y³- 7y²+ 15y - 19 is divided by y- 3.
4) State and draw the locus of a point eqidistance from two parallel lines.
5) The given figure, the medians QS and RT of a ∆ PQR meet at G. prove that:
a) ∆ TGS~ ∆ RGQ
b) QG= 2 GS from (a) above.
6) Solve the following inequation and graph the solution on the number line:
2x -5≤ 5x +4 < 11, x belongs to R.
7) The marks of 20 students in a test were as follows : 5, 6, 8, 9, 10, 11, 11, 12, 13,13, 14, 14, 15,15, 16,16 18, 19 20. Calculate:
a) the mean
b) the median
c) the mode
8) If the matrix
A= 1 -4 & B= -3 2 & C= 4 0
4 1 4 0 0 -3 find
a) A² b) BC c) A²+ BC .
8) The point A(3,4) is reflected to A' in the x-axis, and O' is the image of O(the origin) when reflected in the AA'. Using graph paper, give
a) the coordinates of A' and O'.
b) the lengths of the segments AA' and OO'.
c) the perimeter of the quadrilateral AOA'O'.
d) the geometrical name of the figure AOA'O'.
9) Prove the following identity:
1/(sinA + cosA) + 1/(sinA - cosA)= 2sinA/(2 sin²A -1).
10) In the given figure, AB is the diameter of a circle with centre O. Angle BCD is 130°. Find
a) angle DBA
b) angle BAD.
11) Find the equation of a line passing through the point (-4,6) and having the x-intercept of 8 units.
12) A man wants to buy 72 shares available at Rs 150 (per value of Rs 100).
a) How much should he invest ?
b) if the dividend is 7.5%, what will be his annual income ?
c) if he wants to increase his annual income by Rs 300, how many extra shares should be buy ?
13) The following table gives the weekly wages of workers in a factory:
weekly wages (Rs). No. of workers
150-150 5
155-160 20
160-165 10
165-170 10
170- 175 9
175-180 6
180-185 12
185- 190 8 Calculate
a) the mean
b) the model class
c) the numbers workers getting weekly wages, below Rs 180.
d) the number of workers getting Rs 165 or more, but less than Rs 185 as weekly wages.
14) A hollow sphere of internal and external diameters 8 cm and 16 cm respectively , is melted into a cone of base diameter 16 cm. Find the height of the cone.
15) The shadow of a vertical tower AD on level ground is increased by 30m, when the altitude of the sun changes from 45° to 30° as shown in the given figure.
Find the height of the tower and give your answer correct to 1/10 of a metre.
16) The marks obtained by 240 students in a mathematics test is given below:
Marks No. if students
00-10 10
10-20 18
20-30 32
30-40 44
40-50 52
50-60 26
60-70 22
70-80 12
80-90 16
90-100 8
Draw an ogive for the given distribution on a graph sheet. Use a suitable scale for your ogive and using ogive, estimate:
a) the median
b) the lower quartile
c) the number of student who obtained more than 75% in the test :
d) the number students who did not passing inthe test if the pass percentage was 40.
17) P(2,4), Q(3,3) and R(7,5) are the vertices of a ∆ PQR. Find
a) the coordinates of the centroid G of ∆ PQR.
b) the equation of a line, through G and parallel to PQ.
18) An aeroplane travelled a distance of 800 km at an average speed of x kmph. On the return journey, the speed was increased by 40 kmph. Write down an expression for the time taken for:
a) the onward journey .
b) the return journey .
If the return journey took 40 minutes less then the onward journey, write down an equation in x and find its value.
