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1.

`(log_(27) 9 xx log_(16) 64)/(log_(4) sqrt2) ` का मान क्या है ?A. `1//6`B. `1//4`C. 8D. 4

Answer» Correct Answer - D
दिया है , `(log_(27) 9 x log _(16) 64)/(log _(4)sqrt2)`
` = (2 log_(27) 3 xx 6 log_(16) 2)/(1/2 log_(4) 2) = (2/3 log_(3) 3 xx 6/4 log _(2)2)/(1/4 log_(2) 2)`
` = (2//3 xx 6//4)/(1//4)=(2//3xx3//2)/(1//4) = 4`
Discussion
2.

यदि `2^(x) = 3^(y) = 12^(z)` हो , तब `(x +2y)//(xy)` किसके बराबर है ?A. zB. 1/zC. 2zD. z/2

Answer» Correct Answer - B
माना ` 2^(x) = 3^(y) = 12^(z) = k`
` rArr x = log_(2) k, y = log_(3) k, z = log_(12) k`
अब , ` (x + 2y)/(xy) = (log_(2) k+2 log_(3) k)/(log_(2) k*log_(3) k) = 1/(log_(3) k) + 2/(log_(2) k) `
` = log_(k) 3 + 2 log_(k) 2 = log_(k) 3 + log_(k) 4 = log_(k) 12 `
` = 1/(log_(12) k) = 1/z`
Discussion
3.

`(log_(sqrt(alpha beta))(H))/(log_(sqrt(alpha beta gamma))(H))`का मान क्या है ?A. ` log_(alpha beta) ( alpha)`B. `log_(alpha beta gamma) ( alpha beta)`C. ` log_(alpha beta)( alpha beta gamma)`D. `log_(alpha beta)(beta)`

Answer» Correct Answer - C
` (log_(sqrt(alphabeta))H)/(log_(sqrt(alphabetagamma))H) = (log H)/(logsqrt(alpha beta))*(logsqrt(alphabetagamma))/(log H)`
` = (1//2 log alphabetagamma)/(1//2log alphabeta) = log_(alphabeta) alphabetagamma`
Discussion
4.

यदि `log_(30) 3 = a` तथा `log_(30)5 = b` हो , तब `log_(30)8` का मान बताइए?A. 3 (1 + a + b)B. 3 (1 - a+b)C. 3(1 + a - b)D. 3(1 -a -b)

Answer» Correct Answer - D
दिया है , `log_(30) 3 = a` तथा `log_(30) 5 = b`
अब , ` log_(30) 8 = 3 log_(30)(2 xx 15/15)`
` = 3[ 1 - {log_(30) 5 + log_(30) 3}]`
` = 3 (1 - a - b)`
Discussion
5.

`log_(y) x^(5) * log _(x) y^(2) * log_(z) z^(3)`का मान क्या है ?A. 10B. 20C. 30D. 60

Answer» Correct Answer - C
Discussion
6.

`(yz)^(log y - log z) xx (zx) ^(log z - log x) xx (xy) ^(log x - log y)` बराबर हैA. 2B. 1C. 4D. 3

Answer» Correct Answer - B
माना `k = (yz)^(log y - log z)`
` (zx)^(log z - log x )*(xy)^(log x - log y)`
` rArr log k = log (yz)^(log y - log z)`
`+ log (zx) ^(log z - log x)+log (xy)^(log x - log y)`
` = (log y - log z) (log y + log z)`
` + (log z - log x) xx(log z + logx)`
` + (log x - log y) (log x + log y)`
` = (log y)^(2) - (log z)^(2) + log (z)^(2)`
` - (log x)^(2) + (log x)^(2) - (log y)^(2)`
` = 0 rArr log k =0 rArr k = 1`
Discussion
7.

`log_(10)tna 1^(@) + log_(10) tan 2^(@) + ...+ log _(10) tan 89^(@)` बराबर है

Answer» Correct Answer - A
`log tan 89^(@) = log cot 1^(@) = - log tan 1^(@)`
` :. log tan 1^(@) + log tan 2^(@)`
` + ...+ log tan 44^(@) + log tan 45^(@)`
` - log tan 44^(@) - ...- log tan 2^(@)`
` - log tan 1^(@) = log tan 45^(@) = log 1 = 0`
Discussion
8.

`x^([log_(3) x^(2)+(log _(3) x)^(2) -10]) = 1/x^(2)` हो , तो x का मान बराबर हैA. 3B. 9C. 27D. 81

Answer» Correct Answer - B
`x^([(2 log _(3) x )+(log_(3) x)^(2) - 10])=x^((-2))`
` rArr 2 log_(3) x + (log_(3) x)^(2) - 10 = - 2`
` rArr y^(2) + 2y - 8 = 0 , ` जहाँ ` y = log_(3) x`
` rArr (y -2) (y+4) = 0 rArr y = 2`
अथवा ` y = - 4`
` rArr log_(3) x = 2 ` अथवा ` log_(3) x = - 4`
` rArr x = 3^(2)` अथवा ` x = 3^(-4)`
अर्थात x = 9 और ` x = 1/81`
Discussion
9.

