CBSE Class 10 Mathematics Pair of Linear Equations in Two Variables MCQs Set 24

Answer: (a) Rs. 1700

Explanation:
1. Let a table cost x and a chair cost y: \(2x+3y=2000\) and \(3x+2y=2500\)
2. Adding: \(5(x+y)=4500\), so \(x+y=900\); subtracting: \(x-y=500\)
3. \(x=700\), \(y=200\)
4. \(x+5y=700+1000=1700\)

Teacher's Note:
1. Adding and subtracting the two equations is faster than substitution when coefficients are swapped.
2. Answer the exact quantity asked: one table and five chairs.

Answer: (c) 12 and 16

Explanation:
1. Let the numbers be x and \(x+4\): \(x(x+4)=192\)
2. \(x^2+4x-192=0\), so \((x+16)(x-12)=0\)
3. Positive root x = 12, so the numbers are 12 and 16

Teacher's Note:
1. Reduce the difference-and-product condition to a quadratic.
2. Check: 16 - 12 = 4 and 12 x 16 = 192.

Answer: (a) 40 years and 10 years

Explanation:
1. Let son = S, father = 4S
2. \(4S+5=3(S+5)\)
3. \(S=10\), so father = 40

Teacher's Note:
1. Express the father's age in terms of the son's to get a single equation.
2. Check: 45 = 3 x 15.

Answer: (a) \(\dfrac{5}{13}\)

Explanation:
1. \(x+y=18\) and \(\dfrac{x}{y+2}=\dfrac13\), so \(3x=y+2\)
2. \(3x=20-x\), so \(x=5\) and \(y=13\)

Teacher's Note:
1. Substitute \(y=18-x\) into the second equation.
2. Check: 5/15 = 1/3.

Answer: (b) 20 cm and 12 cm

Explanation:
1. Let width = w and length = \(w+8\): \(w(w+8)=240\)
2. \(w^2+8w-240=0\), so \((w+20)(w-12)=0\)
3. \(w=12\), length = 20

Teacher's Note:
1. Reject the negative root since a length cannot be negative.
2. Check: 20 x 12 = 240.

Answer: (c) 12 cm and 16 cm

Explanation:
1. Let the sides be s and \(s+4\): \(s^2+(s+4)^2=400\)
2. \(s^2+4s-192=0\), so \((s+16)(s-12)=0\)
3. \(s=12\), other side 16

Teacher's Note:
1. Form one quadratic from the area condition.
2. Check: 144 + 256 = 400.

Answer: (a) 70 degrees and 20 degrees

Explanation:
1. Let the angles be a, b and \(c=a+b\). The angle sum gives \(2(a+b)=180\), so \(a+b=90\)
2. With \(a-b=50\): \(a=70\), \(b=20\)

Teacher's Note:
1. The third angle is then 90 degrees, a right angle.
2. Always use the angle sum property of 180 degrees.

Answer: (b) 13 years

Explanation:
1. Ages are \(2k\) and \(3k\): \(5k=65\), so \(k=13\)
2. Ages are 26 and 39
3. Difference = 13 years

Teacher's Note:
1. The difference is simply k.
2. Check: 26 + 39 = 65.

Answer: (c) 1635 and 270

Explanation:
1. Larger = L and smaller = S: \(L-S=1365\) and \(L=6S+15\)
2. \(5S+15=1365\), so \(S=270\)
3. \(L=1635\)

Teacher's Note:
1. Division with remainder gives dividend = divisor x quotient + remainder.
2. Check: 1635 - 270 = 1365.

Answer: (d) \(\dfrac23\)

Explanation:
1. Fraction \(\dfrac{x}{x+1}\)
2. \(\dfrac{x-1}{x}=\dfrac12\), so \(2x-2=x\), \(x=2\)
3. Fraction \(\dfrac23\)

Teacher's Note:
1. The new denominator becomes the old numerator.
2. Check: 1/2.

Answer: (a) 47, 59 and 74 degrees

Explanation:
1. \(x+(x+12)+(x+27)=180\)
2. \(3x+39=180\), so \(x=47\)
3. The angles are 47, 59 and 74

Teacher's Note:
1. Use the angle sum of a triangle.
2. Check: 47 + 59 + 74 = 180.