Paper - 1
1) The point P(a,b) is reflected in the x-axis to obtain the point Q(3,-4). Find a and b. (1)
2) If A= a 3a & B= 2 & C= 5
b 4b 1 12 find a and b when the relation AB= C. (1)
3) The mean of the number 6, y, 7, x and 14 is 8. Express y terms of x. (1)
4) Solve using the quadratic formula, x²- 5x -2=0. Give your answer correct to 3 significant figures. (2)
5) If (8a + 5b)/(8c + 5d)= (8a - 5b)/(8c - 5d), prove that a/b = c/d. (1)
6) Find the value of k, if x - k is a factor of x³- kx²+ x + 4. (1)
7) Solve 1< 3x -3≤ 11, x ∈ R and mark it on a number line. (1)
8) Calculate the mean, median and mode of the following numbers : 12, 11, 10, 11, 12, 13, 14, 13, 15, 13. (2)
9) In the diagram,
chords AB and CD of the circle are produced to meet at O. Given that CD= 4cm, DO= 12cm and BO= 6cm, calculate AB . (2)
10) If cosA= 4/5 and cosB= 24/25; evaluate
a) cosec²A
b) cotA + cotB. (2)
11) on a map drawn to a scale 1:125000, a triangular plot of land has the following measurements :
PQ=10cm, QR= 8cm, angle PRQ= 90°. Calculate
a) the actual length of PQ in km.
b) the area of the plot in square kilometres. (2)
12) The work done by (2x -3) men in (3x +1) days and work done by (3x +1) men in (x +8) days are in the ratio of 11:15. Find the value of x. (2)
13) Find the mean of the following frequency distribution:
Class interval frequency
00-30 3
30-60 7
60-90 15
90-120 14
120-150 7
150-180 4 (3)
14) A man invests Rs 30800 in buying shares of nominal value Rs 56 at 10% premium . The dividend on the shares is 18% per annum. Calculate
a) The number of shares he buys.
b) The dividend he receives annually.
c) The rate of interest he gets on his money. (3)
15) prove that: sinx/(1- cotx) + cosx/(1- tanx)= sinx + cosx. (2)
16) A straight line passes through the points A(-2,8) and B(10,-4). It intersects the coordinate axes at points E and F. P if the midpoint of the segment EF. 
Find
a) the equation of the line.
b) the coordinate of E and F.
c) the coordinates of the point P. (3)
17) In an auditorium, seats were arranged in rows and columns . The number of rows was equal to the number of seats in each row. When the number of rows was doubled and the number of seats in each row was reduced by 15, the total number of seats increased by 400. Find
a) The number of rows in the original arrangement.
b) the number of seats in the auditorium after rearrangement. (3)
18) Draw a histogram and hence estimate the mode for the following frequency distribution:
Class frequency
00-20 3
20-40 8
40-60 10
60-80 6
80-100 4
100-120 3 (3
19) A man standing on the bank of a river observes that the angle of elevation of a tree on the opposite bank is 60°. When he moves 40m away from the bank, he finds the angle of elevation to be 30°. Calculate:
a) the width of the river and
b) the height of the tree. (3)
20) Find a and b, if
a= 3 -2 & B= 2a & C= 4 & D= 2
-1 4 1 5 b with the relation AB + 4C = 3D. (2)
21) A vessel is in the form of an inverted cone. Its height is 15cm and the diameter of its top which is open, is 5cm. It is filled with water up to the rim. When lead shots, each of which is a sphere of diameter 5mm are dropped into the vessel, 1/3 of the water flows out. Find the number of lead shots dropped into the vessel. (3)
22) In the given circle
with diameter AB, find the value of x. (2)
23) Find the value of k for which the lines kx - 7y + 5=0 and 6x - 2y +9=0 are perpendicular to each other. (3)
1)a) Find the rate of GST levied on a car that was sold at a price 3 times its marked price. (1)
b) If the sum of the series 2+5+8+11......is 60100, then the number of terms are
a) 100 b) 200 c) 150 d) 250
2) When 7x²- 3x + 8 is divided by (x -4), find the remainder (using remainder theorem). (1)
3) If 2 cosx x = 2/5, find sinx. (1)
4) Calculate the length of the tangent drawn to a circle of diameter 8cm from a point 5cm away from the centre of the circle. (1)
4) If x²,4 and 9 are in continued proportion , find the value of x. (1)
5) If x ∈Z, find the solution set for the inequation 5< 2x -3≤ 14 and graph the solution on a number line. (1)
6) Find p and q if g(x)= x +2 is a factor of f(x)= x³- px + x + q and f(2)= 4. (2)
7) 1 -2 0
If X= -3 4 & Y= 1
a) Find the matrix Z such X + Z is a zero matrix.