यदि `(log_(3) x)^(2) + (log_(3) x) lt 2` हो , तो निम्नलिखित में से कौन- सा सही है ?A. ` 0 lt x lt 1/9`B. ` 1.9 lt x lt 3`C. ` 3 lt x lt infty`D. ` 1/9 le x le 3`

Answer» Correct Answer - B
`(log_(3) x)^(2) + (log _(3) x) lt 2`
` rArr (log_(3) x)^(2) + (log_(3) x) - 2 lt 0`
` rArr (log_(3) x + 2) (log_(3) x -1) lt 0`
` rArr -2 lt log_(3) x lt 1`
` rArr 1/9 lt x lt 3`
Discussion
10.

यदि `x = log_(a) (bc) , y = log_(b) (ac)` तथा `z = log_(c) (ab)` हो , तब निम्न प्रश्नो के उत्तर दीजिए । x + y + z का मान हैA. `log_(e) abc(log_(a) e + log_(b) e + log_(c) e) - 3`B. ` log_(e) abc (log_(e) a + log_(e) b + log_(e) c) - 3`C. ` log_(e) abc ( log_(abc) e) - 3 `D. इनमे से कोई नहीं

Answer» Correct Answer - A
` x + y + z = log_(a) (bc)+ (log)_(b)(ac)+log_(c)(ab)`
` = ((log b + log c))/(log a) + ((log a+ log c))/(log b)+((log a + log b))/(log c) `
` = ((log abc)/(log a) -1) +((log abc)/(log b) -1) + ((log abc)/(log c) -1)`
` = log abc (log _(a) e + log_(b) e + log_(c)e) - 3`
Discussion
11.

निम्नलिखित कथनों पर विचार कीजिए I. यदि ` x = log_(3) 5` तथा ` y = log_(17) 25` तो x तथा y में संबंध `x lt y` होगा । II. यदि `log_(2) x + log_(x) 2 = 10/3 = log_(2) y + log 2 ` तथा ` x ne y ` तो x +y का मान ` 8 + 2^(1//3)` होगा । उपरोक्त कथनों में से कौन - सा /से कथन सही है /है ?A. केवल IB. केवल IIC. I और II दोनोंD. न तो I और न ही II

Answer» Correct Answer - B
`I. 1/x = log_(5) 3 = 1/2 log_(5) 9` ...(i)
तथा ` y = log_(17) 25 rArr y = 2 log _(17) 5`
` rArr 1/y = 1/2 log_(5) 17` ..(ii)
समी (i) तथा (ii) से स्पष्ट है कि
` 1/y gt 1/x rArr x gt y`
अतः कथन I सही नहीं है ।
II. चूँकि ` 10/3 = 3 + 1/3`
अतः दिया गया समीकरण निम्न रूप का है
` p + 1/p = 3 + 1/3 = q + 1/q`
जहाँ , ` p ne q` क्योंकि ` x ne y`
` :. log _(2) x = 3, log_(2) y = 1/3`
` rArr x = 2^(3) , y = 2^(1//3)`
` rArr x + y = 2^(3) + 2^(1//3) = 8 + 2^(1//3)`
अतः कथन II सही है ।
Discussion
12.

यदि `x = log_(a) (bc) , y = log_(b) (ac)` तथा `z = log_(c) (ab)` हो , तब निम्न प्रश्नो के उत्तर दीजिए । `(1 + x)^(-1) + (1+y)^(-1) + (1+z)^(-1)` का मान है

Answer» Correct Answer - B
Discussion
13.

यदि `log_(2) (x-1) = 2 log_(2) (x - 3)` हो , तो x बराबर हैA. 2B. 5C. 6D. 7

Answer» Correct Answer - B
Discussion
14.

` (log x + log x^(4) + log x^(9) +...+ log x^(n^(2)))/(log x + log x^(2) + log x^(3) + ...+ log x^(n))`बराबर हैA. `(2n+1)/3`B. 2n+1C. `(3(n+2))/2`D. `(3(n-1))/2`

Answer» Correct Answer - A
दिया है , `((1+4 + 9 +...+n^(2))log x)/((1+2+3+...+n)logx)`
` = (sumn^(2))/(sumn) =((n(n+1)(2n+1))/6)/((n(n+1))/2)=(2n+1)/3`
Discussion
15.

यदि `log_(0.3)(x-1) lt log_(0.09) (x-1)` हो , तो x स्थित हैA. `(2 , infty )`B. `(-2, -1)`C. (1, 2)D. इनमे से कोई नहीं

Answer» Correct Answer - A
`log_(0.3)(x-1) lt log_((0.3)^(2))(x-1)`
` = 1/2 log_(0.3)(x-1)`
` :. 1/2 log_(0.3)(x-1) lt 0`
अथवा `log_(0.3)(x-1) lt 0 = log 1`
अथवा ` (x-1) gt 1` अथवा ` x gt 2`
Discussion