Answer: (d) 13 pens and 27 pencils

Explanation:
1. Pens = p and pencils = q: \(p+q=40\) and \(q+5=4(p-5)\)
2. \(q=4p-25\), so \(5p-25=40\), \(p=13\)
3. \(q=27\)

Teacher's Note:
1. Apply the changes (+5 and -5) before forming the second equation.
2. Check: 32 = 4 x 8.

Answer: (c) 24

Explanation:
1. \(10a+b=4(a+b)\) gives \(b=2a\)
2. \(10a+b+18=10b+a\) gives \(b=a+2\)
3. \(a=2\), \(b=4\), number 24

Teacher's Note:
1. Two conditions give two equations in the digits.
2. Check: 24 = 4 x 6 and 24 + 18 = 42.

Answer: (d) \(\dfrac{7}{18}\)

Explanation:
1. Fraction \(\dfrac{n}{2n+4}\)
2. \(2n+4-6=12(n-6)\), so \(2n-2=12n-72\)
3. \(n=7\), denominator 18

Teacher's Note:
1. Write the denominator in terms of the numerator first.
2. Check: 12 = 12 x 1.

Answer: (a) 40 units and 30 units

Explanation:
1. \((l-5)(b+2)=lb-80\) gives \(2l-5b=-70\)
2. \((l+10)(b-5)=lb+50\) gives \(-5l+10b=100\), i.e. \(l=2b-20\)
3. \(4b-40-5b=-70\), so \(b=30\), \(l=40\)

Teacher's Note:
1. Expand and cancel lb in each condition.
2. Check: 35 x 32 = 1120 = 1200 - 80.

Answer: (b) 82, 73 and 25 degrees

Explanation:
1. \(y-9-x=48\) so \(x=y-57\)
2. Angle sum: \(y+(y-9)+(y-57)=180\), so \(y=82\)
3. A = 82, B = 73, C = 25

Teacher's Note:
1. Two relations between three angles are enough with the angle sum.
2. Check: 73 - 25 = 48.

Answer: (c) 20, 40 and 120 degrees

Explanation:
1. Largest = L and others sum to \(180-L\): \(L=2(180-L)\), so \(L=120\)
2. Smallest \(=\dfrac{120}6=20\)
3. Third \(=180-120-20=40\)

Teacher's Note:
1. Use the angle sum to find the largest angle directly.
2. Check: 120 = 2 x 60.

Answer: (c) 5 and 10

Explanation:
1. \(x+y=15\) and \(\dfrac{x+y}{xy}=\dfrac3{10}\), so \(xy=50\)
2. \(x\) and \(y\) are roots of \(t^2-15t+50=0\), i.e. 5 and 10

Teacher's Note:
1. Combine the reciprocals over a common denominator.
2. Check: 1/5 + 1/10 = 3/10.

Answer: (b) \(x=1,\ y=1\)

Explanation:
1. Multiply the second equation by 5: \(35x-5y=30\)
2. Add to the first: \(47x=47\), so \(x=1\)
3. \(y=7-6=1\)

Teacher's Note:
1. Equalise one coefficient and add or subtract.
2. Check in both equations.

Answer: (c) \(x=2,\ y=-3\)

Explanation:
1. Multiply the first by 2 and the second by 3: \(34x+24y=-4\) and \(45x+24y=18\)
2. Subtract: \(11x=22\), so \(x=2\)
3. \(12y=-2-34=-36\), so \(y=-3\)

Teacher's Note:
1. Make the coefficients of y equal in magnitude (24).
2. Check: 30 - 24 = 6.

Answer: (d) \(x=1,\ y=-1\)

Explanation:
1. Multiply the first by 19 and the second by 17: \(437x+323y=114\) and \(663x-323y=986\)
2. Add: \(1100x=1100\), so \(x=1\)
3. \(17y=6-23=-17\), so \(y=-1\)

Teacher's Note:
1. Choose multipliers that make the y coefficients 323 in magnitude.
2. Check: 39 + 19 = 58.