b) Find the matrix M such that X + M = X.
c) Find XY. (3)
8) a) If 7 is the mean of 5, 3, 0.5, 4.5, b, 8.5, 9.5, find b.
b) If each observation is decreased in value by 1 unit, what would the new mean be ? (2)
9) In the figure below,
AB is a chord of the circle with centre O and BT is tangent to the circle at B, if angle OAB= 32°, Find the value of x and y. (2)
10) Construct a regular pentagon of side 3cm. Draw the lines of symmetry. (2)
11) The volume of a cylinder 14cm long is equal to that of a cube having an edge 11cm. Calculate the radius of the cylinder. (3)
12) A piece of butter 3cm by 5cm by 12cm is placed on a hemispherical bowl of radius 3.25cm. Will the butter overflow when it melts completely. (3)
13) A company with 10000 shares of Rs 50 each declares an annual dividend of 5%.
a) What is the total amount of dividend paid by the company ?
b) What would be the annual income of a man who has 72 shares in the company?
c) if he receives only 4% on his investment, find the price he paid for each share. (3)
14)a) State the equation of the mirror line, if point A(5,0) on reflection is mapped as A'(-5, 0).
b) State the equation of the the mirror line, if point B(4,-3) on reflection is mapped as B'(4,3).
c) Point C(-3,5) on reflection in y=2 is mapped as C'. Find the coordinates of C. (3)
15) Tanya standing on a vertical cliff in a jungle observes two rest-horses in a line with her on opposite sides deep in the Jungle below. If their angles of depression are 30° and 45° and the distance between them is 200mp, find the height of the cliff. (3)
16) Find the equation of a line that passes through (1,3) and is parallel to the line y= -3x +2. (2)
17) In the given figure,
calculate
a) angle APB
b) angle AOB. (2)
18) The midpoint of the line joining A(2,p) and B(q,4) is (3,5). Find the numerical values of p and q. (2)
19) From the following table, find:
a) average wage of a worker. Give your answer, to the nearest paise
b) Modal class.
Wages in Rs No of workers
Less than 10 15
Less than 20 35
Less than 30 60
Less than 40 80
Less than 50 96
Less than 60 127
Less than 70 190
Less than 80 200 (3)
20) Examine the ogive given below
which shows the marks obtained out of 100 by a set of students in an examination and answer the following questions:
a) How many students are there in the set ?
b) How many students obtained 40% marks ?
c) How many students obtained 90% and above ?
d) What is the median marks? (4)
21) Show that: √{(1+ cosx)/(1- cosx)}= cosecx + cot x. (2)
22) if the sum of the first four terms of an AP is 4 and the second term is -5 then find the common difference
LINEAR INEQUATIONS
1) If 2x - 7 < 4, where x is a natural number less than 8, than the solution set is:
a) {0,1,2,3,4} b) {1,2,3,4,5} c) {1,2,3,4,5,6} d) {0,1, 2,3,4,5,6}
2) If - x ≥ -3 then:
a) x≤ -3 b) x ≥ 3 c) x = 3 d) x ≤ 3
3) if 2 + 4 x< 2 x - 5 ≤ 3x ∈Z, then the solution set is :
a) {5,4} b) {- 5,-4} c) {- 5, -4, -3} d) {- 4,-3,-2,-1}
4) if 2≤2x - 3 ≤ 5, x∈ R, then the solution set is:
a) {2.5≤ x ≤ 4, x∈ R} b) {2≤ x ≤ 5, x∈ R} c) {3≤ x ≤ 5, x∈ R} d) {2< x < 4, x∈ R}
5) If a> b, then:
a) a - c ≤ b - c b) a - c≥ b - c c) a - c = b - c d) a - c > b - c
6) If x≥ 5 and- ax ≥ 5a, then :
a) a > 0 b) a < 0 c) both a and b d) neither a nor b
7) If x+1≥ 13 - 5x, x ∈{1,2,3,4.....10}, then the solution set is:
a) {1,2,3,4,5,6} b) {6,7,8,9,10} c) {7,8,9,10} d) {6,7,8.....}
8) If 7 - 5x ≥ 3x -1, then the solution set, when x ∈ W is:
a) {0,1} b) {0} c) {1} d) {0,1,2}
9) Given a >0, b >0, c >0 and d <0, then a < b implies :
a) a+ d> b + d b) a - d < b - d c) a - d > b - d d) a + d = b + d
10) Given 2x - 5≤ 5x +4 < 11. If x ∈ I, the solution set is:
a) {-2,-1,0,1} b) {-3,-2,-1,0,1} c) {-3,-2,-1,0} d) {-2,-1,0,1}
11) For the inequation -12< 3 - 4x ≤ 11, x ∈ N, the solution set on the number line can be shown as:
a) 5 b) 4 c) 3 d) 2
13) If 2x - 5≤ 5x + 4 < 11, x ∈ I, then the solution set can be represented as:
a) -2 b) -1 c) 0 d) 1
15) If -9(x -7)≥ 45 - 21x > x +1, x ∈ R, then the solution set is:
a) {-3/2≤ x < 2, x ∈R}
b) {-3/2 < x < 2, x ∈R}
c) {-2/3 ≤ x ≤ 1, x ∈R}
d) {-1/3 ≤ x ≤ 2, x ∈R}
16) If 2x - 5 ≤ 5x + 4 < 11, x ∈ I, then the smallest whole number for x is:
a) 0 b) 1 c) -3 d) 2
17) If 5 - 3x < 11, x ∈ R, then the solution set is:
a) {x> -2, x∈R} b) {x≥ -2, x∈R} c) {x< 2, x∈R} d) {x< -2, x∈R}
18) Given 3x -1 ≤ x +5, x ∈N, then the solution set is:
a) {1,2,3} b) {1,2,3,4} c) {1,2} d) {0,1,2,3}
19) If 8 < 5(x +1) -2 ≤ 18, x ∈R, then the smallest integer value of x is:
a) 1 b) 0 c) -1 d) 2
20) Given a >0, b >0, c >0 and d <0. Then a > b implies:
a) ad >bd b) ad = bd c) ad < bd d) none
SHORT ANSWER TYPE QUESTIONS
1) Find the value of x, which satisfies the inquation -2≤ 1/2 - 2x/3 ≤ 11/6, x ∈N.
Graph the solution set on the number line.
2) Solve the following inequation, write the solution set and represent it on the number line.
-3(x -7)≥ 15 - 7x > (x +1)/3, x ∈R
3) Solve the following inequation, write down the solution set and represent it on the real number line:
-2+ 10x ≤ 13x +10 < 24+ 10x, x ∈ Z
4) Solve the following inequation and write down the solution set:
11x - 4 < 15x +4 ≤ 13x + 14, x ∈ W
Represent the solution on a real number line.
5) Solve the given inequation and graph the solution set on the number line:
2y - 3 < y +1 ≤ 4y +7, y ∈ R
6) Solve the following inequation and represent the solution set on the number line:
2x -5 ≤ 5x +4 < 11, x ∈I
7) Solve the following inequation and write the solution set:
13x -5 < 15x +4 < 7x +12, x∈R
1) Find the mean of the following distribution:
x: 4 6 9 10 15
f: 5 10 10 7 8
2) Following table shows the weights of 12 students:
Weight (in kgs): 67 70 72 73 75
No of students: 4 3 2 2 1
Find the mean weight.
3) Find the mean of the following distribution:
X: 10 30 50 70 89
F: 7 8 10 15 10.
4) If the mean of the following distribution is 6, find the value of p.
X: 2 4 6 10 p+5
F: 3 2 3 1 2
5) Find the value of p, if the mean of the following distribution is 7.5.
X: 3 5 7 9 11 13
F: 6 8 15 p 8 4
6) Find the missing frequencies in the following frequency distribution if it is known that the mean of the distribution is 1.46.
No of accidents(x): 0 1 2 3 4 5 total
Frequency (f): 46 ? ? 25 10 5 200
BOOSTER - C
1) The following table shows the weight of 12 students:
Weight (in kg): 67 70 72 73 75
No of students: 4 3 2 2 1
Find the mean weight. 70.25 kg
2) Find the mean wage from the data given below:
Wages: 800 820 860 900 920 980 1000
F: 7 14 19 25 20 10 5 891.2
3) Apply step-deviation method to find the AM of the distribution:
Variate (x) Frequency(f)
5 20
10 43
15 75
20 67
25 72
30 45
35 39
40 9
45 8
50 6 22.214
3) The weight in kilograms of 60 workers in a factory are given in the following frequency table. Find the mean weight of a worker.
Weight (in kg): 60. 61 62 63. 64 65
No of workers: 5 8 14 16 10 7 62.65
4) The table given the distribution of villages under different heights from sea level in a certain region. Compute the mean height of the region:
Height: 200 600 1000 1400 1800 2200
F: 142 265 560 271 89 16 984.51
CENTRAL TENDENCY
1) The weight of 6 persons in a firm are 64, 66,63,69,75,68 kg respectively. What is their mean weight?
a) 56.7 b) 67.5 c) 76.5 d) 65.7
2) The profit (in Rs) of a small shopkeeper of a week is 207,205,210,221,230,204,218. What is his mean profit per day?
a) 115 b) 225 c) 215 d) 125
3) The AM of 1,3,5,.......,29 is
a) 13 b) 15 c) 14 d) 16
4) The word length in each of the 40 words are as follows:
X: 2 3 4 5 7
F: 6 8 12 10 4
What is the mean of the above distribution?
a) 4.35 b) 4.05 c) 4.25 d) 4.15
5) The AM of a variable x is 40. What will be the mean of the variable y, when y= 4x - 10.
a) 160 b) 165 c) 155 d) 150
6) The mean age of a group of 30 girls is 20 years, and that of a group of 20 boys is 30 years. If the two groups are taken together to form a new group, what is the mean of this group?
a) 22 b) 24 c) 23 d) 25
7) The mean marks of 170 students in a certain class is 75. The mean mark of boys in the class is 85, and of the girls is 65. Find the number of boys and girls in the class.
a) 85,85 b) 65,105 c) 75,95 d) 80,90
8) The mean marks obtained in an examination by two gr of students were found to be 75, 85 respectively. What will be the ratio of students in the two groups, if the mean marks for all students was 100.
9) Suppose x takes the value 2,4,6,8,10,12. Then what will be the value of ∑(xᵢ - mean x)?
a) 0 b) 1 c) 2 d) 3
10) Sum of the deviation from mean is
a) 0 b) 1 c) mean of the number d) can't determine
11) For a set of 10 observations, what will be the mean for the values of x: 10,10,10,.....10.
a) 8 b) 9 c) 10 d) 11
12) The AM of 7 items is 10. If one more item is added to the series, then the AM becomes 12. Find the value of 8th item.
a) 24 b) 26 c) 28 d) none
13) What will be the weighted AM of the first n natural numbers when the numbers are weighted by the corresponding numbers?
a) (n+1)/3 b) (2n+1)/2 c) (2n+1)/3 d) (2n+3)/2
14) The AM of a set of values is 15. If 5 is added to each value, the new AM will be
a) 10 b) 15 c) 20 d) none
15) The AM of 1, 2,2²,......2⁹ is
a) 18.341 b) 18.431 c) 18.143 d) 18.413
16) The weighted AM of first 11 natural numbers whose weights are equal to the corresponding numbers, is
a) 6.77 b) 8.55 c) 7.66 d) none
17) The marks of 7 students in a test in statistics are 0,100,12,18,17,10,32. A suitable average of these marks is
a) Mean b) median c) mode d) none
18) If 3,9,6,7,10,8,4,1,5,2 are the observations, then find the median
a) 5 b) 5.5 c) 6.5 d) 6
19) Which quartile is the median
a) 1st quartile b) 3rd quartile c) 2nd quartile d) none
20) If the relation between two variables x and y be 3x + 7y=50, and median of y is 5. Then median of x is
a) 4 b) 5 c) 6 d) 7
21) If the mean and median are 24 and 26 respectively. Find mode
a) 27 b) 28 c) 29 d) 30
22) In usual symbols, mode= 3 median - a x mean, where a is
a) 1 b) 2 c) 3 d) none
23) How many quartiles are there
a) 1 b) 2 c) 3 d) 4
Matrix - Test
) If A= 2 0 & B= 1 2
1 2 2 1 find A+ B
2) If A= 1 2 3 & B= 1 2
4 5 6 3 4
find A+ B
3) If A= 0 2 & B= 7 8
2 1 1 4 find
a) 2A + 3B
b) 3A - B
c) AB
4) If A= 1 3
3 4 and A²- kA - 5=0, then find k.
5) If A= 1 0 & B= 0 1
0 1 -1 0
Then show that (aA+ bB)(cA+ dB)= (ac - bd)A+ (ad + bc)B.
BANKING
Sap-1
1) Amit deposited Rs 150 per month in a bank for 8 months under the recurring deposits. If the rate of interest is 8% p.a. and intrest is calculated at end of every month?
2) Laxmi took a cumulative time deposit account of Rs 240 per month at 10% p.a. she received Rs 3840 on maturity. Find the period for this account.
3) Manoj opened Recurring Deposit Account in a bank and deposited Rs 500 per month for 3 years. The bank on Rs 20220 on maturity. Find the rate of interest paid by the bank.
4) Raju opened a Recurring Deposit Account with the Bank of Rajasthan and deposits Rs 600 per month for 20 months. Calculate the maturity value of this account, if the bank pays interest at the rate of 10% per annum.
5) Miss Anshu Gupta deposited Rs 350 per month for 20 months under Recurring Deposit Scheme. Find the total amount payable by the bank on maturity of the account if the rate of interest is 11% per annum.
6) Mrs. Mathew opened Recurring Deposit Account in a bank with Rs 500 per month for 5/2 years. Find the amount she will get on the maturity. If the interest is paid on monthly balance at 12.5% per annum .
7) Calculate the amount received on maturity of a recurring deposit of Rs 150 per month for 1 year 6 months. if the rate of interest is 11% per annum .
8) Amar deposits Rs 1600 per month in a Recurring Deposit for 3 years at the rate of 9% p.a simple interest. Find the amount Amar will get at the time of maturity.
9) A Recurring Deposit Account of Rs 1200 per month has a maturity value of Rs 12440. If the rate of interest is 8% and the interest is calculated at the end of every month, find the time (in months) of this Recurring Deposit Account .
10) Sujata deposited , a certain sum of money, every months, for 5/2 years in a cumulative Time Deposit Account. At the time of maturity she collected Rs 4965. if the rate of interest was 8% p.a., find the monthly deposit.
Sap-2
1) Sunita paid Rs 300 per month for 2 years. He received Rs 7875 as the maturity amount. Find the rate of interest.
2) Meena has a cumulative Time Deposit Account of Rs 340 per month at 6% per annum. If she get Rs 7157 at the time of maturity, find the time for which the account was held.
3) On depositing Rs200, every month paying 9% p.a interest, a person collected Rs 2517 at maturity. Find the period.
4) Mamta has a cumulative Time Deposit Account in a bank. She deposits Rs 800 per month and gets Rs 15198 as maturity value. If the rate of interest be 7% p.a., find the total time for which the account was held.
5) Karim has a recurring deposit account for 2 years at 10%. If he receives Rs 1900 as interest, find the value of monthly installment paid by him.
6) Saloni has a cumulative time deposit account of Rs 340 per month at 6% p.a., if she get 7157 at the time of maturity, find the total time for which the account was held.
7) A man deposited Rs 150, every month in a bank for 8 months under the recurring deposit scheme. Find the maturity value of his deposits, if the interest is calculated every month and the rate of interest is 8% p.a.
8) Calculate the amount receivable on maturity of
a) recurring deposit of Rs 1200, deposited every month for 24 months at 10% p.a.
b) recurring deposit of Rs 100, every month for 5 years at 11% p.a.
c) recurring deposit of Rs 500, every month for 27 months at 10.5% p.a.
